Advanced Problem in Mechanics II: Proceedings of the XLVIII International Summer School-Conference “Advanced Problems in Mechanics”, 2020, St. ... (Lecture Notes in Mechanical Engineering) [1st ed. 2022] 3030921433, 9783030921439

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Lecture Notes in Mechanical Engineering

D. A. Indeitsev A. M. Krivtsov   Editors

Advanced Problem in Mechanics II Proceedings of the XLVIII International Summer School-Conference “Advanced Problems in Mechanics”, 2020, St. Petersburg, Russia

Lecture Notes in Mechanical Engineering Series Editors Francisco Cavas-Martínez , Departamento de Estructuras, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany Francesco Gherardini , Dipartimento di Ingegneria, Università di Modena e Reggio Emilia, Modena, Italy Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Vitalii Ivanov, Department of Manufacturing Engineering, Machines and Tools, Sumy State University, Sumy, Ukraine Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA Justyna Trojanowska, Poznan University of Technology, Poznan, Poland

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D. A. Indeitsev A. M. Krivtsov •

Editors

Advanced Problem in Mechanics II Proceedings of the XLVIII International Summer School-Conference “Advanced Problems in Mechanics”, 2020, St. Petersburg, Russia

123

Editors D. A. Indeitsev IPME RAS, Peter the Great St. Petersburg Polytechnic University St. Petersburg, Russia

A. M. Krivtsov IPME RAS, Peter the Great St. Petersburg Polytechnic University St. Petersburg, Russia

ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-3-030-92143-9 ISBN 978-3-030-92144-6 (eBook) https://doi.org/10.1007/978-3-030-92144-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The International Conference “Advanced Problems in Mechanics 2020” is the forty-eighth in a series of annual summer schools held by Russian Academy of Sciences. The conference is organized in commemoration of its Founder, Ya.G. Panovko, by the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences (IPME RAS), Peter the Great St. Petersburg Polytechnic University (Institute of Applied Mathematics and Mechanics), Scientific Council on Solid Mechanics (RAS) (Chairman N.F. Morozov), Russian National Committee on Theoretical and Applied Mechanics (Chairman I.G. Goryacheva) under the patronage of the Russian Academy of Sciences (RAS). APM 2020 is partially supported by the Russian Foundation for Basic Research, the Ministry for Science and Education, Gazpromneft. APM 2020 included two events: the XLVIII International Summer School–Conference Advanced Problems in Mechanics, held online on June 21–26, 2020, and the XLVIII International Autumn School–Conference Advanced Problems in Mechanics, held on November 9–13, 2020, in a mixed format. The list of problems under investigation is not limited to questions of mechanical engineering but includes practically all advanced problems in mechanics, which is reflected in the name of the conference. The main attention is given to problems on the boundary between mechanics and other research areas, which stimulates the investigation in domains such as micro- and nanomechanics, material science, physics of solid states, molecular physics, astrophysics and many others. The conference “Advanced Problems in Mechanics” helps us to maintain the existing contacts and to establish new ones between foreign and Russian scientists. One of the major purposes of conference is to transfer scientific experience of well-known scientists to their young colleagues. During the years 1996–2006, Professor Vladimir A. Palmov co-chaired the conference. He had been supervising many generations of students at the Great St. Petersburg Polytechnic University. His scientific discoveries, as well as his questions and remarks, the brilliant sense of humor and gentle way he approached young researchers, had greatly contributed to the atmosphere of APM during those

v

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Preface

years. To our deep sorrow, Prof. Palmov deceased in October 2018. We will remember him and continue the scientific tradition established by him. The first Summer School was organized by Ya.G. Panovko and his colleagues in 1971. In the early years, the main focus of the School was on nonlinear oscillations of mechanical systems with a finite number of degrees of freedom. This focus was set to distinguish the event from the numerous national meetings and conferences that were organized at the time in Russia (USSR) to cover other areas of mechanics. After 1985, many conferences and schools on mechanics in Russia were terminated due to financial problems. In 1994, the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences restarted this Summer School. The traditional name of “Summer School” has been kept, but the topics covered by the School have been much widened. The School has been transformed into an international conference. The topics of the conference cover now all fields of mechanics and their associated interdisciplinary problems.

Organization

Scientific Committee D. A. Indeitsev (Co-chairman) A. M. Krivtsov (Co-chairman) P. A. Dyatlova (Scientific Secretary) H. Altenbach M. B. Babenkov V. A. Babeshko A. K. Belyaev I. E. Berinskii I. I. Blekhman V. A. Bratov A. A. Burenin

A. V. Cherkaev F. Dell’Isola V. A. Eremeyev V. I. Erofeev A. B. Freidin

IPME RAS, Peter the Great St. Petersburg Polytechnic University, Russia Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia Otto-von-Guericke University Magdeburg, Germany IPME RAS, Peter the Great St. Petersburg Polytechnic University, Russia Southern Scientific Center RAS, Rostov-on-Don, Russia IPME RAS, Peter the Great St. Petersburg Polytechnic University, Russia Tel Aviv University, Israel IPME RAS, Mekhanobr-Tekhnika, St. Petersburg, Russia IPME RAS, St. Petersburg, Russia Institute of Metallurgy and Mechanical Engineering Far-Eastern Branch of RAS, Komsomolsk-na-Amure, Russia University of Utah, Salt Lake City, USA Università di Roma La Sapienza and MEMOCS Centre, Italy Gdansk University of Technology, Poland Mechanical Engineering Research Institute of RAS or MERI RAS, Russia IPME RAS, Peter the Great St. Petersburg Polytechnic University, Russia

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M. E. Frolov S. N. Gavrilov I. G. Goryacheva E. F. Grekova N. Gupta H. E. Huppert H. Irschik M. L. Kachanov B. L. Karihaloo D. Koroteev S. V. Kuznetsov V. A. Kuzkin V. A. Levin A. M. Linkov I. I. Lipatov O. S. Loboda E. V. Lomakin G. Mishuris N. F. Morozov W. H. Müller U. Nackenhorst E. Pavlovskaia S. V. Petinov Y. V. Petrov A. V. Porubov J.-N. Roux M. B. Rubin A. I. Rudskoi S. Rudykh S. H. Sargsyan V. V. Sergeev I. Sevostianov

Organization

Peter the Great St. Petersburg Polytechnic University, Russia IPME RAS, St. Petersburg, Russia Institute for Problems in Mechanics RAS, Moscow, Russia IPME RAS, St. Petersburg, Russia; University of Seville, Spain Indian Institute of Technology Delhi, India University of Cambridge, UK Johannes Kepler University of Linz, Austria Tufts University, Medford, USA Cardiff University, UK Center for Hydrocarbon Recovery, Scoltech (Digital Petroleum), Russia IPMech RAS, Russia Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia Lomonosov Moscow State University, Russia IPME RAS, Russia; Rzeszow University of Technology, Poland Moscow Institute of Physics and Technology, Russia Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia Lomonosov Moscow State University, Russia Aberystwyth University, UK St. Petersburg State University, IPME RAS, Russia Technical University of Berlin, Germany Leibniz University of Hannover, Germany University of Aberdeen, UK IPME RAS, Peter the Great St. Petersburg Polytechnic University, Russia St. Petersburg State University, IPME RAS, Russia IPME RAS, Saint Petersburg, Russia Université Paris-Est, Laboratoire Navier, France Israel Institute of Technology, Haifa, Israel Peter the Great St. Petersburg Polytechnic University, Russia University of Wisconsin-Madison, USA Gyumri State Pedagogical Institute, Armenia Peter the Great St. Petersburg Polytechnic University, Russia New Mexico State University, USA

Organization

M. Simonov M. Wiercigroch H. A. Wu P. Venkitanarayanan E. N. Vilchevskaya M. V. Zakrzhevsky

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Gazpromneft Science & Technology Center, Russia Aberdeen University, Scotland University of Science and Technology of China, Chinese Academy of Sciences Indian Institute of Technology, India IPME RAS, Peter the Great St. Petersburg Polytechnic University, Russia Riga Technical University, Latvia Local

Organizing Committee Polina Dyatlova Anna Kuznetsova Mikhail Babenkov Anna Morozova Mariia Fomicheva Maria Loboda Darina Shulepnikova Anastasia Vilchevskaya

Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

The conference is organized with help of our service agency “Monomax PCO”: www.monomax.ru.

Minisymposia MS1 “Mathematical Modeling in Petroleum Engineering” Organizers Liliana Rybarska-Rusinek Alexander M. Linkov Vitaly A. Kuzkin

Rzeszow University of Technology, Poland IPME RAS, Russia; Rzeszow University of Technology, Poland Peter the Great St. Petersburg Polytechnic University; IPME RAS, Russia

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Organization

MS2 “New Approaches for Oil and Gas Reservoirs Simulation” Organizers Maksim Simonov Dmitry Koroteev

Gazpromneft Science & Technology Center, Russia Center for Hydrocarbon Recovery, Scoltech, Russia

MS3 “Geometry, Topology, Fractal and Multifractal Modeling in Geosciences” Organizer Aleksander Kalyuzhnyuk

Gazpromneft Science & Technology Center, Russia

MS4 “Extreme Loading on Structures” Organizers Danila Prikazchikov Nikita F. Morozov Vladimir A. Bratov

Keele University, UK St. Petersburg State University, IPME RAS, Russia IPME RAS, Russia

MS5 “Nonlinear Waves in Continuous Media” Organizers Vladimir I. Erofeev

Alexey V. Porubov

Mechanical Engineering Research Institute of the Russian Academy of Sciences or MERI RAS, Nizhniy Novgorod, Russia IPME RAS, St. Petersburg, Russia

MS6 “Earthquakes and Seismic Protection” Organizers Tzu-Kang Lin Sergey V. Kuznetsov Vladimir Bratov

National Taiwan University, Taiwan IPMech RAS, Russia IPME RAS, Russia

Organization

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MS7 “Scientific and Technical Creativity in Experimental Mechanics” Organizers Mikhail B. Babenkov Polina A. Dyatlova

IPME RAS, Peter the Great St. Petersburg Polytechnic University, Russia Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia

MS8 “Contact Mechanics, Tribology and Technology” Organizers Irina G. Goryacheva Elena V. Torskaya Jeng-Haur Horng

Ishlinsky Institute for Problems in Mechanics RAS, Russia Ishlinsky Institute for Problems in Mechanics RAS, Russia National Formosa University, Taiwan

MS9 “Heat/Energy Transport in Discrete Systems” Organizer Anton M. Krivtsov

Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia

MS10 “Sustainability of the Oil and Gas Industry: Challenges and Opportunities” Organizers Alexandra Boytsova Nadezhda Sheveleva

Peter the Great St. Petersburg Polytechnic University, Russian Gas Society Eco Oil Gas Consulting

Contents

Plenary Lecture Poly-dispersed Droplets in Streaming Flows: Atomization, Evaporation, Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nikolay N. Smirnov

3

Fluid Mechanics Viscoplastic Fluid Flow in a T-shaped Channel Under Given Pressure Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. A. Dyakova and O. Yu. Frolov

15

Mathematical Modeling of the Flow of Viscous Incompressible Fluid with Suspended Particles in Flat Inclined Channel . . . . . . . . . . . . Regina Iulmukhametova, Airat Musin, and Liana Kovaleva

23

Heat Transfer and Wave Motion Solitary Waves in Hyperelastic Tubes Conveying Inviscid and Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vasily Vedeneev

35

A Note on the Propagation of Antiplane Love Waves in Microstructured Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valentina Volpini

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Acoustic Control of Cracks Self-healing in Plates: An Impulse Induced Wave Field in a Plate with an Initially Closed Crack Parallel to Its Free Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladislav V. Vershinin Simulation of the Operation of a Detonation Engine . . . . . . . . . . . . . . . E. V. Mikhalchenko, V. F. Nikitin, and V. D. Goryachev

82 98

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Contents

Reduced Order Modeling for Thermo – Electric Processes . . . . . . . . . . 108 Alexey Lukin, Ivan Popov, and Pavel Udalov Experimental and Numerical Study of the Water-in-Oil Emulsion Thermal Motion in Rectangular Cavity with a Heated Bottom . . . . . . . 117 Vilena Valiullina, Yuriy Zamula, Almir Mullayanov, Regina Iulmukhametova, Airat Musin, and Liana Kovaleva Nonlinear Thermal Elastic Diffusion Problems Applicable to Surface Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A. G. Knyazeva and E. S. Parfenova Nano-, Micro- and Mesomechanics Models of Nanosystems and Methods of Their Investigation, Connected with Orthogonal Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Victor L. Leontiev Model of a Micromechanical Accelerometer Based on the Phenomenon of Modal Localization . . . . . . . . . . . . . . . . . . . . . . 159 Vasilisa Igumnova, Lev Shtukin, Alexey Lukin, and Ivan Popov Phase Transitions and Nonlinear Elasticity Reducing of Residual Stresses in Metal Parts Produced By SLM Additive Technology with Selective Induction Heating . . . . . . . . . . . . . . 175 Sergei A. Lychev and Montaser Fekry Solids and Structures Critical Velocities and Stability of the Axially Moving Panels . . . . . . . . 197 Nikolay Banichuk and Svetlana Ivanova Features of Applying HEDE Model to Description of the Destruction of Materials Induced by Hydrogen . . . . . . . . . . . . . . 202 Yulia Sedova, Vladimir Polyanskiy, and Nikolay Bessonov Nonlinear and Multibody Dynamics, Chaos and Vibration Approximating Unstable Operation Speeds of Automatic Ball Balancers Based on Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 223 Lars Spannan and Elmar Woschke Shape Control and Modal Control Strategies for Active Vibration Suppression of a Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Aleksandr V. Fedotov Axially Symmetric Oscillations of Circular Cylindrical Shell with Localized Mass on Winkler Foundation . . . . . . . . . . . . . . . . . . . . . 245 George V. Filippenko and Tatiana V. Zinovieva

Contents

xv

Mechanical and Civil Engineering Applications Statistical Quality Analysis of Bag-in-Box Packaging for Food Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 S. A. Atroshenko Modeling the Air Mixture Flow in an AC Plasma Torch . . . . . . . . . . . . 278 Nikolay Bykov, Anton Kobelev, Nikita Obraztsov, Alexander Surov, and Alexey Borovskoy Theoretical Basis of the Mechanism of Synchronous Rotation in Reverse Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Sergey Dragunov Acoustoelastic Effect During Plastic Deformation of Anisotropic Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Alexander Belyaev, Aliya Galyautdinova, and Dmitriy Tretyakov Minisymposium “Contact Mechanics, Tribology and Technology” Wear of Composite Materials in Full Contact with a Viscoelastic Body. The Steady-State Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Anastasia Lyubicheva Pressure Concentration in 2D Elastic Regular Rough Contacts: The Effect of Asperity Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Ivan Y. Tsukanov Effect of Friction in Sliding Contact of Layered Viscoelastic Solids . . . . 320 Elena V. Torskaya and Fedor I. Stepanov The Simulation the Contact Interaction of the Needle and Brain Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Tatiana Lycheva and Sergey Lychev Modeling of Fatigue Wear in Rolling Contact of Elastic Bodies . . . . . . . 350 Almira Meshcheryakova and Irina Goryacheva Minisymposium “Scientific and Technical Creativity in Experimental Mechanics” Dynamics of Hollow Cylindrical Magnets . . . . . . . . . . . . . . . . . . . . . . . . 363 Wilhelm Rickert and Wolfgang H. Müller Experimental Study on Hydraulically-Driven Fracture Initialization and Propagation in the Gelatin Mixture . . . . . . . . . . . . . . . . . . . . . . . . . 381 M. B. Babenkov, E. A. Belousova, B. A. Islamov, S. F. Lebedev, V. A. Timoshenko, and A. U. Vasilyeva

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Contents

Correlation Model of Fracturing Fluid Viscosity with Regard to Proppant Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 D. A. Nikiforov, A. A. Rybakovskaya, I. S. Senkin, and O. I. Tsykunov Minisymposium “Nonlinear Waves in Continuous Media” Solution of Equations for Plane Deformation of Nonlinear Model of Complex Crystal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Anatolii N. Bulygin and Yurii V. Pavlov Solitary Acoustic Pulses Propagating at the Tip of an Elastic Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Pavel D. Pupyrev, Alexey M. Lomonosov, and Andreas P. Mayer Minisymposium “Extreme Loading on Structures” Free Vibrations of Thin Elastic Orthotropic Cantilever Cylindrical Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 G. R. Ghulghazaryan and L. G. Ghulghazaryan Stoneley Waves in Media with Microstructure . . . . . . . . . . . . . . . . . . . . 463 Andrea Nobili Assessment of the Behaviour of Low-Modulus Polyurethane Foams Subjected to Severe Shear Deformation Conditions . . . . . . . . . . . . . . . . 476 Cesare Signorini Minisymposium “Earthquakes and Seismic Protection” Numerical Methods of Structures Seismic Analysis . . . . . . . . . . . . . . . . 489 I. K. Shanshin, A. V. Lukin, and I. G. Svyatogorov Minisymposium “New Approaches for Oil and Gas Reservoirs Simulation” Modeling of Liquid Displacement from the Porous Medium Taking into Account the Presence of Hydraulic Fracture . . . . . . . . . . . . . . . . . . 503 E. I. Kolenkina Skryleva, N. N. Smirnov, V. F. Nikitin, R. R. Fakhretdinova, and M. N. Makeeva Application of a* Algorithm for Tortuosity and Effective Porosity Estimation of 2D Rock Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Filippo Panini, Eloisa Salina Borello, Costanzo Peter, and Dario Viberti

Contents

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Minisymposium “Geometry, Topology, Fractal and Multifractal Modeling in Geosciences” Representative Elementary Volume via Averaged Scalar Minkowski Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 M. V. Andreeva, A. V. Kalyuzhnyuk, V. V. Krutko, N. E. Russkikh, and I. A. Taimanov The Fractal Model of Fractured Reservoirs Based on Pareto Distribution and Integrated Investigations . . . . . . . . . . . . . . . . . . . . . . . 540 A. V. Petukhov Multifractal Interpretation of Images of Coal Specimen Surfaces to Assess the Degree of Coal Tectonic Disturbance . . . . . . . . . . . . . . . . 550 Malinnikova Olga, Malinnikov Vasiliy, Uchaev Denis, and Uchaev Dmitry Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

Plenary Lecture

Poly-dispersed Droplets in Streaming Flows: Atomization, Evaporation, Combustion Nikolay N. Smirnov1,2(B) 1 Faculty of Mechanics and Mathematics Moscow M.V, Lomonosov State University,

Leninskie Gory 1, Moscow 119992, Russia [email protected], [email protected] 2 Federal Science Center “Scientific Research Institute for System Analysis of Russian Academy of Sciences, Nakhimovskiy pr. 36-1, Moscow 117218, Russia

Abstract. The paper presents the survey of results of theoretical, numerical and experimental investigations of combustion and detonation initiation in heterogeneous polydispersed mixtures. The problems of fuel droplets atomization, evaporation and combustion and the non-equilibrium effects in droplets atomization and phase transitions are discussed. The effects of droplets size non-uniformity and spatial distribution non-uniformity on mixture ignition and flame acceleration were investigated for strong and mild initiation of detonation: by a shock wave and spark ignition followed by deflagration to detonation transition (DDT). Peculiarities of jet injection and ignition in reaction chamber are studied. Keywords: Combustion · Aerosol · Poly-dispersed · Non-equilibrium effects · Non-uniformity

1 Mathematical Model The mathematical models for simulating turbulent chemically reacting flows in heterogeneous mixtures were described in details in [1, 2]. Combustion processes in heterogeneous mixtures differ greatly from that in homogeneous mixtures, because they are governed not only by chemistry but also by physical processes of combustible mixture formation, such as droplet atomization [3, 4], evaporation and diffusive mixing of fuel vapor with an oxidant. The model applies both deterministic methods of continuous mechanics of multiphase flows to determine the mean values of parameters of the gaseous phase and stochastic methods to describe the evolution of poly-dispersed particles in it and fluctuations of parameters. Thus the influence of chaotic pulsations on the rate of energy release and mean values of flow parameters can be estimated. The transport of kinetic energy of turbulent pulsations at the same time obeys the deterministic laws being the macroscopic characteristic. Averaging by Favre with the αρ weight (α – volumetric fraction of the gas phase, ρ – gas density) we obtain the following system for the gas phase in a multiphase flow [5] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 3–11, 2022. https://doi.org/10.1007/978-3-030-92144-6_1

4

N. N. Smirnov

(the averaging bars are removed for simplicity): ˙, ∂t (αρ) + ∇ · (αρu) = M

(1)

˙ k + ω˙ k , ∂t (αρYk ) + ∇ · (αρYk u ) = −∇ · Ik + M

(2)

˙ ∂t (αρu) + ∇ · (αρu ⊗ u ) = αρ g − α∇p + ∇ · τ + K,

(3)

˙ ∂t (αρE) + ∇ · (αρEu) = αρu · g − ∇ · pu − ∇ · Iq + ∇ · (τ · u ) + E.

(4)

The Eqs. (1)–(4) include mass balance in the gas phase, mass balance of k-th component, momentum balance and energy balance respectively (p – pressure, u – fluid velocity ˙ – specific momentum flux vector, g – gravity acceleration vector, E – specific energy, K ˙ – specific mass flux, E˙ – specific energy flux, τ – turbulent stress tento gas phase, M sor), Ik is the turbulent diffusion flux of the k-th component. We have the following relationships between the terms in the Eqs. (1) and (2):  k

Yk = 1,

 k

˙, ˙k = M M



Ik = 0,

k



ω˙ k = 0.

k

The state equations for gaseous mixture are the following:  p = Rg ρT Yk /Wk , k

E=

 k

Yk (cvk T + h0k ) +

u 2 + k, 2

(5)

where E – gas energy, k – turbulent kinetic energy, Wk - molar mass of k-th gas component, h0k – specific chemical energy, cpk ,cvk – specific heat capacity, Yk - mass concentration of k-th gas component, T – gas temperature, Rg - universal gas constant. The term responsible for chemical transformations, ω˙ k is very sensitive to temperature variations, as it is usually the Arrhenius law type function for the reactions’ rates. To take into account temperature variations the source term ω˙ k in the Eq. (2) was modeled using the Gaussian quadrature technique. Let us regard the temperature being a stochastic function T with mean T and mean squared deviate θ = T  T  . Then, the mean value of a function having T as independent variable could be determined as follows:  √ f (T ) = f (T + ζ θ )Pd (ζ )d ζ , where ζ is a random value with zero expectation and unit deviate; its probability density function is Pd (ζ ). To estimate the integral, the minimal number of terms is used(namely, three) and Pd (ζ ) is assumed to be even. In this case, the formula for f(T) averaging is:   √ √ 1 1 1 f (T f (T ) = f (T − χ θ ) + 1 − ) + f (T + χ θ). 2χ 2 χ2 2χ 2

Poly-dispersed Droplets in Streaming Flows

5

√ In case of normal (Gaussian) deviate, the value of χ is equal to 3 (Gauss-Hermite case). Therefore, the formula above could be transformed as follows: f (T ) =

√ √ 1 2 1 f (T − 3θ ) + f (T ) + f (T + 3θ ). 6 3 6

In our case, the function f(T) is the Arrhenius temperature dependence; the whole average for ω˙ k is constructed using combinations of these dependencies. Averaged magnitudes for mass fractions and density were used in the Arrhenius law for ω˙ k as the dependence of these functions is not as strong as the dependence of temperature. The turbulent heat flux Iq in the Eq. (4) is a sum of two terms:  Iq = Jq + (cpk T + h0k )Ik , (6) k

where Jq could be interpreted as turbulent conductive heat flux. The eddy kinematic 2 viscosity ν t is expressed according to k-epsilon model as ν t = Cμ kε . The turbulent fluxes are: τ = α(μ + ρν t )(∇ u + ∇ uT − (2/3)(∇ · u )U ) − (2/3)αρkU ,

(7)

Ik = −αρ(D + (ν t /σd ))∇Yk ,

(8)

Jq = −α(λ +



cpk ρ(ν t /σt ))∇T .

(9)

k

The model is closed then by the equations for k, θ and ε: ∂t (αρk) + ∇ · (αρuk) = ∇ · (α(μ + ρ(ν t /σk ))∇k) + τ t :∇ u − αρε, ∂t (αρε) + ∇ · (αρuε) = ∇ · (α(μ + ρ(ν t /σε ))∇ε) + (ε/k)(C1ε τ t :∇ u − C2ε αρε),  ∂t (αρθ ) + ∇ · (αρuθ ) = ∇ · (α(µ + cpk ρ(ν t /σk ))∇θ ) + Pθ + Wθ − Dθ ,

(10) (11) (12)

k

and the dissipation term Dθ are: where the production terms Pθ ,Wθ  Pθ = 2αρ c˜ p σνkt |∇T |2 , Wθ = − k ωk T  h0k , Dθ = Cg αρ



cpk

k

ε θ . k θm − θ

The constants take the following values [5]: Cμ = 0.09,C1ε = 1.45,C2ε = 1.92,σd = 1,σt = 0.9, 2

σk = 1,σε = 1.13, θm = T /4,Cg = 2.8.

(13)

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N. N. Smirnov

One could see that the Eqs. (10)–(12) do not contain terms responsible for particulate phase contribution to turbulence energy growth. This is due to the direct stochastic modeling of the particulate phase: the influence of the last on the gas phase leads to ˙ These terms affect the averaged stochastic behavior of the momentum source terms K. gas phase velocity in the stochastic manner and therefore the source term τ t :∇ u is also affected. The motion of polydispersed droplets (particles) is modeled making use of a stochastic approach. A group of representative model particles is distinguished. Motion of these particles is simulated directly taking into account the influence of the mean stream of gas and pulsations of parameters in gas phase. The procedure is described in details in papers [1, 2, 5]. The non-equilibrium effects in gas – droplet interaction for evaporating droplets and for burning droplets were taken into account based on a single fuel droplet interaction with heated streaming flow of oxidant [6–8]. The effectiveness of numerical scheme was linear: computation time was directly proportional to the increase of cells in a gas phase and number of model droplets.

2 Results of Numerical Investigations Using a single droplet dynamics model, we simulated the interaction of a strong shock wave in the air with an aerosol consisting of fuel droplets. Aerosol droplets are small in size and spherical in shape. The initial state of the aerosol is poly-disperse, that is, it is possible to consider the simultaneous existence of droplets of different initial diameters [4, 5]. When considering the interaction of a shock wave with an aerosol, the following hierarchy of models is considered sequentially: 1 2 3 4

taking into account only the resistance force, taking into account the resistance force and heat exchange, accounting for resistance, heat transfer and evaporation, taking into account the resistance force, heat transfer, evaporation and atomization of droplets, 5 accounting for resistance, heat transfer, evaporation, droplet atomization and combustion. Numerical simulation of two-dimensional flow in a cylindrical tube was performed. The pipe consists of two parts. Pressure and temperature on the left side x ≤ b are elevated (P1 = 10 ÷ 100 bar, T 1 = 1500 K,) as compared with conditions in the right hand side (P0 = 1 bar, T 0 = 300 K). Left hand side is filled with air, while the right hand side for b < x < x 0 is also filled with air, and for x ≥ x 0 aerosol is added (Fig. 1). The tube length L = 2 m, diameter D = 8 cm. Aerosol density 0.8 kg/m3 , which corresponds to a volume concentration 10−3 .

Poly-dispersed Droplets in Streaming Flows

7

Fig. 1. A schematic representation of the problem statement.

In Figs. 2, 3, 4, 5 and 6 the diagrams for the shock wave velocity along the tube axis are presented for different values of initial pressure ratios P1 /P0 . Different curves numbered from 1 to 5 correspond to different models of gas – droplet interaction described above. Figure 2 presents results for initial pressure ratio P1 /P0 = 10. It can be seen that when the shock wave enters the area filled with aerosol, the wave speed decreases. The most intensive speed reduction occurs when only resistance and heat transfer are taken into account (curve 2), the least intensive wave speed reduction – when only resistance is taken into account (curve 1). Other, more complex models correspond to intermediate results. Accounting for evaporation brings to lower reduction of shock wave velocity as compared with accounting for only thermal and mechanical losses. The reason is the following: evaporation brings to formation of additional gas volume behind the shock wave, which supports its propagation (curve 3). With an increase in the initial intensity of the shock wave (P1 /P0 = 25, Fig. 3), the greatest decrease in its intensity also occurs when taking into account resistance and heat exchange (curve 2), but the smallest – when taking into account resistance, heat exchange and evaporation (curve 3). This is due to the fact that the intensity of evaporation of drops increases the volume of gas. Accounting for droplets atomization brings to a more rapid decrease of shock wave velocity as compared to accounting for thermos-mechanical interaction and evaporation (curve 4). The explanation of this fact is the following: atomization of droplets brings to formation of new free surface, which increases momentum and energy exchange between gas and condensed phase; small droplets decelerate and warm up much faster, thus, increasing the momentum and energy losses in the gas flow, which brings to shock wave slowing down. When combustion is taken into account (curve 5), near the right end of the pipe, the wave velocity increases to 1500–1700 m/s, which corresponds to the ignition of the gas mixture and the transition of combustion to the detonation mode. In Fig. 4, the results correspond to the intensity of the shock wave P1 /P0 = 50. The general character of the shock wave intensity decrease in the aerosol-filled area remains similar to the previous results, but when taking combustion into account (curve 5), the transition to detonation occurs at a distance of 1.2–1.5 m from the beginning of the pipe; after that, the wave intensity decreases slightly. This behavior can be interpreted as the transition of deflagration to detonation via an overdriven regime with successive slowing down to a self-sustained stationary velocity.

8

N. N. Smirnov

Figure 5 corresponds to initial pressure ratio P1 /P0 = 75, and Fig. 6 corresponds to initial pressure ratio P1 /P0 = 100. From these figures, we can see the nature of the decrease in the intensity of the initial shock wave for interaction models of various complexity. When combustion is taken into account (curve 5), the acceleration of combustion and the transition to detonation occurs at 1.2–1.3 m (P1 /P0 = 75) – at 0.7–1.1 m (P1 /P0 = 100). Figure 6 shows that the detonation propagates in an overdriven mode (strong detonation) at 1.2–1.3 m, and then slows down to the stationary mode (Chapman-Jouget detonation) Fig. 7 shows examples for droplets fragmentation scenarios on interaction with shock waves moving from left to right.

Fig. 2. Evolution of shock wave velocity along the tube axis on entering dispersed mixture for different models of gas – droplet interactions. P1 /P0 = 10.

Fig. 3. Evolution of shock wave velocity along the tube axis on entering dispersed mixture for different models of gas – droplet interactions. P1 /P0 = 25.

Poly-dispersed Droplets in Streaming Flows

9

Fig. 4. Evolution of shock wave velocity along the tube axis on entering dispersed mixture for different models of gas – droplet interactions. P1 /P0 = 50.

Fig. 5. Evolution of shock wave velocity along the tube axis on entering dispersed mixture for different models of gas – droplet interactions. P1 /P0 = 75.

Fig. 6. Evolution of shock wave velocity along the tube axis on entering dispersed mixture for different models of gas – droplet interactions. P1 /P0 = 100.

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Fig. 7. An example of shock wave-droplet interaction.

3 Conclusions Investigating different models for gas – droplet interaction made it possible evaluating the role of different effects on shock wave behavior: mechanical losses, thermal losses, evaporation, of droplets, atomization, ignition and burning of fuel vapor. It was demonstrated that when the shock wave enters the area filled with aerosol, the wave speed decreases. The most intensive speed reduction occurs when only resistance and heat transfer are taken into account, the least intensive wave speed reduction – when only resistance is taken into account. Accounting for evaporation brings to lower reduction of shock wave velocity as compared with accounting for only thermal and mechanical losses. The reason is the following: evaporation brings to formation of additional gas volume behind the shock wave, which supports its propagation. Accounting for droplets atomization brings to a more rapid decrease of shock wave velocity as compared to accounting for thermos-mechanical interaction and evaporation. The explanation of this fact is the following: atomization of droplets brings to formation of new free surface, which increases momentum and energy exchange between gas and condensed phase; small droplets decelerate and warm up much faster, thus, increasing the momentum and energy losses in the gas flow, which brings to shock wave slowing down. When combustion is taken into account, after some period of deceleration in the aerosol, the shock wave velocity increases to 1500–1700 m/s, which corresponds to the ignition of the gas mixture and the transition of deflagration to the detonation mode. The transition of deflagration to detonation takes place via an overdriven regime (strong detonation mode characterized by an elevated velocity and pressure) with successive slowing down to a self-sustained stationary velocity of the Chapman-Jouget mode. Acknowledgements. The author gratefully acknowledges financial support from the Russian Foundation for Basic research (Grant 20–03-00297).

Poly-dispersed Droplets in Streaming Flows

11

References 1. Smirnov, N.N., Nikitin, V.F., Legros, J.C.: Ignition and combustion of turbulized dust – air mixtures. Combust. Flame 123(1/2), 46–67 (2000) 2. Smirnov, N.N., Nikitin, V.F., Khadem, J., Aliari Shourekhdeli, S.: Onset of detonation in polydispersed fuel-air mixtures. Proc. Combust. Inst. 31, 832–841 (2007) 3. Smirnov, N.N.: Combustion and detonation in multiphase media. Initiation of detonation in dispersed-film systems behind a shock wave. Int. J. Heat Mass Transfer 31(4), 779–793 (1988) 4. Khadem, J.: Ignition, combustion and detonation in poly-dispersed fuel – air mixtures. In: Dissertation. Lomonosov State University, Moscow M.V. (2005) 5. Betelin, V.B., Smirnov, N.N., Dushin, V.R., Nikitin, V.F., Kushnirenko, A.G., Nerchenko, V.A.: Evaporation and ignition of droplets in combustion chambers modeling and simulation. Acta Astronaut. 70, 23–35 (2012) 6. Dushin, V.R., et al.: Mathematical simulation for non-equilibrium droplet evaporation. Acta Astronaut. 63, 1360–1371 (2008) 7. Tyurenkova, V.V.: Non-equilibrium diffusion combustion of a fuel droplet. Acta Astronaut. 75, 78–84 (2012) 8. Tyurenkova, V.V., Smirnova, M.N., Nikitin, V.F.: Two-phase fuel droplet burning in weightlessness. Acta Astronautica 176, 672–681 (2020)

Fluid Mechanics

Viscoplastic Fluid Flow in a T-shaped Channel Under Given Pressure Boundary Conditions O. A. Dyakova

and O. Yu. Frolov(B)

National Research Tomsk State University, Tomsk, Russia

Abstract. A numerical simulation of a non-Newtonian incompressible fluid flow in a planar T-shaped channel is performed. A mathematical basis of the problem includes the momentum and continuity equations written in the dimensionless form. The rheology of the fluid is specified by the Shvedov-Bingham law. No-slip boundary conditions are assigned on the solid walls. The motion of the fluid is caused by a given pressure difference between inlet and outlet boundaries. The flow is assumed to be laminar and steady. A numerical solution to the problem is obtained using the finite volume method and the SIMPLE procedure. The rheological equation is regularized to provide both a finite value of the viscosity in the regions of low strain rates and the stability of the computational algorithm. Hydrodynamic characteristics of the viscoplastic fluid flow are studied at various dimensionless criteria and pressure values assigned at the inlet/outlet sections. The boundaries of the unyielded regions, occurring in the flow, are determined. The flow patterns with the unyielded region overlapping one of three boundary sections of the channel are presented. Keywords: Laminar flow · Non-Newtonian fluid · T-shaped channel · Boundary conditions · Numerical simulation

1 Introduction Pipeline systems, which are used to transport liquids and gases in different technologies, consist of many elements. One of the elements is a T-shaped channel serving for branching and mixing the flows. Studies on the flows in such structural elements are of practical importance for irrigation systems, chemical and petroleum industries, and other industries [1, 2]. Over the past decade, a large number of the results related to a fluid flow in T-microchannels have been obtained. These data are used in biomedicine for transportation of nanoparticles, bacteria, DNA molecules, as well as in a cooling technique for microelectronic devices [3, 4]. One of the main features of a T-shaped channel is that the design includes three boundaries which can serve either as inlet or as outlet sections. There is no commonly accepted opinion on the preferred conditions to be preassigned on the inflow/outflow boundaries of the channel of such geometry, neither from mathematical nor from physical standpoints [5]. By present time, many studies on the Newtonian [6–11] and non-Newtonian © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 15–22, 2022. https://doi.org/10.1007/978-3-030-92144-6_2

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[11–15] fluid flows in T-shaped channels with a given velocity profile have been carried out. However, in many cases the velocity profile at the inflow/outflow boundaries is unknown. Therefore, it is more advantageous to specify pressure values instead of velocity profile, and afterwards to determine the flow rate and flow directions. Nowadays, a limited number of works devoted to viscous fluid flows use pressure as boundary conditions [16–22]. In this paper, a steady flow of an incompressible Shvedov-Bingham viscoplastic fluid in a planar T-shaped channel under the given pressure boundary conditions is considered. The problem is solved numerically using a self-developed software package. The computed solution is analyzed depending on main parameters of the problem.

2 Problem Formulation A laminar steady flow of an incompressible viscoplastic fluid in a planar T-shaped channel is considered. Figure 1 demonstrates the flow region geometry; the origin of the Cartesian coordinate system is located at point A. The fluid flows into or out of the channel through boundary sections AM, FE, and BC under the given pressure drop. Mathematical formulation of the problem includes the momentum and continuity equations written in the dimensionless vector form, as follows: (u · ∇)u = −∇p + ∇ · (2ηE),

(1)

∇ · u = 0.

(2)

Here, u = (u, v) is the dimensionless velocity vector, p is the dimensionless pressure, and E is the dimensionless strain-rate tensor.

Fig. 1. Flow region

The rheological behavior of viscoplastic fluids is described by the Shvedov-Bingham model. The corresponding apparent viscosity in the dimensionless form is determined

Viscoplastic Fluid Flow in a T-shaped Channel

17

by the following expression: η=

Bn + A , A

(3)

 0.5 where A = 2eij eji is the dimensionless intensity of the the strain-rate tensor, eij =  ∂uj τ0 L 1 ∂ui 2 ∂xj + ∂xi are the components of the strain-rate tensor, and Bn = μU0 is the Bingham number. At Bn = 0, the Shvedov-Bingham model describes the rheological behavior of the Newtonian fluid. The following values are used to scale the length and the velocity: L (the width of boundary section AM) and U0 = μ/ρL, respectively. The dimensionless pressure is calculated by formula p=

P − PFE . μ2 /ρL2

(4)

Here, μ is the plastic viscosity of the fluid, τ0 is the yield stress, ρ is the density of the fluid, P is the dimensional pressure, PFE is the dimensional pressure specified at boundary FE. An essential feature of the Shvedov-Bingham model is the presence of a “yield stress” [23], i.e. a critical value of the shear stress below which the fluid retains a rigid structure and moves like a solid. The region of this motion is termed “unyielded”. When the yield stress is exceeded, the destruction of the solid structure immediately occurs, and the fluid moves in accordance with Newton’s law of viscosity. On the solid walls, no-slip boundary conditions are imposed: u = 0, v = 0.

(5)

In boundary sections AM, FE, and BC, zero tangential components of the velocity vector and the given pressure values are specified. These boundary conditions can be written in terms of dimensionless variables as follows: v = 0, pAM = p1 , x = 0, 0 ≤ y ≤ 1; u = 0, pFE = 0, L1 ≤ x ≤ L1 + 1, y = L3 + 1; v = 0, pBC = p3 , x = L1 + L2 + 1, 0 ≤ y ≤ 1.

(6)

Here, L 1 , L 2 , and L 3 are the dimensionless geometric parameters of the flow region (Fig. 1). The considered T-shaped channel has branches of unit width and of the same length (L 1 = L 2 = L 3 = 3). For such a problem formulation, the flow characteristics are dependent on the Bingham number and on the pressure values p1 and p3 assigned at boundaries AM and BC, respectively.

3 Numerical Method and Verification A self-developed software package is used to solve the formulated problem. A finite volume method [24] is applied to rewrite the initial system of equations in the discrete

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O. A. Dyakova and O. Yu. Frolov

differential form. The correction of the velocity and pressure fields is carried out utilizing the SIMPLE procedure [25]. Steady fields of the velocity and pressure are obtained using the false transient method [26]. The Shvedov-Bingham model has a feature of “infinite” apparent viscosity as the intensity of the strain-rate tensor approaches zero (A → 0). Using a through calculation of the viscoplastic fluid flow without explicit separating of unyielded regions, the rheological model is regularized to eliminate the singularity in the regions of zero values of A. In this study, a modified rheological model [27] is utilized according to which the apparent viscosity is defined by formula √ Bn + A2 + ε2 , (7) η= √ A2 + ε2 where ε is the small regularization parameter. The unyielded regions are distinguished in the flow by the following expression: ηA ≤ Bn,

(8)

which represents a dimensionless analog of the condition for separation of the flow regions where the shear stress is below the yield stress. During the software development, test calculations have been carried out. The results of the approximation convergence verification for the computational algorithm are presented in [28]. A comparison with experimental and numerical data of other authors is also shown in [28].

4 Results and Discussion Figure 2 demonstrates characteristics of the flow at p1 = 300, p3 = 160, and Bn = 1. According to Fig. 2 (a), illustrating streamline distributions, the fluid flows into the channel through section AM and flows out through section FE; a negative value of the flow rate corresponds to the case when the fluid flows out of the T-shaped channel. There is no flow in the right branch of the T-shaped channel due to a dead zone formation in this part of the region where the shear stress is below the yield stress. As a consequence, the flow rate through boundary section BC is equal to zero. One-dimensional motion of the viscoplastic fluid with a fully-developed velocity profile is observed in the vicinity of boundaries AM and FE. The flow structure and the viscosity field at p1 = 300, p3 = 160, and Bn = 1 are shown in Fig. 3. The unyielded regions are painted in black and formed in each branch of the T-shaped channel. The unyielded regions in the left and middle branches are of the same size, and the largest one is observed in the right branch. Further results are related to a study of the influence of the viscoplastic fluid rheology on the flow pattern in the T-shaped channel. It is found that an increase in the Bingham number leads to the growth of the unyielded regions. Figure 4 and Fig. 5 demonstrate the flow structures observed for the Bingham number varying in the range of 0 ≤ Bn ≤ 8. Figure 4 (a) shows that there is no unyielded region at Bn = 0, when the fluid flows in accordance with Newton’s law of viscosity. An unyielded region is formed in

Viscoplastic Fluid Flow in a T-shaped Channel

19

Fig. 2. Characteristics of the flow at p1 = 300, p3 = 160, Bn = 1: (a) – the streamlines, (b) – the field of pressure p, (c) – the field of velocity u, and (d) – the field of velocity v

Fig. 3. Flow structure and a viscosity field at p1 = 300, p3 = 160, Bn = 1

the right branch of the T-shaped channel with an increase in the Bingham number, and it completely overlaps this part of the channel at Bn = 1. As a consequence, a viscoplastic plug is observed in the right branch of the channel. The flow structure at Bn > 1 is presented in Fig. 5. Further increase in the Bingham number leads to the formation of two unyielded regions in the left and middle branches of the channel, where the one-dimensional motion of the fluid is observed. The higher the Bingham number is, the larger unyielded regions occur; while the dead zone boundaries remain the same in the right branch of the T-shaped channel.

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Fig. 4. Flow structures at p1 = 300, p3 = 160: (a), (b), (c), (d) – Bn = 0, 0.25, 0.5, and 1

Fig. 5. Flow structures at p1 = 300, p3 = 160: (a), (b), (c) – Bn = 2, 4, and 8

5 Conclusion In this work, a numerical simulation of a steady laminar flow of an incompressible Shvedov-Bingham viscoplastic fluid in a planar T-shaped channel under the given pressure boundary conditions has been carried out.

Viscoplastic Fluid Flow in a T-shaped Channel

21

The effect of dimensionless parameters (the Bingham number, the pressure values p1 and p3 specified at boundaries AM and BC, respectively) on the flow pattern as well as on the kinematic and dynamic characteristics of the flow has been studied. The flow patterns with the formation of unyielded regions and a dead zone in one of the channel branches have been demonstrated at the Bingham number varying from 0 to 8. Acknowledgements. The research is implemented at the expenses of the Russian Science Foundation (project No. 18–19-00021).

References 1. Grace, J.L., Priest, M.S.: Division of flow in open channel junctions. Engineering Experiment Station Auburn, Alabama (1958) 2. Neofytou, P., Housiadas, C., Tsangaris, S.G., Stubos, A.K., Fotiadis, D.I.: Newtonian and power-law fluid flow in a T-junction of rectangular ducts. Theoret. Comput. Fluid Dyn. 28(2), 233–256 (2014) 3. Kovalev, A.V., Yagodnitsyna, A.A., Bilsky, A.V.: Flow hydrodynamics of immiscible liquids with low viscosity ratio in a rectangular microchannel with T-junction. Theor. Found. Chem. Eng. 352, 120–132 (2018) 4. Lobasov, A.S., Minakov, A.V., Rudyak, V.Y.: Flow modes of non-Newtonian fluids with power-law rheology in a T-shaped micromixer. Theor. Found. Chem. Eng. 52(3), 393–403 (2018) 5. Sani, R.L., Gresho, P.M.: Résumé and remarks on the open boundary condition minisymposium. Int. J. Numer. Meth. Fluids 18(10), 983–1008 (1994) 6. Liepsch, D., Moravec, S., Rastogi, A.K., Vlachos, N.S.: Measurement and calculations of laminar flow in a ninety degree bifurcation. J. Biomech. 15(7), 473–485 (1982) 7. Khodadadi, J.M., Nguyen, T.M., Vlachos, N.S.: Laminar forced convective heat transfer in a two-dimensional 90 bifurcation. Numer. Heat Transf. 9, 677–695 (1986) 8. Neary, V.S., Sotiropoulos, F.: Numerical investigation of laminar flows through 90-degree diversions of rectangular cross-section. Comput. Fluids 25(2), 95–118 (1996) 9. Shamloo, H., Pirzadeh, B.: Investigation of characteristics of separation zones in T-junctions. WSEAS Transaction on Mathematics. 7(5), 303–312 (2008) 10. Beneš, L., Louda, P., Kozel, K., Keslerová, R., Štigler, J.: Numerical simulations of flow through channels with T-junction. Appl. Math. Comput. 219(13), 7225–7235 (2013) 11. Miranda, A.I.P., Oliveira, P.J., Pinho, F.T.: Steady and unsteady laminar flows of Newtonian and generalized Newtonian fluids in a planar T-junction. Int. J. Numer. Meth. Fluids 57(3), 295–328 (2008) 12. Khodadadi, J.M., Vlachos, N.S., Liepsch, D., Moravec, S.: LDA measurements and numerical prediction of pulsatile laminar flow in a plane 90-degree bifurcation. J. Biomech. Eng. 110(2), 129–136 (1988) 13. Khodadadi, J.M.: Wall pressure and shear stress variations in a 90-deg bifurcation during pulsatile laminar flow. J. Fluids Eng. 113(1), 111–115 (1991) 14. Matos, H.M., Oliveira, P.J.: Steady and unsteady non-Newtonian inelastic flows in a planar T-junction. Int. J. Heat Fluid Flow 39, 102–126 (2013) 15. Khandelwal, V., Dhiman, A., Baranyi, L.: Laminar flow of non-Newtonian shear-thinning fluids in a T-channel. Comput. Fluids 108, 79–91 (2015) 16. Kuznetsov, B.G., Moshkin, N.P., Smagulov, S.: Numerical investigation of viscous incompressible fluid flows in channels of complicated geometry with given pressure. Proc. Inst. Theor Appl. Mech. Siberian Branch USSR Acad. Sci. 14(15), 87–99 (1983)

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17. Hayes, R.E., Nandakumar, K., Nasr-El-Din, H.: Steady laminar flow in a 90 degree planar branch. Comput. Fluids 17(4), 537–553 (1989) 18. Heywood, J.G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids 22(5), 325–352 (1996) 19. Kelkar, K.M., Choudhury, D.: Numerical method for the prediction of incompressible flow and heat transfer in domains with specified pressure boundary conditions. Numer. Heat Transfer, Part B 38(1), 15–36 (2000) 20. Fernandez-Feria, R., Sanmiguel-Rojas, E.: An explicit projection method for solving incompressible flows driven by a pressure difference. Comput. Fluids 33(3), 463–483 (2004) 21. Barth, W.L., Carey, G.F.: On a boundary condition for pressure-driven laminar flow of incompressible fluids. Int. J. Numer. Meth. Fluids 54(11), 1313–1325 (2007) 22. Moshkin, N., Yambangwi, D.: Steady viscous incompressible flow driven by a pressure difference in a planar T-junction channel. Int. J. Comut. Fluid Dyn. 23(3), 259–270 (2009) 23. Balmforth, N.J., Frigaard, I.A., Overlez, G.: Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu. Rev. Fluid Mech. 46, 121–146 (2014) 24. Klapp, J., Medina, A., Cros, A., Vargas, C.A.: Fluid Dynamics in Physics, Engineering and Environmental Applications. Springer-Verlag, Berlin (2013) 25. Patankar, S.: Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York (1980) 26. Godunov, S.K., Ryabenkii, V.A.: Difference Schemes. Nauka, Moscow (1977).(in Russian) 27. Bercovier, M., Engelman, M.: A finite-element method for incompressible non-Newtonian flows. J. Comput. Phys. 36(3), 313–326 (1980) 28. Borzenko, E.I., Dyakova, O.A., Shrager, G.R.: Laminar power-law fluid flow in a T-shaped channel at given pressure differences. Fluid Dyn. 54(4), 501–509 (2019)

Mathematical Modeling of the Flow of Viscous Incompressible Fluid with Suspended Particles in Flat Inclined Channel Regina Iulmukhametova(B) , Airat Musin, and Liana Kovaleva Bashkir State University, Ufa, Russian Federation

Abstract. In this paper, we simulate the flow of a disperse system, consisting of solid spherical particles and viscous incompressible fluid, in a flat channel at different angles of inclination. The mathematical model written in the one-fluid approximation includes the equation of continuity for mixture, the equation of motion of the mixture, and the balance equation in the form of a convective-diffusion equation for the transfer of the volume concentration of particles. The system of equations of the mathematical model was solved with the control volume method in the OpenFoam software package. It was found that the limiting liquid injection rate into the channel, at which the channel is filled with a solid phase, depends on the angle of inclination according to the harmonic law. Keywords: Disperse system · Single-fluid approximation flow · Solid particles · Flat inclined channel

1

· Suspension

Introduction

A flow of suspensions in the channel is found in many areas of industry. For example, when applying one of the most popular stimulation technologies - hydraulic fracturing. The issue of transporting proppant in a fracture is associated with gravitational deposition of proppant, using the correct fluid to transport it deep into the fracture, leakage of fluid from the fracture to the formation, and plugging of the fractures [1]. However, despite their widespread use, the properties of suspensions still can’t be properly predicted using a single numerical model. In the case where the suspension contains particles with a density higher than liquid, they tend to settle and accumulate along the length of the channel. When working with such suspensions, it is necessary to take into account many variables, such as flow characteristics, flow behavior in channels of various geometries, as well as particle concentration, shape, size, and size distribution [2]. For concentrated suspensions, during numerical modeling, it is also important to take into account phenomena such as the interaction between particles, between particles and walls, particle migration, flow regime, as well as lifting forces. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 23–32, 2022. https://doi.org/10.1007/978-3-030-92144-6_3

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Considering approaches to modeling the flow of suspensions, two approaches can be distinguished: single-fluid and two-continuum [3–5]. The differences between the one-liquid model and the two-continuum model are significant. However, for most practical purposes, solving a complete system of two-continuum models not only requires computational costs but is also not necessary. The authors of [5] noted that when stable flows are characterized by a short dynamic particle relaxation time compared to hydrodynamic time or a small Stokes number, particle transport has a diffusion character. The diffusion model of the suspension is applicable for modeling various laminar flows of the suspension in the Stokes approximation.

2 2.1

Numerical Research Problem Statement

We study the flow of viscous incompressible fluid with suspended solid spherical particles in flat inclined channel. It is believed that the fluid is incompressible, solid spherical particles have the same size and shape, flow is laminar. Figure 1 shows a schematic of the computational domain.

Fig. 1. Computational domain

2.2

Mathematical Model

The mathematical model written in the one-fluid approximation, includes the equation of continuity for mixture, equation of motion of the mixture, the balance equation in the form of a convective-diffusion equation for the transfer of the volume concentration of particles [5,6]: ∂ρ + ∇ · (ρu) = 0 ∂t ∂(ρu) + ∇ · (ρuu) = −∇p + ρg + ∇ · Σ − ∇ · ∂t

(1) 

ρp ρf (1 − C)Cur ur ρ

 (2)

Mathematical Modeling of the Flow of Viscous Incompressible Fluid

25

∂C + ∇ · (Cu) = −∇ · (C(1 − fp )ur ) (3) ∂t where ρ = ρp C + ρf (1 − C) – mixture density; u – mixture velocity; p – average mixture pressure; C – volume concentration; fp = ρp C/ρ – mass fraction of dispersed phase; g = (−g sin α, 0, −g cos α) – gravity acceleration; α – angle of inclination; Σ = μ[∇u + (∇u)T − 2/3(∇ · u)I] – stress tensor in mixture; ur – relative velocity between phases; μ - effective viscosity ratio of the mixture; I – identity matrix. Hereinafter index p denotes parameters related to solid dispersed phase, f - to liquid continuous phase. The coefficient of effective viscosity of the mixture is calculated according to the empirical relationship proposed by Krieger [7]:  μ = μf

1−

−β

C

(4)

Cmax

where μf – dynamic viscosity coefficient of liquid continuous phase; Cmax – ultimate packing density of particles; β – empirical coefficient. Relative velocity between phases is determined by the formula [5]: ur = f h (ust −

d2 d2 (ρp − ρf ) du + ∇ · Σp ) 18μf dt 18μf C

ust =

d2 (ρp − ρf )g 18μf

(6)

∇ · Σp = −γ∇(μf an ) − Kγ μf an ∇γ  Kγ =

Kη 2− Kc

 1−

C Cmax

(5)

p +

Kη Kc

(7) (8)

where ust – sedimentation velocity according to Stokes; d – particle diameter; Σp – stress tensor in the medium of particles; γ – mixture flow shear rate; an = μ/μf − (1 + 2.5C) – empirical function; Kc and Kη – empirical coefficients, the ratio of which according to [8] is defined as Kc /Kη = 1.042C + 0.1142. The constrained sedimentation function is determined by the formula:   2   β C C h 1− (9) f = 1− Cmax Cmax Initial conditions: C(x, y, z, t = 0) = 0

(10)

u(x, y, z, t = 0) = 0

(11)

26

R. Iulmukhametova et al.

Boundary conditions For the volume concentration on solid walls, the condition of the absence of flow into the wall is set: ∂C =0 (12) ∂n A constant concentration at the channel entrance Cin is set: C(x, y, z, t = 0) = Cin

(13)

At the exit from the channel, a flow condition is specified. For the tangent component of the mixture velocity to the wall, the condition of partial slip on the wall is specified [5]:   C μ ∂u =u (14) βw d 1 − C Cmax μf ∂n where βw – slip parameter depending on the sphere radius [9]. For the normal component of the mixture velocity to the wall, the no-flow condition on the solid wall is specified: unw = 0

(15)

where nw – normal to the wall surface. Constant velocity at the entrance to the channel: u = (uin , 0, 0)

(16)

At the exit from the channel, the flow condition is set: ∂u =0 ∂n

3

(17)

Results

The problem in the three-dimensional formulation was solved numerically in an open-source CFD software OpenFOAM using the control volume method. To validate the program code, we compared the calculation results with experimental data [10]. The authors of the experiment studied fully developed profiles of particle velocity and concentration during suspension flow in a flat channel. Satisfactory agreement between results of numerical calculation and experiment was obtained. The results of the numerical simulation were obtained with the following system parameters: channel length – 0.5 m, width – 0.002 m, height – 0.04 m, 3 3 ρf = 1000 kg/m , μf = 0.001 Pa×s, ρp = 2650 kg/m , d = 0.0004 m, Cin = 0.3, Cmax = 0.64, parameter p = 8, β = 1.4. Comparative modeling of the distribution dynamics of the solid spherical particles concentration the fluid flow and their sedimentation along the channel,

Mathematical Modeling of the Flow of Viscous Incompressible Fluid

27

depending on the fluid injection rate (uin ) and the angle of inclination of the channel relative to the horizon (α) (Fig. 1), has been carried out. The value of the liquid injection rate into the channel was chosen so that the flow rate in the channel was comparable to the Stokes sedimentation rate (6). The channel tilt angle varied from 0 to π/2. The tilt angle 0 corresponds to the case when the injection velocity vector is perpendicular to the gravity vector, and the tilt angle π/2 corresponds to the case when these vectors are oppositely directed. The simulation showed that at low injection rate the fluid in the channel at a certain non-zero angle of inclination the particles stop moving along the channel, which is associated with excess gravitational particle sedimentation rate on the flow rate of the mixture. Figure 2 shows the graphs of the distribution of the concentration of solid spherical particles along the center of the channel at an injection rate of uin = 0.02 m/s at tilt angles a) 0, b) π/6 and c) π/3, plotted at the time t = 40 s. It can be seen from the figure that in the horizontal channel (Fig. 2a) during this time the particles reach the end of the channel. The height of the formed sediment decreases almost linearly from the entrance to the channel to its exit and practically doesn’t change over time. At the channel tilt angle π/6, a different picture is observed (Fig. 2b). The left part of the channel is almost completely occupied by the sediment; during this time, the particle front reaches only the middle of the channel. This is because this injection rate is comparable to the sedimentation rate of particles and the particles hardly overcome gravitational forces. In the left part of the channel, an area with a low concentration of particles, in which liquid movement is possible, is observed only in the near-wall region in the upper part of the channel. In the lower part, the concentration is close to the limit value of close packing. With an increase in the channel tilt angle to π/3 (Fig. 2c), the situation is aggravated. During the considered time, the particles have time to overcome only a small part of the path and accumulate at the entrance to the channel.

Fig. 2. Distribution of concentration of solid particles along the length of the channel at uin = 0.02 m/s at an angle of inclination: a - 0; b - π/6; c - π/3;

28

R. Iulmukhametova et al.

At higher injection rates, the pattern of sediment distribution in the channel changes significantly. Figure 3 shows the graphs of the distribution of the concentration of solid spherical particles along the center of the channel at different angles of inclination of the channel at an injection rate of 0.04 m/s, which is twice the value of the rate at which the previous results were obtained. Accordingly, the time at which the distributions were plotted was chosen two times less and is t = 20 s. It can be seen that in the horizontal channel the pattern of sediment distribution along the channel (Fig. 3a) practically doesn’t differ from the results obtained at a lower injection rate (Fig. 2a). The height of the formed sediment decreases almost linearly from the entrance to the channel to its exit. At an angle of inclination of the channel π/6, the pattern of the sediment distribution (Fig. 3b) is also identical to the pattern obtained at a lower velocity (Fig. 2b). However, at large tilt angles of the channel (π/3), a significantly different picture of the sediment distribution is observed (Fig. 3c). In the lower part of the channel, a small area with sediment is formed, the concentration of which is close to the limiting one. The area of low concentration in the upper part of the channel is wider than at lower tilt angles. In this case, the displacement front in this case slightly lags behind the front at an angle of π/6 and reaches almost the middle of the channel (Fig. 3c), which is much further than at an injection rate of 0.02 m/s (Fig. 2c).

Fig. 3. Distribution of concentration of solid particles along the length of the channel at uin = 0.04 m/s at an angle of inclination: a - 0; b - π/6; c - π/3;

This is due to the fact that the flow rate in the channel is much higher than the rate of gravitational sedimentation of particles. In addition, in this case, a significant blurring of the displacement front is observed. The structure of the front has the shape of an incident wave; in the upper and lower parts of the channel, the front of particles leads the front in the middle of the channel. For clarity, Fig. 4 shows the concentration distribution along the channel in the upper, lower and middle parts of the channel, plotted for the tilt angles π/6 and π/3 at an injection rate of 0.04 m/s. It is seen that at an angle of π/6, the front of particles in the lower part of the channel leads the front in the middle and upper parts, and at an angle of π/3, the front in the upper part of the

Mathematical Modeling of the Flow of Viscous Incompressible Fluid

29

channel begins to lead. This is due to the fact that at high velocities with an increase in the angle of inclination of the channel behind the displacement front, a vortex flow of the liquid arises, which increases with an increase in the injection rate.

Fig. 4. Concentration distribution along the channel in the upper (line), lower (dashed line with dot) and middle (dashed line) parts of the channel at uin = 0.04 m/s at an angle of inclination: a - π/6; b - π/3;

Figure 5 shows the velocity field and streamlines in the channel at different slope angles (0, π/6 and π/3) at an injection rate of 0.04 m/s, which corresponds to the concentration distribution pattern in Fig. 3.

Fig. 5. Velocity field and streamlines for uin = 0.04 m/s at an angle of inclination: a 0; b - π/6; c - π/3;

It can be seen that in the horizontal channel (Fig. 5a) streamlines are directed mainly along the channel. The vortex flow occurs only at the entrance to the channel due to the formation of a particle roll and doesn’t have a strong effect on the formation of the particle front in the depth of the channel. With an increase in the angle of inclination up to π/6 (Fig. 5b), the size of the vortex arising at the entrance to the channel increases.

30

R. Iulmukhametova et al.

The flow velocity of the mixture in the upper part of the channel becomes noticeably higher than in the rest of the channel due to the formation of a layer with a dense packing of particles, in which there is practically no movement. At an angle of inclination π/3, the vorticity of the flow behind the displacement front is observed, which leads to the blurring of the front, which is observed in Fig. 3c. During computational studies, it was found that the limiting rate of liquid injection into the channel at which the channel begins to fill with a solid phase depends on the angle of inclination according to the harmonic law and is described by the following equation: uin = uπ/2 · sin(α)

(18)

where uπ/2 is the limiting velocity at which the filling of the channel with a solid phase begins for the angle π/2. Figure 6 shows this dependence. It can be seen that with an increase in the angle of inclination, the limiting velocity increases and has a maximum value at an angle π/2, which corresponds to the movement of the liquid against the action of gravity. Since the channel is symmetric about the center, it is quite obvious that the curve in Fig. 6 can be extended symmetrically to the angle π. The part of the curve from π/2 to π will be described by the same dependence and correspond to the channel tilt to the other side.

Fig. 6. Dependence of the limiting liquid injection rate into the channel at which the channel starts filling with the solid phase on the channel tilt angle

4

Conclusion

The mathematical model of the laminar flow for incompressible fluid with suspended particles in the one-fluid approximation in the Open Foam software

Mathematical Modeling of the Flow of Viscous Incompressible Fluid

31

package was implemented. The algorithm has been tested on experimental data. Satisfactory agreement between results of numerical calculation and experiment was obtained. The study of the particles distribution dynamics in the fluid flow, as well as their sedimentation along the length of the channel, depending on the values of the fluid injection rate and the angle of inclination of the channel relative to the horizon was carried out. It is shown that at a high injection rate in a horizontal channel, the vortex flow arising at the entrance to the channel doesn’t have a strong effect on the formation of the front of particles in the depth of the channel. With an increase in the angle of inclination to π/6, the size of the vortex flow arising at the entrance of the channel increases. The flow rate of the mixture in the top of the channel becomes noticeably higher than in the rest of the channel where, due to the formation of a layer with a dense packing of particles, their movement is practically absent. When an angle of inclination equal to π/3, a vorticity of the flow behind the displacement front appears, which leads to a blurring of the front. It was found that at a low rate of liquid injection into the channel at a certain non-zero angle, the particles cease to move along the channel, which is associated with the excess of the gravitational sedimentation of particles over the mixture flow rate. It was also found that the limiting liquid injection rate into the channel, at which the channel is filled with a solid phase, depends on the angle of inclination according to the harmonic law. Acknowledgments. The reported study was funded by RFBR, project number 1931-90157.

References 1. Aksakov, A.V., et al.: Corporate fracturing simulator: from a mathematical model to software implementation. Oil Ind. 11, 35–40 (2016). (in Russian) 2. Sokovnin O.M., Zagoskin N.V., Zagoskin S.N. Hydrodynamics of Motion of Particles, Droplets and Bubbles in Non-Newtonian Fluids, p. 215. Nauka (2019). (in Russian) 3. Nigmatulin, R.I.: Dynamics of Multiphase Media, vol. I and II, p. 464. Nauka, Moscow (1987) 4. Boronin, S.A., Osiptsov, A.A.: Two-continuous model of a suspension flow in a hydraulic fracture. Rep. Acad. Sci. 431(6), 758–761 (2010). (in Russian) 5. Gavrilov, A.A., Shebelev, A.V.: Single-fluid model of a mixture for laminar flows of highly concentrated suspensions. Fluid Dyn. 53(2), 255–269 (2018) 6. Manninen, M., Taivassalo, V., Kallio, S.: On the mixture model for multiphase flow, p. 67. VTT Publications 288. Technical Research Centre of Finland (1996) 7. Krieger, I.M.: Rheology of monodisperse lattice. Adv. Colloid Interface Sci. 3, 111–136 (1972) 8. Tetlow, N., Graham, A.L., Ingber, M.S., Subia, S.R., Mondy, L.A., Altobelli, S.A.: Particle migration in a Couette apparatus: experiment and modeling. J. Rheol. 42, 307–327 (1998)

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9. Ingber, M.S., Graham, A.L., Mondy, L.A., Fang, Z.: An improved constitutive model for concentrated suspensions accounting for shear-induced particle migration rate dependence on particle radius. Int. J. Multiph. Flow 35, 270–276 (2009) 10. Lyon, M.K., Leal, L.G.: An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 1 monodisperse systems. J. Fluid Mech. 363, 25–56 (1998)

Heat Transfer and Wave Motion

Solitary Waves in Hyperelastic Tubes Conveying Inviscid and Viscous Fluid Vasily Vedeneev(B) Lomonosov Moscow State University, Moscow, Russia [email protected]

Abstract. We study possible steady states of an infinitely long tube made of a hyperelastic membrane and conveying either an inviscid, or a viscous fluid with power-law rheology. The tube model is geometrically and physically nonlinear; the fluid model is limited to smooth changes in the tube’s radius. For the inviscid case, we analyse the tube’s stretch and flow velocity range at which standing solitary waves of both the swelling and the necking type exist. For the viscous case, we show that a steadystate solution exists only for sufficiently small flow speeds and that it has a form of a kink wave; solitary waves do not exist. For the case of a semi-infinite tube (infinite either upstream or downstream), there exist both kink and solitary wave solutions. For finite-length tubes, there exist solutions of any kind, i.e. in the form of pieces of kink waves, solitary waves, and periodic waves.

Keywords: Tubes conveying fluid material · Non-Newtonian fluid

1

· Solitary wave · Hyperelastic

Introduction

Nonlinear waves in fluid-filled elastic tubes play an important role in problems of the cardiovascular system [5,33]. Solitary wave solutions are used for the analysis of pulse waves as well as for the study of the formation of aneurysms [1,7,14]. In connection with solitary waves, several experimental studies of bulge formation and propagation in elastic tubes have been conducted [18,28,31,38]. Theoretical analysis of nonlinear solitary waves in fluid-filled elastic tubes was initially performed with a number of simplified assumptions, such as neglecting axial displacement, weakly nonlinear waves, and a long-wave approximation [8,40], in applications to pulsatile blood flow in arteries. However, it was shown [6] that the exact equations of motion for a hyperelastic membrane tube conveying an inviscid fluid have two first integrals and, consequently, can be analysed directly without any additional assumptions. In particular, the existence of solitary waves was proved analytically for the exact membrane tube model. Using the exact model, steady solitary waves for different models of hyperelastic tube c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 35–68, 2022. https://doi.org/10.1007/978-3-030-92144-6_4

36

V. Vedeneev

material were studied [12]. The relation between the approximate and exact formulations and between the corresponding weakly and fully nonlinear solitary waves was established [9]. The bifurcation diagram shows that a travelling solitary wave separates at zero amplitude from a linear wave that has a finite travelling speed [23]. When the solitary wave amplitude grows, its travelling speed decreases, and the solitary wave branch ends, depending on the material model and properties, as either a standing solitary wave (i.e. a static localised bulge) or a kink wave [23]. A standing solitary wave solution can be considered to be a mathematical model of aneurysm in a blood vessel. As a more adequate description of aneurysm formation, an initial localised wall thinning was introduced into the tube model in [11] and [22], where bulged solutions and their stability were studied. It was shown that there are two types of bulged tube states, one with a smaller and one with a larger amplitude. For the vanishing tube imperfection, the first state tends to a uniform tube, while the second state tends to a standing solitary wave. It was shown that the first state is stable, while the second state is not. However, the presence of a fluid flow stabilises the standing solitary wave [10,21] so that both solutions can be stable. In this series of studies, the membrane tube was modelled by an exact, geometrically and physically nonlinear model, but the fluid model was simplistic, with the fluid assumed to be inviscid with a constant velocity distribution in each cross-section. With respect to biomechanical applications, the tube model also had several limitations. First, actual blood vessel properties [2] are anisotropic so that hyperelastic blood vessel models should include an anisotropic part [4,36]. Next, blood vessel walls are sufficiently thick so that their bending stiffness and corresponding shell effects (studied, e.g., in [26]) can be important. Finally, axisymmetry of deformations was assumed, which is not always the case in the cardiovascular system. Nevertheless, this series of studies is an important step towards understanding the nonlinear dynamics of real blood vessels. Another type of studies, that of fluid flow in collapsible tubes [17,19], deals with viscous fluids, but the tube model, in its 1-dimensional formulation, is either linear or extremely simplified nonlinear in the form of the ‘tube law’ [24,25,32,39]. Although they are more advanced in terms of fluid mechanics, this type of models does not admit of a bifurcation of a uniform tube to a bulged solitary wave solution. Also, all studies of collapsible tubes consider Newtonian fluid flows; however, it is known that blood in small vessels has essentially nonNewtonian rheology [3,13,16,27,30]. [41] improved the 1-dimensional fluid model to include non-Newtonian power-law rheology, which was used for the analysis of steady states [35] and stability [37] of linearly elastic tubes conveying fluid, under the assumption of long-wave and low-frequency motions. The goal of the present paper is to combine the two approaches used in the two series of studies, to analyse a geometrically and physically nonlinear hyperelastic membrane tube conveying a viscous non-Newtonian fluid. In Sect. 2, we upgrade the exact tube model [6] by including non-Newtonian fluid viscosity under the assumptions used in [41]. The rest of the paper deals with steady-state solutions of this system. In Sect. 3, we analyse the first integrals of the system

Solitary Waves

37

of equations and introduce the phase plane used in the subsequent analysis. Section 4 is devoted to solitary wave analysis, including a non-constant crosssection velocity distribution but neglecting fluid viscosity. In Sect. 5, we include fluid viscosity in the analysis. We prove that for the Gent tube material and sufficiently small fluid velocities, there exists a unique steady-steady solution for an infinitely long tube, of which the tube radius changes monotonically so that no solitary waves are possible. For semi-infinite tubes (infinite either upstream or downstream), a second solitary wave solution exists. For finite-length tubes, several solitary wave, periodic wave, and monotonic solutions exist. Section 6 summarises the results and concludes the paper.

Fig. 1. Cylindrical membrane tube in the initial and the deformed state.

Fig. 2. The undeformed dx and the deformed dx∗ element of the tube (a); fluid forces (b).

2 2.1

Equations of Motion of an Elastic Tube Containing a Flowing Viscous Fluid Formulation of the Problem and Preliminary Relationships

We consider a cylindrical membrane tube with a circular cross-section with a thickness of h and a radius of R, made of hyperelastic material (Fig. 1). The

38

V. Vedeneev

Fig. 3. Forces acting on a deformed shell element.

ratio h/R is sufficiently small for the bending stresses to be neglected compared to the membrane stresses. The tube conveys a non-Newtonian viscous fluid whose rheology obeys a power law. We restrict ourselves to axisymmetric motion with two components of the displacement vector, longitudinal (axial) u and radial w. [6] gave a self-contained derivation of the exact equations of motion for the case of an inviscid fluid. Here we briefly revisit this derivation to take fluid viscosity and rheology into consideration. The equations of motion are derived in Lagrangian cylindrical coordinates corresponding to the undeformed state of the tube. The x axis is directed along the tube’s axis, and the angle θ is the circumferential direction. The axial and circumferential length elements are denoted by dx and ds; the area element is dA = dxds (Fig. 1). In the deformed state, the lengths and area of the same elements, as well as other values, will be denoted by a star. In what follows, we will need a relationship between the length of the element dx before and after deformation and the angle ϕ between them. From Fig. 2a we have    w , dx∗ = (1 + u )2 + w2 dx, ϕ = arctan 1 + u where the prime denotes differentiation with respect to the Lagrangian coordinate x. The lengths of the element ds and the sizes of the area dA before and after deformation are related as    w w ds, dA∗ = (1 + u )2 + w2 1 + dA. ds∗ = 1 + R R 2.2

Equations of the Tube Motion

Consider fluid forces acting in the axial direction on a tube element with an undeformed area dA (Figs. 2b, 3a). After simple algebra, for pressure and friction forces we have   w w −p sin ϕ dA∗ = −pw 1 + dA, τ cos ϕdA∗ = τ (1 + u ) 1 + dA. R R

Solitary Waves

39

The force resulting from elastic tensile stress (Fig. 3a) is expressed as follows:   σ1 h∗ (1 + w/R)  F (x + dx) cos(ϕ + dϕ) − F (x) cos ϕ =  (1 + u ) dA, (1 + u )2 + w2 where F = σ1 h∗ ds∗ and σ1 is the longitudinal physical component of Cauchy stress. Here we assumed a uniform distribution of stresses over the tube thickness, which corresponds to the membrane model. ¨dA∗ = ρh¨ udA, where ρ is Balancing these forces with the inertial force ρ∗ h∗ u the material density, we obtain the longitudinal equation of motion:       ∗ w σ h (1 + w/R) w 1 +τ (1+u ) 1 + +  (1 + u ) . (1) ρh¨ u = −pw 1 + R R (1 + u )2 + w2 Similarly, for fluid forces acting in the radial direction (Figs. 2b, 3a), we obtain   w w dA, τ sin ϕdA∗ = τ w 1 + dA. p cos ϕ dA∗ = p(1 + u ) 1 + R R The radial force from elastic stresses is the sum of the longitudinal stress at ends x and x + dx (Fig. 3a),   w ∗ F (x + dx) sin(ϕ + dϕ) − F (x) sin ϕ = σ1 h (1 + w/R)  dA, (1 + u )2 + w2 and the circumferential stress at ends s and s + ds (Fig. 3b),  (1 + u )2 + w2 ∗ dA, −G(θ + dθ) sin(dθ/2) − G(θ) sin(dθ/2) = −σ2 h R where σ2 is the circumferential physical component of the Cauchy stress. Balancing the radial elastic and fluid forces with the intertial force ¨ = ρhdAw, ¨ we obtain the radial equation of motion: ρ∗ h∗ dA∗ w   w w + τ w 1 + ρhw ¨ = p(1 + u ) 1 + R R     (1 + u )2 + w2 w ∗ . (2) − σ 2 h∗ + σ1 h (1 + w/R)  R (1 + u )2 + w2 2.3

Equations of Fluid Motion

Next, we consider the motion of the fluid. We will assume that its rheology obeys the Ostwald-de Waele power law, which for pure shear reads  n dv1 12 τ =μ dx2

40

V. Vedeneev

and for the general case is τ ij = 2μ

√

n−1 2I2 (e) eij ,

I2 =

eij eij ,

(3)

where τ ij and eij are the components of the viscous stress tensor and strain rate tensor. The special case of n = 1 corresponds to a Newtonian viscous fluid, and μ = 0 corresponds to an ideal fluid. Assuming that the motion is slow (quasi-stationary) and that the wavelengths are large, a Poiseuille velocity distribution is established in each tube crosssection at each moment:     n+1 n r 3n + 1 , (4) 1− vx (x, r) = vf (x) n+1 R+w where vf (x) is the average velocity in the section. Under this assumption, the Navier-Stokes equations integrated over the cross-section give a spatially onedimensional system of equations, as shown by [41]. A different form of this system was obtained in [37]; Eqs. (2.10) and (2.11) of that paper in the present notations have the form ∂w R + w ∂vf ∂w + vf + = 0, ∂t ∂q 2 ∂q ∂vf 3n + 1 ∂vf 2n vf ∂w 1 ∂p + vf − + ∂t 2n + 1 ∂q 2n + 1 R + w ∂t ρf ∂q vfn μ 2(3n + 1)n = 0, + ρf nn (R + w)n+1

(5)

where q is the Eulerian coordinate of the tube axis. To have both tube and fluid equations formulated in the same coordinate system, we now switch to Lagrangian coordinates in the fluid equations (5). The transformation from Eulerian to Lagrangian coordinates is expressed as ∂f f = , ∂q 1 + u

u˙ ∂f = f˙ − f  ∂t 1 + u

for any function f . Substituting these into system (5), and slightly transforming the resulting equations, we obtain 1 w˙ + wu ˙  − w u˙ + vf w + (R + w)vf = 0, 2

ρf

  vf 3n + 1 2n vf vf − (w˙ + wu ˙  − w u) ˙ v˙ f + v˙ f u − vf u˙ + 2n + 1 2n + 1 R + w vfn 2(3n + 1)n + p + μ (1 + u ) = 0. nn (R + w)n+1

(6)

(7)

Solitary Waves

2.4

41

Expression for Viscous Friction τ

To close the system, let us obtain an expression for the viscous friction τ included in the equations of the tube motion. For the Poiseuille velocity distribution (4), the wall friction comes only from the component τ = τrx , which is expressed as  n ∂vx . τ =μ ∂r Using the distribution (4), we obtain the friction at the tube wall  n  n vf (x, t) 1 + 3n τ (x, t) = μ . R + w(x, t) n 2.5

(8)

Closed System of Equations

Hereunder we will assume the incompressibility of the tube material, which excludes the deformed thickness from Eqs. (1) and (2): h∗ = 

h (1 +

u )2

+ w2 (1 + w/R)

.

(9)

Then the system consisting of Eqs. (1), (2), (6), (7), and (8), supplemented by a hyperelastic model of the tube’s material, is a closed system of equations based on the geometrically and physically nonlinear theory of the membrane tube [6] and the approximate one-dimensional equations of motion of a power-law fluid derived under the assumptions used in [41] and [37]. Thus, this system generalises both the equations of [6] by taking into account the viscosity and rheology of the fluid and the equations of [41] and [37] by taking into account the geometric and physical nonlinearity of the tube. In particular, for n = 0 (a uniform velocity profile) and μ = 0 (an inviscid fluid), the system of equations coincides with the system of [6].

3

Steady-State Equations

Next, we will study the possible steady states of the tube conveying fluid, by setting all time derivatives equal to zero. Then the system consisting of Eqs. (1), (2), (6), and (7), taking into account Eq. (9), will take the form     σ1 h w w  + τ (1 + us ) 1 + + (1 + us ) = 0, (10) − pw 1 + R R (1 + us )2 + w2     w w w σ2 h  +τw 1 + + σ1 h = 0, (11) − p(1 + us ) 1 + 2 2 R R (1 + us ) + w R+w 1 vf w + (R + w)vf = 0, (12) 2 3n + 1 2τ ρf vf vf + p + (1 + us ) = 0, (13) 2n + 1 R+w where us (x) ≡ u (x) (the function u(x) itself is not present in the steady-state equations).

42

3.1

V. Vedeneev

Integration of the Fluid Equations

Note that Eq. (12) is integrated and gives the relationship between fluid velocity and radial displacement (conservation of fluid mass): vf (w) = vf 0

(R + w0 )2 . (R + w)2

With the use of this relationship, Eq. (13) is also integrated in the absence of viscosity (generalised Bernoulli equation):   4  R + w0 3n + 1 vf2 0 pinv (w) = p0 + ρf 1− . 2n + 1 2 R+w In the presence of viscosity, the pressure takes the form

x 2τ (1 + us )dx, f (x) = − p(x) = pinv (x) + f (x), x0 R + w

(14)

where f (x) is a monotonically decreasing function reflecting viscous pressure loss. 3.2

First Integrals of the Tube Equations

Let us now consider Eqs. (10) and (11). It is known that for an inviscid fluid, they have two first integrals [6,9]. Let us denote the principal (axial and circumferential) stretches: λ1 =



(1 + us )2 + w2 ,

λ2 = 1 +

w . R

Then these equations can be rewritten in the form   σ1 R τ (1 + u ) − p λ2 λ2 + (1 + us )λ2 = 0, s λ21 h h  σ1

λ2 λ21



p = pinv + f,

−

σ2 1 τ (1 + us )λ2 + λ2 λ2 = 0, +p 2 R λ2 Rh h   4  λ20 3n + 1 vf2 0 1− pinv = p0 + ρf . 2n + 1 2 λ2

(15)

(16)

Note that hereunder, we use the notations of [6] for the principal stretches, which differ from the notations of [9] in that the axial and circumferential stretches are switched. Next, we define a hyperelastic material model that for the incompressible case reads ∂W , i = 1, 2, σi = λi Wi , Wi = ∂λi

Solitary Waves

43

ˆ (λ1 , λ2 , (λ1 λ2 )−1 ) and W ˆ (λ1 , λ2 , λ3 ) is the strain energy where W (λ1 , λ2 ) = W function (see [12] for details). Representing the pressure in the form of the sum of ‘inviscid’ pressure pinv and the viscous pressure loss f , we integrate Eq. (15) and obtain   4  λ20 R λ22 R 3n + 1 vf2 0 2 W1 − ρf λ 1+ (1 + us ) − p0 = C1 (x) + F (x), λ1 h 2 h 2n + 1 4 2 λ2

C1 (x) = −

x x0



τ (1 + us )λ2 dx, h

F (x) =

x

x0

f

R λ2 λ2 dx. h

Let us consider in more detail the behaviour of F (x), integrating by parts:

F (x) =

f

R d h dx



λ22 2

 dx = f

R λ22 − h 2



f

R λ22 dx h 2

R λ22 τ =f + (1 + us )λ2 dx. h 2 h

It is seen that the first term can be combined with p0 on the left-hand side, and the second (integral) term is cancelled by the same term in C1 (x). Finally, we get   4  λ20 W1 R 3n + 1 vf2 0 2 R λ22 − ρf λ 1+ (1 + us ) − p0 (x) (17) = C1 , λ1 h 2 h 2n + 1 4 2 λ2 where C1 = const. Hence, the effect of viscous friction consists of a monotonic pressure decrease, p0 (x) = p0 + f (x), due to friction losses f (x). To derive the other first integral, consider the sum of Eq. (15) multiplied by (1 + us ) and Eq. (16) multiplied by R2 λ2 : 

   σ1 λ2 σ2 λ2 τ τ (1 + us ) (1 + us ) + σ1 2 R2 λ2 − = − (1 + us )2 λ2 − R2 λ2 2 λ2 . 2 λ1 λ1 λ2 h h

After simple algebra, the resulting equation is rewritten as τ σ1 − W  = − λ21 λ2 . h Integrating, we get

W − σ1 = C2 (x),

C2 (x) =

x

x1

τ 2 λ λ2 dx, h 1

(18)

where C2 (x) is a monotonously growing function. In the absence of friction, p0 (x) = const, C2 (x) = const, and expressions (17) and (18) are the first integrals. In the presence of friction, p0 (x) is a monotonously decreasing function, and C2 (x) is a monotonously increasing function; moreover,

44

V. Vedeneev

they themselves depend on the solution. Strictly speaking, expressions (17) and (18) are no longer the first integrals, but we will call them so for brevity. Both first integrals (17) and (18) have a clear physical meaning. Equation (17) reflects the conservation of the resultant force in the axial direction at each cross-section. Equation (18) is the equilibrium equation of a membrane element, in projection onto the deformed element; in particular, fluid pressure is not present in Eq. (18), because it is cancelled at the projection. Note that the fluid friction acts separately in these equations: only through pressure losses in Eq. (17) and only through increasing upstream (and decreasing downstream) traction force in Eq. (18). 3.3

Non-dimensionalisation and Transition to Variables λ1 and λ2

We proceed to dimensionless quantities by choosing the non-deformed tube radius R as the length scale, the shear modulus of the tube material G as the stress scale, and the fluid density ρf as the density scale. In addition, to get rid of the factors R/h, for pressure p and friction τ we choose the scale  P = Gh/R, and for the fluid speed vf the scale P/ρf . Also, we switch from the unknowns us (x) and w(x) to the principal stretches λ1 (x) and λ2 (x), which have the dimensionless form  λ1 = (1 + us )2 + w2 , λ2 = 1 + w. Expressing us and w through λ1 and λ2 , the first integrals can be rewritten in the form λ2

 = λ 1 1 −

1 W1 (λ1 , λ2 )2



3n + 1 vf2 0 2 λ2 λ p0 (x) 2 + 2 2n + 1 4 2



 1+

λ20 λ2

4 

2 + C1

, (19)

W (λ1 , λ2 ) − λ1 W1 (λ1 , λ2 ) = C2 (x).

(20)

Expression (20) is the algebraic relationship between λ1 , λ2 , and x. If C2 (x) is known, then from this expression we find implicitly λ1 = λ1 (λ2 , C2 (x)). Substituting into Eq. (19), we obtain an ordinary differential equation for the function λ2 (x). To simplify the non-dimensional expression (8) for the friction, we introduce the Reynolds number of the power-law fluid in the definition of [29]: Re =

ρf (R + w)n vf2−n μ

8nn . (3n + 1)n

Then the expression for dimensionless friction is written as 8vf2 . (21) Re Here, both the dimensionless average velocity vf and the Reynolds-Metzner-Reed number Re are not constants; i.e. they are functions of x. τ=

Solitary Waves

3.4

45

Phase Plane

To study the possible types of solutions, it is useful to increase the order of the differential Eq. (19) to the second order, thus getting rid of C1 , and to investigate the phase plane of the resulting equation, as in [35]. Differentiating Eq. (19), we get λ λ λ1 λ2 = 1 2 −  λ1 λ 2 W1  +

λ2 λ2



 λ21 − λ2 2

3n + 1 vf2 0 p0 + 2n + 1 2

p0 

λ22 2

1−



λ20 x

4 

λ2 − λ2 2 W1 − 1 λ1

 . (22)

To find the expression λ1 (λ1 , λ2 , λ2 ), we differentiate Eq. (20) to obtain λ1 = λ2

W2 − λ1 W12 C2 − , λ1 W11 λ1 W11

where Wij = ∂ 2 W/∂λi ∂λj . Next, to compute W1 (λ1 , λ2 ), we write W1 = W11 λ1 + W12 λ2 = λ2

W2 − λ1 W12 C λ C − 2 + W12 λ2 = 2 W2 − 2 . λ1 λ1 λ1 λ1

Substituting these expressions into Eq. (22) and noting that the term with p0 is cancelled by the term −C2 /λ1 in the expansion of W1 , we obtain ⎧ ⎪ X  = Y, ⎪ ⎪ ⎪ W2 − λ1 W12 XY ⎪ ⎪ ⎪Y  = − τ +Y2 ⎪ ⎪ W λ21 W 11 ⎨   11  4  2 v λ λ1 3n + 1 (23) 20 f 0 ⎪ 1− − X λ21 − Y 2 p0 (x) + ⎪ ⎪ W 2n + 1 2 X 1 ⎪ ⎪  ⎪ ⎪ λ2 − Y 2 ⎪ ⎪ ⎩ − 1 2 W2 . λ1 where X ≡ λ2 , λ1 (X, x) is the function obtained by solving algebraic equation (20) for a given value of C2 (x), and the axial Lagrangian coordinate x acts as an analogue of time in a dynamic system.

4

Steady States in the Case of an Inviscid Fluid

In this section, we set τ = 0, i.e. consider an inviscid fluid, but keep a nonconstant cross-sectional velocity distribution (4). Therefore, p0 (x) = const, C2 (x) = const, and, consequently, λ1 = λ1 (X). We obtain the following autonomous system of equations:

46

V. Vedeneev

⎧ ⎪ X  = Y, ⎪ ⎪ ⎪ W2 − λ1 W12 ⎪ ⎪ ⎪ Y =Y2 ⎪ 2 ⎪ 11 ⎨ λ1 W    4  2 v λ λ1 3n + 1 20 f 0 ⎪ − X λ21 − Y 2 p0 + 1− ⎪ ⎪ W1 2n + 1 2 X ⎪ ⎪  ⎪ 2 2 ⎪ λ −Y ⎪ ⎪ ⎩ − 1 2 W2 . λ1

(24)

It can immediately be seen that for the inviscid fluid, the non-constant velocity distribution is expressed only in the factor (3n + 1)/(2n + 1), which tends to 1 as n → 0, i.e. as the velocity distribution tends to a constant. Hence, the effect of the cross-sectional velocity distribution consists of increasing the effective mean flow velocity by a factor (3n + 1)/(2n + 1) compared to a constant distribution. In particular, for a regular parabolic velocity profile (n = 1), the effective velocity is increased by ≈15% with respect to the uniform profile. 4.1

Phase-Plane Structure

As can be seen, the phase plane of Eq. (24) is symmetric about the X axis, is two-sheeted (due to the square root), and is defined only for |Y | < |λ1 (X)|. We will call the lines Y = ±λ1 (X) the limit lines: they are the transition lines of the integral curves from one sheet of the Riemann surface to the other. It is easy to verify that the limit lines are integral trajectories themselves, but the uniqueness theorem for an integral trajectory passing through a given point is not valid for limit lines. Physically, the points lying on the limit lines correspond to the vertical tangent to the tube surface (us = −1). In this case, the assumption of flow laminarity inside the tube is not correct: the flow will detach from the walls and switch to a complex unsteady motion. We will call such a situation the tube’s collapse; it is obvious that the steady state without collapse corresponds to only one sheet of the phase plane, corresponding to the positive value of the square root. Further, we restrict ourselves to considering this sheet of the phase plane. The stationary points of the phase plane (Xs , Ys ) are determined by Ys = 0, and Xs are the roots of the equation    4  λ20 W2 (λ1 (X), X) 3n + 1 vf2 0 1− = 0. (25) + S(X) = −X p0 + 2n + 1 2 X λ1 (X) They correspond to possible uniform states of an infinitely long tube. The type of stationary points is determined by the value of dS(Xs )/dX: for W1 > 0 (tensile axial stress in the tube wall), they are of a centre type when dS(Xs )/dX < 0 and of a saddle type when dS(Xs )/dX > 0; for W1 < 0 (compressive axial stress), the types are reversed. The integral curves emerging from the saddle point (separatrices) and returning back to the same point correspond to standing solitary waves.

Solitary Waves

47

The phase plane of Eq. (24) is characterised by three parameters: p0 , vf 0 , and C2 . The parameter C1 defines only a specific integral curve in the phase plane. To determine the physical meaning of these parameters, we assume that the tube state is homogeneous as x → ±∞; i.e. all its parameters tend to constants. This far-field state is characterised by stretches λ10 and λ20 . After setting these stretches, the constant C2 and, therefore, the function λ1 (X) are uniquely determined from Eq. (20). The fluid pressure p0 is expressed from the equilibrium condition using Eq. (16): W2 (λ10 , λ20 ) p0 = . (26) λ10 λ20 The fluid velocity at infinity, vf 0 , can be set arbitrarily. 4.2

Material Model

Since further study without specification of the material model is impossible, we will consider the Gent model of incompressible hyperelastic material [15,20]: ˆ (λ1 , λ2 , (λ1 λ2 )−1 ), W (λ1 , λ2 ) = W   2 2 2 ˆ (λ1 , λ2 , λ3 ) = − 1 GJm ln 1 − λ1 + λ2 + λ3 − 3 , W 2 Jm with the shear modulus G = 106 Pa and Jm = 97.3 corresponding to rubber properties. By direct calculations, we find   ∂W (λ1 , λ2 ) = Ga λ2i − (λ1 λ2 )−2 λ−1 i = 1, 2, i , ∂λi   2 −2 2 ) ∂W1 (λ1 , λ2 ) 2λ−2 −4 −2 1 (λ1 − (λ1 λ2 ) = = Ga a + (1 + 3λ1 λ2 ) , ∂λ1 Jm  ∂W1 (λ1 , λ2 ) 2(λ1 λ2 )−1 (λ21 − (λ1 λ2 )−2 )(λ22 − (λ1 λ2 )−2 ) = = Ga a ∂λ2 Jm  −1  λ2 + λ22 + (λ1 λ2 )−2 − 3 a= 1− 1 + 2(λ1 λ2 )−3 , . Jm

Wi = W11 W12

The Gent model is chosen for analysis because, although being sufficiently simple, it correctly reproduces real nonlinear properties of many hyperelastic materials, including rubber (Jm ∼ 100) and arterial wall tissues (Jm ∼ 1) [20]. Also, the Gent material has limited stretch that reflects the limited extensibility of molecular chains. Because of the latter feature, the phase plane is defined for those values of X ≡ λ2 for which there is at least one λ1 such that a(λ1 , λ2 ) > 0. This gives the range Xlim− < X < Xlim+ , where Xlim− and Xlim+ are the positive roots of the equation X 3 + (Jm + 3)X + 2 = 0. It can be proved that in this range of X, Eq. (20) always has a solution λ1 = λ1 (X, C2 ) for any real value of C2 . Thus, this stretch range X is the region of the material model’s validity and, accordingly, the region of definition of the phase plane. For Jm = 97.3, we have Xlim− = 0.01994 and Xlim+ = 10.0050.

48

V. Vedeneev S

(a)

S

(b)

0.4

0.4

0.2

0.2 λ20=1.11 0

0 1

3

4

5

7

8 X

1

3

4

5

6

7

8 X

-0.2

-0.2

-0.4

6

λ20=1.5

-0.4

λ20=1.5 λ20=1.9

Fig. 4. Plot of the function S(X) for λ10 = 1 and λ20 = 1.5, 1.1763, 1.15, 1.14, 1.11 (a), for λ20 = 1.5, 1.69, 1.9 (b).

4.3

Phase Plane and Solitary Wave Solutions in the Case of a Quiescent Fluid

Let us first consider a phase plane with vf 0 = 0 and far-field stretches λ10 = 1 and λ20 = 1.5 (an axially unstretched but inflated state), which, according to Eq. (26), correspond to p0 ≈ 0.808. In this case, there are three stationary points: Xs = 1.5, 1.85, and 7.35 (Fig. 4a). The first point is of the saddle type (corresponding to a homogeneous far-field state), the second point is of the centre type, and the third point is of the saddle type. The vector field corresponding to the phase plane is shown in Fig. 5. Hereunder, we use the following notation of stationary points: the first letter is the type of point (c is the centre, and s is the saddle), and the second digit is the number of the stationary point; the far-field state always corresponds to number 1. As can be seen, there exists a solitary wave solution: the separatrix of the saddle s1, enveloping the centre c2 and returning to the original saddle. The solutions inside the saddle separatrix loop correspond to periodic tube swellings; any solution outside the separatrix loop tends to X → 0 and Y → −∞ and approaches the limit line; i.e. such solutions exist only for a finite tube length. The separatrices of the saddle s3 are not closed: the separatrices going to the left cross the limit line; the separatrices going to the right reach the value Xlim+ , at which the material reaches its stretch limit. Thus, with the parameters considered, there exists, in addition to the uniform state, the only standing solitary wave in the form of a localised tube swelling. Reducing stretch λ20 causes the centre c2 and saddle s3 to come closer together (Fig. 4a). The solitary wave solution disappears at λ20 ≈ 1.1763, when the separatrix of the first saddle s1 becomes the separatrix of the saddle s3 (Fig. 6a), and the solitary wave transforms into a kink. With a further decrease in λ20 , the initial swelling solitary wave disappears, but a standing necking soli-

Solitary Waves 2

(a)

(b)

1

1

0

2

49

s3 1

2

3

4

5

6

7

8

s1 0

-1

-1

-2

-2

0.5

1.0

1.5

c2 2.0

2.5

Fig. 5. Vector field of system (24) and separatrices of the stationary saddle points at λ10 = 1 and λ20 = 1.5. General view (a), enlarged view in the area of the separatrix loop (b).

tary wave (corresponding to a much more inflated far-field state) appears, in which the left separatrix of the saddle s3 goes around the centre c2 (Fig. 6b). For λ20 ≈ 1.14, the saddle s3 and the centre merge (Fig. 4a), and the amplitude of the necking solitary wave tends to zero, after which there remains the only stationary point, the saddle s1, corresponding to the original far-field state. Obviously, for lower values of λ20 solitary wave solutions do not exist (Fig. 7). Let us now consider a change in the phase plane with an increased λ20 from 1.5 and higher. The saddle point s1, corresponding to the uniform state, moves to the right, and the centre c2 moves to the left; for λ20 = 1.69, they pass through each other (Fig. 4b). For λ20 > 1.69, the type of stationary points changes: the saddle point s1 becomes the centre c1, and the centre c2 becomes the saddle s2. In this case, the solitary wave solution now corresponds to a less inflated far-field tube state (Fig. 8), while for the original far-field state, there remains only the uniform-tube solution. The parameter range in which there exists a family of standing swelling solitary waves for a quiescent fluid was first obtained by [34]. Another bifurcation of the phase plane occurs at λ20 = 1.73. In this case, a range of X appears for which W1 (λ1 (X), X) < 0 (Fig. 9); i.e. the region in which the axial stress becomes compressive. In this range, the vector field (24) turns around, and at the points at which W1 = 0, it has a singularity (note that W11 (λ1 (X), X) is always positive; i.e. there are only singularities associated with zero longitudinal stress). Thus, solutions starting in the region W1 > 0 cannot penetrate the area left of the line W1 = 0 but end on the limit line (Fig. 8). With a further increase in λ20 , the range in which W1 < 0 expands and captures the saddle s2, which then again becomes the centre c2 (Fig. 10). There are no standing solitary wave solutions.

50

V. Vedeneev

2

2

(a)

1

1

0

(b)

s1 1

2

c2 3

s3 4

5

1

8

7

-1

-1

-2

-2

c2

s1

0

2

3

s3 4

6

7

8

Fig. 6. Vector field of system (24) and separatrices of the stationary saddle points for λ10 = 1 and λ20 = 1.1763 (a), λ20 = 1.15 (b). 2

1

0

s1 1

2

3

4

5

6

7

8

-1

-2

Fig. 7. Vector field of system (24) and separatrices of the stationary saddle point for λ10 = 1 and λ20 = 1.11.

4.4

Phase Plane and Solitary Waves in the Case of a Moving Fluid

To be specific, we put n = 0 (a uniform fluid velocity distribution); as noted above, other values of n yield rescaling of vf 0 , but the qualitative picture will obviously remain the same. With far-field parameters λ10 = 1.5, λ20 = 0, and vf 0 < 0.0624, the structure of the phase plane is the same as for a quiescent fluid. For vf 0 ≥ 0.0624, in addition to the three stationary points, two more points appear in the vicinity of X = 0.038: the centre c4 and the saddle s5. As the fluid velocity increases, the saddle moves to the left and, for vf 0 > 0.37, leaves the phase plane through the left boundary; we will not consider it further. The centre moves to the right (Fig. 11) and is located to the left of the saddle s1, which corresponds to the far-field state. Due to the presence of a centre, the vector field

Solitary Waves 2

2

(a)

1

0

51 (b)

1

1

2

3

4

5

6

7

s2 c1

s3 8

0

-1

-1

-2

-2

0.5

1.0

2.0

2.5

Fig. 8. Vector field of system (24) and separatrices of the stationary saddle points for λ10 = 1 and λ20 = 1.9. General view (a), enlarged view in the area of the separatrix loop (b). The vertical dashed line is the singularity of the vector field W1 (X) = 0. 5

W1

4 3 2 1 0 -1

λ20=1.5 0.5

1.0 λ20=2.1

2.0 X 2.5

Fig. 9. Plot W1 (λ1 (X), X) for λ20 = 1.5, 1.7, 1.9, 2.1.

turns around for small X (Fig. 12): if for vf 0 = 0 it was directed downwards, now it is directed upwards, and there are closed trajectories enveloping a new stationary centre. There are two standing solitary waves simultaneously emerging from the saddle s1 corresponding to the uniform state: the localised swelling in which the separatrix loop envelops the right centre c2 and the localised necking in which it envelops the left centre c4 (Fig. 12). Both solutions are shown in Fig. 13 as functions λ2 (x). An increase in the flow velocity vf 0 leads to changes in the phase plane, as with an increase in λ20 for the quiescent fluid. Namely, the right centre c2 moves to the left, and at vf 0 ≈ 0.58, it merges and passes through the saddle s1,which corresponds to a homogeneous state (Fig. 11). Then the types of both stationary

52

V. Vedeneev 2

1 c1

c2 0

0.5

1.0

1.5

2.0

2.5

-1

-2

Fig. 10. Vector field of system (24) and integral trajectories enveloping the centres for λ10 = 1 and λ20 = 2.1. The vertical dashed line is the singularity of the vector field W1 (X) = 0. S 0.4

(a)

1

3

4

5

6

7

X

-0.2 -0.4

vf0=1.0

(b)

0.05

0.2 0

S

0

0.5

1.0

X 2.0

-0.05 vf0=0.4

vf0=0.4

vf0=1.0 -0.10

Fig. 11. Plot of the function S(X) at λ10 = 1, λ20 = 1.5, and vf 0 = 0.4, 0.58, 0.64, 1.0 (a), enlarged view in the region of small X (b). The arrows show the direction of movement of the roots with increasing vf 0 .

points are reversed. The swelling and necking solitary waves still exist, but their conditions at infinity correspond to a smaller λ20 than in the original far-field state (Fig. 14a). At vf 0 ≈ 0.64, the two left stationary points, the saddle s2 and the centre c4, approach, merge, and disappear (Fig. 11). For larger values of the velocity, two stationary points remain: the centre c1, which corresponds to the far-field state (Fig. 14b), and the saddle s3, which corresponds to much larger stretches; no standing solitary wave solutions exist. With an increase in speed, the saddle s3 gradually moves to the right, and for vf 0 > 9, it leaves the phase plane through its right boundary. At higher speeds, the only stationary point remains, the centre c1.

Solitary Waves 2

(a)

2

(b)

1

1

0

53

1

2

3

4

5

6

7

s3 8

s1

c4 0

-1

-1

-2

-2

0.5

1.0

c2

1.5

2.0

2.5

Fig. 12. Vector field of system (24) and separatrices of the stationary saddle points for λ10 = 1, λ20 = 1.5, and vf 0 = 0.4. General view (a), enlarged view in the area of the separatrix loops (b). λ2

λ2

(a)

(b)

1.6

1.6 1.2

-0.08

-0.04

0

0.8

0.8

0.4

0.4

0.04

0.08 x

-0.08

-0.04

0

0.04

0.08 x

Fig. 13. Swelling (a) and necking (b) solitary waves λ2 (x) for λ10 = 1, λ20 = 1.5, and vf 0 = 0.4.

5

Steady States in the Case of a Viscous Fluid

As can be seen from system (23), for τ = 0 the vector field becomes nonsymmetric with respect to the X axis due to the term −τ XY /W11 . For the Gent model W11 > 0; therefore, each centre point becomes a stable focus; each saddle point remains a saddle, but its separatrices rotate somewhat clockwise. In addition, the values of p0 and C2 , which were associated with the tube state at infinity for an inviscid fluid, become non-constant in the viscous case: p0 (x) is a decreasing function, C2 (x) is a growing function, and therefore λ1 = λ1 (X, x). Note that the functions p0 (x) and C2 (x) themselves depend on the solution; i.e. system (23), strictly speaking, is not a system of differential equations.

54

V. Vedeneev

2

(a)

1

(b)

1 c4

0

2

0.5

c1

s2 c1 1.5

1.0

2.0

2.5

0

-1

-1

-2

-2

0.5

1.0

1.5

2.0

2.5

Fig. 14. Vector field of system (24) and separatrices of the stationary saddle points for λ10 = 1, λ20 = 1.5, and vf 0 = 0.63 (a), vf 0 = 1.0 (b).

Parameters λ20 and vf 0 , which also corresponded to the tube state at infinity for the case of an inviscid fluid, can now refer to any tube cross-section. The stationary points of the phase plane, i.e. the solutions of Eq. (25), now also depend on x. The motion of the integral curve can be represented as the motion along the vector field, which itself changes with x; moreover, for each solution the vector field changes in its own way. 5.1

Stretch Limit States as x → ±∞

First, consider possible deformed steady states, i.e. λ1 and λ2 independent from x as x → ±∞. In this case, we have p0 (x) = −2τ

λ1 x = −P x, λ2

P = 2τ

λ1 >0 λ2

C2 (x) = τ λ21 λ2 x.

The constants in p0 (x) and C2 (x) for large values of |x| can be neglected. Consider the solution of Eq. (20) for the Gent material model (Sect. 4.2): W − λ 1 W1 =

  1 GJm ln a − Ga λ21 − (λ1 λ2 )−2 = C2 (x) = τ λ21 λ2 x. 2

Obviously, this equation can be satisfied as x → ±∞ only if a(x) ∼ Ax → ∞ with A = const, i.e. a−1 = 1 −

λ21 + λ22 + (λ1 λ2 )−2 − 3 → 0. Jm

Neglecting the first term (ln a a), we obtain A=−

−1 τ 2  2 λ1 λ2 λ1 − (λ1 λ2 )−2 . G

(27)

Solitary Waves

55

Since a > 0, it is also necessary that the following inequalities are satisfied: λ21 − (λ1 λ2 )−2 < 0,

x → +∞;

λ21 − (λ1 λ2 )−2 > 0,

x → −∞.

(28)

Expression (27) gives the relationship between the possible limit values of λ1 and λ2 . The limit value of W2 has the form   W2 = Ga λ22 − (λ1 λ2 )−2 λ−1 2 = Bx,

B = −τ λ21

λ22 − (λ1 λ2 )−2 . λ21 − (λ1 λ2 )−2

Next, consider Eq. (25) as x → ±∞. Leaving only the leading terms, we have − P λ2 x −

Bx =0 λ1

⇒

2=

λ22 − (λ1 λ2 )−2 . λ21 − (λ1 λ2 )−2

(29)

Let us prove that there always exist unique limit states satisfying Eqs. (27) and (29) and inequalities (28). Rewrite them, denoting λ21 = l1 and λ22 = l2 : l12 l2 + l1 l22 − (3 + Jm )l1 l2 + 1 = 0, 2(1 − l12 l2 ) = 1 − l1 l22 > 0,

1−

(30)

l1 l22 ,

x → +∞;

(31) 1−

l1 l22

< 0,

x → −∞.

(32)

Equations (30) and (31) are equivalent to the system l1 l2 (l1 + l2 − (3 + Jm )) = −1 = l1 l2 (l2 − 2l1 ). From here we obtain 3 + Jm l1 = , 3

 l2 = l1 ±

l12 −

1 . l1

As l1 > 1, the value of l2 is always real and positive. It is easy to verify that the ‘plus’ sign before the root satisfies inequality (32) as x → −∞ and the ‘minus’ sign as x → +∞. As x → −∞, the tube is swollen and axially stretched (λ1 > 1, λ2 > 1): far upstream, the stretched state tends to the limit state, and the tensile stresses tend to infinity. In this case, the axial stress in the tube wall σ1 is balanced by the fluid viscous force, which axially stretches the tube sections lying upstream, and the circumferential stress σ2 is balanced by pressure, which grows unlimitedly upstream. As x → +∞, the tube is also axially stretched but compressed in the circumferential direction (λ1 > 1, λ2 < 1). In this case, both axial and circumferential stresses are compressive and tend to infinity. The axial stress σ1 downstream is balanced by the viscous force, which compresses the tube axially, and the circumferential stress σ2 is balanced by pressure, which decreases unlimitedly due to viscous losses.

56

V. Vedeneev

We note that the limit states do not depend neither on the fluid properties nor on the values of the constants but depend only on parameter Jm of the tube material. By direct calculation, taking into account Eq. (29) and a large value of a, it can be proved that for both limit states dS/dX > 0, which means that the limit stationary point of a ‘frozen’ vector field at large |x| is saddle point as x → −∞ and stable focus point as x → +∞. For Jm = 97.3, which corresponds to rubber, we have limit axial stretch λ1 = 5.782156, and limit circumferential stretches λ2 = 0.021149 as x → +∞ and λ2 = 8.177176 as x → −∞. 5.2

Evolution of the Tube When Moving from Infinity

Let us now investigate the possibility of combining limit states as x → −∞ and x → +∞ by a single integral curve, i.e. constructing a steady solution for the entire infinitely long tube conveying a viscous fluid. To do this, we divide the tube into three sections. Two sections are neighbourhoods of infinities as x → −∞ and x → +∞, where the limit stationary points continuously move when x changes, but they remain the only stationary points of the phase plane (or, if other stationary points appear, they do not interact with the limit stationary point). The third section is the central section, where new stationary points appear and can interact with stationary points that came from neighbourhoods of infinity. In a certain cross-section of the central segment, we put the origin of the x axis, and specify stretches λ1 = λ10 and λ2 = λ0 and fluid velocity vf 0. We do not specify pressure in this cross-section; below we will show that there is only one pressure value that provides the existence of a solution for −∞ < x < ∞. In this section, we study the behaviour of integral trajectories with a motion of limit stationary points. Evolution of the Tube as x → −∞. First, consider the limit saddle point as x → −∞. As the absolute value of x decreases, the value of Xs corresponding to this stationary point decreases; in the central part of the tube, this saddle continuously passes to the left saddle s3 in Fig. 5. Moreover, it is easy to see from the local structure of the vector field in the vicinity of the moving saddle that there always exists an integral curve, which for each x is located near the moving saddle point and does not ‘fall’ onto the separatrix leaving it. Moreover, for x → −∞, such an integral curve tends to the limit position of the saddle (a limit state of the tube). Thus, the moving saddle ‘leads’ such an integral curve. When the saddle moves to the left, the integral curve follows below it; the lower the curve the higher the speed of the saddle. When the saddle moves to the right, the integral trajectory, on the contrary, follows the saddle above it. The existence of an integral trajectory following the moving saddle can also be seen from the explicit solution of the model problem X  + 2cX  = A(X − X0 (x)),

A > 0,

c > 0,

Solitary Waves

57

where X0 (x) is the variable position of the saddle and X(x) is an unknown function. The solution of this equation has the form X(x) = X0 (x)  x √  

x √ √ √ 1 √ e− κξ β(ξ)dξe κx − e κξ β(ξ)dξe− κx + e−cx 2 κ x0 x0  √ √ β(x) = (−X0 − 2cX0 )ect . + c1 e κx + c2 e− κx , κ = A + c2 , It is easy to see that if the saddle position has moved along a finite segment x from one fixed position to another (in this case, β(x) = 0 only on a finite segment x, and the integrals tend to constants as x → +∞), then there exists a solution that asymptotically tends to the initial state as x → −∞ and to the final state as x → +∞. Thus, before reaching the central section of the tube, where the interaction of the stationary points occurs, there exists an integral trajectory coming from −∞ following the saddle. Evolution of the Tube as x → +∞. Now consider the integral trajectory following from x → +∞ to the central section. It will be shown below that it is necessary that in the central part of the tube the stationary point continuously passes to the right saddle s1 in Fig. 5. For such a continuous evolution, it is necessary that during the motion from x → +∞, no new stationary point emerges. If such a point arises (it must be a focus that in the inviscid case corresponds to the centre c4 in Fig. 12), it will separate two stationary points: the analogues of the right saddle s1 in Fig. 12 and the saddle point s5 lying near the left boundary of the phase plane and corresponding to the limit state at x → +∞. A continuous transition of one stationary point to another with a change in x takes place only for sufficiently small vf 0 , for which there is no ‘intermediate’ stationary point. In particular, the calculation shows that for λ1 (0) = λ10 = 1, λ2 (0) = λ20 = 1.5, and the initial condition p0 (0) defined by the equilibrium condition in inviscid flow (26), the transition to x → +∞ occurs without the formation of an intermediate stationary point for vf 0 ≤ 0.0635, which is close to the value at which the intermediate stationary point is absent for x = 0 (Sect. 4.4). Below, we will assume that this condition is satisfied; otherwise, as will be shown below, it is impossible to connect the limit states of the tube as x → ±∞. The limit stationary point for x → +∞ is a stable focus. It is easy to see from the structure of the vector field that when the focus moves, there is a trajectory that remains in its vicinity and rotates around it; when the focus movement stops, the trajectory asymptotically tends to it. As a result, as in the case of a saddle at x → −∞, there is an integral trajectory ‘following’ the focus motion. The existence of such a trajectory is also evident from the explicit solution of the model problem X  + 2cX  = −B(X − X0 (x)),

B > 0,

c > 0,

58

V. Vedeneev

where X0 (x) is the variable focus position and X(x) is an unknown function. The solution of this equation has the form X(x) = X0 (x)  

√ √ √ √ e−cx + √ cos( δξ)β(ξ)dx sin( δx) − sin( δξ)β(ξ)dx cos( δx) δ √ √ + c1 cos( δx)e−cx + c2 sin( δx)e−cx , δ = B − c2 . It is seen that if the location of the focus tends to a constant as x → ∞, then all the trajectories, including those coming out of the vicinity of its initial position, asymptotically approach its final position. Hereunder we will assume that the central section of the tube, where we must connect the trajectory coming out from x → −∞ and going to x → +∞, is inflated: for the inviscid case, there are several stationary points, and there exist standing solitary waves. Then the axial stress in the central section is tensile so that W1 > 0. However, it was shown above that as x → +∞, the axial stress is compressive, i.e. W1 < 0, which means that there is a point Xc at which W1 = 0; along the vertical line X = Xc the vector field (23) has a singularity. In addition, when passing through a singularity, the type of the stationary point changes: a saddle becomes a focus and vice versa. Only one integral trajectory passes through the singularity line; all other trajectories end at the limit line Y = λ1 (X, x). Namely, when a stationary point passes through a line of zero axial stress below the stationary point, the vectors are horizontal. If we follow the vector field at a distance from the stationary point at which the length of the horizontal vector is equal to the stationary point’s speed, a smooth passage through the singularity is ensured; all other integral trajectories turn up or down before the singularity and end at the limit line. The following model equation is an illustration of the transition through a singularity: X +x . (33) X  = A X The equation’s stationary point is a saddle for x < 0 and a centre for x > 0. The vector fields for x = −0.5 and x = 0.5 are shown in Fig. 15. Its exact solution X(x) = −x for each x corresponds to a stationary point; on the phase plane, due to the motion of the stationary point, the integral trajectory moves under it along the line Y = −1. Any other solution, starting from x < 0, cannot penetrate the line X = 0 and remains to its right, turning up or down before this line. It is important that this unique trajectory corresponds to the transition to x → +∞ only when the point Xc at which the change in sign of W1 takes place is unique. If there are more such points, then it is impossible to have a trajectory penetrating several singularities: the condition of passing through the first one selects a single integral curve that can no longer, except for special cases, pass through other singularities. Further, we will assume that point Xc is unique. For the parameters λ1 (0) = λ10 = 1 and λ2 (0) = λ20 = 1.5 and the initial condition p0 (0) determined from Eq. (26), the transition to x → +∞ occurs with a single

Solitary Waves 1.5

(а)

1.5

1.0

1.0

0.5

0.5

-1.5 -1.0 -0.5 0

0.5

1.0

1.5

59

-1.5 -1.0 -0.5 0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5

(b)

0.5

1.0

1.5

Fig. 15. Phase plane of the ‘frozen’ Eq. (33) for x = −0.5 (a) and x = 0.5 (b). The arrow shows the direction of the stationary point’s motion with increasing x.

Xc for vf 0 ≤ 0.0565. In particular, this condition is satisfied in the example constructed below in Sect. 5.4. Note that for fluid velocities exceeding the critical values (at which either an intermediate stationary point or more than one Xc appear) and for realistic values of fluid viscosity, the inability to continue the trajectory as x → +∞ arises at sufficiently large values of x, exceeding hundreds or thousands of tube radii. As a result, although it is mathematically impossible to continue the solution to infinity, the region of its existence exceeds any lengths of tubes conveying fluid that are encountered in applications. 5.3

Connection of Integral Trajectories from Infinities in the Central Part of the Tube

Thus, from both left and right infinities, at a sufficiently low speed vf 0 , it is possible to continue an integral trajectory to the central region of the tube, where both stationary points are saddles. Expecting solutions with a solitary wave form, we will assume that in the central part, the structure of the ‘frozen’ phase plane qualitatively corresponds to Fig. 5 (except that the centre c2 becomes a stable focus). When increasing x (moving downstream), the focus and the right saddle merge and disappear; when decreasing x (moving upstream), the focus and the left saddle merge and disappear; in both cases, there is a homoclinic bifurcation of the vector field. The remaining saddle evolves into a limit state, as shown in the previous section. For such a configuration, in which at a certain x a stable focus exists between the two saddles, it is impossible to transform the limit saddles into each other with a change in x. However, under certain conditions, there is an integral trajectory connecting the saddles and bypassing the focus from below. More precisely, from the vicinity of the right saddle with increasing x, the integral trajectory comes into the vicinity of the left saddle. For this, it is necessary that the focus

60

V. Vedeneev (a) λ2

2

(b)

5 1 4 0 1 -1

-2

2

3

4

5

7

3 2

1

1

2

4

6

8

x

12

Fig. 16. Transition in the central part of the tube from the vicinity of the stationary point as x → −∞ to the neighbourhood of the stationary point as x → +∞ for the initial (x = 0) parameters λ10 = 1, λ20 = 1.5, vf 0 = 0.05, and p0 ≈ 0.484. The integral curve in the phase plane (the vector field corresponds to the state at x = 0, the arrows show the direction of motion of the stationary points with increasing x) (a); plot λ2 (x) (b).

is located between the saddles in such a way that the integral trajectory from the neighbourhood of one stationary point comes into the neighbourhood of the other (Fig. 16). To realise such a configuration, there are two free parameters in the problem, since the pressure p0 and C2 are defined to within a constant. In the absence of viscosity, these constants are determined by the parameters at infinity, but in the presence of viscosity, these constants can be chosen arbitrarily, since both pressure and C2 tend to infinity at x → ∞. Only one of these parameters is essentially arbitrary, since their simultaneous change in a certain combination corresponds only to a shift in the origin of the x axis (or, equivalently, a shift of the entire tube to the left or to the right). Thus, due to this one free parameter, e.g. pressure, it is possible to locate the focus so that the integral trajectories for x → −∞ and x → +∞ are connected in the central part of the tube with a focus bypass from below. Small changes in pressure will lead to small changes in the integral trajectory, up to the moment of intersection with the singularity (the line of change in the sign of W1 , i.e. a change of tensile longitudinal stress to compressive stress). As shown in Sect. 5.2, there is only one trajectory that penetrates this singularity, while the other trajectories on the phase plane turn up or down and end at the limit line. Thus, the parameter (e.g. pressure) has the only value at which the integral trajectory passes through the singularity. After that, the trajectory moves in an uncontrolled way. However, since it moves in the vicinity of a stable focus (which moves to the left with increasing x), it does not leave its vicinity and asymptotically tends to the limit state as x → +∞.

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61

Now we can explain why it is necessary that during the evolution of a stationary point as x → +∞, an intermediate stationary point should not occur (Sect. 5.2). As in the central tube section, at a certain position of the three stationary points, it would be possible to connect the two outer points by passing the middle point from below (unlike in the central part, here the saddles and focuses are switched if the interaction occurs at W1 < 0; however, this does not preclude the possibility of their connection by an integral trajectory). To do this, the middle stationary point must be located between the outer points in such a position that the trajectory from the neighbourhood of the right point comes into the neighbourhood of the left point. However, we do not have other free parameters to organise such a connection. Therefore, the only way to continue the trajectory to x → +∞ is to prevent the occurrence of the focus, which is ensured by the requirement that the fluid velocity in the central tube segment is sufficiently small: vf 0 < vf 0cr . The only special case may be a fluid velocity vf 0 > vf 0cr such that the desired location of the stationary points occurs simultaneously with penetration of the singularity; however, this exceptional case is not of general interest, since it cannot be realised in reality: an arbitrarily small deviation from this value vf 0 will lead to the end of the trajectory at the limit line and the tube’s collapse. 5.4

An Example of a Solution for an Infinitely Long Tube

Consider an example of a solution for an infinitely long tube. The calculations were performed numerically, separately for the central tube section and the neighbourhoods of infinity. In the central region, the full differential equation (19) was solved by the Euler method taking into account the algebraic relation (20). The calculation took into account the change in p0 (x) and C2 (x) according to formulas (14) and (18) (in dimensionless form). The values of the integrals were updated at each x-step by the rectangle method simultaneously with the numerical integration of Eq. (19). In the vicinity of infinities, the numerical integration of the full problem is practically impossible, because the integral trajectories tend to ‘fall’ onto the separatrices coming out of the saddles and, therefore, are extremely sensitive to the initial conditions. For example, to keep the trajectory near a stationary point at a distance of ∼1 tube radius, an accuracy of setting the initial conditions of ∼10−6 is required; and the required accuracy increases exponentially with increasing tube length. Thus, although a solution remaining in the vicinity of the saddle exists, it is almost impossible to obtain it numerically on a long x-interval. To calculate the evolution of the position of the stationary points, it was assumed that their motion is rather slow and that the derivative λ2 in Eq. (19) can be neglected. Then, replacing it by zero, this equation becomes algebraic; it was solved numerically for given values of p0 (x) and C2 (x). A segment of the x axis of a sufficiently large length (directed either to +∞ or −∞) was divided into a sufficiently fine grid, along which the integrals p0 (x) and C2 (x) and, accordingly, the position of the stationary point λ2 (x) changed. Although this approach is approximate, it yields a rather accurate calculation of the stationary points’

62

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S 0.4

(a)

(b)

1

0.2

0

2

1

2

3

6

4

0

X

-0.2

1

3

2

4

5

7

-1

-0.4 -2

Fig. 17. Plot S(X) (a) and phase plane and integral curve in the absence of viscosity (b) for the parameters λ1 (0) = λ10 = 1, λ2 (0) = λ20 = 1.5, vf 0 = 0.05, and p0 ≈ 0.477.

X

(a)

(b)

X 1.0

7.6 0.8 7.2 0.6 6.8 0.4 6.4 0.2 6.0 −10 x × 10-3 −6

−4

−2

2

6

10

14

x × 10-4

Fig. 18. The calculated evolution of the position of the stationary point to infinity for x < 0 (a) and for x > 12 (b) for the initial (x = 0) parameters λ10 = 1, λ20 = 1.5, vf 0 = 0.05, and p0 ≈ 0.484. The dashed lines are the limit values.

motion, because outside the central section they move slowly; for x → ±∞ their positions tend to fixed values, while the speed of motion λ2 (x) tends to zero. In the calculations, for simplicity, the power-law index n = 0 was taken; however, it is clear that for any other value, the solution will be qualitatively the same. Since the fluid friction in this case is constant, formula (21) can be rewritten as follows: τ = 8vf2 0 / Re0 , where the index ‘0’ corresponds to an initial section of the tube. In the calculations, the initial Reynolds-Metzner-Read number Re0 = 100 was set.

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63

We set the parameters λ1 (0) = λ10 = 1, λ2 (0) = λ20 = 1.5, and vf 0 = 0.05 and selected a pressure p0 = 0.477 so that the focus of the ‘frozen’ vector field is located between two saddles, which have common separatrices (Fig. 17). We take this position for x = 0 and consider this value to be the left border of the central tube section. However, due to viscosity, the vector field shown in Fig. 17 will evolve: the right saddle and centre approach each other, while the left saddle moves to the left. Therefore, taking into account viscosity, the integral trajectory at this initial value p0 will come to the right of the neighbourhood of the left saddle, and a pressure correction is necessary. Calculations show that for p0 ≈ 0.484, the trajectory, taking into account viscosity, comes into the vicinity of the left saddle (Fig. 16). Thus, this and very similar initial pressure values provide the connection of the neighbourhoods of the stationary points that come from infinity by a single integral curve. The numerically calculated motion of the stationary points when moving to infinity for x < 0 and x > 12 is shown in Fig. 18. As can be seen, they are continuously moving to their asymptotic values. Note that at x ≈ 580, the value of W1 (x) becomes negative and then retains its sign for an unlimited increase in x; at the change in sign, the stationary point’s location is X ≈ 1.13. Since after changing the sign, the saddle becomes a stable focus, the monotonic decrease in X(x) is replaced by an oscillatory motion. In dimensional terms, taking a ratio h/R = 0.1, the shear modulus of rubber G = 106 Pa, and a fluid density ρf = 1000 kg/m3 , the constructed solution corresponds to a fluid velocity vf 0 = 0.5 m/s and a pressure p0 = 48.4 kPa. A Reynolds number Re = 100 corresponds to a fluid friction τ = 20 Pa. 5.5

Existence of Solitary-Wave-Like Solutions

The solution constructed above corresponds to a monotonic downstream change from the inflated limit state to the compressed limit state of the tube. Let us show that a solitary wave solution, i.e. a solution enveloping focus, does not exist. It can be seen from the calculations that, with an increase in x in the central part of the tube, the focus moves to the right, and both saddles move to the left. Suppose that p0 (0) is chosen so that the trajectory, starting from the vicinity of the right saddle point, just does not reach the left saddle point and makes one revolution around the focus point. However, due to the movement of the focus to the right, and the movement of the right saddle to the left, the position of the trajectory after the revolution will be above the right saddle. The trajectory will then follow the upper right separatrix to the boundary of the phase plane, where the tube will collapse. Thus, solutions in the form of a standing solitary wave, i.e. with a single local swelling or necking of the tube, do not exist for an infinitely long tube. However, they exist for a semi-infinite tube. For a tube that is unbounded as x → −∞, an example is given in the previous paragraph; it is possible to select p0 (0) and a final value x > 0 to make one revolution around the focus; as a result, we have a necking solitary wave. For a tube that is unbounded as x → +∞, we can choose p0 (0) and start the path above the initial position of the upper right separatrix

64

V. Vedeneev (a) λ2

2

(b)

8 1 6 0

1

2

3

4

5

6

7 4

-1 2 -2

0

2

(c)

1

0

-1

4

8

12

16 x

λ2

(d)

4

1

2

3

4

5

6

7

3 2

1 -2

0

4

8

x 12

Fig. 19. Solitary-wave-like solutions for an infinitely long tube as x → −∞ (a, b) and for an infinitely long tube as x → +∞ (c, d). The integral trajectory on the phase plane (the vector field corresponds to x = 0) (a, c), and the solution λ2 (x) (b, d) are shown.

of the left saddle, after which it continues indefinitely as x → +∞. The result is a swelling solitary wave. Both examples of solitary waves at semi-infinite tubes are shown in Fig. 19. Obviously, for a tube of finite length, there are solutions that monotonically connect two states and make a certain number of revolutions around the focus (but always a finite number because sooner or later the focus will disappear from the phase plane due to the influence of viscosity), each revolution corresponding to swelling or necking of the tube.

Solitary Waves

6

65

Conclusions

In this paper, we analysed the possible steady states of an elastic tube made of an incompressible hyperelastic Gent material (rubber), conveying a viscous fluid with power-law rheology. It is proved that for a quiescent fluid (or, equivalently, if a constant pressure is set in the tube) in a tube that is axially unstretched at infinity (λ10 = 1), a standing solitary wave in the form of a localised swelling exists for a range of far-field circumferential stretches 1.18 < λ20 < 1.69. This result was previously obtained in [34]. In the case of the motion of an inviscid fluid (generally, with a non-uniform cross-sectional velocity distribution) for λ10 = 1, λ20 = 1.5, and a dimensionless velocity 0.063 ≤ vf 0 ≤ 0.58, there exists, simultaneously with the standing swelling solitary wave, a standing necking solitary wave. At a lower fluid velocity, there is only a swelling solitary wave; for larger velocities, no solitary waves exist. Note that in a model of a geometrically and physically linear tube, in which only the nonlinearity of the flow was taken into account [35], there always exists, for any nonzero flow velocity, only a standing necking solitary wave. Thus, both the existence of a standing swelling solitary wave and the limited range of fluid velocities for which a standing necking solitary wave exists are consequences of the physical and geometrical nonlinearities of the tube model. When a viscous fluid moves, there are limit stretch states of the tube as x → −∞ and x → +∞, with the stretches λ1 and λ2 tending to constants but the stresses tending to infinities to compensate for the fluid pressure and the longitudinal stress caused by the fluid viscosity, which are infinitely growing upstream and infinitely decreasing downstream. The transition between these limit states occurs in the central section of the tube and exists only if the fluid velocity is sufficiently small. In this case, for given stretches λ1 and λ2 and flow speed vf in a chosen cross-section, there is a unique solution linking the states at infinity in the form of a monotonic decrease in the radius downstream, i.e. a kink-like solution. Localised swelling or necking solutions for a tube that is infinitely long in both directions do not exist. However, such solutions exist if the tube is infinitely long in only one direction, either downstream or upstream. But solutions in which a semi-infinite tube has multiple neckings or swellings do not exist. For finite-length tubes, there exist ‘pieces’ of both swelling and necking solitary waves, as well as close-to-solitary-wave solutions with a finite number of successive swellings or neckings. The principal point of constructing a solution in an infinitely long tube conveying a viscous fluid is the existence of a limited material stretch that reflects the limited extensibility of polymeric molecular chains, which is a principal feature of Gent material [15,20]. For other conventional hyperelastic models, such as Ogden material, there is no limited stretch so that the tube will infinitely swell upstream and narrow downstream. However, for realistic fluid viscosity, the difference in the tube’s limit behaviour will manifest itself at thousands of diameters upstream and downstream from the central segment so that for practical applications, the results of the present study can be transferred to other hyperelastic rubber models without any changes.

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Finally, we note that the stability of the obtained solutions is not analysed in this study, and this could be a topic of a separate investigation. I thank A.T. Il’ichev for stimulating discussions on the problem. This work was supported by a grant of the Russian Foundation for Basic Research No. 1829-10020.

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18. Guo, Z., Wang, S., Li, L., Ji, H., Wang, Z., Cai, S.: Inflation of stressed cylindrical tubes: an experimental study. In: Proceedings of the SPIE, vol. 9234, p. 92340H (2014) 19. Heil, M., Hazel, A.L.: Fluid-structure interaction in internal physiological flows. Ann. Rev. Fluid Mech. 43, 141–162 (2011) 20. Horgan, C.O.: The remarkable gent constitutive model for hyperelastic materials. Int. J. Non-linear Mech. 68, 9–16 (2015) 21. Il’ichev, A.T., Fu, Y.B.: Stability of aneurysm solutions in a fluid-filled elastic membrane tube. Acta Mechanica Sinica. 28(4), 1209–1218 (2012). https://doi. org/10.1007/s10409-012-0135-2 22. Il’ichev, A.T., Fu, Y.B.: Stability of an inflated hyperelastic membrane tube with localized wall thinning. Int. J. Eng. Sci. 80, 53–61 (2014) 23. Il’ichev, A.T., Shargatov, V.A., Fu, Y.B.: Characterization and dynamical stability of solitary waves in a fluid-filled hyperelastic membrane tube (2020, in press) 24. Jensen, O.E., Pedley, T.J.: The existence of steady flow in a collapsed tube. J. Fluid Mech. 206, 339–374 (1989) 25. Jensen, O.E.: Instabilities of flow in a collapsed tube. J. Fluid Mech. 220, 623–659 (1990) 26. Karagiozis, K.N., Paidoussis, M.P., Amabili, M.: Effect of geometry on the stability of cylindrical clamped shells subjected to internal fluid flow. Comput. Struct. 85, 645–659 (2007) 27. Ku, D.N.: Blood flow in arteries. Annu. Rev. Fluid Mech. 29, 399–434 (1997) 28. Kyriakides, S., Chang, Y.-C.: The initiation and propagation of a localized instability in an inflated elastic tube. Int. J. Solids Struct. 27, 1085–1111 (1991) 29. Metzner, A.B., Reed, J.C.: Flow of non-Newtonian fluids-correlation of the laminar, transition, and turbulent-flow regions. AIChE J. 1(4), 434–440 (1955) 30. Moore, J.E., Jr., Maier, S.E., Ku, D.N., Boesiger, P.: Hemodynamics in the abdominal aorta: a comparison of in vitro and in vivo measurements. J. Appl. Physiol. 76(4), 1520–1527 (1985) 31. Pamplona, D.C., Goncalves, P.B., Lopes, S.R.X.: Finite deformations of cylindrical membrane under internal pressure. Int. J. Mech. Sci. 48, 683–696 (2006) 32. Pedley, T.J., Luo, X.Y.: Modelling flow and oscillations in collapsible tubes. Theor. Comput. Fluid Dyn. 10, 277–294 (1998). https://doi.org/10.1007/s001620050064 33. Pedley, T.J.: Arterial and venous fluid dynamics. In: Pedrizzetti, G., Perktold, K. (eds.) Cardiovascular Fluid Mechanics. International Centre for Mechanical Sciences (Courses and Lectures), vol. 446, pp. 1–72. Springer, Vienna (2003). https:// doi.org/10.1007/978-3-7091-2542-7 1 34. Pearce, S.P., Fu, Y.B.: Characterization and stability of localized bulging/necking in inflated membrane tubes. IMA J. Appl. Math. 75, 581–602 (2010) 35. Poroshina, A.B., Vedeneev, V.V.: Existence and uniqueness of steady state of elastic tubes conveying power law fluid. Russ. J. Biomech. 22(2), 169–193 (2018) 36. Vassilevski, Y.V., Salamatova, V.Y., Simakov, S.S.: On the elasticity of blood vessels in one-dimensional problems of hemodynamics. Comput. Math. Math. Phys. 55(9), 1567–1578 (2015). https://doi.org/10.1134/S0965542515090134 37. Vedeneev, V.V., Poroshina, A.B.: Stability of an elastic tube conveying a nonNewtonian fluid and having a locally weakened section. Proc. Steklov Inst. Math. 300, 34–55 (2018). https://doi.org/10.1134/S0081543818010030 38. Wang, S., Guo, Z., Zhou, L., Li, L., Fu, Y.: An experimental study of localized bulging in inflated cylindrical tubes guided by newly emerged analytical results. J. Mech. Phys. Solids 124, 536–554 (2019)

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39. Whittaker, R.J., Heil, M., Jensen, O.E., Waters, S.L.: Predicting the onset of highfrequency self-excited oscillations in elastic-walled tubes. Proc. Roy. Soc. A. 466, 3635–3657 (2010) 40. Yomosa, S.: Solitary waves in large blood vessels. J. Phys. Soc. Jpn. 56(2), 506–520 (1987) 41. Yushutin, V.S.: Stability of flow of a nonlinear viscous power-law hardening medium in a deformable channel. Moscow Univ. Mech. Bull. 67(4), 99–102 (2012). https://doi.org/10.3103/S002713301204005X

A Note on the Propagation of Antiplane Love Waves in Micro-structured Media Valentina Volpini(B) Research Centre CRICT, Via Vivarelli 10, 41125 Modena, Italy [email protected]

Abstract. This note focuses on the propagation of antiplane Love waves localised at the free surface of a thin layer, supposed to be perfectly bonded to a half-space. As a main novelty, both the thin layer and the half-space, which follow an elastic linear isotropic behaviour, consist of micro-structured media, i.e. they are characterised by a micro-structure. We adopt the couple stress polar theory with micro-inertia, by considering the kinematical field of micro-rotations, neglected in the context of classical elasticity. We establish a convenient analytical expression of the dispersion equation and determine the frequency spectra, by investigating the effect of some key parameters. We find that Love waves can propagate under broad conditions, but only beyond a so-called cuton frequency, i.e. the frequency starting from which wave propagation is admitted. Keywords: Antiplane Love waves · Couple stress Dispersion equation · Cuton frequency

1

· Microstructure ·

Introduction

The problem of antiplane waves propagating at the free surface of a thin layer bonded to a half-space was firstly theoretically investigated in the classical elasticity context by Love, in his pioneering work of 1911 [1]. More recently, further developments along this line of research aim to take into account the effects of material micro-structure, which have proven to highly alter the wave propagation features in different boundary value problems [2,3]. Couple Stress (CS) theory, that is one of the simpler strain gradient theory [4,5], has proven to suitably accomplish this task through the introduction of a micro-rotations field, which, differently from micro-polar theories, is supposed to be directly related to the classical displacement field [6]. Among others, in a recent contribution Fan and Xu [7] adopt the CS theory to describe the behaviour of a surface layer supported by a classical linear elastic half-space. In the present note, we extend this work by considering both the thin layer and the half-space consisting of micro-structured media, characterised by different material properties. We show that antiplane Love waves are supported in CS media under very general conditions, but the propagation is only permitted c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 69–81, 2022. https://doi.org/10.1007/978-3-030-92144-6_5

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starting from a cuton frequency, representing a threshold beyond which waves always exist. We organise the note as follows. In Sect. 2, we introduce the main features of CS theory and present the modelling framework for the analysis of the propagation of Love waves. Specifically, we illustrate the governing equations (Sect. 2.1), the main assumptions (Sects. 2.2 and 2.3), and the boundary/interface conditions (Sect. 2.4) characterising the problem at hand. In Sect. 3, we particularise the CS theory to the peculiar case of two media characterised by different material properties and present an analytical form of dispersion equation (Sect. 3.1). We finally illustrate and discuss the results in Sect. 4, whereas conclusions and open issues are summarised in Sect. 5.

2

Problem Statement and Couple Stress Theory

We introduce a right-handed orthonormal coordinate system (O, x1 , x2 , x3 ) and consider a supporting half-space BA = {(x1 , x2 , x3 ) : x2 < 0} perfectly bonded to a surface thin layer BB = {(x1 , x2 , x3 ) : 0 < x2 < h} (see Fig. 1), both consisting of isotropic linear elastic micro-structured media, which are modelled through the Couple Stress (CS) theory. Unlike Classical Elasticity (CE), in this context the principle of virtual power may be written as the sum of two contributions as (see [6])    σ · ∇u˙ + μ · ∇ϕ˙ T dV, (1) B

where · denotes the inner product of tensors, i.e. A · B = Aij Bij , a superposed dot indicates the time differentiation, and the superscript T is the transpose operator. Here and henceforth, the Einstein summation convention for repeated indices is adopted. On the one hand, the first addend in (1) involves the symmetric part of the classical Cauchy stress tensor t,   σ = 12 t + tT ,

Fig. 1. Schematics of the half-space A supporting the thin layer B, with mechanical and microstructural properties.

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whose corresponding skew-symmetric part is   τ = 12 t − tT . The symmetric tensor σ is work-conjugate to the classical strain tensor ε, that is   ε = 12 ∇u + ∇uT , (2) where (∇u)kj = uk,j = ∂uk /∂xj indicates the gradient of the displacement field u. On the other hand, the second contribution in (1) takes into account the polar behaviour of CS media by means of the introduction of the CS tensor μ. For any oriented surface characterised by the unit normal n, in fact, μ expresses the internal reduced couple vector r = μn, (3) which acts across the surface. As made clear in (1), the CS tensor is workconjugate with the transpose of the so-called torsion-flexure or wryness tensor, χ = ∇ϕ,

(4)

expressed as a function of the micro-rotation field ϕ, vanishing in the context of CE. Unlike Cosserat micro-polar theories, the following relation between displacement and micro-rotation fields is assumed [6]: ϕ = 12 ∇ × u,

or componentwise

ϕi = 12 IEijk uk,j ,

(5)

in which IEijk is the third-order permutation tensor (see, e.g., [8]). In view of relations (4) and (5), the torsion-flexure tensor turns out to be purely deviatoric, such that tensor μ may be replaced by its deviatoric part, μD = μ − μS , with μS = 13 (μ · 1)1 indicating the spherical part, and being 1 the second-order identity tensor. For ease of notation, here and henceforth we drop the superscript denoting the deviatoric part of the CS tensor, the latter being simply indicated as μ. 2.1

Governing Equations

In the absence of body forces, both half-space A and thin layer B are governed by the following equations of motion: ¨ ∇ · t = ρu, ¨ 2 axial τ + ∇ · μ = J ϕ

(6a) ⇒

τ =

− 12 IE (∇

¨ , · μ − J ϕ)

(6b)

in which (∇ · t)j = tij,i = ∂tij /∂xi indicates the divergence of stress tensor t, and parameters ρ and J ≥ 0 are the mass density and the rotational inertia,

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respectively. In Eq. (6b), which is trivially satisfied in the CE context where μ = τ = 0 and J = 0, (axial τ )i = IEijk τjk denotes the axial vector attached to a skew-symmetric tensor. For what concerns the constitutive prescriptions, we assume for both halfspace A and thin layer B a hyperelastic isotropic behaviour, which is governed by a stored energy potential U = U (ε, χ). Therefore, in the small deformation framework, at the leading order the following equations hold (see [6]): σ=

∂U ∂ε

⇒

σ = 2Gε + Λ(tr ε)1,

(7a)

  ∂U ⇒ μ = 2G2 χT + ηχ , (7b) ∂χ where Λ and G are the classical Lam´e constants, whereas parameters  > 0 and −1 < η < 1 characterise the microstructure. More specifically,  is referred to as characteristic length, and the dimensionless scalar η plays a role similar to that of Poisson’s ratio. Both parameters may be related to the material characteristic length in bending and torsion, b and t , experimentally evaluated by Lakes [9] for polyurethane and syntactic foams. The connecting relations read μ=

 b = √ , 2  t =  1 + η. 2.2

Assumption of Antiplane Shear Deformations

The hypothesis of antiplane shear deformations implies u = (0, 0, u3 ),

where

u3 = u3 (x1 , x2 , t),

that is, the sole relevant component of the displacement field u is the out-ofplane component u3 , which is independent of the coordinate x3 . Under this assumption, by referring to the constitutive prescriptions (7), we immediately observe that tr ε vanishes, such that the elastic constant Λ become completely irrelevant in the problem at hand. More specifically, by referring to Eqs. (2), (5), and (4), the sole relevant components of ε, ϕ, and χ are ε13 = 12 u3,1 , ϕ1 =

ε23 = 12 u3,2 ,

1 2 u3,2 ,

χ11 = −χ22 =

ϕ2 = 1 2 u3,12 ,

χ21 =

− 21 u3,1 , − 21 u3,11 ,

(8a) (8b) χ12 =

1 2 u3,22 .

(8c)

whereas the equations of motion (6) may be rewritten as σ13,1 + σ23,2 + τ13,1 + τ23,2 = ρ¨ u3 , μ11,1 + μ21,2 + 2τ23 = J ϕ¨1 ,

(9a) (9b)

μ12,1 + μ22,2 − 2τ13 = J ϕ¨2 ,

(9c)

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73

with σ13 = Gu3,1 , μ11 = −μ22 = G (1 + η)u3,12 , 2

σ23 = Gu3,2 ,

(10a)

μ21 = G (u3,22 − ηu3,11 ),

(10b)

2

μ12 = −G (u3,11 − ηu3,22 ). 2

(10c)

Then, combination of Eqs. (6b), (8b), and (10) allows the determination of the components (see [10]) ˆ 3,1 + τ13 = − 12 G2 u

J u ¨3,1 , 4

ˆ 3,2 + τ23 = − 12 G2 u

J u ¨3,2 , 4

(11)

ˆ indicates the 2-D Laplace operator in the x1 , x2 coordiin which the notation  nates. As a final step, substitution of Eqs. (10a) and (11) into (9a) leads to the expression of the governing equations in terms of the sole displacement field   ˆ 3 − J ¨ ˆ u ˆ 3 − u ˆ u3 + ρ¨ u3 = 0, (12) G 12 2  4 which correctly particularises to the expression presented in [7]. Now, we may conveniently reformulate Eq. (12) in a nondimensional form. For what concerns the spatial variables, we scale the coordinate x by introducing the scalar Θ (see [2]), to be specified in the following, such that ξ = (ξ1 , ξ2 , ξ3 ) =

1 (x1 , x2 , x3 ) Θ

is the new dimensionless coordinate system. Consistently, here and henceforth  indicates the Laplace operator in ξ. However, we let the dimensionless thickness of the thin layer H = h/ and, by following [10], we introduce the parameter 0 =

d , 

which represents the dimensionless form of  √ J d = 12 ∝ ld = 2 6d , ρ

(13a)

(13b)

in which ld is the dynamic characteristic length (see [11]). For whatconcerns the time variable t, we use T = /cs as a reference time, where cs = G/ρ is the classical shear wave speed of CE, such that τ=

t T

is the new dimensionless variable. Similarly, being ω = t−1 the time frequency, Ω = ωT > 0 is the frequency dimensionless form. In the light of the introduced variables, equilibrium equation (12) becomes 2

0 u − u u3 − 2Θ2 u3 − 2Θ4 = 0. (14) 3,τ τ 3,τ τ Θ2

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Assumption of Time-Harmonic Solution

We consider time-harmonic and straight-crested waves propagating in plane (ξ1 , ξ2 ). Consequently, the relevant component of the displacement field may be expressed as the product of a time-independent function W and a spatialindependent exponential function, that is u3 = W (ξ1 , ξ2 ) exp(−ıΩτ ), in which ı indicates the imaginary unit. Therefore, the equilibrium equation (14) may be expressed as (see [12])    (15)  − 2 1 − 20 Ω 2 Θ2  − 2Ω 2 Θ4 W = 0, which represents a bi-harmonic homogeneous partial differential equation in W . Factoring out (15) leads to    + δ 2 ( − 1) W = 0, (16) with δ being the wavenumber of shear horizontal travelling bulk waves and  (1 − 20 Ω 2 )2 + 2Ω 2 − 1 + 20 Ω 2 2 (17) Θ = 2Ω 2 a limited function of Ω, satisfying the bi-quadratic equation (see [2]) 2Ω 2 Θ4 + 2(1 − 20 Ω 2 )Θ2 − 1 = 0. More specifically, δ may be expressed as δ = 2δcr Θ2 ,

with

δcr = 0cr Ω,

(18)

whence, by referring to Eq. (17), we finally have

  1 (1 − 20 Ω 2 )2 + 2Ω 2 − 1 + 20 Ω 2 . δ= 2δcr 2.4

Boundary and Interface Conditions

At the top layer surface x2 = h, characterised by the unit normal vector n = (0, 1, 0), we impose stress free conditions through the definition of p = tT n + 12 ∇μnn × n,

(19a)

with μnn = n · μn = r · n (see Eq. (3)) and q = μT n − μnn n.

(19b)

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75

Vectors p and q are referred to as reduced force traction and CS traction, respectively (see [6]). At the layer/half-space joining surface x2 = 0, we impose the continuity of both kinematics and static fields, such that

3

uA = uB ,

(20a)

ϕ A = ϕB , pA = pB ,

(20b) (20c)

qA = qB .

(20d)

Love Waves Propagating in Micro-structured Media with Different Material Properties

We particularise the CS theory developed in Sect. 2 by assuming that the supporting half-space A and the thin layer B consist of media characterised by different material properties. For convenience, we select the dimensional quantities  and T corresponding to those which pertain to the half-space A, namely 

and T = TA = A /csA ,

 = A

where cs A,B = GA,B /ρA,B is the shear wave speed of CE in the relevant medium. Moreover, we conveniently introduce the ratios (see [3]) β = B /A

and

υ = TA /TB

⇒

υβ = cs B /cs A ,

the last remaining bounded also in the in the absence of microstructure for B, i.e. υ → +∞, and β → 0. Therefore, in each medium the governing Eq. (16) reads    + δ 2 ( − 1) WA = 0, (ξ1 , ξ2 ) ∈ BA ,     + δ12  − δ22 WB = 0, (ξ1 , ξ2 ) ∈ BB , where δ12 =

δψ β 2 υ2

and δ22 =

δ . β2ψ

Here, by referring to (13), we let  dB = and

1 2

JB , ρB

0B = dB /B ,

 (υ 2 − 0 2B Ω 2 )2 + 2υ 2 Ω 2 − υ 2 + 0 2B Ω 2 √ ψ= . 2Ω

(21a) (21b)

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For what concerns the thin layer B, in the light of the arguments developed so far, the sole relevant components of the reduced force traction (19a) and the CS traction (19b) are p3B and q1B (see [6]), respectively, such that stress free conditions may be imposed as follows: p3B = t23B + 12 μ22,1B

   δ ψ2 GB = − 3 β 2 (ηB + 2) WB ,112 + WB ,222 + − 1 W = 0,(22a) B ,2 2Θ ψ υ2  GB  2  β WB ,22 − ηB WB ,11 = 0. 2 Θ Similarly, interface conditions (20) become

(22b)

q1B = μ21B =

3.1

WA = WB , ∂WA ∂WB = , ∂ξ2 ∂ξ2 p3A = p3B

(23b)

q1A = q1B .

(23d)

(23a)

(23c)

Dispersion Equation

We assume guided waves propagating along the longitudinal direction ξ1 , such that we may conveniently express the time-independent function W in each medium as WA,B (ξ1 , ξ2 ) = A wA,B (ξ2 ) exp (ıκξ1 ) . Here and henceforth κ = ΘK = ΘkA , with K denoting the dimensionless spatial wavenumber. Therefore, the general solutions of Eqs. (21), respectively holding in the half-space A and in the thin layer B, are wA (ξ2 ) = exp (A1 ξ2 ) s1 + exp (A2 ξ2 ) s2 ,

ξ2 ∈ BA ,

(24a)

wB (ξ2 ) = cosh (B1 ξ2 ) e1 + cosh (B2 ξ2 ) e2 + B1−1 sinh (B1 ξ2 ) o1 + B2−1 sinh (B2 ξ2 ) o2 , where A1 = B1 =

 

κ2 − δ 2 ,

κ2 − δ12 ,

 κ2 + 1,  B2 = κ2 + δ22 . A2 =

ξ2 ∈ BB , (24b)

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The six unknown amplitudes s1,2 , e1,2 , o1,2 may be determined through application of boundary/interface conditions (22) and (23), which in nondimensional form may be expressed as (25a) p3B (θ−1 H) = 0, q1B (θ−1 H) = 0,

(25b)

wA (0) = wB (0),   wA (0) = wB (0),

(26a) (26b)

q1A (0) = q1B (0), p3A (0) = p3B (0).

(26c) (26d)

and

Therefore, by plugging solutions (24) into conditions (23) and (26) we obtain a homogeneous system of linear equations in the unknown amplitudes. This system admits non-trivial solutions if the determinant of the linear system Δ vanishes, such that the secular equation Δ(κ) = 0 (27) holds. By introducing the coefficient Γ = GB /GA , the general form of the secular equation may be written as  Δ(κ) = (A1 − A2 ) d1 RA (κ) + Γ d2 + Γ 2 DB (κ) , where

(28)

2  2  RA (κ) = (1 + ηA )κ2 + 1 A1 − (1 + ηA )κ2 − δ 2 A2 ,

is the Rayleigh function holding for the antiplane Rayleigh waves propagating in the sole half-space and DB (κ) is the dispersion relations for symmetric and antisymmetric Rayleigh-Lamb modes (see [2]). The quite complicated expressions of coefficient d1 and d2 , the last expressing the coupling between half-space and thin layer, are here omitted for brevity.

4

Results and Discussion

To provide an insight on the role of material micro-structure, here we study the influence of two key parameters on the propagation of Love waves. Specifically, in Fig. 2 we plot the frequency spectrum for different values of the ratio Γ between the shear moduli of media A and B, whereas in Fig. 3 we vary the nondimensional parameter −1 < ηA < 1 characterising the half-space. For the sake of compliteness, the plots here reported are obtained by selecting 0A = 0.3, 0 B = 0.5, β = υ = 1.1, ηB = 0.5, and H = 0.1.

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Fig. 2. Frequency spectrum for different values of the ratio Γ between the shear moduli of media A and B (here ηA = 0.5): Γ = 0.5 (blue dotted line), Γ = 5 (red dashed line), and Γ = 10 (black solid line).

Comparison of Figs. 2 and 3 makes clear that propagation of Love waves is mostly influenced by ratio Γ , while parameter ηA seems not to largely affect the response. Curves in the frequency spectra, which show a monotonic increasing behaviour, significantly diverge in Fig. 2 for wavenumbers κ > 1. On the other hand, little variations are observed in Fig. 3. This behaviour can be ascribed to the fact that parameter ηA is only present in the applied boundary/interface conditions. Interestingly, for all the parameters here investigated, Love waves start to propagate beyond a threshold value, the so-called cuton frequency Ωcuton . This feature, similar to that of Stoneley waves propagating in CS media [3], is quite clear in Fig. 3: for Ω < 1.5 (κ < 0.65), in fact, the dispersion equation (28) does not admit any solution and waves cannot propagate. Moreover, Fig. 4 shows that the cuton frequency is a monotonic increasing function of Γ , which exhibits a horizontal asymptote at Ωcuton,asym ≈ 2. Therefore, the propagation of Love waves turns out to be possible for any Γ .

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Fig. 3. Frequency spectrum for different values of the parameter −1 < ηA < 1 of the half-space (here Γ = 5): ηA = −0.9 (blue dotted line), ηA = 0 (black solid line), and ηA = 0.9 (red dashed line).

Fig. 4. Cuton frequency as a function of the ratio Γ between the shear moduli of media A and B for different values of the parameter −1 < η < 1 of the half-space A: ηA − 0.9 (blue dotted line), ηA = −0.5 (green dash-dotted line), ηA = 0.5 (black solid line), and ηA = 0.9 (red dashed line).

80

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Conclusions and Open Issues

In this note, we investigate the role of material micro-structure on the propagation of antiplane Love waves in micro-structured media. Specifically, we adopt the couple stress theory with micro-rotational inertia to model a thin layer perfectly bonded to a half-space with different material properties, both characterised by micro-structure. Determination of an anlytical expression of the dispersion equation allows us to easily plot the frequency spectra by varying the microstuctural half-space parameter, ηA , and the ratio between shear moduli of the two media, Γ . These two key parameters differently influence the propagation of Love waves, which turns out to be always admitted beyond a cuton frequency. This threshold, in fact, is a monotonic increasing function of Γ and shows a horizontal asymptotic behaviour. In order to enrich this preliminary contribution, future works on this topic should focus on the establishment of rigorous existence conditions, which may better clarify the role of each material parameter involved in the propagation of Love waves. Moreover, the determination of effective boundary conditions (see [14] and references therein) may pave new ways to simplify the investigation of wave propagation in micro-structured media. Acknowledgment. This research was supported under the grant POR FESR 2014-2020 ASSE 1 463 AZIONE 1.2.2 awarded to the project “IMPReSA” CUP E81F18000310009.

References 1. Love, A.E.H.: Some Problems of Geodynamics. Cambridge University Press, Cambridge (1911). https://doi.org/10.1038/089471a0 2. Nobili, A., Radi, E., Signorini, C.: A new Rayleigh-like wave in guided propagation of antiplane waves in couple stress materials. Proc. Roy. Soc. A 476(2235), 20190822 (2020). https://doi.org/10.1098/rspa.2019.0822 3. Nobili, A., Volpini, V., Signorini, C.: Antiplane Stoneley waves propagating at the interface between two couple stress elastic materials. Acta Mech. 232(3), 1207– 1225 (2021). https://doi.org/10.1007/s00707-020-02909-y 4. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. An. 11(1), 415–448 (1962). https://doi.org/10.1007/BF00253946 5. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. An. 11(1), 385–414 (1962). https://doi.org/10.1007/BF00253945 6. Koiter, W.T.: Couple-stress in the theory of elasticity. In: Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, vol. 67, pp. 17–44 (1964) 7. Fan, H., Xu, L.: Love wave in a classical linear elastic half-space covered by a surface layer described by the couple stress theory. Acta Mech. 229(12), 5121–5132 (2018). https://doi.org/10.1007/s00707-018-2293-1 8. Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. PrenticeHall Inc., Englewood Cliffs (1969) 9. Lakes, R.S.: Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22(1), 55–63 (1986). https://doi.org/10.1016/0020-7683(86)90103-4

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10. Mishuris, G., Piccolroaz, A., Radi, E.: Steady-state propagation of a mode III crack in couple stress elastic materials. Int. J. Eng. Sci. 61, 112–128 (2012). https://doi. org/10.1016/j.ijengsci.2012.06.015 11. Shodja, H.M., Goodarzi, A., Delfani, M.R., Haftbaradaran, H.: Scattering of an anti-plane shear wave by an embedded cylindrical micro-/nano-fiber within couple stress theory with micro inertia. Int. J. Solids Struct. 58, 73–90 (2015). https:// doi.org/10.1016/j.ijsolstr.2014.12.020 12. Georgiadis, H.G., Velgaki, E.G.: High-frequency Rayleigh waves in materials with micro-structure and couple-stress effects. Int. J. Solids Struct. 40(10), 2501–2520 (2003). https://doi.org/10.1016/S0020-7683(03)00054-4 13. Nobili, A., Radi, E., Vellender, A.: Diffraction of antiplane shear waves and stress concentration in a cracked couple stress elastic material with micro inertia. J. Mech. Phys. Solids 124, 663–680 (2019). https://doi.org/10.1016/j.jmps.2018.11.013 14. Auld, B.A.: Acoustic Fields and Waves in Solids, 2nd edn. Krieger Publishing Company, Malabar (1990)

Acoustic Control of Cracks Self-healing in Plates: An Impulse Induced Wave Field in a Plate with an Initially Closed Crack Parallel to Its Free Surfaces Vladislav V. Vershinin1,2(B) 1 Moscow State University of Civil Engineering, Moscow 129337, Russia

[email protected] 2 Ishlinsky Institute for Problems in Mechanics RAS, Moscow 119526, Russia

Abstract. A homogeneous isotropic linearly elastic plate constantly staying in a plane strain state and having a zero initial stress field and a plane crack parallel to plate free surfaces is considered. Wave fields induced in the plates with and without the crack by an impulse distributed surface load are simulated through the finite element and the finite difference methods to determine a normal displacement field on the plate free surfaces and to identify key problem parameters. Totally 26 different combinations of the problem parameters are considered. It is shown that a distributed surface source of acoustic signals of a small magnitude and positioned on its axis a transducer of diffracted and reflected signals allows one to uniquely detect in a homogeneous isotropic plate an initially closed plane crack located right beneath the source and being parallel to the plate free surfaces, but, at the same time, this technique with such an instrumentation arrangement is essentially local and requires high sampling frequencies of the devices. Such parameters of the considered problem as the plate material Poisson’s ratio, a width of a major peak of an impulse load spectrum, and a width of the loaded area are ascertained to have evident quantitative effects on the recorded waveforms, but exert either no or insignificant qualitative effects onto them, so that these problem parameters may be precluded from consideration when numerically investigating the acoustic control capability to follow a cracks self-healing process. Keywords: Crack self-healing · Wave field · Acoustic control · Numerical simulation

1 Introduction Self-healing, i.e. partially or fully autonomous recovery of initial properties degraded due to defects or cracks formation, is an extremely important capability of structural materials since it provides material integrity and, hence, in-service structural reliability. Initially inspired by self-healing in biological systems, researches on the issue of selfhealing materials have experienced an exploding growth over the last 30 years. At present a self-healing property exists in or has been introduced into almost all types of structural © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 82–97, 2022. https://doi.org/10.1007/978-3-030-92144-6_6

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materials: metals and alloys [5], ceramics [27], concrete and engineered cementitious composites [22, 26], polymeric matrix composites [9, 19], etc. Evidently, self-healing should be somehow assessed to evaluate efficiency of a particular self-healing approach and to monitor in-situ a progress of a self-healing process. Despite a variety of types of the self-healing materials, assessment techniques and methods applied in practice are pretty much unified [5, 9, 22, 26, 27]. To evaluate the efficiency of some particular self-healing approach a full range of available methods is employed. They are mechanical tests, various visualization techniques, methods to determine material physical properties, etc. The list of self-healing monitoring techniques that are applicable in-situ is much more limited since such techniques must impart minimal or no damage to a structure and require portable equipment that should be relatively simple in operation, i.e. do not require highly specialized skills and specific pre- and post-test arrangements. Ultrasonic testing is among a limited number of methods that are applied to both evaluation of self-healing approach efficiency and in-situ monitoring of a self-healing process, yet it is not very popular in the former case. For instance, in [3] an ultrasonic pulse velocity (UPV) method was employed to monitor a self-healing process in concrete and mortar samples loaded at the age of 8, 16, 24, or 72 h to different fractions of their compressive strengths and then cured in air or in water up to the age of 90 days. Aldea et al. [6] utilized measurements of elastic waves transmission in the frequency range 30–60 kHz to quantify a self-healing phenomenon in cylindrical concrete specimens with a low height-to-diameter ratio and through thickness radial cracks of various width which were subjected to a low-pressure water permeability test for 100 days. Watanabe et al. [31] evaluated effect of fly ash onto autogenous healing of microcracks developed in concrete during freezing-thawing cycles measuring relative changes of speed and amplitude of ultrasonic elastic waves. Abd Elmoaty [2] investigated a self-healing of cracks in cubic specimens made of polymer modified concrete using UPV method and performed a parametric study to assess effects of polymer content, polymer type, watercement ratio, cement content (under fixed w/c ratio), cement type, and specimen age when it had been loaded to cracks formation. In [17] diffuse ultrasound and its such relevant parameters as diffusivity and arrival time of maximum energy were used to monitor and evaluate a self-healing process in prestressed concrete specimens made of mixes with three different cementitious material compositions, in which cracks along either a whole cross-section or its part were induced and which were then immersed in sea water for 120 days to simulate operation conditions of prestressed concrete piles in marine environment. Using coda wave interferometry (CWI) analysis, Liu et al. [20] made an effort to establish a correlation between a relative change of coda waves velocity in cracked neat and biomimetic mortars and a degree of self-healing of internal cracks quantitatively assessed through compressive strength tests. The ultrasonic testing is a special case of a more general non-destructive evaluation technique – an acoustic control using elastic stress waves. The acoustic control technique is based onto an analysis of diffracted and scattered fields of the elastic waves. Within this method the elastic stress waves can have an arbitrary spectrum and, thus, can be generated by various means. Yet, it seems that this more general method is less popular than the ultrasonic testing.

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In the present paper the acoustic control will be numerically investigated in regard to cracks detection in homogeneous plates keeping in mind its potential application as an in-situ monitoring technique of self-healing processes. Researches on the stress waves as a mean to identify inhomogeneities in elastic media have being been conducted for more than half a century. One can find in literature a vast amount of theoretical and numerical considerations of relevant direct (see, for instance, [7, 8, 11–15, 18, 21, 23, 25, 28–30] and references therein) and inverse (see, for instance, [4, 24] and references therein) problems. Yet, it should be noted that in the cited papers rather general problem formulations were considered so that results obtained therein cannot be directly applied to the problem stated in the present paper. Moreover, a numerical simulation of interactions of the elastic stress waves with a crack in a homogeneous plate will allow one (1) to determine problem parameters that may be excluded from further considerations since they don’t have any significant influence onto obtained results and (2) to form benchmarks for a subsequent numerical investigation of an acoustic control capability to follow a cracks self-healing process.

2 Problem Formulation A homogeneous isotropic linearly elastic plate with a zero initial stress field, infinite in-plane dimensions and a thickness of H is considered (Fig. 1). In the plate exists a crack with a width of 2l and an infinite length located at a depth of h beneath the plate front surface. At the time instant t = 0 the crack is closed, but with zero contact stresses between its faces. The plate front surface is exposed to a transient load p(t) of small magnitude uniformly distributed over an area with a width of s and an infinite length so that the plate is constantly in a plane strain state. A center of the loaded area is located at a distance of x from a crack center projection line.

Fig. 1. Problem formulation

A plate material has the following properties: a density of ρ, the Young’s modulus of E, and the Poisson’s ratio of ν. The transient load p(t) is defined for t ≥ 0, has a form of a triangle impulse with a magnitude of A and duration of τ , and is symmetric with respect to the time instant t = τ/2.

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Friction between the crack faces is not considered. Determined is a normal displacement field u2 (x, t) on the plate free surfaces for t ≥ 0. Monitoring points are spaced at intervals of 0.1H symmetrically with respect to the crack center projection line (see Fig. 2; the points are numbered symmetrically with respect to the crack center projection line, but those to the left also have asterisks).

Fig. 2. Monitoring points

3 Governing Equations The formulated problem is described by the following equations: ˆ x, t) = ρ u¨ (x, t) (x ∈ ), ∇ · σ(

(1)

u (x, 0) = 0 (x ∈ ),

(2)

u˙ (x, 0) = 0 (x ∈ ),

(3)

 (x, t) = −p(t) p n(x, 0) (x ∈ load ),

(4)

ˆˆ ˆ x, t) = C( σ( x):ˆ(x, t) (x ∈ ),

(5)

ˆ (x, t) = Sym(∇ u (x, t)) (x ∈ )

(6)

ˆˆ where σˆ is the Cauchy stress tensor, C is the stiffness tensor,  is a spatial domain occupied by the plate, load ∈  is a loaded area of the plate free surfaces, n is an outer unit normal to any outer or inner surface of the plate. Boundary conditions on the crack faces during their contact are the following:   (1) (2) (7) u (y, t) · n (y, t) = u (z, t) · n (z, t) y ∈ crack , z ∈ crack (1)

(2)

where crack ∈  and crack ∈  are the crack faces.

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4 Problem Parameterization and Dimensional Analysis The considered problem is multiparameter. To reduce a number of parameters and to make development of a numerical model more convenient, one can perform all calculations for a model problem basing on the Buckingham π theorem (see, for instance, Chap. 6 in Ref. [16] or any other handbook on the dimensional analysis). According to this theorem, results obtained for the model problem are valid for any practical problem (governed by the same set of equations) which has the same values of characteristic dimensionless parameters as the model problem. So, having a solution of the model problem, one can recalculate it for the real one. For the considered problem basic parameters which are used for a construction of governing dimensionless combinations are the following. All linear dimensions are normalized by the thickness H of the plate. A basic time unit is the duration τ of the applied load. The plate material Young’s modulus E is a basic parameter expressed in a unit of stress. A speed cP of longitudinal (primary) waves in the plate is used as a velocity unit. Hence, a speed cS of transversal (secondary) waves and a speed cR of Rayleigh waves in the plate together with the speed cP form dimensionless combinations cP /cS and cP /cR , respectively. Taking the well-known expressions for the speeds of elastic waves (see, for instance, Chap. 5 in Ref. [10]), one can easily obtain the relationships  cP 2(1 − ν) , (8) = cS 1 − 2ν  cP 2(1 − ν) 1+ν (9) ≈ cR 0.87 + 1.12ν 1 − 2ν which show that the dimensionless parameters cP /cS and cP /cR are uniquely defined by another non-dimensional parameter – the plate material Poisson’s ratio ν. Default values of the problem parameters are given in Table 1. In Table 2 combinations of the problem parameters considered in the present research are presented. To refer to a particular parameters combination a triplet of numbers of the form i − j − k is used throughout the present paper. The first number i takes values from 1 to 2 where i = 1 corresponds to the case of the plate without a crack, i = 2 corresponds to the case of the closed crack without initial contact stresses on its faces. The second number j takes values from 1 to 6 and is referred to different values of the crack parameters such as the crack width 2l and the crack center depth h beneath the plate front surface. The third number k takes values from 1 to 12 and indicates different values of the plate and load parameters such as the load impulse duration τ , the width s of the loaded area, the position x of the loaded area center relative to the crack center projection line, and the plate material Poisson’s ratio ν. The problem parameters combinations considered within the present research (see Table 2) allows one to obtain normal displacement field u2 (x, t) on the free surfaces of the plate without a crack and of the plate with the crack under different values of specific parameters. A comparative analysis of obtained results may enable to decrease

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Table 1. Default values of the problem parameters

Parameter

Default value 1.0

H, m , kg m3

1.0

E, Pa

1.0 0.0

x, m

0.0

A, Pa

0.0001

,s

0.005

s, m

1.0

l, m

0.5

h, m

0.5

Table 2. Considered combinations of the problem parameters

2

10 11 12

×

×

×

4

5

6

×

×

×

×

×

×

×

1

Default l = 0.1H l = 0.2H l=H h = 0.25H h = 0.75H

×

×

×

×

×

×

2 3 4 5 6

×

×

×

s = 0.2H, x = 0.9H

s = 0.2H, x = 0.3H 9

3

–

s = 0.2H, x = 0.7H

s = 0.2H, x = 0.1H 8

2

1

s = 0.2H, x = 0.5H

s = 2H

s = 0.2H

ν = 0.4

ν = 0.2

τ = 0.05s

7

1 – Rectilinear segment

1

Crack parameters

Default

Crack opening shape

τ = 0.01s

Plate and load parameters

×

×

× ×

×

× ×

the number of the problem parameters and their combinations that will be necessary to consider when investigating a possibility of monitoring of cracks self-healing through the acoustic control. It is also worth noting that the plane crack which is initially closed with no contact stresses on its faces can open and close under a stress waves action. As it has been stated in Sect. 2, within the present research the transient load p(t) is defined for t ≥ 0, has the form of the triangle impulse with the magnitude of A and the duration of τ , and is symmetric with respect to the time instant t = τ/2. Such a

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time history is commonly used when modeling impact loading of various nature. One can easily obtain a spectrum of the load p(t) using the Fourier transform and see that the triangle impulse is an anharmonic loading. The longer is the impulse duration τ , the narrower is its spectrum. Setting τ to different values, it is possible to determine an influence of a width of a load spectrum major peak onto normal displacement field u2 (x, t) on the free surfaces of the plates with and without a crack.

5 Numerical Models The problem formulated in Sect. 2 and parametrized in Sect. 4 is solved using a commercial finite element analysis package SIMULIA Abaqus, in particular its module Abaqus/Explicit. In Abaqus/Explicit [1] spatial discretization of a problem is performed with finite elements while time discretization is realized via central-difference scheme. Solution discontinuities are not distinguished. An obtained solution is of the second order of accuracy. The discretization scheme implemented in Abaqus/Explicit is conditionally stable. A time increment is determined autonomously by the program to ensure the calculation stability [1]. An artificial bulk viscosity containing a linear term to reduce solution oscillations behind waves fronts and a quadratic term to prevent finite elements from collapsing to zero volume under extremely high velocity gradients is also implemented in Abaqus/Explicit [1]. Within the present research parameters of these linear and quadratic terms were set to their default values b1 = 0.06 and b2 = 1.2, respectively. Despite the infinite plate is stated in Sect. 2, within numerical models plates of finite widths are created. But their widths are set large enough for the fastest (longitudinal) waves reflected from lateral boundaries not to be able to come back to the monitoring points during simulated time periods. Finite elements of the type CPE4R (a 4-node quadrilateral bilinear plane strain element with reduced integration and an hourglass control) are used to mesh the plate within the numerical models. A characteristic mesh size is set equal to 0.0005H . Numerical models with asymmetric loading are meshed with totally 32 000 000 finite elements. Numerical models with symmetric loading contain totally 16 000 000 finite elements for ν = 0, 18 800 000 finite elements for ν = 0.2, and 24 800 000 finite elements for ν = 0.4. An example of the numerical model with defined boundary conditions and loads correspondent to particular problem parameters combinations (see Table 2) is presented in Fig. 3.

Fig. 3. Numerical model in the cases of the plate with the initially closed plane crack and the asymmetric loading (x/H > 0)

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6 Simulation Results and Their Analysis The distributed surface acoustic source generates a single plane compression wave which reverberates either between the plate free surfaces or between the plate free surfaces and the faces of the crack opened due to the wave action. This wave experiences compressiontension conversion with each reflection. Boundaries of the loaded area are sources of longitudinal and vertically polarized transversal waves with cylindrical fronts, and of Rayleigh waves which propagate along the plate free surface both ways. All the bulk waves excited in the plate experience scattering at crack tips (edges) which act as sources of secondary waves, viz. bulk waves radiated into the plate and Rayleigh waves propagating along the crack faces. Also all the bulk waves propagating in the plate and obliquely arriving at any free surface existing in the plate at that instant of time split into longitudinal and vertically polarized transversal waves when reflecting from this surface (see, for instance, Chap. 5 in Ref. [10]). For a reflected wave of the same type as an incident wave an angle of reflection equals to an angle of incidence. When a reflected wave is of the type other than the type of an incident wave, valid is the Snell’s law (see, for instance, Chap. 5 in Ref. [10]): cos(θS ) cos(θP ) = cP cS

(10)

where θP is an angle between a direction of propagation of a longitudinal wave and that free surface at which it reflects, θS is an angle between a direction of propagation of a vertically polarized transversal wave and that free surface at which it reflects. To illustrate some of the above statements, histories u2 (t) of vertical displacements obtained numerically for the problem parameters combination 1–1-1 at the monitoring points on the plate front and rear surfaces are shown in Fig. 4 and Fig. 5, respectively. One can see that even in the plate without a crack the wave field being initially simple complicates more and more with time. In the case of a plate with an initially closed plane crack with zero contact stresses between its faces stress waves generated by a source cause oscillations of the crack faces. These oscillations can evolve out-of-phase along the crack width so that partial opening or closure of the crack can take place. This causes a significant distortion of a diffracted and reflected acoustic signal recorded at a front (relative to the acoustic source) surface of the plate. Monitoring of a normal displacement field u2 (x, t) at both free surfaces of the plate significantly shorten the list of structures that can be inspected using the acoustic control since it requires an access to both sides of the plate, but, at the same time, it may facilitate a blind treatment of simulation (and, hence, testing) results. A representative example is presented in Figs. 6 and 7. From Fig. 7 it is possible to draw some conclusions on crack size and position, while Fig. 6 alone barely provides any useful data. In Figs. 8 and 9 it can be seen that a distributed surface source of acoustic signals and positioned at its center a transducer of diffracted and reflected signals allows one to uniquely detect in a homogeneous isotropic linearly elastic plate an initially closed plane crack located right beneath the source. Yet, such instrumentation arrangement yields weak sensitivity to crack parameters, right in accordance with the discussion above. Monitoring only the front (relative to the acoustic source) surface of the plate

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11

6

5 4 3 2 1

Fig. 4. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combination 1–1-1 at the monitoring points on the plate front surface

22

12

Fig. 5. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combination 1–1-1 at the monitoring points on the plate rear surface

makes initially closed cracks with different widths and located at different depths almost indistinguishable between each other since there are no characteristic features in the corresponding waveforms. It is also worth noting that it is sufficiently to consider a time span during which a longitudinal wave covers the distance of 4H to uniquely detect a crack and identify its parameters through the acoustic control. Moreover, it is no use analyzing larger time spans due to a gradual complication of the wave field in the plate with time, as described in the beginning of the present section. An influence of the existing crack on a diffracted and reflected acoustic signal recorded at the center of the distributed surface acoustic source is negligible already at the distance x = 0.4H between the monitoring point and a projection of a crack tip

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11

6

5 4 3 2 1

Fig. 6. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combination 2–6-1 at the monitoring points on the plate front surface

22

12

Fig. 7. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combination 2–6-1 at the monitoring points on the plate rear surface

onto the front plate surface (see Fig. 10). This result indicates that the acoustic control with the arrangement of instrumentation only on one of the plate free surfaces is an essentially local monitoring technique. From Fig. 11 it is clear that an increase of the plate material Poisson’s ratio ν leading to a significant raise of values of the ratios cP /cS and cP /cR changes a wave field (viz. order in which waves of different types arrive at a particular monitoring point) in the plate caused by the distributed surface load p(t), but still allows one to uniquely detect an initially closed plane crack right beneath the acoustic source. A variation of the width of the major peak of the distributed surface load spectrum does not result in significant qualitative changes of the excited normal displacement field u2 (x, t) at the free surfaces of the plate without a crack as well as of the plate with an

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2-1-1 2-4-1

2-3-1

1-1-1

2-2-1

Fig. 8. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combinations 1 − 1 − 1 and 2 − j − 1 (j = 1, ..., 4) at the center of the loaded area

2-5-1

2-6-1 2-1-1

1-1-1

Fig. 9. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combinations 1 − 1 − 1 and 2 − j − 1 (j = 1, 5, 6) at the center of the loaded area

initially closed plane crack (see Figs. 12 and 13). Pretty much the same conclusion can be drawn concerning an effect of the width of the loaded area (see Figs. 14 and 15).

7 Discussion and Conclusion The formulation of the problem given in Sect. 2 implies that instrumentation used to detect cracks in a homogeneous isotropic plate and to follow cracks self-healing process comprises of a distributed surface source of acoustic signals of various waveforms and durations and a transducer of diffracted and reflected signals. An analyzing device may be also comprised. Optimal is believed to be such an instrumentation design in which the

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2-1-6

2-1-12

Fig. 10. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combinations 2 − 1 − k (k = 6, 8, ..., 12) at the center of the loaded area

2-1-4

2-1-5

2-1-1

Fig. 11. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combinations 2 − 1 − k (k = 1, 4, 5) at the center of the loaded area

signals source and the diffracted and reflected signals receiver are located at the same side of a tested plate with devices axes being coincident and the devices themselves having no unwanted action onto the plate. Yet, arrangement of the source and the receiver on opposite sides of the plate may facilitate treatment of in-situ measurement results. The smaller is a characteristic size of the receiver, the higher is a resolution of the instrumentation and the more subtle effects can be investigated. Basing on the performed investigation in which totally 26 different combinations of the problem parameters have been considered, it can be concluded that a distributed surface source of acoustic signals of a small magnitude and positioned on its axis a transducer of diffracted and reflected signals allows one to uniquely detect in a homogeneous

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1-1-1

1-1-2

1-1-3

Fig. 12. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combinations 1 − 1 − k (k = 1, ..., 3) at the center of the loaded area

2-1-1

2-1-2

2-1-3

Fig. 13. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combinations 2 − 1 − k (k = 1, ..., 3) at the center of the loaded area

isotropic plate an initially closed plane crack located right beneath the source and being parallel to plate free surfaces. It has been revealed that the acoustic control with a coaxial arrangement of a source and a transducer is an essentially local monitoring technique. To determine a crack location it is necessary to move the instrumentation with small steps. A size of the steps should be of the order of a crack width. Also the instrumentation must have high enough sampling frequency since a wave field induced in a plate by a distributed surface load gradually complicates with time and, hence, very short time spans (comparable with time of a wave travel through a plate thickness) have to be considered. It can be stated that such parameters of the considered problem as the plate material Poisson’s ratio, the width of the major peak of the impulse load spectrum, and the width

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1-1-6

1-1-1 1-1-7

Fig. 14. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combinations 1 − 1 − k (k = 1, 6, 7) at the center of the loaded area

2-1-6

2-1-1

2-1-7

Fig. 15. Histories u2 (t) of normal displacements obtained numerically for the problem parameters combinations 2 − 1 − k (k = 1, 6, 7) at the center of the loaded area

of the loaded area have evident quantitative effects on the recorded waveforms, but exert either no or insignificant qualitative effects onto them. The latter fact is very important when we consider blind treatment of in-situ measurements since usually in this case one doesn’t have any reference values, but can analyze observing trends and their changes. So, the above specified problem parameters may be precluded from consideration when numerically investigating the acoustic control capability to follow a cracks self-healing process. Finally, it should be emphasized that prior to proceeding with simulations of monitoring of a crack self-healing process with the acoustic control an influence of several problem parameters not considered within the present paper should be assessed. These parameters are (1) a stress field σˆ 0 (x) initially existing in a plate so that a plane crack may

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be initially either opened or closed with non-zero contact stresses between its faces, (2) friction between the crack faces, and (3) an inclination angle of the plane crack relative to plate free surfaces. This should be a goal of a subsequent research. Acknowledgments. This work is supported by the Russian Science Foundation under grant 19– 19-00616.

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18. Kundu, T., Boström, A.: Elastic wave scattering by a circular crack in a transversely isotropic solid. Wave Motion 15(3), 285–300 (1992). https://doi.org/10.1016/0165-2125(92)90012-Q 19. Li, G., Meng, H.: Overview of crack self-healing. In: Li, G., Meng, H. (eds.) Recent Advances in Smart Self-healing Polymers and Composites. Woodhead Publishing Series in Composites Science and Engineering, vol. 58, pp. 1–19. Elsevier, Amsterdam, The Netherlands (2015) 20. Liu, S., Bundur, Z.B., Zhu, J., Ferron, R.D.: Evaluation of self-healing of internal cracks in biomimetic mortar using coda wave interferometry. Cem. Concr. Res. 83, 70–78 (2016). https://doi.org/10.1016/j.cemconres.2016.01.006 21. Möhring, W.: Wave momentum and scattering of elastic waves by two-dimensional thin objects. Wave Motion 4(4), 339–347 (1982). https://doi.org/10.1016/0165-2125(82)90003-8 22. Muhammad, N.Z., et al.: Tests and methods of evaluating the self-healing efficiency of concrete: a review. Constr. Build. Mater. 112, 1123–1132 (2016). https://doi.org/10.1016/j.con buildmat.2016.03.017 23. Persson, G., Olsson, P.: 2-D elastodynamic scattering from a semi-infinite cracklike flaw with interfacial forces. Wave Motion 13(1), 21–41 (1991). https://doi.org/10.1016/0165-212 5(91)90003-7 24. Sedov, A., Schmerr, L.W., Jr.: The time domain elastodynamic Kirchhoff approximation for cracks: the inverse problem. Wave Motion 8(1), 15–26 (1986). https://doi.org/10.1016/01652125(86)90003-X 25. Shah, A.H., Wong, K.C., Datta, S.K.: Surface displacements due to elastic wave scattering by buried planar and non-planar cracks. Wave Motion 7(4), 319–333 (1985). https://doi.org/ 10.1016/0165-2125(85)90003-4 26. Tang, W., Kardani, O., Cui, H.: Robust evaluation of self-healing efficiency in cementitious materials – a review. Constr. Build. Mater. 81, 233–247 (2015). https://doi.org/10.1016/j.con buildmat.2015.02.054 27. Tavangarian, F., Hui, D., Li, G.: Crack-healing in ceramics. Compos. Part B: Engrg. 144, 56–87 (2018). https://doi.org/10.1016/j.compositesb.2018.02.025 28. Visscher, W.M.: Calculation of the scattering of elastic waves from a penny-shaped crack by the method of optimal truncation. Wave Motion 3(1), 49–69 (1981). https://doi.org/10.1016/ 0165-2125(81)90011-1 29. Visscher, V.M.: Theory of scattering of elastic waves from flat cracks of arbitrary shape. Wave Motion 5(1), 15–32 (1983). https://doi.org/10.1016/0165-2125(83)90003-3 30. Viswanathan, K., Sharma, J.P., Datta, S.K.: On the matched-asymptotic solution to the diffraction of a plane elastic wave by a semi-infinite stress-free boundary of finite width. Wave Motion 4(1), 1–13 (1982). https://doi.org/10.1016/0165-2125(82)90010-5 31. Watanabe, T., Fujiwara, Y., Hashimoto, C., Ishimaru, K.: Evaluation of self-healing effect in fly-ash concrete by ultrasonic test method. Int. J. Mod. Phys. B 25(31), 4307–4310 (2011). https://doi.org/10.1142/S0217979211066830

Simulation of the Operation of a Detonation Engine E. V. Mikhalchenko1(B) , V. F. Nikitin1 , and V. D. Goryachev2 1 Scientific Research Institute for System Analysis of Russian Academy of Sciences,

Moscow, Russia 2 Department of Mathematics, Tver State Technical University, Tver, Russia

Abstract. The paper presents results of three-dimensional numerical simulations of an annular rotating detonation engine using the author’s software package. The program was based on a mathematical model of multicomponent gas dynamics with chemical reactions considering transport processes in a turbulent mode and external sources of mass and energy. The following species were used to model the kinetics of hydrogen-air mixture combustion: H2 O, OH, H, O, HO2 , H2 O2 , O2 , H2 , N2 . The chemical kinetics used elementary reactions; their number depended on the mechanism. Six kinetic mechanisms were considered: the Maas-WarnatzPope mechanism (1992), the Hong mechanism (2010), the Williams mechanism (2004), the Gri-Mech 3.0 mechanism (1992), the Li – Zhao – Kazakov – Dryer mechanism (2004) and the author’s kinetic mechanism, were compared. Two types of feed mixture were considered: a rich mixture and a stoichiometric one. The results were depicted in a visualization system designed for graphical processing of large data calculations. Different engine operating modes were obtained. Keywords: Combustion chamber · Detonation · RDE · Visualization system

1 Introduction A study and development of detonation engines is of a great interest nowadays. The efficiency of modern engines based on the traditional design is now close to its technological limit. The engines performance may be increased only with the use of new technical solutions. There are two modes of flame propagation in combustible mixtures: slow combustion or deflagration, and rapid detonation. The first method of propagation is the phenomenon of transport: diffusion and thermal conductivity. The detonation mode is supported by a high-intensity leading shock wave which heats the combustible mixture and compresses it. The combustion propagation velocity in this mode is several kilometers per second. Pressure increases several times, in contrast to conventional combustion. It was shown that an engine based on detonation in a fuel mixture can be significantly more efficient than an engine based on a classical scheme [1, 2]. Currently, there are two main types of detonation engines: a pulse detonation engine and a rotating detonation wave engine. Among the engines with a rotating detonation wave, there are many types distinguished by geometry: cylindrical annular with an internal body [3–5], © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 98–107, 2022. https://doi.org/10.1007/978-3-030-92144-6_7

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cylindrical hollow (without an inner body) [6], elliptical (cross-section is an ellipse) [7], and disk [8, 9]. Multiple factors like the reactants type, combustor geometry, thrust, oxidizer and fuel flow rates, and the detonation wave speed at these conditions have been studied in recent years. However, significant challenges still remain. One of these issues is to consider the influence of the kinetic mechanism choice on the numerical simulation results of detonation processes in the combustion chamber. The purpose of this study is to investigate the influence of the kinetic mechanism choice on the results of computational modeling. The paper also presents the results of a visualization system designed for graphical processing of large data calculations.

2 Mathematical Model The mathematical model contains the governing equations, differential and algebraic, boundary and initial conditions. Details of the numerical realization together with the computational mesh design and the placement of variables on the mesh are subjects of the numerical model.  ∂Jk,j ∂  ∂ρk + ρk uj − = ω˙ k ∂t ∂xj ∂xj

(1)

 ∂p ∂τi,j ∂ρui ∂  + ρui uj + − =0 ∂t ∂xj ∂xi ∂xj

(2)

  ∂ET ∂  ∂  ˙ + JT ,j + ui τi,j = Q (ET + p)uj − ∂t ∂xj ∂xj

(3)

In the system (1)–(3) the following notations are used. The ρ k is partial density of a species k, J k,j is a diffusion vector component j of a species k, ω˙ k is intensity of component k formation in chemical reactions, ρ is gas mixture density, uj is a velocity vector component j, p is gas mixture pressure, τ i,j is a tensor component (i,j) of viscous and turbulent stresses, E T is total energy of a species k per volume unit: the sum of thermal, chemical, kinetic, and turbulent energy, J T, j is a component j of the heat and ˙ s the intensity of the energy inflow from an internal chemical energy flow vector, Q source. In Eqs. (1)–(3), and everywhere else below, unless otherwise stated, the repeated indexes denote the summation. In the system (1)–(3), the index k runs through the range 1…N c (number of components), indexes i,j are values 1,2,3 (list of coordinates). Total there are N c +4 equations in a three-dimensional system, not including the turbulence model equations. The latter define transport flows J k,j and J T,j , and the stress tensor in the gas mixture, which is represented as the sum of the ball part and the deviator as (−pδ i,j +τ i,j ). When the turbulence is not modeled, p stands for the thermodynamic pressure of gas. Equations (1)–(3) are supplemented by algebraic relations [10]. Turbulence modeling was performed using the Wilcox k-ω turbulence model [11].

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3 Numerical Scheme The calculations are based on the system of Eqs. (1)–(3) along with the equations for turbulence modeling and the chemical kinetics. All variables were placed in cells centers, and the finite volume method was used to construct the numerical scheme. The MUSCL interpolation of variables from the cells centers to the cell faces was used combined the explicit AUSM-plus method stabilizing the scheme. The method was 2nd order both in space and time, except for tiny zones of strong gradients of variables including shock and detonation waves. Calculations were performed on a uniform grid of cubic cells. The shape of the calculation area was set by a user function in the software package, as well as the distribution of boundary conditions (wall, input and output) by its surface, and initial conditions by domain. The software package consists of the following main parts: mesh generation, construction of the initial state, calculation, and various postprocessing of results. The calculation module processed transitions of variable sets to new layers over time with periodic hard-disk recording of intermediate and final data, and with a possibility ability to resume the calculations from a saved state.

4 Kinetic Mechanism The paper considered the detailed chemical kinetics of the hydrogen-oxygen mixture. When combustion hydrogen with oxygen, in addition to the main product of the reaction, water vapor, several radicals are formed (a component that does not exist under normal conditions in a stable state). The fuel mixture consists of the following species, the table shows the species formula, name, and molar weight in grams per mole (Table 1): Table 1. Fuel mixture composition. Symbol

Title

W, g/mol

H2 O

Water vapor

18.0152

OH

Hydroxyl

17.0073

H

Atomic hydrogen

O

Atomic oxygen

15.9994

HO2

Perhydroxyl

33.0067

H2 O2

Hydrogen peroxide

34.0146

H2

Hydrogen

O2

Oxygen

31.9988

N2

Nitrogen

28.0134

1.0079

2.0158

A chemically reacting mixture is described by the same number of components for all kinetic mechanisms considered. The paper discusses the following kinetic mechanisms: the mechanism Maas-Warnatz-Pope mechanism (19 reactions) [12], the Hong

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mechanism (20 reactions) [13], the Williams mechanism (22 reactions) [14], the GriMech 3.0 mechanism (20 reactions) [15], the Li – Zhao – Kazakov – Dryer mechanism (18 reactions) [16] and the author’s kinetic mechanism (20 reactions) [17]. All the reactions in those mechanisms are assumed reversible. The algebraic relations that determine the component production rate in a kinetic mechanism are complex, but also make the resulting system of equations stiff. To solve the obtained system, we used a semi-implicit Novikov (4,2) method [18] with automatic accuracy control, specially designed for stiff systems of equations.

5 Setting a Model Problem A combustion chamber of a ramjet detonation engine was considered consisting of a coaxial cylinder with an internal cylindrical body ending with a cone. The fuel is supplied from the injectors located on the bottom end. The ratio [H2 ]:[O2 ] = 3:1 is set for a rich mixture, and the ratio [H2 ]:[O2 ] = 2:1 for a stoichiometric mixture. The stagnation pressure was considered 15 bar, and the stagnation temperature T = 258 K. On inner and outer walls of the combustion chamber, there were also injectors supplying the same mixture but with 10 bar stagnation pressure. At the initial time, the chamber is filled with air at a pressure of 1 bar and a temperature of 300 K. The ignition is simulated by adding extra to a ball-shaped area with a radius of rign = 2.5 mm during tign = 1 μs with power Q = 20 kW/sm3 after t = 10 μs from the beginning of the process. The length of the entire domain was L = 10 cm, the domain radius was R = 5 cm, the radius of the inner body Rb = 3 cm, the length of the inner body cylindrical part Lb = 3 cm, the length of the cone at the tip was Lc = 3 cm, number of bottom injectors Nr = 72, the radius of an injector r = 0.2 cm, the number of injectors on the inner and outer wall is equal to Nr1 = 24, Nr2 = 28, the radius of the lateral injectors r = 0.15 cm. Figure 1 shows the geometry of the combustion chamber being investigated.

Fig. 1. The chamber geometry: side view (1 – injectors on the outer wall, 2 – injectors on the inner wall, 3 – the inner body).

6 Results Figure 2 shows the pressure distribution in the OXY section at a time of 150 μs for the case of a rich mixture. This distribution is different for all the kinetic mechanisms tested.

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For M-P-W, Gri-Mech and the author at the time of 150 μs, a relatively stable singlewave mode is observed, with small internal fluctuations and, as a result, the formation of self-ignition foci. For the case of Hong’s kinetics, the time 150 μs is the moment of reflection of two colliding waves of detonation. The picture is similar in the case of Williams’ kinetics, but here only the moment of contact of the oncoming detonation waves. In the case of LZKD kinetics, it is the moment of collision of the head wave with the explosion from the center of self-initiation, the collision with the oncoming wave and the departure of the wave in the direction of the head detonation are observed.

Maas-Warnatz-Pope

Hong

Williams

Gri-Mech 3.0

Li – Zhao – Ka-zakov – Dryer

User

Fig. 2. Rich mixture. Pressure field in the cross section of the chamber near the bottom (where the rotating detonation is localized), 150 μs.

Figure 3 shows the pressure distribution in the OXY cross section at the time of 150 μs for the case of stoichiometric mixture supply. The pressure level is generally lower for all cases of kinetics than for a rich mixture, and the wave pattern is also different for all the mechanisms. At this time, a prominent single-mode mode is observed only in the case of the author’s kinetic mechanism. In the case of Hong, there are two waves propagating towards collision with each other. In the Williams case the moment of collision with the oncoming wave and the departure of the wave in the direction of the main detonation observed. During the simulation process, at random times, the wave can change direction (clockwise to counterclockwise and vice versa) without any visible changes in the operating conditions. This behavior of the wave is associated with a complex and mostly random process, including initiation, quenching and spontaneous reinitiation of the detonation wave, which, when formed, tends to move in all the directions, consuming fresh reagents, thereby causing a complete damping of the detonation wave, with its possible re-initiation after restoring the necessary level of concentration of reagents required for reactivation.

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Maas-Warnatz-Pope

Hong

Williams

Gri-Mech 3.0

Li – Zhao – Kazakov – Dryer

User

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Fig. 3. Stoichiometric mixture. Pressure field in the cross section of the chamber near the bottom (where the rotating detonation is localized), 150 μs.

Fig. 4. Rich mixture. Change in thrust over time

Figure 4 shows the thrust force in Newtons for cases with different kinetics for the case of a rich mixture supply. Despite small differences in detonation rates, the average thrust value for the considered calculation time differs slightly: for the Hong mechanism 1490 N, for the Li – Zhao – Kazakov – Dryer mechanism 1466 N, for the Maas-WarnatzPope mechanism 1452 N, for the User 1530 N, for the Williams mechanism 1430 N, and for the Gri-Mech 3.0 mechanism, this number is 1420 N. Figure 5 shows the thrust force in newton for cases with different kinetics for the case of feeding a stoichiometric mixture. The average value of the thrust: mechanism Hong – 1403 N, for the mechanism of Li – Zhao – Kazakov – Dryer −1393 N, for mechanisms mA Maas-Warnatz-Pope – 1380 N, for the author’s – 1383 N. Williams mechanism for – 1396 N, and mechanism Gri-Mech 3.0, that number is 1375 N.

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Fig. 5. Stoichiometric mixture. Change in thrust over time

7 Postprocessing in High Definition Visualization System When performing predictive modeling in fluid mechanics, it is necessary to take into account the effects of instability and perturbations of different time levels and spatial scales. These may be shock waves in supersonic flows that occur in the calculations of zones with large temperature gradients during combustion, detonation, etc. A new visualization system designed for analysis and complex processing of calculations for modeling various-scale physical problems with big data output, as part of a system to support labor-consuming calculations is in demand in the scientific community. Figure 6 shows how the authorized system HDVIS (High Definition Visualization System) [19, 20] can be used. The distribution of the temperature field on the outer and inner walls of the chamber is presented for the case of a rich and stoichiometric mixture supply. The kinetic mechanism used is The GRI-Mech 3.0 kinetic mechanism. The Fig. 6 (a) shows the distribution of temperature at the inner and outer walls of the annular chamber at t = 50 μs. The detonation wave position is determined by a portion of increased temperature. For the rich mixture, the wave is more distinctive. There are portions of low temperature at the bottom end of the chamber near the fresh mixture supply. The Fig. 6 (b) shows an isothermal surface for T = 3130 K. The Fig. 7 shows distributions of fuel and oxidizer for rich and stoichiometric gas mixtures at t = 50 μs. The detonation wave is propagating near the supply end of the chamber, and it is marked by lower concentration of both. The visualization system with combining different data representation of primitive and derived functions allows us to consider both the position of the wave, the areas of maximum temperature values, and the places of heating and vorticity, where areas of additional ignition or re-initiation of detonation waves can be formed. This system allows enhancing treatment for a large amount of data much faster than other existing ones, without losing appropriate functions by generating combined cross-sections, pathlines, iso-surfaces, high relief maps of visualized objects, fast their animation etc.

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b

Fig. 6. a) Temperature distribution of fuel near the annular channel walls, t = 50 μs. 1) The stoichiometric mixture H2 :O2 = 2:1. 2) The rich mixture H2 :O2 = 3:1.b) Isotherm T = 3130 K and temperature distribution map on wall surfaces are shown for case: H2 :O2 = 2:1.

Fig. 7. Fuel and oxidizer concentration distribution near the annular channel walls, 50 μs 1) The stoichiometric mixture H2 :O2 = 2:1. 2) The rich mixture H2 :O2 = 3:1.

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8 Conclusions Six kinetic mechanisms of hydrogen combustion with oxygen and air were tested. All the mechanisms contained from 18 to 22 reversible reactions between 8 active components, including radicals. Nitrogen is considered neutral. The mechanisms differed in the time of self-ignition delay, the time of reaching the chemical equilibrium, and the type of the most active radicals involved in the development of the process. Three-dimensional test calculations were performed for all the studied mechanisms with identical parameters. The species set and the thermodynamic properties of the species were the same for all the kinetics used in tests. In the course of the computational experiment, a single-wave mode was established for all the considered kinetics and geometry over time for the case of rich mixture supply, and the transition to the combustion mode was obtained for the case of stoichiometric mixture supply. For the parameters under consideration, traction characteristics are obtained for both types of the mixture. The thrust force values for all kinetic mechanisms are slightly higher in the case of a rich mixture than in the case of a stoichiometric mixture. Changing to the combustion mode generally leads to a certain reduction in thrust, compared to detonation modes. The chain character of the reaction kinetics strongly affects complex processes in the engine chamber with a rotating detonation wave, including the stability of processes in it, though the formation enthalpy and entropy of the species involved do not change. A kinetic mechanism used in calculations of such systems should be chosen by a criterion of the best agreement between the calculated and experimental delay times of self-ignition, and the times of reaching a state of chemical equilibrium in a wide range of parameters (temperature, pressure, etc.) A new author’s visualization system was tested on the task of visualizing a large data obtained during computational modeling of processes occurring in the combustion chamber of an engine with a rotating detonation wave. HDVIS made it possible to significantly simplify the interpretation of the simulation results by quickly constructing a large amount of graphical data describing the distribution of physical quantities in the considered time and spatial area with high resolution. Acknowledgements. The reported study was partially funded by RFBR project number 19-2909070.

References 1. Wola´nski, P.: Detonative propulsion. Proc. Combust. Inst. 34, 125–158 (2013) 2. Anand, V., Gutmark, E.: Rotating detonation combustors and their similarities to rocket instabilities. Prog. Energy Combust. Sci. 73, 182–234 (2019) 3. Bykovskii, F.A., Zhdan, S.A., Vedernikov, E.F.: Continuous spin detonation in annular combustors. Combust. Explos. Shock Waves 41(4), 449 (2005) 4. Kindracki, J., Kobiera, A., Wola´nski, P., Gut, Z., Folusiak, M., Swiderski, K.: Experimental and numerical study of the rotating detonation engine in hydrogen-air mixtures. Prog. Propul. Phys. 2, 555–582 (2012)

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5. Frolov, S.M., Dubrovskii, A.V., Ivanov, V.S.: Three-dimensional numerical simulation of operation process in rotating detonation engine. Prog. Propul. Phys. 4, 467–488 (2013) 6. Tang, X.-M., Wang, J.-P., Shao, Y.-T.: Three-dimensional numerical investigations of the rotating detonation engine with a hollow combustor. Combust. Flame 162(4), 997–1008 (2015) 7. Wen, H., Xie, Q., Wang, B.: Propagation behaviors of rotating detonation in an obround combustor. Combust. Flame 210, 389–398 (2019) 8. Bykovskii, F.A., Vedernikov, E.F., Zholobov, Y.A.: Detonation combustion of lignite with titanium dioxide and water additives in air. Combust. Explos. Shock Waves 53(4), 453–460 (2017) 9. Nakagami, S., Matsuoka, K., Kasahara, J., Kumazawa, Y., Fujii, J., Matsuo, A.: Experimental visualization of the structure of rotating detonation waves in a disk-shaped combustor. J. Propul. Power 33, 80–88 (2016) 10. Smirnov, N.N., Nikitin, V.F., Stamov, L.I., Mikhalchenko, E.V., Tyurenkova, V.V.: Threedimensional modeling of rotating detonation in a ramjet engine. Acta Astronaut. 163, 168–176 (2019) 11. Wilcox, D.C.: Turbulence Modeling for CFD. DCW Industries, Inc., La Canada (1993) 12. Maas, U., Pope, S.B.: Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combust. Flame 88, 239–264 (1992) 13. Hong, Z., Cook, R.D., Davidson, D.F., Hanson, R.K.: A shock tube study of OH + H2O2 → H2O + HO2 and H2O2 + M → 2OH + M using laser absorption of H2O and OH. J. Phys. Chem. A 114(18), 5718–5727 (2010) 14. Williams, F.A.: Short chemical mechanisms for deflagrations and detonations. Western States Section. The Combustion Institute. Spring meeting. No. 04-S-1 (2004) 15. GRI-Mech Version 3.0, 7/30/99 CHEMKINII format. http://www.me.berkeley.edu/gri_mech/ 16. Li, J., Zhao, Z.W., Kazakov, A., Dryer, F.L.: An updative comprehensive kinetic model of hydrogen combustion. Int. J. Chem. Kinet. 36, 566–575 (2004) 17. Smirnov, N.N., Nikitin, V.F., Stamov, L.I., Mikhalchenko, E.V., Tyurenkova, V.V.: Rotating detonation in a ramjet engine three-dimensional modeling. Aerosp. Sci. Technol. 81, 213–224 (2018) 18. Novikov, E.A.: Research (m, 2) – methods of solving rigid systems. Comput. Technol. 12(5), 103–115 (2007) 19. Rybakin, B., Goryachev, V.: Modeling of density stratification and filamentous structure formation in molecular clouds after shock wave collision. Comput. Fluids 173, 189–194 (2018) 20. Abramov, A.G., Smirnov, E.M., Goryachev, V.D.: Temporal direct numerical simulation of transitional natural-convection boundary layer under conditions of considerable external turbulence effects. Fluid Dyn. Res. 46(4), 1–17 (2014)

Reduced Order Modeling for Thermo – Electric Processes Alexey Lukin(B) , Ivan Popov, and Pavel Udalov Department of Mechanics and Control Processes, Saint Petersburg Polytechnical University, Polytechnicheskaya, 29, Saint Petersburg 195251, Russia {lukin_av,popov_ia}@spbstu.ru

Abstract. In the study of any objects and devices, the first step is the construction of the geometric structure, which is divided into finite elements with large number of nodes. Such model with millions of degrees of freedom is inappropriate to use in control and data processing problems. Therefore, to solve this problem, a discrete model is constructed with a smaller number of degrees of freedom than in the original one. But with this approach to modeling related tasks, problems arise that need to be addressed. In this paper we will consider the ROM of electrothermal problems and effective methods for recording, isolating and storing elemental matrices obtained by CAD systems. Keywords: Reduced order modeling · Thermoelectric · Component Mode Synthesis · Ansys APDL · Matlab · CAD · MEMS

1 Introduction Modeling of electro-thermal processes occupies one of the important parts in the developing of micro-electro-mechanical systems (MEMS) [1–3]. With a decrease in size and an increase in the complexity of MEMS it is no longer possible to neglect the mutual thermal effect between adjacent circuit elements [2]. Self-heating of each element in MEMS, for example, self-heating of transistors, will change not only the characteristics of the transistor itself, but also the whole system. The dominant effect – the Joule effect in MEMS will be a parasitic effect of heating and dissipation of heat (the transition of electrical energy to heat), as a result of which the characteristics of the MEMS will change and the system itself may become technologically unusable. In order to study MEMS qualitatively, it is needful to go through a certain path in its construction. First, it is necessary to construct the geometry of this system, after which it must be divided into a sufficiently accurate finite element mesh. The order of the number of nodes can easily exceed 100000, but for the immersion of a given system into a system in state spaces, such a size of the system will be unsuitable for research. This is due to the fact that such a size of the problem will not provide time-optimal signal processing, tuning and verification of the system itself in the state space. Therefore, it is necessary to carry out the procedure for reducing the order of the settlement system, significantly reduced its order, but at the same time not losing anything in terms of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 108–116, 2022. https://doi.org/10.1007/978-3-030-92144-6_8

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system reaction. So we will get a system suitable for research. This is what this work will be devoted to – we will analysis the thermoelectric processes and the construction of compact models.

2 Research of Thermoelectric Processes Thermal conductivity is used to describe the thermal properties of any material. Thermal conductivity at the macroscopic level means that in the presence of a difference, a temperature gradient in a solid, thermal energy will move from a higher temperature region to a lower temperature region. This phenomenon is described by Fourier’s law: q = −K∇T ,

(1)

where q is the heat flux vector, T is the temperature field, K is the thermal conductivity coefficient. The conductivity equation describes the flow of electric current in the presence of an electric potential gradient, if in Eq. (1) the temperature is replaced by the electric potential, and the vector of the heat flux is replaced by the vector of the electric current density j, then the resulting equation is the equation of electrical conductivity: j = −σ ∇ϕ,

(2)

where σ – electrical conductivity, ϕ – electrical potential. In the case of a coupled thermoelectric problem the expressions for the heat flux q and the electric current density vector j are modified according to the following effects: The Peltier effect is the effect of energy transfer when an electric current passes at the point of contact between two dissimilar conductors. The Thomson effect is a phenomenon, which consists in the fact that in a homogeneous unevenly heated conductor where an electric current flows, not only heat will be released due to the Joule-Lenz law, but also Thomson’s heat will be released or absorbed depending on the direction of the current flow. Seebeck effect is the phenomenon of EMF at the ends of series-connected dissimilar conductors the contacts of which are at different temperatures. Thus, let us turn to a mathematical description of the problem of a coupled thermoelectric problem.

3 Mathematical Model 3.1 Problem Formulation We write the coupled equations of heat conduction and the law of conservation of electric charge [1, 4] as ∂T + ∇ · q = Q, ∂t    ∂ D ∇ · j + = 0, ∂t

ρC

(3.a)

(3.b)

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which are related through the definition of the heat flux consisting of the heat flux described by the Fourier law and the flux caused by the Peltier effect. q = αT j − K∇T ,

(3.c)

j = σ E  − σ α∇T .

(3.d)

The relationship between the vector of electrical permeability and vector of electric field intensity  = εE,  D

(3.e)

where ρ – density, C – specific heat, T – temperature, Q – bulk density of heat  – electric field vector, generation, q – heat flow vector, j – current density vector, E  D – electric induction vector, K – thermal conductivity, σ – electrical conductivity, α – Seebeck coefficient,  = T α – Peltier coefficient, ε – permittivity, ∇ – del operator, t – time. 3.2 Finite Element Formulation Further, according to [1, 5], we write down the finite element supply of the coupled thermoelectric setting in order to understand which matrices and load vectors need to be found and uploaded to build a reduced model.    TT      TT 0 0 T˙ e K Te Q + QP + QJ C + = , (4) ϕ˙e ϕe I 0 C ϕϕ K ϕT K ϕϕ where: N – vector of element shapes functions, T e – vector of nodal temperatures, ϕ e – vector of nodal electric potentials, K TT = ∫ ∇N· ≤ [K] · ∇NdV – thermal stiffness matrix, V

K ϕϕ = ∫ ∇N · [σ ] · ∇NdV – electric stiffness matrix, V

K ϕT = ∫ ∇N · [σ ] · [α] · ∇NdV – Seebeck stiffness matrix, V

C TT = ρ ∫ CNNdV – thermal damping matrix, V

C ϕϕ = ∫ ∇N · [ε] · ∇NdV – dielectric damping matrix, V

QP = ∫ ∇N · [] · JdV – Peltier heat load vector, V

QJ = ∫ NE · JdV – electric power load vector, V

Q – vector of combined heat generation loads, I – electric current load.

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4 Construct of Reduced Order Models of Thermoelectric System At the first step of model reduction a geometric model of the object is constructed. After that we create a detailed finite element model with a large number of degrees of freedom, and in the end we formed a discrete model of the object with a least number of degrees of freedom than in the original finite element model using mathematical methods of reduction [2]. 4.1 Finite Element Model We create a computational domain for the problem under consideration. For such an area we will choose a rectangular area consisting of two different materials. The geometric model and finite element model of Peltier element are shown in Fig. 1. The construction and study of the finite element model will be carried out in the ANSYS APDL. The physical and geometric parameters of the system are shown in Table 1.

(a)

(b)

Fig. 1. (a) Geometric model of the Peltier element, (b) finite element model of the Peltier element

Characteristics of the finite element model: number of nodes – 181, number of elements – 51, element type: plane77.

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Descriptions/Symbol

First material

Second material

Summary length, a

0.15 m

Width, b

0.02 m

Seebeck coefficient, αi

−165*10−6 V /K

210*10−6 V /K

Thermal conductivity, K

1.3 W /(m*K)

1.2 W /(m*K)

Resitivity, r i

1.05 ∗ 10−5 Ohm ∗ m

0.98 ∗ 10−5 Ohm ∗ m

Specific heat, C

150 J/(kg*K)

Permittivity, ε

15

Density, ρ

2210 kg/m3

Current, I

1A

According to Eq. (4) we consider only the governing equations for the heat equation. For further progress it is necessary to somehow determine and export from the ANSYS itself the vector of thermal loads Joule QJ and the Peltier load vector QP . To do this, consider two auxiliary problems. At the first stage we consider a problem in which a Peltier heat flux vector QP exists without boundary conditions and is specified at the junction of two materials. The boundary conditions in this case are shown in the Fig. 2.

Fig. 2. Boundary conditions for determining the Peltier load vector

The peculiarity of this stage is that for the unambiguous determination of this vector, it is necessary to know the value of the nodal temperatures, which correspond to the junction of the two materials. This problem can be solved, for example, as follows – to carry out a direct calculation of the electrothermal problem [4] and extract from it the values of the corresponding nodal temperatures at the junction of two materials. At the second stage, we will apply to the computational domain the heat sources responsible for heat generation due to Joule’s law. The boundary conditions in this case are shown in the Fig. 3. Next we must pay attention to how we can perform calculations and unloading matrices in problems without boundary conditions. For this in the ANSYS APDL software

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Fig. 3. Boundary conditions for determining the Joule load vector

package we will use the method of synthesizing the “component” forms (CMS – Component Mode Synthesis) [5]. This method is based on dividing a separate, complex and large model into a group of smaller problems [2]. Using this approach to the development and solution of almost any problem, the developer and researcher saves time and computing resources. Also, the motivation for using the CMS was the fact that it is possible to unload element matrices and load vectors in it without performing a complete calculation as such. So, to upload the mass matrix M, the thermal conductivity matrix K and the Peltier heat load vector, we use the following ANSYS APDL built-in commands. It is also worth remembering that for each thermal effect, it is worth creating a separate calculation in order to avoid the imposition of load effects on each other in any nodes of the finite element model. From the performed calculation, the matrices of mass, thermal conductivity and the Joule thermal load vector were written, which are saved in the files JouleMass.hb, JouleStiff.hb, and the load vector can be saved in any of these matrices [5]. Likewise for Peltier load vector. Next we will consider the very procedure of reduction and comparison of results in the Matlab software package. 4.2 Reduced Order Modeling in Matlab Earlier we obtained the matrices of masses M and thermal conductivity K, the Joule load vectors QJ and Peltier QP , which correspond to the considered finite element problems presented in Figs. 1, 2 and 3 without specifying boundary conditions, if they are available, then it is possible to write system (4) in the state space. E T˙ = AT + Bu, T0 y = CT

(5)

The inclusion of boundary conditions in the problem shown in Figs. 1, 2 and 3 will change not only the matrices M and K, but also the load vectors QJ and QP themselves. Next, the question arises of how to take into account the presence of boundary conditions from the resulting system in the state space (5) without specifying them in the problems of Figs. 1, 2 and 3. For this, for example, consider the following approach. Suppose that zero temperature is set at the boundaries of the problem under consideration T |x=0 = T |x=a = 0

(6.a)

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Then the system (5) can be written in the form e11 T˙ 1 + e12 T˙ 2 + . . . = a11 T1 + a12 T2 + . . . + B11 u1 + B12 u2 + . . .

(6.b)

We assume that the border corresponds to the nodes corresponding to the components e11 and e12 . From a constant and zero value of the boundary conditions, it means that T1 = 0, T˙ 1 = 0, T2 = 0, T˙ 2 = 0

(6.c)

This means excluding from consideration the nodes corresponding to the boundaries.

5 Results We will show the reaction of the initial system (5) taken from the ANSYS APDL and reduced by the 18th order model.

(а)

(b)

Fig. 4. System responses to (a) stepwise action, (b) impulse action at zero initial and boundary conditions of the complete system (black line) and a reduced 18th order model (red dotted line)

Next we show Bode diagrams for the case of a complete system and a reduced system of the 18th order in Fig. 5. From Figs. 4 and 5 it can be seen that the outputs of the complete system to typical influences fully meet our expectations, which indicates the correctness of the application of the CMS procedure. It was also found that the reduced 18th order model in its behavior completely coincides with the behavior of the complete system from ANSYS, which opens a further way in modeling control systems using already reduced models, which will significantly save calculation time.

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Fig. 5. Bode plots for the complete system (black line) and the reduced 18th order model (red dashed line)

6 Conclusion Modeling of the thermoelectric processes in MEMS takes one of the important stages in their construction and operation. In finite element modeling the number of degrees of freedom can reach large values, even 10,000 nodes in this model is a problem in terms of compiling control systems for microsensors and microactuators, etc. It is necessary to use methods of model ordering order to reduce the computer time with the same exit from the object under study. Therefore, it is important to consider and verify the reduction techniques for electrothermal models. In this paper, a method was considered for unloading the matrices of masses, stiffnesses and load vectors using superelement methods (CMS) in the Matlab software system. Then, using the built-in reduction methods, a reduced 18th order model was built. The outputs on typical actions and Bode diagrams of the complete and reduced models were compared and analyzed – they completely coincided. Thus, the reduced models open the way to the construction of the most convenient in terms of reducing the dimensionality of control objects for control systems, thereby saving calculation time and computing power of the computer.

References 1. Antonova, E.E., Looman, D.C.: Finite elements for thermoelectric device analysis in ANSYS. In: IEEE ICT 2005. 24th International Conference on Thermoelectrics, 2005. – Clemson, SC, USA (2005.06.19–2005.06.23)] ICT 2005. 24th International Conference on Thermoelectrics (2005) 2. Bechtold, T., et al.: System-Level Modelling of MEMS. Wiley-VCH (2013) 3. Korvink, J., Bechtold, T.: Fast Simulation of Electro-Thermal MEMS. Springer, Efficient Dynamic Compact Models (2006)

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4. Landau, L.D., Lifshitz, E.M.: Electrostatics of conductors. In: Electrodynamics of Continuous Media, pp. 1–33. Elsevier (1984). https://doi.org/10.1016/B978-0-08-030275-1.50007-2 5. Seifert, W., Ueltzen, M., Strmpel, C., Mueller, E.: One – dimensional modeling of a peltier element. Phys. Stat. Sol. 194(1), 277–290 (2002)

Experimental and Numerical Study of the Water-in-Oil Emulsion Thermal Motion in Rectangular Cavity with a Heated Bottom Vilena Valiullina(B) , Yuriy Zamula, Almir Mullayanov, Regina Iulmukhametova, Airat Musin, and Liana Kovaleva Bashkir State University, Ufa, Russia

Abstract. This paper presents the results of experimental and mathematical modeling of the “water-in-oil” emulsion movement dynamics under thermal influence in convection conditions. Cases of contact heating of an emulsion inside a rectangular vessel were considered. Images of emulsion system motion dynamics was obtained as a result of the research. Numerical calculations of emulsion system dynamics were carried out at different values of temperature difference. The results of modeling showed that thermal convection takes place in the emulsion system can negatively affect the process of gravitational separation of emulsions. Keywords: Water-in-oil emulsion · Contact heating convection · Experimental and numerical modeling

1

· Thermal

Introduction

One of the most common phenomena in nature and technology is convection. It can be observed both in simple processes, such as boiling, and in more complex ones. The study of fluid flows in rectangular cavities is widely used in the engineering industry, such as pulp and paper, food, as well as in oil drilling, slurry transport, etc. [1,2] A formation of a highly-stable water-in-oil emulsion during oil products extraction and processing is one of the main reasons for oil losses and increase of oil processing costs. The problem of water-in-oil emulsion destruction is related with the process of liquidation of accumulated oil sludge and oil products utilization based on it. There is a great number of numerical and experimental works in the field of studying the development of convection in the emulsion system. For example, in [3] the effect of thermal convection on the emulsion inside a rectangular vessel heated from the vertical wall and cooled c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 117–125, 2022. https://doi.org/10.1007/978-3-030-92144-6_9

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from the opposite wall was studied. There were used the “oil-in-water” emulsion as a test sample and silicon oil as the disperse phase of the emulsion. Emulsions with different sizes of oil particles were obtained, volume fraction of oil and influence of these parameters on stratification conditions and heat transfer characteristics of emulsions were studied. The temperature difference between the walls of a rectangular vessel was also changed and the effect of the temperature difference between the walls on these characteristics was studied. In [4] a numerical study of the natural convection of an abnormally viscous liquid in a flat cell with specified temperatures at the top and bottom of the cell and thermally insulated side walls was conducted. It is shown that there is an isolated mode of convection for abnormally viscous liquids, which is not typical for liquids with constant viscosity and with monotonically increasing (decreasing) viscosity dependence on temperature. In addition, [5] has numerically investigated a stable two-dimensional natural convection in rectangular two-dimensional cavities filled with non-Newtonian fluids of the Boussinesque Step Law. In article [6] it was shown that the thermal convective flows arising within the droplet and the surrounding liquid can both promote and prevent the destruction of the emulsion. Therefore, it is very important to choose the correct regime of thermal influence on the water-in-oil emulsion. In this article the results of experimental and mathematical modeling of the water-in-oil emulsion dynamics in the closed vessel for cases of contact heating from below are presented.

2

Experimental Research

Experimental Procedure. The object of research in this paper is an inverse “water-in-oil” emulsion. Medical vaseline oil was used as a disperse phase. Dispersion medium was ultrapure water (Milli-Q). A non-ionic surfactant sorbitan Span 80 monooleate was used to stabilize the emulsion droplets. The manufacturing procedure was as follows: Span 80 was added to the vaseline oil in the amount of 0.1 wt% and stirred for 5 min at 300 rpm using a top-drive stirrer

Fig. 1. Photo of emulsion droplets (a) and Particle-size distribution of emulsion droplets (b)

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(ES 8300D, Ekros), after that 1% of distilled water was added during the mixing process, the process of mixing water and oil lasted for 5 min. Studies to determine the size of emulsion droplets were carried out on an inverted Olympus IX71 microscope with subsequent image processing in the ImageJ program (Fig. 1a). Based on the results of digital image processing, histograms of the size distribution of emulsion droplets were obtained (Fig. 1b). Histograms of the size distribution of emulsion droplets were obtained based on digital image processing. The histograms were approximated by the Lognormal distribution function, which satisfactorily describes the distribution of particle frequencies by their sizes during random splitting. The average droplet size of the original emulsion was 6 µm, μ = 1.63, and σ = 0.57. The process of formation and development of convective flows of the emulsion system occurred in an experimental polycarbonate cell (Fig. 2), which is described in [7]. However, the cell had been improved. The front and back walls of the cell were made of mineral glass and thermocouples (TP) were installed on the sides of the inner cavity of the cell in this modification. The size of the inner cavity of the cell where the convective movement of the test sample occurs was 50 × 50 × 12 mm. Input and output were made on the side faces in order to place the test sample in the cell. Heating and cooling of the emulsion system was carried out with the help of aluminum tubes of rectangular cross-section 12 × 12 mm, which are installed in the cell at the bottom and top respectively. Water is pumped through the tubes using two thermostats (LAUDA Alpha A6 and LOIP LT-117 b), the temperature of which can vary in the range of 0–70 ◦ C. The emulsion medium in the cell is heated from the bottom and cooled from the top.

Fig. 2. Scheme of the experimental cell (left) and transverse section (right)

The experimental cell area was shot using Canon’s EOS 1100D with the Tamron SP 90 mm macro lens in manual mode at 1/100s, f/9 aperture, ISO 200. The frame rate was 1 frame every 30 s. Backlighting of the cell was carried out with the Sumita LS-M250 light source.

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The Results of Experimental Modeling. The images that were processed using the MatLab software package were obtained as a result of the experiment. Figure 3 shows the photo with motion tracks of emulsion medium particles under the influence of thermal influence. It can be seen that the process has the character of a steady movement. Four streams of emulsion medium can be clearly seen in the picture, as well as the particles, which are gas bubbles. These gas bubbles were formed during heating of the emulsion medium and settled on the surfaces, which is undesirable for the experiment.

Fig. 3. Processing result of 25 images to detect the tracks of the emulsion system droplet movement

Fig. 4. Velocity fields depending on time in the presence of convective flow in the emulsion system

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In Fig. 4 presents photos of the results of digital image processing to determine the velocity field depending on time in the presence of convective flow in the emulsion system. Images were processed in the Matlab software package with the help of the PIVlab application, designed to determine the velocity of particles from images. It can be seen that over time the speed of emulsions increases in the central part of the cell. Figure 5 shows the dependence of the velocity of droplets of emulsion medium on the width of the convective region. It can be seen that in the center the flow speed is maximum.

Fig. 5. Schedule of speed distribution over the width of the observed zone of the cell

3

Mathematical Modeling

Mathematical Model. The problem of the temperature effect on the water-inoil emulsion dynamics in the field of gravitational forces was considered. It was assumed that: medium was Newtonian, uniform, incompressible and isotropic; liquid flow was laminar; thermophysical parameters of the medium didn’t depend on temperature (small temperature differences were considered); the concentration of drops was low; spherical droplet didn’t change its shape and size; coalescence wasn’t taken into account; diffusion approximation was valid. The mathematical model written in the one-fluid approximation, includes the equation of motion of the emulsion, the equation of continuity for emulsion, the heat equation, the balance equation in the form of a convective-diffusion equation for the transfer of the volume concentration of droplets. 1 ∂v + (v∇)v = − ∇p + ∇(ν∇v) − gβT ∂t ρ

(1)

∂T + ρcp (v∇)T = ∇(λ∇T ) ∂t

(2)

∂C + ((v + vsed. )∇)C = DΔC ∂t

(3)

ρcp

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∂ρ + ∇(ρv) = 0 (4) ∂t where ρ, cp , λ – density, specific heat and coefficient of thermal conductivity of the water-in-oil emulsion respectively; ν – coefficient of kinematic viscosity; v – velocity vector of the thermal motion of the emulsion; vsed – sedimentation velocity of water droplets; g – gravity acceleration; p – emulsion pressure; T – temperature; C – concentration of water droplets in the emulsion. Thermophysical parameters of the water-in-oil emulsion such as ρ, cp , λ were calculated as additive values: (5) φ = φ1 C − φ2 (1 − C) Hereinafter index 1 denotes water parameters, 2 - oil parameters. To calculate the dynamic viscosity of the water-in-oil emulsion, the empirical equation of Einstein was used: η = η2 (1 + 2.5C)

(6)

The Hadamard-Rybczynski formula expresses the sedimentation velocity of water droplets in the water-in-oil emulsion: vsed =

2 2 (ρ1 − ρ2 ) η2 + η1 r g 3 0 η2 2η2 + 3η1

(7)

where r0 – radius of droplet. Initial conditions: v(t = 0) = 0

(8)

C(t = 0) = C0

(9)

T (t = 0) = T0

(10)

Boundary conditions: For the velocity field at all boundaries of vessel: v|W = 0

(11)

The walls of the vessel are considered to be impervious to the phases: (∇C)nW = 0

(12)

For the temperature the side walls of the vessel were thermally insulated, and a temperature difference between the bottom and top of the vessel: T (y = 0) = T0 + T

(13)

T (y = 0.05) = T0

(14)

where T – temperature difference. The simulation of the problem was carried out in an open integrated platform OpenFOAM. A modified built-in solver was used for the solution of the mathematical model. The dynamics of the water-in-oil emulsion stratification was calculated for case when the bottom of the vessel is heated. Geometric dimensions and physical parameters are set in accordance with experimental data.

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Numerical Modeling Results. When the liquid was heated from below the convective flows occur, and their structure varies depending on the value difference of temperature. To analyze the effect of the structure and intensity of thermoconvective flows on the process of the water-in-oil emulsion stratification, a complex of computational studies was conducted for various combinations of the difference of temperature and droplets size. Numerical calculations showed that for small temperature differences a stratification of the emulsion system practically always was observed. This is due to the fact that the velocity of thermal convection was so low that it practically has no effect on the process of water-in-oil emulsion stratification. When temperature differences increases, convective flow forms in the medium, and it could prevent the stratification of the emulsion systems. At the initial time, water droplets gradually settled down in the emulsion. The concentration of water droplets on the top of the vessel decreased, while at the bottom it increased. Partial stratification of the emulsion was observed. However, the velocity of thermal convection of the liquid increased during the time and convective flows began to prevent sedimentation of water droplets in the emulsion. The convective flow of the liquid began to mix the emulsion system intensively. The droplets of the emulsion were drawn into the flow of the liquid and perform translational motion along the velocity vector of the liquid thermal motion. The only exception was the area along the walls of the vessel, which wasn’t covered by the flow. Figure 6a shows the temperature field in the water-in-oil emulsion at temperature differences 3◦ . The results of numerical and experimental simulations show good convergence.

Fig. 6. Temperature field in water-in-oil emulsion at T = 3 ◦ C

It is seen that the temperature field in the water-in-oil emulsion is significantly distorted and repeats the structure of the convective flow. A double-vortex flow structure is formed in the liquid. The results of numerical and experimental simulations show good convergence. The distribution of the water droplet concentration in the emulsion is shown in Fig. 7. It can be seen that a uniform distribution of the concentration of water

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droplets in the emulsion took place in the middle of the vessel. The concentration value practically didn’t differ from the initial value. The complete stratification of the emulsion wasn’t observed. This is due to the fact that at given values of the parameters of the medium, the velocity of the thermal motion of the liquid exceeds the velocity of the droplet sedimentation.

Fig. 7. Distribution of concentration of water droplets in the emulsion at T = 3 ◦ C

4

Conclusion

Experimental and mathematical modeling of the “water-in-oil” emulsion movement dynamics under thermal influence in convection conditions in the rectangular cell was carried out. Images of emulsion system motion dynamics were obtained, in which flows are clearly observed. The obtained images perfectly match the pictures of currents obtained during mathematical modeling. The mathematical model of water-in-oil emulsion dynamics under the thermal effect taking into account the thermal convection of liquid was formulated. Studies of the water-in-oil emulsion stratification were carried out in case contact heating of water-in-oil emulsion. At the initial time, water droplets gradually settle down in the emulsion. The concentration of water droplets on the top of the vessel decreases, while at the bottom it increases. Partial stratification of the emulsion was observed. However, the velocity of thermal convection of the liquid increases with time and convective flows begin to prevent sedimentation of water droplets in the emulsion. The convective flow of the liquid begins to mix the emulsion system intensively. The droplets of the emulsion are drawn into the flow of the liquid and perform translational motion along the velocity vector of the liquid thermal motion. The only exception is the area along the walls of the vessel, which isn’t covered by the flow. Thus, it is shown that thermal convection leads can have a negative effect on the process of destruction of the water-in-oil emulsion because emulsions become more stable.

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Acknowledgments. The reported study was funded by the grant of the Russian Science Foundation (project №. 19-11-00298).

References 1. Kang, C., Okada, M., Hattori, A., Oyama, K.: Natural convection of water-fine particle suspension in a rectangular vessel heated and cooled from opposing vertical walls (classification of the natural convection in the case of suspension with a narrowsize distribution). Int. J. Heat Mass Transf. 44(15), 2973–2982 (2001) 2. Gitli, T., Silverstein, M.S.: Emulsion templated bicontinuous hydrophobichydrophilic polymers: Loading and release. Polymer 52(1), 107–115 (2011) 3. Morimoto, T., Ikeda, T., Kumano, H.: Study on natural convection characteristics of oil/water emulsions inside a rectangular vessel with vertical heating/cooling walls. Int. J. Heat Mass Transf, 127, 616–628 (2018) 4. Kuleshov, V.S., Moiseev, K.V., Urmancheev, S.F.: Isolated convection modes for the anomalous thermoviscous liquid in a plane cell. Fluid Dyn. 54(7), 983–990 (2019). https://doi.org/10.1134/S0015462819070097 5. Khezzar, L., Siginer, D., Vinogradov, I.: Natural convection of power law fluids in inclined cavities. Int. J. Therm. Sci. 53, 8–17 (2012) 6. Kovaleva, L.A., Musin, A.A., Fatkhullina, Y.I.: Microwave heating of an emulsion drop. High Temp. 56(2), 234–238 (2018). https://doi.org/10.1134/ S0018151X18020141 7. Tukhbatova, E.R., Zamula, Y.S., Valiullina, V.I., Musin, A.A., Kovaleva, L.A.: Experimental and numerical study of the natural convection in dispersed systems in a heated rectangular cell. J. Phys. Conf. Ser. 1359(1), 012112 (2019). IOP Publishing

Nonlinear Thermal Elastic Diffusion Problems Applicable to Surface Modification A. G. Knyazeva(B) and E. S. Parfenova Institute of Strength Physics and Materials Science, pr. Academicheskii, 2/4, Tomsk 634055, Russia

Abstract. This is a brief historical excursion into the development of the theory of thermoelastic diffusion. The difference between classical and generalized theories is noted. The method of derivation of constitutive relations on the basis of thermodynamics of irreversible processes leading to nonlinear theory is presented. Examples of application of nonlinear theory to modeling of interaction of diffusion, thermal and mechanical waves under ion implantation conditions are presented. The simplest isothermal task; the task for non-isothermal conditions as well as the task on interaction with the surface of the combined particle beam are analyzed. All tasks are formulated in the approximation of uniaxial loading. Keywords: Thermal elastic diffusion · Coupling problem · Thermodynamics · Nonlinear equations · Wave interrelation · Surface treatment · Implantation

1 Introduction One of the traditional areas of mechanics of deformable solids is the study of thermoelastic diffusion in a solid. This is connected with the theoretical study of the problems of interaction of fields of different physical nature, with the construction of models of multicomponent and multiphase media and with the tasks of controlling physical and chemical processes in modern technologies. Thermoelastic diffusion is diffusion in an elastic body under nonisothermal conditions together with accompanying effects. The interaction of stresses and diffusion was first studied in the work [1]. Variants of the theory of varying degrees of generality were proposed in different years. The theory of thermoelastic diffusion makes it possible to theoretically investigate modern technological processes of material processing, taking into account the interrelation of different fields. For example, in [2] shock wave propagation in thermoelastic medium with point defects and their interaction among each other have been studied. Investigations of transient characteristics of a multilayer sample subjected to symmetrical thermal and chemical loading are presented in [3]. It is shown that in the transition from layer to layer, the change in the characteristic parameters of the material has little effect on temperature, displacement and stress, but strongly affects the distribution of chemical potential and concentration. The work [4] is devoted to redistribution of elements at thermal actions. When solving the coupled problems within the framework of generalized theory of thermoelastic diffusion, mathematical difficulties associated with the complexity of obtaining analytical © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 126–147, 2022. https://doi.org/10.1007/978-3-030-92144-6_10

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solutions arise, which leads to the need to develop special approaches [5, 6]. Numerical methods for solving related problems are also actively used in literature: finite-difference methods, finite-element methods, boundary element methods. There are many works that combine both numerical methods and analytical ones, for example [7–10]. This paper presents some details of the general theory in a historical context; shows the relationship between nonlinear theory and methods of thermodynamics of irreversible processes and presents examples of applying the theory to modeling wave phenomena in the surface layer of an elastic body under conditions of processing by particle beams.

2 Preliminary Details Investigation of mathematical models of continuum mechanics taking into account the interaction between physical fields of various physical natures is important both from theoretical and from the practical viewpoint due to numerous applications. The best known theory in mechanics with interrelating fields is the theory of thermoelasticity, which has a long history. Thermal elasticity theory deals with the influence of mechanical and thermal disturbances on the elastic body. The appearance of the thermal plasticity theory in the 19th century is associated with the names of Duhamel (1837) and Neumann (1885). For an isotropic body, the relations between the stress tensor components, deformation tensor components and temperature in the theory of thermoelasticity can be written in the form of σij = 2μεij + δij [λεkk − Kw]

(1)

where w corresponds to temperature strain, w = 3αT (T − T0 ) = 3αT .

(2)

Here λ, μ - Lame coefficients, K - is bulk module, αT - is linear thermal expansion coefficient, T0 - is the temperature for non-deformed state, T − T0 = . In the classical version of the stress tensor theory, the stress tensor is symmetrical σij = σji ; small deformations satisfy Cauchy relations   ∂uj 1 ∂ui ; (3) εij = + 2 ∂xj ∂xi where ui - are the displacement vector u components, i = 1, 2, 3; the temperature change 0 is small, T −T T0 0. Using (45), from (43) we can present motion equation for stress ρ ∂ 2σ ∂ 2C ∂ 2σ + ρα = E ∂t 2 ∂t 2 ∂x2 From (42), (44) the diffusion equation of hyperbolic type follows:       ∂ ∂C ∂ ∂σ ∂ 2C ∂C + tD 2 = ρD − BC . ρ ∂t ∂t ∂x ∂x ∂x ∂x

(46)

(47)

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In contrast to traditional theory, this equation is nonlinear. Although the density can still be considered constant [51]. In variables S=

σ t x ε ; τ = ; ξ = ; e = , C, σ∗ t∗ x∗ ε∗

where σ∗ = 3Eα0 ; ε∗ = 3α0 ; x∗ =



t∗ D0 ; t∗ = ρD0 E,

suitable for numerical investigation of this problem, the equations take the form ∂ 2S ∂ 2C ∂ 2S + γ = ; ∂τ2 ∂τ2 ∂ξ2     ∂C ∂C ∂S ∂ 2C ∂ ∂ + τD 2 = f (C) − ωD γ C . ∂τ ∂τ ∂ξ ∂ξ ∂ξ ∂ξ Boundary and initial conditions correspond to one-axis loading: ξ=0: J=m ˜ 0 ϕ(τ); S = σ˜ 0 ϕ(τ); ξ→∞:

∂C = 0; S = 0; ∂ξ

τ = 0 : C = C0 ; S = 0;

∂S ∂C = 0; = 0, ∂τ ∂τ

where ∂J ∂C ∂S J D0 + ωD γC − τD ; J = ; J∗ = J = −f (C) ≡ ∂ξ ∂ξ ∂τ J∗ x∗

 E . ρ

Here the non-dimensional complexes appear: γ = (α − α0 ) α0 is the relative difference between the coefficients of concentration expansion of the material being introduced and the base; τD = tD E ρD0 is the relative relaxation time of mass flux equals to a square waves propagation;  of the ratio of velocities of mechanical and diffusion

m ˜ 0 = m0 ρ E is the particle beam density; ωD = 3 mEα20 (RT ρ) is the coupling coefficient for this model. This is the ratio of the mechanical energy required to move the elements under a stress gradient to the thermal energy required for the same purpose. A typical example of the evolution of the concentration profile and the mechanical wave shape in the case when all the characteristic times are comparable with each other is shown in Fig. 1. The parameters were evaluated for the system Al (Ti). The element to be implanted is placed in brackets. Note that in contrast to thermal elastic problems, here coupling coefficient is a greater than one. Curves 2 and 4 correspond times before and after impulse. Then diffusion wave attenuate. The depth of impurity penetration is of finite value. The position of the front of the diffusion wave for different time points

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Impurity concentration

0.03

3

0.02

1

2

0.01

0.00

4

0

0.0000

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0.02

5 0.01

8

6 7

0

2

4

6

8

0.0000

1

-0.0005

2

-0.0010

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0.00

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Impurity concentration

is shown with gray dots. Obviously, the area under the curve characterizes the amount of impurities introduced during the pulse time. The areas under all curves starting from 3 (which corresponds to τ = τimp = 8) are the same. The maxima on the concentration curves (black points) correspond to the minima on the deformation wave, the front of the concentration wave - an inflection on the deformation wave. Then the deformations change sign.

3 4

-0.0015

7

-0.0004

-0.0012

-0.0020

8

6

-0.0008

5

-0.0016 0

4

8

12

0

5

a

10

15

20

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b

Fig. 1. Concentration and deformation distribution for different time points τ: 1 – 3.0; 2 – 6.0; 3 – 8.0; 4 – 10.0 (a); 5 – 13.0; 6 – 16.0; 7 – 20.0; 8 – 24.0 (b). Parameters used for the calculation ˜ 0 = 0.01, σ˜ 0 = 0.2 γ = −0.05, ω = 100, τD = 13, τimp = 8, m

Non-isothermal Model. In most cases, the impact of the particle beam on the target surface is accompanied not only by mechanical action but also by heat as a result of partial energy loss by the particles as they approach to the surface. Variants of models and numerical calculations can be found in [54–56]. In this case, the equation of thermal conductivity associated with the stresses ∂Jq ∂T ∂σ + αT T =− ∂t ∂t ∂x is added to Eqs. (42) and (43). The equation for the heat flux ρcσ

Jq = −λT

∂Jq ∂T − tq , ∂x ∂t

(48)

(49)

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is added to the equation for the component flux (44), and new term appears in relation (45): σ = E[ε − αT (T − T0 ) − α(C − C0 )]

(50)

There is still one component of the displacement vector u, associated with the deformation of ε in the direction of loading by the ratio of ε = ∂u ∂x. Diffusion coefficient depend on temperature by Arrhenius law:   ED . D = D0 exp − RT Initial and boundary condition take the form x = 0 : J = m0 ϕ(t); Jq = q0 ϕ(t); σ = σ0 ϕ(t) x→∞:

∂T ∂C = 0; = 0, σ = 0; ∂x ∂x

t = 0 : C = C0 ; T = T0 ; σ = 0;

∂T ∂σ ∂C = 0; = 0; = 0. ∂t ∂t ∂t

Using relations (44), (49) and (50) in approximation of one-axis loading we obtain the system of three equation: for concentration of implanted admixture C – (47), stress σ in the loading direction and temperature T: ρ ∂ 2σ ∂ 2T ∂ 2C ∂ 2σ + ρα + ρα = T E ∂t 2 ∂t 2 ∂t 2 ∂x2  2      ∂ ∂ T ∂T ∂σ ∂T ∂σ ∂σ = λT − αT T αT T ρcσ tq 2 + − tq ∂t ∂t ∂x ∂x ∂t ∂t ∂t

(51) (52)

In dimensionless variables σ t x ε T − T0 ;τ = ;ξ = ;e = ,  = , σ∗ t∗ x∗ ε∗ T∗ − T0



where σ∗ = EαT (T∗ − T0 ); ε∗ = αT (T∗ − T0 ); T∗ = q0 D0 λT ρ E, and spatial and time scales are calculated with prior formulae, the problem takes the form     ∂ ∂C ∂ F() ∂S ∂ 2C ∂C = F() − M ωγ C ; τD 2 + ∂τ ∂τ ∂ξ ∂ξ ∂ξ  + σ ∂ξ   1 ∂ 2 ∂ 2C ∂C ∂S ∂ ∂S τq 2 + = − τq ω ; − ω( + σ) ( + σ) ∂τ ∂τ Le ∂ξ2 ∂τ ∂τ ∂τ S=

∂ 2S ∂ 2 ∂ 2C ∂ 2S + +γ 2 = 2. 2 2 ∂τ ∂τ ∂τ ∂ξ

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We will also present boundary and initial conditions in dimensionless variables: ξ=0: J=m ˜ 0 ϕ(τ); Jq = ϕ(t); S = σ˜ 0 ϕ(τ); ξ→∞:

∂ ∂C = 0; = 0, S = 0; ∂ξ ∂ξ

τ = 0 : C = C0 ; S = 0;  = 0 ;

∂C ∂ ∂S = 0; = 0, = 0. ∂τ ∂τ ∂τ

The constitutive relations in dimensionless variables are as follows J = −F()

F() ∂S ∂J ∂C + CγM ω − τD ; ∂ξ +σ ∂ξ ∂τ Jq = −

∂Jq ∂ − τq ; ∂ξ ∂τ

S = e −  − γ(C − C0 ).

 The function F() = exp −1 [β(σ + )] is dimensionless diffusion coefficient. This problem contains following dimensionless parameters γ=

α2 E D0 α − α0 σ0 D0 ρ tD E

; Le = ω = T (T∗ − T0 ); ; τD = ; σ˜ 0 = αT (T∗ − T0 ) σ∗ E D0 ρ ρcσ λT (ρcσ ) τD =

tq E mcσ R(T∗ − T0 ) m0 T0 ; M = ; β= ; m ˜0 = √ ; σ = . D0 ρ R ED T∗ − T0 ρE

Depending on the ratio of characteristic times, the interaction of waves manifests itself differently. In Fig. 2, characteristic points corresponding to the same coordinates are marked in the same way. For Mo(Ni) system the estimates are obtained: τq = 0.006 and τD = 0.03. If τimp = 0.025, the interaction of concentration and diffusion waves is quite distinct. In contrast to the previous one, not all curves of the front position of the concentration wave meet the maximum deformation value. For example, for curves 3, the maximum in concentration corresponds to a point behind the deformation front, while the front edge of the concentration wave rather corresponds to the maximum temperature. For the 4 curves, the front edge of the concentration wave corresponds to the maximum deformation, and the minimum deformation corresponds to the local convexity on the temperature curve. The kinks on the temperature curves can also be detected when the deformation sign changes. For the time values of the pulse duration, the traces of the concentration wave are detected in the tail of the deformation wave. The temperature deformation curves continue to be “affected” by kinks and local extrema, despite the fact that the thermal relaxation time is the minimum of all times.

0,3

1

3

0,2

4

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0,1 0,0 0,00

0,2

5 6

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7 8

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0,0 0,00

0,02

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0,06

0,08

Deformation

e

4

0,005

2 1

3

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0,000

5

-0,005 0,00

0,04

-0,005 0,0

0,08

0,003

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0,008

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7

6

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0,000 0,00

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f

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0,2

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2

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c Temperature

0,04

b

0,010

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d

0,4

0,000

141

a

0,5

Impurity concentration

Impurity concentration

Nonlinear Thermal Elastic Diffusion Problems Applicable

8

0,1

0,2

0,3

Fig. 2. Example of coupling problem solution for the system Mo(Ni): (a), (d) – distribution of diffusant concentration, Ni; (b), (e) – forms of deformation waves; (c), (f) – temperature distribution. Time points are τ: 1 – 0.02, 2 – 0.025, 3 – 0.03, 4 – 0.05, 5 – 0.06, 6 – 0.09, 7 – 0.12, 8– 0.15. τimp = 0.025.

Mo(Ni)

γ

Le

ω

M

μ

S0

τD

τq

β

σ

−0,003

40,7

0,001

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0,0018

110,0

Surface Treatment of Metal by Combined Particle Beam. It is of interest to practice the simultaneous influence of two or more elements on the target surface. This significantly enhances the capabilities of the ion implantation method and allows obtaining unique properties. An example model for implantation of two elements is presented in [57]. In this case two balance equations for implanted particles of type (42) are necessary: ρ

∂Cj ∂Jj =− ∂t ∂x

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The Eqs. (43) and (48) do not change. The relation for heat fluxes has a form (49), and relations for mass fluxes are similar to (44): Jj = −ρD0j

∂σ ∂Cj ∂Jj + Bj Cj kk − tDj , j = 1, 2, ∂x ∂x ∂t

D m

0j j where Bj = RT αj , j = 1,2, are transfer coefficients under stresses gradient; D0j are the self-diffusion coefficients, mj are molar masses; tDj are the relaxation times of mass flux; tq are the relaxation time of heat fluxes; αj = αj − α0 are the differences between the coefficients of concentration expansion of the implanted elements α1 , α2 and the green material α0 . We see, one more relaxation time, one more nonlinear diffusion coefficient and concentration deformations from particles of two types appear:

σ = E[ε − αT (T − T0 ) − α1 (C1 − C10 ) − α2 (C2 − C20 )]. The boundary and initial conditions are similar to previous ones: x = 0 : J1 = m01 ϕ(t), J2 = m02 ϕ(t), Jq = q0 ϕ(t); σ = σ0 ϕ(t), x → ∞ : C1 = 0, C2 = 0, σ = 0, t = 0 : C1 = C10 , C2 = C20 , σ = 0, T = T0 ,

∂C2 ∂T ∂σ ∂C1 = 0, = 0, =0 = 0. ∂t ∂t ∂t ∂t

Dimensionless variables are similar to previous also. The equation system in dimensionless variables is ∂ 2 C1 ∂ 2 C2 ∂ 2S ∂ 2 ∂ 2S + + γ1 2 + γ2 = 2, 2 2 2 ∂τ ∂τ ∂τ ∂τ ∂ξ     ∂ 2 Cj Fj () ∂S ∂Cj ∂Cj ∂ ∂ + τDj 2 = Fj () − Mj ωγj Cj · , j = 1, 2, ∂τ ∂τ ∂ξ ∂ξ ∂ξ  + σ ∂ξ   1 ∂ 2 ∂ 2  ∂ ∂S ∂ ∂S τq 2 + = − τ , − ω(σ + ) ω + ] [σ q ∂τ ∂τ Le ∂ξ2 ∂τ ∂τ ∂τ ξ = 0 : J1 = μ1 ϕ(τ), J2 = μ2 ϕ(τ), Jq = ϕ(τ), S = S0 ϕ(τ), ∂ ∂J1 ∂J2 = 0, = 0, = 0, S = 0, then e =  + γ1 (C1 − C10 ) + γ2 (C2 − C20 ), ∂ξ ∂ξ ∂ξ ∂C1 ∂C2 ∂ ∂S τ = 0 : C1 = C10 , C2 = C20 ,  = 0, S = 0, = 0, = 0, =0 = 0. ∂τ ∂τ ∂τ ∂τ

ξ→∞:

The functions F1 () and F2 () are dimensionless diffusion coefficients:     1 ε , F2 () = D exp − . F1 () = exp − β(σ + ) β(σ + )

Impurity concentration, С1

Impurity concentration, С1

Nonlinear Thermal Elastic Diffusion Problems Applicable

1

0,15

0,04

2

0,10

0,02

0,05

3 0,02

4

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0,06

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0,00 0,00

0,08

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Impurity concentration, С2

0,00 0,00

0,15

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0,05

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3

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0,06

1

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2 3

0,15

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0,10

0,15

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0,001

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0,001

8

0,10

8

0,00 0,00

0,08

0,002

0,05

7

6

6 7

0,02

0,00 0,00

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143

5 0,000

6 7

8

0,000

-0,001 0,04

0,08

0,12

0,00

0,002

0,001

0,000 0,00

0,05

0,10

0,15

0,20

0,25

5

0,003

Temperature

Temperature

0,00

1

2 0,04

3

0,0010

6

0,0005

7

4 0,08

0,12

0,0000 0,00

0,05

0,10

0,15

8

0,20

0,25

Fig. 3. Example of the coupling problem solution for the system Mo(Ni,Cu), τimp = 0.025: Time points, τ: 1 – 0.01, 2 – 0.025, 3 – 0.03, 4 – 0.05, 5 – 0.06, 6 – 0.09, 7 – 0.12, 8 – 0.15.

The parameters included in the model: γ1 =

α 1 − α3 α2 − α3 t1 E t2 E D10

; τD1 = ; γ2 = ; Le = ; τD2 = ; αT (T∗ − T0 ) αT (T∗ − T0 ) D10 ρ D10 ρ λT (ρcσ )

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ω=

α2T E tq E Ea2 D20 R(T∗ − T0 ) ; ε= ; D= ; ; β= (T∗ − T0 ); τq = ρcσ D10 ρ Ea1 Ea1 D10

m01 m02 m1 cσ m2 cσ T0 μ1 = √ ; μ2 = √ ; σ = ; M1 = ; M2 = . T∗ − T0 R R ρE ρE An example illustrating the interaction of all waves is shown in Fig. 3. The calculation was performed for Mo(Ni,Cu) system. Even with such a rather soft ratio of model parameters, when the pulse duration exceeds all characteristic times of the problem, the wave interaction is noticeably quite distinct, both on temperature and concentration curves. ε

γ1 , γ2

Le

ω

M1 , M2 μ1 , μ2 S0

1.2 −0,003 40,0 0,001 10,5 −0.002 5.0

0,8 0.4

τD1 , τD2 τq

0,001 0,015 0.01

β

σ

D

0,006 0,0034 80,0 1.25

5 Conclusion So, nonlinear theory of thermal elastic diffusion can be constructed and generalize in term of thermodynamics of irreversible processes. This theory allows constructing the coupled models of surface treatment by particle beams. The coupled models of surface layer modification in the conditions of ion implantation in the approximation of uniaxial loading are described in the work. The interaction of nonlinear thermal, diffusion and mechanical waves is analyzed using particular models as an examples. It was demonstrated that the inhomogeneous distribution of the impurities in the surface layer can be explained by the influence of mechanical stress on diffusion. The interrelation between waves of different physical nature depends on rations of characteristic times. Acknowledgements. The work was performed according to the Government research assignment for ISPMS SB RAS, project FWRW-2022-0003.

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Nano-, Micro- and Mesomechanics

Models of Nanosystems and Methods of Their Investigation, Connected with Orthogonal Splines Victor L. Leontiev(B) Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya Street, 29, 195251 Saint Petersburg, Russia [email protected]

Abstract. The world’s first orthogonal splines (OS) were created in the works of the author of this article and his postgraduate students, starting in 1993. Orthogonal splines form basis of Sobolev functional space. In contrast to Daubechies’s orthogonal wavelets with compact supports, orthogonal splines have an analytical shape, higher smoothness, and properties of symmetry, so orthogonal splines have proved effective in mathematical modeling and in mixed numerical methods. The compact supports of OS on grids that have steps comparable to the distance between atoms in a nanosystem makes these functions a mathematical tool that adequately takes into account the localization of atoms in nanosystems, so OS allowed creating a new potential for interatomic interaction forces. The use of this potential led to the appearance of models and the numerical-analytical method and a corresponding set of programs that allow analyzing the molecular dynamics of nanosystems without reducing the accuracy of calculations with lower computational costs compared to other nanomechanics programs that use molecular dynamics methods. Integral OS-transformations allow studying all components of dynamic exposure or radiation associated with nanosystems. The results of realized physical experiments confirmed high adequacy of used models and accuracy of methods of their research. The novelty of this scientific direction is connected with starting of the application of orthogonal splines in nanomechanics. Keywords: Orthogonal splines · Models of nanomechanics · Numerical methods · Spectral analysis

1 Introduction One of the directions of nanomechanics development is considered: models of nanomechanics and methods of their research that connected with the use of orthogonal splines (OS) [1]. The compact supports of OS on grids that have steps comparable to the distance between atoms in a nanosystem makes these functions a mathematical tool that adequately takes into account the localization of atoms in nanosystems. Orthogonal splines allowed creating a new potential for interatomic interaction forces [2]. The use of this potential led to the appearance of models and the numerical-analytical method © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 151–158, 2022. https://doi.org/10.1007/978-3-030-92144-6_11

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and a corresponding set of programs that allow analyzing the molecular dynamics of nanosystems without reducing the accuracy of calculations with lower computational costs compared to other nanomechanics programs that use molecular dynamics methods. This potential generates models of molecular dynamics, for which numerical analysis on a computer is characterized by reduced computational costs. Orthogonal splines used in the mixed variational-grid methods related to variational principles of Reissner and Hu-Washizu leads to the elimination of the basic deficiencies of their classical variants, caused by the increase in their number of nodal unknowns compared to variational-grid methods “in displacements” based on the variational principle of Lagrange. When modeling nanosystems (nanotubes, etc.) with a relatively large number of atoms, models of continuum mechanics (models of rods, shells, and elasticity theory) are used also. Such models also are used in the works [3–5] and allowed to obtain numerical results that correspond to the results of the conducted experiments. The use of OS in mixed variational-grid methods in the study of such models gives approximate solutions for both displacements and deformations and stresses that have the same order of accuracy and the same smoothness. To study the dynamics of nanosystems, along with integral Fourier transforms and integral wavelet transformations, integral OS-transformations [6] are used for investigation of the dynamics of nanosystems. Integral OS-transformations have a property similar to the property of wavelet transformations not to lose when performing on a computer discrete spectral analysis of short-time components of exposure or radiation, that is, they allow studying all components of dynamic exposure or radiation associated with nanosystems.

2 Models of Nanomechanics and Methods of Their Investigation The world’s first orthogonal splines were created in the works of the author of this article and his postgraduate students, starting in 1993. Orthogonal splines form basis of Sobolev functional space. In contrast to Daubechies’s orthogonal wavelets [7] with compact supports, orthogonal splines [1] have an analytical shape, higher smoothness, and properties of symmetry, so orthogonal splines have proved effective in mathematical modeling of nanosystems and in the mixed variational-grid methods. The graph of the second-order orthogonal spline Φ 2 is shown on Fig. 1, the orthogonal spline Φ 2 was obtained by a modification of the classical second-order B-spline. Sequence of grid groups of such splines form basis of Sobolev space, and these splines are highly smooth. Tensor products of such functions on rectangular grid have similar properties. Second-order orthogonal splines on each triangle of a triangular grid are a linear combination of six functions of Lagrangian basis for this triangle. Figures 2 and 3 show two characteristic functions of such a linear combination in system of coordinates x, y, z. A compact support of an orthogonal spline is a group of triangles with one common node. For one of these triangles of the compact support of OS, after its transformation to a triangle with vertices P1 (0, 0), P3 (1, 0), P5 (0, 1), Figs. 2 and 3 show the surfaces of the values of basis functions F1 , F4 corresponding to the nodes P1 (0, 0), P4 (0.5, 0.5) of the grid. The figures show the characteristic points of functions at the vertices of the triangle, in the points P2 (0.5, 0), P4 (0.5, 0.5), P6 (0, 0.5) on sides of the triangle and

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at points M1 (1/6, 1/6), M2 (1/3, 1/3), M3 (2/3, 1/6), M4 (1/6, 2/3) of intersection of the medians of triangles P1 P2 P6 , P2 P4 P6 , P2 P3 P4 , P4 P5 P6 . The other basis functions Fi (i = 3, 5, 2, 6) associated with points P3 , P5 , and P2 , P6 , respectively, have a similar appearance.

Fig. 1. The second-order orthogonal spline.

Fig. 2. The function F1 associated with point Fig. 3. The function F4 associated with point P1 . P4 .

The program for numerical research of nanosystem dynamics was created using the potential [2] of interatomic interaction forces based on the application of OS and has

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shown its effectiveness in calculations. The Nanoengineer program uses the LippincottMorse’s potential, consisting of the Morse’s potential at r < r 0 and the Lippincott’s potential at r > r 0 , where r 0 is the distance to the minimum of the potential pit. The OSpotential [2] is close to the harmonic potential and the Morse’s potential. At r > r 0 , the graph of the OS-potential approaches to the line of the energy level of interatomic bond destruction faster, which leads to acceleration of numerical calculations by the method of molecular dynamics without significant loss of accuracy. The simple analytical form of the OS-potential can be directly used in computer modeling, while the Morse, Lippincott, and other potentials in computer modeling are subjected to a preliminary interpolation procedure by cubic polynomials or splines. For example, Nanoengineer program has an interpolator that simplifies the Lippincott-Morse’s potential, but leads to increase in computing costs of calculations. The integral transformations based on OS were created [6] how tool for investigations of electromagnetic signals connected with nanosystems. The efficiency of OS transformations is confirmed by the examples of spectral analysis of different signals: (Signal A) f (t) = sin t + γ δ(t − c), where δ(t) − δ − function; (Signal B) f (t) = sin(2π μ1 t) + sin(2π μ2 t) + γ [δ(t − t1 ) + δ(t − t2 )], where μ1 = 500, μ2 = 1000, γ = 1.5.

Fig. 4. Wavelet transformation of Signal A.

Fig. 5. Wavelet transformation of Signal B.

The study of the efficiency of integral OS-transformations is carried out on the basis of their comparison with integral wavelet transformations. Figures 4 and 5 – wavelet transformations w correspondently of signals A and B based on the use of known MHAT wavelets. Figures 6, 7 – OS-transformation w correspondently of signals A and B. The results of spectral OS-analysis of nanosystems are similar or better than results of the spectral wavelet analysis, with lower computational costs. First, the possibility of using such models and methods to obtain estimates of the frequency of free vibrations of carbon nanotubes (CNT) is justified, and second, the high degree of reliability of the results of theoretical studies of CNT vibrations was shown, with account to experimental results [4, 5], third, it is shown that the obtained conclusions about the possibility of implementing THz mechanical vibrations of CNTs have a high degree of reliability.

Models of Nanosystems and Methods of Their Investigation

Fig. 6. OS-transformation of Signal A.

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Fig. 7. OS-transformation of Signal B.

Dynamic properties of CNT depend on the aspect ratio μ = L/R, where L – a length, R – a radius of CNT. Mixed finite element analysis of free vibrations of CNTs rigidly fixed at one end and free at the other end was performed. The first frequency of free vibrations of CNT (Fig. 8) is equal 3, 6·109 Hz, and the third frequency of free vibrations of CNT (Fig. 9) is equal 6, 5 · 109 Hz, if L/R = 40.

Fig. 8. First form of free vibrations for μ = 40.

Fig. 9. Third form of free vibrations for μ = 40.

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The first frequency of free vibrations of CNT (Fig. 10) is equal 1, 2 · 1010 Hz, and the third frequency of free vibrations of CNT (Fig. 11) is equal 1, 6 · 1010 Hz, if L/R = 6.

Fig. 10. First form of free vibrations for μ = 6.

Fig.11. Third form of free vibrations for μ = 6.

When the aspect ratio μ decreases, the free oscillation frequencies of CNT increase and reach terahertz (THz) values. Therefore, it is possible to implement CNT nanodevices with the presence of mechanical CNT resonances (up to 9THz). To create such nanodevices, as shown by the constructed estimates, it is possible for CNT, for which the aspect ratio is the minimum of their possible values (in the sense of technical implementation). Orthogonal splines used in the mixed variational-grid methods related to variational principles of Reissner and Hu-Washizu leads to the elimination of the basic deficiencies of their classical variants, caused by the increase in their number of nodal unknowns (stresses and strains) compared to variational-grid methods “in displacements” based on the variational principle of Lagrange. The use of orthogonal splines in mixed variationalgrid methods and mixed finite element methods in the study of dynamic models of nanosystems in the form of shells gives approximate solutions for both displacements and deformations and stresses with the same order of accuracy and the same smoothness. Calculations of forced vibrations of CNT using the mixed finite element method, associated with preliminary calculations of free vibrations, give approximate solutions for kinematic and for force factors with such properties. An illustration of accuracy of the same order of approximate solutions for kinematical and force factors obtained by the mixed finite element method in plate and shell models of nanomechanics is provided by the test problem for a square plate rigidly fixed on the boundary and subjected to uniform pressure. A comparison of approximate solutions of such a problem with a known exact solution in the form of a series is shown in Fig. 12 and is carried out in the center of the plate (deflection w − , bending moment M1 − O) and at its boundary in the middle of the side of the square (M1 −). Figure 12 shows the relative errors, the values of which decrease as the number Ki of nodes of uniform grids P1, . . . .P6 increases: K1 = 400, K2 = 484, K3 = 576, K4 = 784, K5 = 900, K6 = 1600. Figure 12 demonstrates high accuracy for deflection and accuracy of the same order for bending moment, while the smoothness of the approximate solutions for deflection and bending moment is the same. At the same time, orthogonal splines allow to exclude unknown values of stresses in nodes of a grid before solving of a system of algebraic equations which appeared in

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Fig. 12. Relative errors of approximate solutions for deflection and bending moment

mixed variational-grid method. Thus, the computational costs in the mixed variationalgrid method are significantly reduced and become comparable to the numerical method based on the Lagrange variational principle. In this case, the smoothness and accuracy of approximate solutions for deformations and stresses, on the contrary, is higher.

3 Conclusion The fact that splines are localized, because have compact supports, and are orthogonal gives possibilities for constructing of adequate models of nanosystems, which have rational for numerical investigations structures. High efficiency of methods of molecular dynamics based on the OS-potential of interatomic interaction forces, algorithms of mixed variational-grid methods, the numerical algorithm of the integral OStransformation for studying of dynamics of nanosystems is connected, first, with good quality of approximate solutions how for displacements so and for force factors and, second, with decrease in computational costs. The results of physical experiments [4, 5] confirmed the high adequacy of the used models and methods of their research. Orthogonal splines make it possible to construct adequate models of nanosystems, as well as to study the stress-strain state of nanosystems based on their models using approximate solutions for displacements, strains, and force factors that are characterized by the same smoothness and accuracy of the same order. The proposed models and methods based on the use of orthogonal splines represent the united new scientific direction that complements the known models and methods of nanomechanics, for example [8–18], and creates additional possibilities for modeling nanosystems and analyzing their models.

References 1. Leontiev, V.L., Lukashanets, N.: Grid bases of orthogonal compactly supported functions. Comput. Math. Math. Phys. 39(7), 1116–1126 (1999)

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2. Leontiev, V.L., Mikhailov, I.S.: On the construction of the interaction potential of atoms based on orthogonal finite functions. Nano- and Microsystem Technology 9, 48–50 (2011). [in Russian] 3. Leontiev, V.L., Bulyarsky, S.V., Pavlov, A.A.: Resonant detectors and antennas based on carbon nanotubes, elastically fixed at both ends. Nano- and Microsystem Technology 10, 595–603 (2016). [in Russian] 4. Bulyarsky, S.V., Dudin, A.A., Orlov, A.P., Pavlov, A.A., Shamanaev, A.A., Leontiev, V.L.: Resonance of a carbon nanotube with a current in an electromagnetic field. Nano- and Microsystem Technology 7, 395–398 (2017). [in Russian] 5. Bulyarsky, S.V., Dudin, A.A., Orlov, A.P., Pavlov, A.A., Leontiev, V.L.: Forced vibration of a carbon nanotube with emission currents in an electromagnetic field. Tech. Phys. 62(11), 1627–1630 (2017) 6. Leontiev, V.L., Rikov, E.A.: Integral transformations related to orthogonal finite functions in problems of spectral analysis of mathematical models of signals. Math. Model. 18(7), 93–100 (2006) 7. Daubechies, I.: Orthonormal bases of compactly supported wavelets. Comm. Pure and Appl. Math. 41, 909–996 (1988) 8. Noé, F., Nüske, F.: A variational approach to modeling slow processes in stochastic dynamical systems. Multiscale Model. Simul. 11, 635–655 (2013) 9. Nüske, F., Keller, B.G., Pérez-Hernández, G., Mey, A.S.J.S., Noé, F.: Variational approach to molecular kinetics. J. Chem. Theory Comput. 10, 1739–1752 (2014) 10. Olsson, S., Wu, H., Paul, F., Clementi, C., Noé, F.: Combining experimental and simulation data of molecular processes via augmented Markov models. Proc. Natl. Acad. Sci. U.S.A. 114, 8265–8270 (2017) 11. Vakhrushev, A.V.: Physical and mathematical models for nanosystems simulation. In: Vakhrushev, A.V. (ed.) Computational Multiscale Modeling of Multiphase Nanosystems: Theory and Applications, pp. 1–60. Apple Academic Press, Toronto; New Jersey: Apple Academic Press, 2017. | Series: Innovations in chemical physics and mesoscopy (2017). https://doi.org/ 10.1201/9781315207445-1 12. Kaviani, M., Di Valentin, C.: Rational design of nanosystems for simultaneous drug delivery and photodynamic therapy by quantum mechanical modeling. Nanoscale 11, 15576–15588 (2019) 13. Biehs, S.-A., Agarwal, G.S.: Dynamical quantum theory of heat transfer between plasmonic nanosystems. J. Opt. Soc. Am. B 30(3), 700–707 (2013) 14. Ta’asan, S.: From molecular dynamics to continuum models. Multigrid methods VI Proceedings of the Sixth European Multigrid Conference Held in Gent 1999, Belgium, pp. 235–241 (1999) 15. Olsson, S., Noé, F.: Dynamic graphical models of molecular kinetics. Proc. Nat. Acad. Sci. U. S. A. 116(30), 15001–15006 (2019) 16. Nguyen, T.D., Plimpton, S.J.: Aspherical particle models for molecular dynamics simulation. Comput. Phys. Commun. 243(October), 12–24 (2019) 17. Okunbor, D.I., Skeel, R.D.: Canonical numerical methods for molecular dynamics simulations. Comput. Chem. 15(1), 72–79 (1994) 18. Aktulga, H.M., Fogarty, J.C., Pandit, S.A., Grama, A.Y.: Parallel reactive molecular dynamics: numerical methods and algorithmic techniques. Parallel Comput. 38(4–5), 245–259 (2012)

Model of a Micromechanical Accelerometer Based on the Phenomenon of Modal Localization Vasilisa Igumnova(B) , Lev Shtukin, Alexey Lukin, and Ivan Popov Peter the Great St. Petersburg Polytechnic University (SPbPU), Saint Petersburg, Russia igumnova [email protected]

Abstract. In the present work, a model of a microelectromechanical accelerometer with two movable beam elements located between two stationary electrodes is proposed. The action of portable inertia forces in the longitudinal direction leads to a change in the spectral properties of the system, which is a useful output signal of the sensor. The dynamics of the system in the presence of a weak electrostatic coupling between the sensitive elements is characterized by the phenomenon of modal localization - a significant change in the amplitude relationships for the forms of in-phase and antiphase oscillations with small changes in the measured component of the object’s acceleration vector. The diagrams of the equilibrium positions and the dependences of the natural frequency are constructed with varying the potential difference V and ΔV . The dependences of the frequencies and the ratio of the components of the eigenvector on the external disturbance are investigated. It is shown that the sensitivity of a sensor based on modal localization can be orders of magnitude higher than the sensitivity of known systems based on measuring the shift of natural frequencies.

Keywords: Resonant accelerometer localization

1

· Weakly coupled system · Modal

Introduction

Currently, micro-electromechanical systems (MEMS) are widely used in various technical applications, as well as for the study of fundamental physical phenomena. In weakly coupled resonators, the phenomenon of modal localization of oscillations is known. The mode localization phenomenon is defined as the limitation of the vibration energy of one element of a coupled system when disturbances appear in the system in the form of a change in the rigidity of the structure. Another phenomenon of coupled systems in which mode localization is manifested is a change in the eigenvalue curve [7]. Veering occurs when the frequencies of the two modes approach and deviate from each other when the c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 159–171, 2022. https://doi.org/10.1007/978-3-030-92144-6_12

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external control parameter changes. The sensitivity of sensors in which mode localization is implemented can be 1–4 orders of magnitude higher than that of sensors based on measuring the frequency shift [1,2], that is, it becomes possible to create ultra-high sensitivity sensors. Also, sensors of this type, to a small extent with respect to sensors based on the measurement of frequency shift, are sensitive to environmental factors: temperature, pressure, etc. In work [3], various approaches are presented for increasing the sensitivity of the resonator for sensory applications. An analytical and numerical study of a MEMS resonance accelerometer is presented in [4]. The use of weakly coupled systems that implement the modal localization phenomenon increases the sensitivity of the sensor by orders of magnitude compared to the sensitivity of a sensor with a single resonator. A review of the use of such MEMS resonators is given in [5]. The article [6] describes a resonant accelerometer with four degrees of freedom, which has an electromechanical weak coupling. This makes it possible to achieve high sensor sensitivity over a wide range of acceleration measurements. In [8], a theoretical and experimental study was made of the phenomenon of mode localization for both mechanically and electrically coupled two microbeams. In addition to studying the problem of eigenvalues and the effect of lateral electrode displacement on the position of the pivot point, it was found that a decrease in damping at one of the resonators of a coupled system can lead to an increase in the quality factor of the system. In this work, we study the dynamics of a resonant accelerometer, consisting of two beam elastic elements located between two stationary electrodes (Fig. 1).

Fig. 1. Scheme accelerometer

When acceleration occurs in the system, inertial masses move, which in turn creates a longitudinal compressive force for one resonator and a tensile force for the other resonator. The action of portable inertia forces in the longitudinal direction leads to a change in the spectral properties of the system. As shown below, in the presence of a weak electrostatic coupling between the sensitive elements, the dynamics of the system is characterized by the phenomenon of modal localization - a significant change in the amplitude relationships for the

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forms of in-phase and antiphase oscillations with small changes in the measured component of the object’s acceleration vector.

2

Derivation of the Equations of Motion

The equations of motion of the first and second inertial mass (IM), respectively:  −M y¨1 − cy1 − c(y1 − u1 (l)) − M W = 0, (1) M y¨2 + cy2 + c(y2 − u2 (l)) − M W = 0, where y1 , y2 - longitudinal movement of the first and second IM, respectively, M - mass, c - spring stiffness, W - hull acceleration, u1 (l), u2 (l) - displacement of the end of the first and second beam element. Neglecting relative acceleration, we can obtain the law of motion of the first and second MI, respectively:  1 (cu1 (l) − M W ), y1 = 2c (2) 1 y2 = 2c (cu2 (l) + M W ). The equation of longitudinal vibrations has the form: 1 m¨ u − ESu = ES( w2 − u w2 ) + EI[w (w − u w − 2u w − 3u w )] , (3) 2 where u - longitudinal displacements of the beam, w - transverse displacements, m - mass of the beam, E - Young’s modulus, S - cross-sectional area, I - moment of inertia. The term 12 w2 associated with elastic restorative force. Neglecting the dynamics and small terms, we obtain a simplified equation of longitudinal vibrations with the corresponding boundary conditions: 1 ES(u + ( w2 )) = 0, 2 u(0) = 0, N (l) = c[u(l) − y],

(4) (5)

where N (l) - longitudinal force in the end section. The axial inertial forces of the beam are not taken into account, therefore, the longitudinal force is constant along the entire length of the beam. Expressions for longitudinal forces N1 and N2 will be of the form: ⎧ l ⎪ 1 1 ⎪ ⎨ N1 = 2 [M W − 2 c w12 dx], 0

l ⎪ ⎪ ⎩ N2 = 12 [−M W − 12 c w22 dx].

(6)

0

Substituting the expressions for N1 and N2 in the equation of bending vibrations taking into account electrostatic forces in the interelectrode gaps, we obtain:

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⎧ l 2 ⎪ ⎪ ε0 bV 2 1 1  ⎪ EIw + c w ˙ + [ M W − c w1 dx]w1 + ρS w¨1 − 12 (d−w 2 d 1 ⎪ 1 2 4 1) ⎪ ⎪ 0 ⎪ 2 ⎪ ⎨ + 1 ε0 bΔV = 0, 2 (d+w1 −w2 )2

(7)

l ⎪ ε0 bV 2 ⎪ ⎪ EIw2 + cd w˙2 + [− 12 M W − 14 c w22 dx]w2 + ρS w¨2 + 12 (d+w 2 ⎪ 2) ⎪ ⎪ 0 ⎪ 2 ⎪ ε bΔV 1 ⎩ 0 − 2 (d+w 2 = 0. 1 −w2 )

where ε0 - is the dielectric constant of the gap medium, b - beam width, V voltage between the stationary electrode and the beam, ΔV - voltage between two beam elements. Model Parameters: Table 1. System parameters Parameter

Value

Parameter

Value

Beam length (l)

600 µm

Air gap (d)

3 µm

Beam width (b)

120 µm

Spring stiffness (c)

65.4 N/m

Beam thickness (h)

3 µm

Mass of the weight (M) 5.02e–9 kg 2328 kg/m3

Young’s modulus of silicon (E) 109 GPa Silicon Density (ρ) Dielectric permittivity (ε)

1

Dielectric constant (ε0 ) 8.85e–12 F/m

Enter dimensionless parameters: 6ε0 l4 . (8) Eh3 d3  ρSl4 ˜ w1 = w˜1 d, . (9) w2 = w˜2 d, x=x ˜l, t = tT, T = ET Omitting the sign ∼, we obtain the system of equations in the dimensionless form: ⎧ 1 ⎪ αV 2 αΔV 2 ⎪ ⎨ w1 + Cnon w˙1 + w¨1 + [Pnon − Csp w12 dx]w1 − (1−w1 )2 + (1+w1 −w2 )2 = 0, Cnon =

12cd l4 , ET bh3

Csp =

3cd2 l , Ebh3

Pnon =

6M W l2 , Ebh3

α=

0

1 ⎪ ⎪ ⎩ w2 + Cnon w˙2 + w¨2 + [−Pnon − Csp w22 dx]w2 + 0

αV 2 (1+w2 )2

−

αΔV 2 (1+w1 −w2 )2

= 0. (10)

3

Finding Static Deflections

We study the dependence of the equilibrium position of the system on the strength of the electric field. The first Eq. (10) is multiplied by the denominator (1 − w1 )2 (1 + w1 − w2 )2 , and the second equation is multiplied by

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(1 + w2 )2 (1 + w1 − w2 )2 and apply the Galerkin method in the expansion in its own forms of an articulated beam: w1 (x, t) =

n

Ci (t)φi (x),

w2 (x, t) =

i=1

n

Di (t)φi (x).

(11)

i=1

Nonlinear equations of statics will look like: ⎧ n n n n 1 1 1

1



⎪ ⎪ ⎪ Ci φ Ci φi dx−Csp Ci φ2 Ci φi dx ⎪ φj i dx+Pnon φj i dx φj ⎪ ⎪ i=1 i=1 i=1 i=1 0 0 0 0 ⎪ ⎪ ⎪ 1 1 ⎪ αV 2 αΔV 2 ⎪ − φj + φj = 0, ⎪ n n n    ⎪ 2 ⎨ (1− Ci φi ) (1+ Ci φi − Di φi )2 0 0 i=1

i=1

i=1

i=1

i=1

i=1

n n n n 1 1 1

1



⎪ ⎪ ⎪ φj Di φ dx−Pnon φj Di φi dx−Csp Di φ2 dx φj Di φi dx ⎪ i i ⎪ ⎪ i=1 i=1 i=1 i=1 0 0 0 0 ⎪ ⎪ ⎪ 1 1 ⎪ αV 2 αΔV 2 ⎪ + φj − φj = 0. ⎪ n n n    ⎪ ⎩ (1+ Di φi )2 0 (1+ Ci φi − Di φi )2 0

(12) Figure 2 shows diagrams of equilibrium positions depending on the potential differences V and ΔV . The solid line indicates stable equilibrium position, and the dashed line indicates unstable.

Fig. 2. Deflection in the middle of the upper beam from potential differences V and ΔV .

For greater clarity of the results, Fig. 3 shows a bifurcation diagram of the equilibrium positions in 3D with varying parameters.

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Fig. 3. Equilibrium diagram for various values of V and ΔV . Blue shows the deflection for the upper beam, red - the deflection of the lower beam.

4

Comparison of Equilibrium Diagrams for Two Models

If we initially assume that the field is concentrated in the middle of the beam, then Eqs. (10) will have the form: ⎧ ⎪ ⎪ ⎪ ⎨

1 w1 + Cnon w˙1 + w¨1 + [Pnon − Csp w12 dx]w1 − 0 1

⎪ ⎪  2  ⎪ ⎩ w2 + Cnon w˙2 + w¨2 + [−Pnon − Csp w2 dx]w2 + 0

αV 2 δ(x− 1 ) 2 (1−w1 )2 2

αV δ(x− 1 ) 2 (1+w2 )2

+ −

αΔV 2 δ(x− 1 ) 2 (1+w1 −w2 )2 2

αΔV δ(x− 1 ) 2 (1+w1 −w2 )2

= 0, = 0.

(13) We apply the Galerkin method to Eq. (13), leaving one form in the expansion: w1 (x, t) = q1 (t)φ1 (x),

w2 (x, t) = q2 (t)φ1 (x).

System of equations can be obtained:  q¨1 + (97.4 − 9.87Pnon )q1 + 48.7Csp q13 − q¨2 + (97.4 + 9.87Pnon )q2 +

48.7Csp q23

+

2αV 2 (1−q1 )2 2αV 2 (1+q2 )2

+ −

2αΔV 2 (1+q1 −q2 )2 2αΔV 2 (1+q1 −q2 )2

(14)

= 0, = 0.

(15)

The bifurcation diagram for the exact and approximate equations has the form:

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Fig. 4. Comparison of equilibrium diagrams

From Fig. 4 it can be seen that the exact and approximate solutions have a similar form of the bifurcation curve, but have differences in the value of the voltage corresponding to the “pull-in” effect. It is also seen that as the number of terms in the expansion increases, the solution approaches the exact one.

5

Analysis of the Spectral Properties of the System

Let us return to the system of equations in the dimensionless form: ⎧ 1 ⎪ αV 2 ⎪ w1 + Cnon w˙1 + w¨1 + [Pnon − Csp w12 dx]w1 − (1−w 2 ⎪ 1) ⎪ ⎪ 0 ⎪ 2 ⎪ αΔV ⎨ + (1+w 2 = 0, 1 −w2 ) 1  ⎪  αV 2 ⎪ ⎪ w + Cnon w˙2 + w¨2 + [−Pnon − Csp w22 dx]w2 + (1+w 2 ⎪ 2) ⎪ 2 ⎪ 0 ⎪ ⎩ αΔV 2 − (1+w1 −w2 )2 = 0.

(16)

Next, we decompose the deflections of the beam elements into a static and dynamic part: w1 (x, t) = w1s (x) + w1d (x, t),

w2 (x, t) = w2s (x) + w2d (x, t).

(17)

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where w1s (x) and w2s (x) are the result of solving the static problem. We obtain the equations for the dynamic component: ⎧ 1 2 ⎪   ⎪ w + C w ˙ + w ¨ + [P − C w1d dx]w1d non 1d 1d non sp ⎪ 1d ⎪ ⎪ 0 ⎪ ⎪ 2αV 2 2αΔV 2 ⎨ + (1−w 3 + (1+w −w )3 (w2d − w1d ) = 0, 1s ) 1s 2s 1 2 ⎪   ⎪ ⎪ ¨2d + [−Pnon − Csp w2d dx]w2d w2d + Cnon w˙ 2d + w ⎪ ⎪ ⎪ 0 ⎪ ⎩ 2αV 2 2αΔV 2 − (1+w 3 − (1+w −w )3 (w2d − w1d ) = 0. 2s ) 1s 2s

(18)

For the analysis of free vibrations, we also apply the Galerkin method, taking into account only the lower vibration modes of two beams: w1d (x, t) = u1 (t)φ1 (x),

w2d (x, t) = v1 (t)φ1 (x).

(19)

The equations of motion in matrix form can be written as: ¨ + C U˙ + KU = 0, MU (20)



10 Cnon 0 where M = - mass matrix, C = - dissipation matrix 01 0 Cnon (further consider C = 0), U = [u1 v1 ] - displacement vector, stiffness matrix:  K=

K4 ΔV 2 −K2 Csp + K3 V 2 − K4 ΔV 2 + K1 Pnon 2 K4 ΔV −K5 Csp − K6 V 2 − K4 ΔV 2 − K1 Pnon



The coefficients of the stiffness matrix has the form: ⎧ 1 1 2 1 1   1 ⎪ ⎪  ⎪ K1 = φ1 φ1 dx, K2 = w1s dx φ1 φ1 dx + 2 w1s φ1 dx φ1 w1s dx, ⎪ ⎪ ⎪ 0 0 0 0 0 ⎪ ⎨ 1 1 K3 = (w1s2α K4 = (1+w1s2α−w2s )3 φ1 φ1 dx, 3 φ1 φ1 dx, −1) ⎪ 0 0 ⎪ ⎪ ⎪ 1 2 1 1   1 1 ⎪ ⎪   ⎪ K5 = w2s dx φ1 φ1 dx + 2 w2s φ1 dx φ1 w2s dx, K6 = (w 2α 3 φ1 φ1 dx. ⎩ 2s +1) 0

0

0

0

0

(21) The eigenfrequencies ω1,2 and the eigenvectors U1,2 of the system of Eqs. (21) depend on the dimensionless transport inertia Pnon . An analysis of the nature of these dependencies allows us to evaluate the sensitivity of the proposed sensor model. Figure 5 shows the dependence of the natural frequency of the in-phase and antiphase modes with increasing potential differences ΔV and V .

Model of a Micromechanical Accelerometer

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Fig. 5. The dependence of the natural frequency with increasing potential differences ΔV and V .

Let us compare two measurement methods: with a frequency output and with an amplitude output. Figure 6 shows the dependence of the oscillation frequency on the external disturbance.

Fig. 6. Dependence of the oscillation frequency on the external disturbance Pnon . The number “1” indicates the in-phase mode, and the number “2” indicates the antiphase mode. With parameter V = 0.05V .

Figure 6 shows the dependence of the frequency on the external disturbance at various ΔV values. It can be seen that with an increase in the weak coupling ΔV , the difference in the frequencies of the in-phase and antiphase modes significantly increases. The region near the value Pnon = 0 is called the veering zone, because,

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as can be seen from the figure in this zone, the frequencies of the first and second modes repel each other, but do not intersect in the presence of a small coupling ΔV . Figure 7 shows the dependence of the oscillation amplitude on the external disturbance.

Fig. 7. Dependence of the amplitude of oscillations on the external disturbance Pnon . The number “1” indicates the in-phase mode, and the number “2” indicates the antiphase mode. With parameter V = 0.05V .

Figure 7 shows the dependence of the ratio of the components of the eigenvectors of free vibrations on the external disturbance for various values of ΔV . As can be seen from the figure, with a decrease in the weak coupling ΔV , the dependence of the amplitude indices on external acceleration becomes stronger.

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The sensitivity of sensors based on the frequency and based on the amplitude ratio can be calculated by the formulas: Sω = |

ωi − ωi0 |, ωi0

Sα = |

ui − u0i |, u0i

(22)

where ωi0 and u0i are the eigenfrequency and amplitude ratio of the components of the eigenvector in the absence of weak coupling, that is, ΔV = 0, i = 1, 2 denotes the first mode (in-phase) and the second mode (antiphase), respectively. Figure 8 and Fig. 9 show the dependences of the parameters Sω and Sα on the dimensionless axial component of the acceleration Pnon .

Fig. 8. The sensitivity of the sensor based on the frequency from the external disturbance Pnon . With V = 1V .

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Fig. 9. Sensitivity of a sensor based on modal localization from the external disturbance Pnon . The number “1” indicates the in-phase mode and the number “2” indicates the antiphase mode. With parameter V = 1V .

As can be seen from the figures, the sensitivity of the accelerometer using the principle of localization of oscillations is several orders of magnitude higher than in a sensor with a frequency output.

6

Conclusions

In the present work, a model of a microelectromechanical accelerometer with two movable beam elements located between two stationary electrodes is proposed. The diagrams of the equilibrium positions and the dependences of the natural frequency are constructed with varying the potential difference V and ΔV . The dependences of the frequencies and the ratio of the components of the eigenvector on the external disturbance are investigated. It is shown that the sensitivity of a sensor based on the phenomenon of localization of oscillations in weakly coupled systems can be orders of magnitude higher than the sensitivity of the system in the mode of measuring the shift in natural frequencies. The symmetry of the proposed sensor architecture also ensures its high resistance to environmental changes (temperature disturbances, pressure changes). The work was supported by the RFBR grant 20-01-00537.

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References 1. Zhao, C., Wood, G.S., Xie, J., Chang, H., Pu, S.H., Kraft, M.: A three degreeof-freedom weakly coupled resonator sensor With enhanced stiffness sensitivity. J. Microelectromech. Syst. 25(1), 38–51 (2016) 2. Manav, M., Phani, A.S., Cretu, E.: Mode localization and sensitivity in weakly coupled resonators. IEEE Sensors J. 19(8), 2999–3007 (2018) 3. Hajjaj, A.Z., Jaber, N., Ilyas, S., Alfosail, F.K., Younis, M.I.: Linear and nonlinear dynamics of micro and nano-resonators: review of recent advances. Int. J. NonLinear Mech. (2019) 4. Zhang, H., Kraft, M.: An acceleration sensing method based on the mode localization of weakly coupled resonators (2017) 5. Zhao, C., Montaseri, M.H., Wood, G.S., Pu, S.H., Seshia, A.A., Kraft, M.: A review on MEMS coupled resonators for sensing applications utilizing mode localization. Sens. Actuators, A 249, 93–111 (2016) 6. Peng, B., Hu, K., Shao, L., Yan, H., Li, L., Wei, X.: A sensitivity tunable accelerometer based on series-parallel electromechanically coupled resonators using mode localization. J. Microelectromech. Syst. 29(1), 3–13 (2020) 7. Pierre, C., Dowell, E.H.: Localization of vibrations by structural irregularity. J. Sound Vib. 114(3), 549–564 (1987) 8. Ilyas, S., Younis, M.I.: Theoretical and experimental investigation of mode localization in electrostatically and mechanically coupled microbeam resonators. Int. J. Non-Linear Mech. 125, 103516 (2020)

Phase Transitions and Nonlinear Elasticity

Reducing of Residual Stresses in Metal Parts Produced By SLM Additive Technology with Selective Induction Heating Sergei A. Lychev1 1

and Montaser Fekry2,3(B)

Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, Bldg 1, Moscow 119526, Russia [email protected] 2 Department of Mechanics and Control Processes, Moscow Institute of Physics and Technology, Moscow, Russia [email protected] 3 Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt

Abstract. In the present paper a mathematical model for the temperature and residual stress fields evolution in growing thermoelastic cylinder is investigated. It is based on the idea of analyzing a sequence of boundary value problems describing the steps of the growth process. The main goal is to give qualitative clarification and modeling for residual stress accumulation and distortion in the final geometric shape, which appears in additive manufacturing, particularly in SLM or SLS technological processes. We proposed such way to control these unwanted phenomena. The main idea is to apply inhomogeneous inductive heating by skin effect phenomena during the additive process. In so doing one can compensate the incompatibility of thermoelastic deformations caused by sequential addition of heated up to melting temperature material by controlled inhomogeneous thermal expansion resulting from such way of heating. The process can be controlled by changing the frequency of an alternating electric current and the amplitude supplied to the growing body. This controling leads to minimize residual stresses and/or shape distortion of the body during and after additive process completion. For the axisymmetric cylindrical problem investigated below, it is possible to obtain optimal control parameters based on analytical solution of sequence of boundary value problems. This solution is the main result of present paper. Keywords: Additive manufacturing · Residual stresses solids · Thermoelasticity · Analytical solutions

· Growing

c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 175–193, 2022. https://doi.org/10.1007/978-3-030-92144-6_14

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Introduction

Selective Laser Melting and Sintering (SLM and SLS) are promising technology for manufacturing 3D metal parts with complicate shape and specified inhomogeneity. The main theme is that the part is being created sequentially, piece by piece, due to melting and fusing metallic powders together in precise geometric shape [1–4]. There are, nonetheless, some challenges that hamper the practical application of such technologies. One of the most tangible is related with inhomogeneous thermal expansion of added material in the course of technological process. The main reason behind the latter is that the metallic particles are heated to the melting temperature and then attached to manufactured component part, which temperature is less then the temperature of the particles. After temperature equalisation an incompatible deformations arise both in bulk of the body and attached part. This causes distortion of geometrical shape and accumulation of residual stresses. Residual stresses can be defined as those stresses that remain in a material or body after manufacturing in the absence of external forces or thermal gradients. Residual stresses have the same effect on materials and their performance as externally applied stresses [5–12]. It leads to undesirable consequences, such as a shape distortion, local discontinuity, loss of stability. Up to now a variety of ways to reduce residual stresses in SLM manufactured parts are known. Most use the modulation of melting beam or overall heating of the part during additive process [13–17]. These allow to reduce the inhomogeneity of temperature field and, consequently, to reduce residual stresses. In present work we propose to take a further step: to heat the part during SLM process in specific non-uniform manner which upon the technological (melting) heating results in almost constant temperature profiles and hence in low residual stresses. In order to generate such specific heating, an induction with high frequency current modulated in time can be used. One may observe here the similarity with skin-effect induced by alternating current.

2

Growing Process

The concept of a solids growth refers to a new branch of continuum mechanics [18–23], therefore it seems appropriate here to clarify the definition of the growing solid. In a broad sense growing process defines the alteration of the body composition occurring in the course of deformation. The growing process may be accompanied by a change of topological properties of the body. It can be said that the altering of the body composition is the accession of new material points and (or) formation of new constrains between particles already included into the composition. It should also be noted that the change of topological properties can occur without the influx of material and can be caused by the transition of the boundary points into the interior. In modern continuum mechanics there are many different approaches to the studying of the growth phenomenon. For today a large number of papers devoted to mechanics of growing solids have

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been published. References may be found in the review [8]. The works [24,25] are devoted to the development of geometric methods adopted for the mechanics of incompatible strains arising as the result of the growing process. In the works [18–21] the growth is investigated as the continuous process of deposition of strained material surfaces to a deformable 3D body. It is known that under certain additional assumptions on the continuity of functions that define the stress-strain state of adhered material surfaces, the continuous growing process can be considered as the limit of a sequence of discrete processes [26,27]. A discretely accreted body is represented as a finite family of added layers to the initial body (1) B0 ⊂ B1 ⊂ ... ⊂ Bk ⊂ ... ⊂ BN , where B0 is the initial body and Bk is B0 after adding k layers. The sequence (1) is associated with the sequence of numbers 0 ⊂ τ1 ⊂ ... ⊂ τk ⊂ ... ⊂ τN , < ...
ω2 > ω1 , σ = 5.9 × 107 S m−1 ,

μ0 = 1.256629 × 10−6 H m−1 ,

{ω1 , ω2 , ω3 } = {5, 7, 9} × 2π × 105 s−1 . According to the Joule-Lenz law, the power of heating generated by an electrical conductor is proportional to the product of its resistance and the square of the current, thus, the temperature in that conductor will be concentrated near the surface. 1 |J|2 .

= 2σ

6 0

5

0

1

2

3

J

4 3 2 1 0.0

0.2

0.4

0.6

0.8

Fig. 2. The current density in cylinder with different frequencies.

1.0

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3.2

181

Heat Transfer

The distribution of the temperature can be obtained from the solution of the heat conduction problem Λ∇2 Θ − ρκΘ˙ + ∗ = 0, ∂r Θ|r=R = 0,

Θ|t=0 = Θ0 ,

(9)

where Θ is the temperature change above the uniform reference temperature T0 , ρ is the mass density, Λ is the coefficient of thermal conductivity, κ is the specific heat per unit mass at constant strain and ∗ is the heat source. To facilitate the solution, the following non-dimensional variables are used  r ˇ = Θ , tˇ = 1 μ t. (10) rˇ = , Θ R T0 R ρ In the dimensionless variables (10), the equation (9) (after dropping the dimensionless symbol for simplicity) takes the form: ∇2 Θ − B Θ˙ + = 0,

(11)

where the following dimensionless quantities are introduced √ κR μρ R2 ∗

= .

, B= ΛT0 Λ The boundary and initial conditions in the dimensionless variables are stated as ∂r Θ|r=R = 0,

Θ|t=0 = Θ0 .

Note that in the dimensionless form the radius of the initial cylinder takes the value “1”, and since the radius of the growing cylinder increases in a constant rate during the process we call the non-dimensional radius by R. With the Duhamel’s principle [30], the solution of the heat equation (11) can be represented as the sum of particular solution Θp of the inhomogeneous equation and the solution Θh of the homogeneous one Θ = Θh + Θp . The solution Θh meets the boundary value problem ∇2 Θh − B Θ˙ h = 0, ∂r Θh |r=R = 0, Θh |t=0 = Θ0 .

(12)

By separation of variables one can get the solution for (12) as follows 2

−γ ¯ B t, Θh = Θ(r)e

(13)

¯ is non-trivial solution of Sturm Liouville problem where Θ ¯ ¯ ∂2Θ 1 ∂Θ ¯ = 0, + γ2Θ + 2 ∂r r ∂r

¯ r=1 = 0. ∂r Θ|

(14)

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The bounded solution of eq. (14) at r = 0 (14) can be represented as follows ¯ = J0 (γr), Θ The boundary condition at r = R gives a sequence values for γ γ ∈ {γm , m = 0, 1, ...∞}, where γm are the roots of the equation J1 (γR) = 0, here are some roots for R = 1, {0, 3.83, 7.01, 10.17, 13.32, 16.47, 19.61, ...}. Note: The first root of J1 (γR) = 0 is γ0 = 0 and the other roots are numerically calculated in “Mathematica” by the order γm =

1 BesselJZero[1, m]. R

Thus, the non-trivial solutions (eigenfunctions) for Sturm Liouville problem (14) can be represented as follows ¯ 0 = 1, Θ

¯ m = J0 (γm r), Θ

m, = 1, 2, ...∞.

Due to the self-conjugate property of differential operator, defined with Eq. (14) and corresponding boundary conditions, all these solutions together constitute an orthogonal system. Hovewer, they are determined up to an arbitrary multipliers. It is appropriate to take them such that the system becomes normalized. To this end we calculate the normalization factors   R √ [J0 (γm r)]2 rdr = R πL|J0 (γm )|, Nm = 2πL 0

and divide solutions obtained above by them. Finally we get the orthonormal eigenfunctions system ¯0 = Θ

1 √ , R πL

¯m = Θ

J (γ r) √0 m , R πL|J0 (γm )|

m, = 1, 2, ...∞.

(15)

Table 1 shows graphs for three eigenfunctions with its derivatives with respect to r, it’s clear that solutions satisfy the boundary conditions.

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¯ m and their derivatives. Table 1. Some of eigenfunctions Θ ¯ 4 = 0.922J0 (10.17r) Θ

¯ 7 = 1.27J0 (19.61r) Θ

¯ 10 = 1.55J0 (29.04r) Θ 1.5

0.8

1.0

¯m Θ

0.6

1.0

0.4

0.5

0.5

0.2

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2 0.5

0.5

0.4

10

¯m ∂r Θ

2

0.2

2

4

10

5

0.4

0.6

0.8

1.0

0.2 5

10

0.4

0.6

0.8

1.0

10

20

15

Now one can obtain the representation of solution for (12) in terms of expansion √  R ∞ 2 γm 2 πL h − B t h Cm J0 (γm r)e , Cm = Θ0 J0 (γm r)rdr. (16) Θh = R|J0 (γm )| 0 m=0 With Dugamel’s principle one can obtain partial solution Θp for inhomogeneous problem (11) in such a way √  t  R ∞ 2 γm 2 πL p p Θp = J0 (γm r) Cm (s)e− B (t−s) ds, Cm =

(r)J0 (γm r)rdr. J0 (γm ) 0 0 m=0 (17) The sum of the series (16) and (17) provides formal solution for heat problem stated above. All elements in this representation except eigenvalues γm are obtained in closed form. In contrast with them the γ’s are calculated numerically as the roots of transcendental equation. Some omissions may occure in their search that may cause the incompleteness for the solution. In this regard the verification is desirable at this stage. To verify the completeness of eigenfunction-system and convergence of partial sum sequences we provide test expansions with “good and bad” examples. The former is the bump function, which is twice differentiable and obey boundary conditions stated above, and the latter is discontinuous function 

r 2 r 3

r 3 2R 1, R 3r 3 ≤r ≤ 3 . +3 − 1− , G2 (r) = G1 (r) = 64 R R R R 0, otherwise The graphs for partial sums together with original functions are shown on Figs. 3 and 4. The sequences for corresponding Fourier cofficients are represented graphically on Fig. 5

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As it can be seen from the figures, the partial sums of expansions for smooth function and for discontinuous function are significantly different, where convergence to the original smooth function is more satisfied even for small orders of partial sums. Thus, we numerically approve the completeness and covergency for proposed formal series.

1.0

1.0

0.8

0.8

0.6

0.6

0.4 0.4 0.2 0.2 0.2

0.4

0.6

0.8

1.0 0.2

(a) for 4 Eignfunctions

0.4

0.6

0.8

1.0

(b) for 25 Eignfunctions

Fig. 3. Partial sums of different orders for the function G1 (r).

(a) for 25 Eignfunctions

(b) for 100 Eignfunctions

Fig. 4. Partial sums of different orders for the function G2 (r). 1.5 1.5 1.0 1.0 0.5 0.5 5 5

10

15

20

25

10

15

- 0.5

- 0.5 - 1.0 - 1.0

- 1.5

(a) for the function G1 (r)

(b) for the function G2 (r)

Fig. 5. Fourier cofficients for 25 corresponding eignfunctions.

20

25

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3.3

185

Stress-Strain Problem

In the absence of body forces, the basic equations for temperature rate-dependent linear isotropic thermoelastic medium can be written as follows [31,32]: u, μ∇2 u + (λ + μ)∇∇ · u − γ∇Θ = ρ¨

(18)

where β = (3λ + 2μ)α, α is the coefficient of linear thermal expansion, λ and μ are Lame’s constants. To facilitate the solution, the non-dimensional parameters (10) are used beside u . R With dimensionless variables, the equations (18) (after dropping the dimensionless symbol for simplicity) takes the form: ˇ= u

¨, ∇2 u + k∇∇ · u − A∇Θ = u where k=

λ+μ , μ

A=

(19)

γT0 . μ

The boundary and initial conditions in the dimensionless form are stated as σr |r=R = 0,

u|t=0 = u0 ,

˙ t=0 = v0 , u|

the dimensionless stress components take the form σr = (k − 1)∇ · u + 2

∂u − AΘ, ∂r

σθ = (k − 1)∇ · u + 2

u − AΘ. r

The separation of variables once again is used to solve the problem (18). Suppose that u = U(r)T (t). In such a case Eq. (19) can be resolved to the form [∇2 U + k∇∇ · U]T − UT¨ = X ,

(20)

where the notation X = X (r, t) = A∇Θ is introduced for brevity. Now suppose that the following Sturm Liouville problem ∇2 U + k∇∇ · U = −η 2 U,

(21)

has solutions Ui , i = 1, 2, .., ∞, which are corresponding to the eigenvalues ηi , i = 1, 2, .., ∞. Due to the fact that the differential operator, defined with Eq.(20) with the corresponding is self conjugate, the system of its eigenfunctions, that are the solutions of Sturm Liouville problem (21) is complete and orthonormal. After appropriate normalization it becomes orthonormal. We will

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address this issue in more detail in next section. Here we suppose that such eigensystem is already known and one can obtain with it formal solution for (19), i.e. ∞ Ui (r)Ti (t). (22) u= i=1

Using such a representation for u the Eq. (20) can be transformed to ∞

[−ηi2 Ui Ti − Ui T¨i ] = X .

(23)

i=1

With the following notation for inner product 

R

Ui , Uj = 2πL 0

 1 i=j Ui · Uj rdr = , 0 i=

j

one can get ordinary differential equations of the time variable t as follow − ηj2 Tj − T¨j = Yj ,

(24)

where Yj (t) = Uj (r), X (r, t) . The initial conditions define the following relations u|t=0 = ˙ t=0 = u|

∞ i=1 ∞

Ui (r)Ti (0) = u0 ,

Tj (0) = Uj , u0 = u0j ,

Ui (r)T˙i (0) = v0 ,

T˙j (0) = Uj , v0 = vj0 .

(25)

i=1

Finally, the general solution of the non-homogeneous equation (24) can be obtained as the sum of the solution of the corresponding homogeneous equation and a particular solution of the nonhomogeneous equation, i.e. Tj = 3.4

u0j

vj0 1 cos(ηj t) + sin(ηj t) − ηj ηj



t

Yj (s) sin [ηj (t − s)] ds.

0

Eigenproblem

Keeping in mind, that all fields in considered problem depend only on radial coordinate r and time, we assume the dynamic displacement vector can be repˇ=u resented as u ˇ er . Then the Eq. (21), where U = U er , takes the form U ∂2U 1 ∂U η2 − 2+ U = 0, + 2 ∂r r ∂r r (k + 1)

(26)

Reducing of Residual Stresses - SLM Additive Technology

the solution of Eq. (26) can be represented in terms of Bessel function  η2 , U = c1 J1 (ar), a = k+1

187

(27)

where c1 is arbitrary constant. Using the boundary conditions for the stresses on cylinder’s surface, we obtain a homogeneous algebraic equation a(k + 1)J0 (ar) −

2J1 (ar) = 0. r

(28)

The roots of Eq. (28) form a sequence of eigenvalues ηq , q = 1, .., ∞. The complete system of eigenfunctions can be given as following  ηq2 J1 (aq r) , , aq = Uq = Nq k+1 where the normalization factor Nq is given in the form  Nq =



2πL

R 0

 1   2 2J0 (aq )J1 (aq ) + RJ1 (aq )2 J1 (aq r)2 rdr = πLR RJ0 (aq )2 − . aq R

Table 2 shows graphs for three eigenfunctions, which represent the displacement component in r direction, and the corresponding stress component σrq , it’s clear that solutions satisfy the boundary condition. Table 2. Some of eigenfunctions Uq and their derivatives. U7 = 2.3J1 (21.19r)

U10 = 2.76J1 (30.62r)

σrq

Uq

U4 = 1.71J1 (11.75r)

Test expansions of the eigenfunctions system Uq , for the discontinuous function G2(r), which is mentioned in Subsect. (3.2) are provided. Figure 6 shows partial sums of different order.

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(a) for 25 Eignfunctions

(b) for 150 Eignfunctions

Fig. 6. Test expansion for different numbers of eignfunctions.

From the stresses-displacement components relations, one can calculate the intensity of normal stresses τ , which is defined by the formula  1 (σr − σθ )2 . τ= 2

4

Computational Analysis and Discussion

In the rest of the paper we provide computational analysis for the thermoelastic growth of circular cylindrical solids, manufactured from one of the two metallic materials, copper which is diamagnetic, and titanium, purported to be paramagnetic. The considered dimensions of the cylinder are radius R = 0.5 cm and lenght L = 2 cm. Calculations have been carried out with following material data [33,34]:

Titanium Copper

λ, GP a

μ, GP a

ρ, kg/m3

α, K −1

113.8

44

4510

8.6 × 10−6

89.47

40.95

8960

16.4 × 10

Λ, W/(mK) −6

17 385

κ, J/(kgK)

σ, S/m

521

2.38 × 106

385

5.9 × 10

μ0 , H/m 7

1.26 × 10−6 1.256629 × 10−6

In a dimensionless form, the process starts at r = 1 as the radius of the initial cylinder, and then grows at 0.01, which represents the thickness of the new layer, every 4 s (duration of a step). It is assumed that there are no mass forces and at the initial time moment, the growing body was free from stresses and at rest. In case of inhomogeneous heating, the metallic particles are heated to a temperature slightly lower than the melting temperature (the melting temperature of Copper and Titanium are 1360 K and 1940 K respectively) and then attached to the main part, which has temperature is less then the temperature of the particles, And because the body retains most of the heat during the process, the heat source must be controlled at every step as showen in Fig. 7 so that the body does not melte complitlly.

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120

150

100

80 100 60

40 50 20

0

0

(a) for Titanium

(b) for Copper

Fig. 7. The distribution of the heat source power during the process.

(a) without heating

(b) with in-homogeneous heating

Fig. 8. The temperature distribution inside the body (Titanium).

Figures 8 and 9 show the temperature distribution in the whole body show the temperature distribution in the whole body during the process for two cases: without heating and with external in-homogeneous heating of the growing body. Figures 10 and 11 illustrate the temperature distribution on the moving boundary for the two cases. It’s clear that, by using external heating, the gradient of temperature on the growing surface significantly decrease, thereby reducing residual stresses.

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(a) without heating

(b) with in-homogeneous heating

Fig. 9. The temperature distribution inside the body (Copper).

123.5

120

123.0

100

122.5

80 60

122.0

40

121.5

20 10

20

30

40

50

10

60

20

30

40

50

60

(b) with in-homogeneous heating

(a) without heating the main body

Fig. 10. The temperature distribution on the moving boundary for Copper. 200 192 190

150

188 100 186 184 10

20

30

40

50

(a) without heating the main body

60

10

20

30

40

50

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(b) with in-homogeneous heating

Fig. 11. The temperature distribution on the moving boundary for Titanium.

Figures 12 and 13 present the stress intensity distribution at some moment after ending the process for two cases: without heating and with external inhomogeneous heating of the growing body. The main finding of the study can be outlined as follows. The inductive heating significantly affect on the residual stress distribution and being judiciously applied can substantially reduce them. The solution of corresponding optimiza-

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stress intensity 0.00080

0.0095 0.00075

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1.00

1.05

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Fig. 12. The stress intensity distribution at some moment after ending the process for Copper.

tion problem are requested to determine the most viable option. This problem will be the subject of further research. stress intensity

stress intensity 0.0016

0.025

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Fig. 13. The stress intensity distribution at some moment after ending the process for Titanium.

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Acknowledgements. The study was partially supported by the Russian Government program (contract #AAAA − A20 − 120011690132 − 4) and partially supported by RFBR (grant N o. 18 − 08 − 01346 and grant N o.18 − 29 − 03228).

References 1. Levy, G.N., Schindel, R., Kruth, J.P.: Rapid manufacturing and rapid tooling with layer manufacturing (LM) technologies, state of the art and future perspectives. CIRP Ann. 52(2), 589–609 (2003) 2. DebRoy, T., et al.: Additive manufacturing of metallic components-process, structure and properties. Progress Mater. Sci. 92, 112–224 (2018) 3. Kruth, J.P., Leu, M.C., Nakagawa, T.: Progress in additive manufacturing and rapid prototyping. CIRP Ann. Manuf. Technol. 47(2), 525–540 (1998) 4. Meiners, W., Wissenbach, K., Gasser, A.: Shaped body especially prototype or replacement part production. DE Patent. 19 (1998) 5. Ciarletta, P., Destrade, M., Gower, A.L., Taffetani, M.: Morphology of residually stressed tubular tissues: beyond the elastic multiplicative decomposition. J. Mech. Phys. Solids 90, 242–53 (2016) 6. Green, A.E.: Thermoelastic stresses in initially stressed bodies. Proc. Royal Soc. London. Ser. Math. Phys. Sci. 266(1324), 1–9 (1962) 7. Johnson, B.E., Hoger, A.: The use of a virtual configuration in formulating constitutive equations for residually stressed elastic materials. J. Elast. 41(3), 177–215 (1995) 8. Klarbring, A., Olsson, T., Stalhand, J.: Theory of residual stresses with application to an arterial geometry. Arch. Mech. 59(4–5), 341–64 (2007) 9. Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51(3), 032902 (2010) 10. Sadik, S., Yavari, A.: Geometric nonlinear thermoelasticity and the time evolution of thermal stresses. Math. Mech. Solids 22(7), 1546–87 (2017) 11. Wang, J., Slattery, S.P.: Thermoelasticity without energy dissipation for initially stressed bodies. Int. J. Math. Math. Sci. 31, 329–337 (2002) 12. Othman, M.I., Fekry, M., Marin, M.: Plane waves in generalized magneto-thermoviscoelastic medium with voids under the effect of initial stress and laser pulse heating. Struct. Eng. Mech. 73(6), 621–9 (2020) 13. Mercelis, P., Kruth, J.P.: Residual stresses in selective laser sintering and selective laser melting. Rapid Prototyping J. (2006) 14. Buchbinder, D., Meiners, W., Pirch, N., Wissenbach, K., Schrage, J.: Investigation on reducing distortion by preheating during manufacture of aluminum components using selective laser melting. J. Laser Appl. 26(1), 012004 (2014) 15. Zaeh, M.F., Branner, G.: Investigations on residual stresses and deformations in selective laser melting. Prod. Eng. 4(1), 35–45 (2010) 16. Vilaro, T., Colin, C., Bartout, J.D.: As-fabricated and heat-treated microstructures of the Ti-6Al-4V alloy processed by selective laser melting. Metall. Mater. Trans. A. 42(10), 3190–3199 (2011) 17. Kruth, J.P., Deckers, J., Yasa, E., Wauthl´e, R.: Assessing and comparing influencing factors of residual stresses in selective laser melting using a novel analysis method. Proc. Inst. Mech. Eng. Part B J. Eng. Manuf. 226(6), 980–91 (2012) 18. Arutyunyan, N.K., Drozdov, A.D., Naumov, V.E.: Mechanics of growing viscoelastoplastic bodies (1987)

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19. Lychev, S.A., Manzhirov, A.V.: The mathematical theory of growing bodies. Finite deformations. J. Appl. Math. Mech. 77(4), 421–32 (2013) 20. Lychev, S.A., Manzhirov, A.V.: Reference configurations of growing bodies. Mech. Solids 48(5), 553–560 (2013). https://doi.org/10.3103/S0025654413050117 21. Lychev, S.A.: Universal deformations of growing solids. Mech. Solids 46(6), 863–76 (2011) 22. Lychev, S., Manzhirov, A., Shatalov, M., Fedotov, I.: Transient temperature fields in growing bodies subject to discrete and continuous growth regimes. Procedia IUTAM. 1(23), 120–129 (2017) 23. Polyanin, A.D., Lychev, S.A.: Decomposition methods for coupled 3D equations of applied mathematics and continuum mechanics: partial survey, classification, new results, and generalizations. Appl. Math. Mod. 40(4), 3298–324 (2016) 24. Lychev, S., Koifman, K.: Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics. De Gruyter (2018) 25. Yavari, A.: A geometric theory of growth mechanics. J. Nonlinear Sci. 20(6), 781– 830 (2010) 26. Lychev, S.A., Manzhirov, A.V.: Discrete and continuous growth of hollow cylinder. Finite deformations. In: Proceedings of the World Congress on Engineering, vol. 2, pp. 1327–1332 (2014 ) 27. Levitin, A.L., Lychev, S.A., Manzhirov, A.V., Shatalov, M.Y.: Nonstationary vibrations of a discretely accreted thermoelastic parallelepiped. Mech. Solids 47(6), 677–89 (2012) 28. Lorrain, P., Corson, D.R.: Electromagnetic Fields and Waves (1970) 29. Weeks, W.L.: Transmission and distribution of electrical energy, Harpercollins, (1981) 30. John, F.: Partial Differential Equations, Springer-Verlag. New York (1982) 31. Nowacki, W.: Theory of Elasticity [in Polish]. PWN, Warszawa (1970) 32. Othman, M.I., Fekry, M.: The effect of initial stress on generalized thermoviscoelastic medium with voids and temperature-dependent properties under Green-Neghdi theory. Mech. Mech. Eng. 21(2), 291–308 (2017) 33. Lychev, S.A., Manzhirov, A.V., Joubert, S.V.: Closed solutions of boundary-value problems of coupled thermoelasticity. Mech. Solids 45(4), 610–23 (2010) 34. Donachie, M.J.: Titanium: a Technical Guide. ASM International (2000)

Solids and Structures

Critical Velocities and Stability of the Axially Moving Panels Nikolay Banichuk(B) and Svetlana Ivanova Ishlinsky Institute for Problems in Mechnics RAS, Prospect Vernadskogo 101, Bld. 1, 119526 Moscow, Russia

Abstract. The problem of stability of elastic continuous panels moving axially with constant velocities and performing separately different types of free vibrations (torsional, longitudinal and transverse) is considered and analyzed. All considered types of free vibrations of small amplitude are described by a general second-order constant-coefficient partial differential equation. The influence of transport phenomena on stability of the panels moving at critical velocities is discussed. It is shown that the elastic panel (rod) if travelling at the critical velocity must stay in a steady-state configuration. The material (modelled by elastic rod) still undergoes axial translation, but this profile will remain static in time. The observer sees a steady-state motion akin to a steady-state fluid flow.

Keywords: Mathematical modeling velocities · Stability

1

· Axially moving panels · Critical

Introduction

Investigation of axially moving materials such as strings, plates, and panels consists of the finding of the reliable range of problem parameters guaranteeing the thermomechanical stability. Thus, the purpose of studies is to evaluate the travelling velocity, dangerous temperature and other important geometrical and mechanical parameters realizing the phenomenon of instability. These studies are important as from theoretical as practical point of view. Investigations of instability parameters of axially moving material began to appear starting with those by Sack [1] and Archibald and Emslie [2]. Wide exposition of obtained results in domain of mechanics of moving materials is presented in the books by Marynowski [3] and Banichuk et al. [4,5]. An extensive literature review can be also found in these books.

2

Basic Notions and Assumptions

In this paper we consider a narrow thin elastic panel moving in x-direction between rollers at x = 0 and x = l and modelled by a continuous one-dimensional c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 197–201, 2022. https://doi.org/10.1007/978-3-030-92144-6_15

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elastic element (rod) having rectangular cross-section of given width and thickness (see Fig. 1). We consider separately three types of free vibrations of small amplitude: torsional, longitudinal, and transverse. All considered cases of panel movement are of the same mathematical form and it is convenient to write them as a general second-order constant-coefficient partial differential equation [5] a

∂2U ∂2U ∂2U + c + 2b = 0, ∂t2 ∂x∂t ∂x2

0≤x≤l

(1)

where U = ϑ, a = ρI0 , b = ρI0 V0 , c = ρI0 V02 − GIk for torsional vibration; U = u, a = ρS, b = ρSV0 , c = ρSV02 − ES for longitudinal vibration; U = w, a = m, b = mV0 , c = mV02 − T0 for transverse vibration. Here ϑ(x, t) is the angle function, which represents the angle of torsion per unit length of the panel (rod); u = u(x, t) is the longitudinal displacement at time instant t that the panel experiences at laboratory coordinate x; w = w(x, t) is the transverse displacement function; m = ρS is the mass per unit length of the panel, ρ is the material density, S is the cross-sectional area of the panel, GIk is the torsional rigidity, G is the shear modulus, E is the Young modulus, Ik is the polar moment of inertia, I0 is the moment of inertia of the panel cross-section around the x-axis.

Fig. 1. The axial movement of elastic panel.

The discriminants of the Eq. (1) are respectively Dϑ = ρI0 GIk > 0,

Du = ρS 2 E > 0,

Dw = mT0 > 0

(2)

and hence, the Eq. (1) is always hyperbolic regardless of the value of the axial velocity V0 . This observation reflects the physical nature of the problem: the introduction of axial velocity should not change the basic vibrational nature of the mechanical response. In the following free vibration analysis, we set only the boundary conditions (zero Dirichlet) (U )x=0 = 0,

(U )x=l = 0

(3)

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and the solution of the Eq. (1) will have two free parameters. Free vibration analysis is only concerned with determining possible motions of the unloaded system, in other words, the nontrivial solution of the homogeneous partial differential Equation (1).

3

Critical Velocities

Now we consider the mechanical response at the velocity V0 , for which the coefficient c cancels out. We will call this special value of V0 the critical velocity, denoted with the symbol C. We have the following expressions for critical velocities presented also in Table 1 for all three types of vibration:  1/2 GIk − in torsional case, C= ρI0   1/2 E C= − in longitudinal case, ρ  1/2 T0 C= − in transverse case. m

Table 1. Critical velocities for torsional, longitudinal, and transverse types of vibration. Vibration type

Torsional  1/2 GIk Critical velocity C= ρI0

Longitudinal  1/2 E C= ρ

Transverse  1/2 T0 C= m

At the critical velocity V0 = C, the Eq. (1) (after dividing by a and noting b/a = V0 = C) simplifies into ∂2U ∂2U = 0. + 2C ∂t2 ∂x∂t

(4)

By introducing the notation v ≡ ∂U/∂t the Eq. (4) can be rewritten as ∂v ∂v + 2C = 0. ∂t ∂x

(5)

Thus, the velocity-like quantity v obeys the (homogeneous) first-order transport equation along the panel (rod) with the transport of the quantity occurring at constant velocity 2C. Hence, we have v(x, t) = g(x − 2Ct)

(6)

for some differentiable function g. Consequently, the function U (x, t) is determined as U (x, t) = f (x) + h(x − 2Ct) (7)

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where some function f (x) is constant in time and a new differentiable function h(x − 2Ct) is related to g(x − 2Ct) by the equation v=

∂U = (−2C)h (x − 2Ct) ≡ g(x − 2Ct). ∂t

(8)

Using the representation v ≡ ∂U/∂t and the boundary conditions (3) we obtain (v)x=0 = 0,

(v)x=l = 0

(9)

for all t. From the boundary conditions (9) we will have v(x, t) = g(x − 2Ct) ≡ 0.

(10)

Hence, the considered panel (rod), if travelling at the critical velocity (see Table 1), must stay in a steady-state configuration. It is possible to find the function f (x) explicitly via a direct approach. Let us integrate (4) with respect to t, obtaining ∂U ∂U + 2C = h(x) ∂t ∂x

(11)

where Eq. (11) is the standard nonhomogeneous transport equation. Its solution is [6]  1 U (x, t) = h(x)dx + g(x − 2Ct). (12) 2C As above, it is a linear superposition of two components: a steady-state one, and one being transported toward the +x-direction at velocity 2C. Following the same argument regarding boundary conditions as above, we find g(x − 2Ct) ≡ 0 (see (10)). By differentiating (12) with respect to x, and then setting t = 0, we determine ∂U (x, 0) (13) h(x) = 2C ∂x and finally, by substituting (13) back into (12) , we obtain U (x, t) = U (x, 0).

(14)

Hence, when the axial motion occurs with the critical velocity C given in Table 1, the initial condition for the position, namely (U )t=0 = χ1 (x)

(15)

completely determines the solution for all t. Here χ1 (x) is a given function. The other initial condition, namely   ∂U (v)t=0 ≡ = χ2 (x) (16) ∂t t=0 must have χ2 (x) ≡ 0 in order to be compatible with the solution (14) .

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Conclusion

The conclusion is that if the compatibility condition (v)t=0 ≡ 0, 0 < x < l holds for our initial conditions, then, upon a quasistatic transition to the limit state V0 = C, the state variable profile of a freely vibrating axially travelling elastic element (panel, rod) will “freeze” into the shape it had when the limit state was reached. By state variable profile we mean the function U (x, t) describing the state U as seen in the laboratory coordinates (that is, by a stationary observer). The material still undergoes axial transition, but this profile will remain static in time. The observer sees a steady-state motion, akin to a steady-state fluid flow. In practice, the compatibility condition (v)t=0 ≡ 0, 0 < x < l is reasonable for many physically admissible situations for the considered model near V0 = C, because it was shown in [5], all eigenfrequencies of the travelling elastic element tend to zero in the limit V0 → C. Nevertheless, the quasistatic analysis has its limitations. For example, consider the case where we would like to initially set V0 = C − ε (with ε < 0 small), (U )t=0 ≡ 0, (v)t=0 = f2 (x) ≡ 0 and then perform a transition to V0 → C. The quasistatic analysis is not applicable, because the given initial condition for v violates the compatibility condition f2 (x) ≡ 0. If this category of cases is to be analyzed, a more general treatment of the dynamics including the effects of accelerating motion is required. Acknowledgements. The study was performed in Ishlinsky Institute for Problems in Mechanics RAS and it is partially supported by the Ministry of Science and Higher Education within the framework of the Russian State Assignment under contract No AAAA-A20-120011690132-4 and partially supported by RFBR Grant 20-08-00082-a.

References 1. Sack, R.A.: Transverse oscillations in travelling strings. Br. J. Appl. Phys. 5, 224– 226 (1954) 2. Archibald, F.R., Emslie, A.G.: The vibration of a string having a uniform motion along its length. ASME J. Appl. Mech. 25, 347–348 (1958) 3. Marynowski, K.: Dynamics of Axially Moving Orthotropic Web. Lecture Notes in Applied and Computational Mechanics, vol. 38. Springer-Verlag, Germahy (2008) 4. Banichuk, N., Jeronen, J., Neittaanm¨ aki, P., Saksa, T., Tuovinen, T.: Mechanics of Moving Materials. Solid Mechanics and Its Applications, vol. 207. Springer International Publishing Switzerland, Cham (2014) 5. Banichuk, N. V., Barsuk, A. A., Jeronen, J., Tuovinen, T., Neittaanm¨ aki, P.: Stability of axially moving materials. In: Solid Mechanics and Its Applications, vol. 259. Springer Nature Switzerland AG, Cham (2020) 6. Polyanin, A.D., Zaitsev, V.F., Moussiaux, A.: Handbook of First Order Partial Differential Equations. Taylor & Francis, London (2002)

Features of Applying HEDE Model to Description of the Destruction of Materials Induced by Hydrogen Yulia Sedova(B)

, Vladimir Polyanskiy , and Nikolay Bessonov

Institute for Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), Saint Petersburg, Russia

Abstract. The presence of hydrogen has a strong destructive effect on the properties of the material. In a hydrogen environment, metals lose their strength, ductility, toughness, and fail at much lower loads than unsaturated with this substance materials. This phenomenon has been known for more than one century and is called hydrogen embrittlement. To date, many articles and studies have been devoted to this problem and several models that describe hydrogen embrittlement have been proposed. One of the most widely accepted approach to modeling this phenomenon is the hydrogen enhanced decohesion model (HEDE). The main idea of this method is concentration of hydrogen in the region of the maximum average normal tensile stress (that is near the stress concentrator) that lead to a decrease in cohesion of the crack surfaces and the crack growth. In the present article, we focused on the detailed research of the features of the hydrogen enhanced decohesion model, studied the dependencies and laws included in the concept of this method, and based on this we concluded about applicability of the HEDE to description the phenomenon of hydrogen embrittlement. Keywords: Hydrogen embrittlement · Decohesion · Destruction · HEDE model

1 Introduction Hydrogen embrittlement is one of the most dangerous causes of material destruction. And this problem is especially relevant for materials with increased strength characteristics, which are more often used in modern industry. As a rule, the possibility of hydrogeninduced destruction cannot be determined in an external study of details and structural elements. In this regard, there is a need to create models and parameter relationships that could predict the effect of hydrogen on the mechanical properties of materials. But the phenomena associated with the interaction of hydrogen with a solid are very diverse. And in addition, there are many varieties of hydrogen embrittlement, the description of the study of some of them is contained in the book [1]. Therefore, the creation of a uniform model describing the interaction of hydrogen with a solid is an extremely difficult task. This problem is a very popular research topic, many articles devoted to this issue are published every year. Nowadays, there are several approaches © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 202–219, 2022. https://doi.org/10.1007/978-3-030-92144-6_16

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to description the effect of hydrogen on the structure and mechanical properties of a material. The most popular of them is the HEDE model, and the present work is devoted to the features of application of this mechanism.

2 HEDE Model Review HEDE model (Hydrogen Enhanced Decohesion) is based on the idea of growth of crack existing in the material. This approach considers fundamentally brittle fracture as a result of the development of hydrogen embrittlement without plastic deformation. According to the HEDE model, hydrogen is concentrated in the region of the maximum average normal tensile stress (that is near the stress concentrator) leading to a decrease in cohesion of the crack surfaces and the crack growth [2]. Within the framework of model, the process of hydrogen transport inside a solid is most often described by a Fick law supplemented by the thermodynamic or chemical potential [3]. Considering the influence of mechanical stress this equation can be written as CVH ∇ 2 p VH ∇C∇p ∂C = D(T )∇ 2 C − D(T ) − D(T ) ∂t RT RT

(1)

where C is hydrogen concentration, VH – partial molar volume of hydrogen, D - diffusion coefficient,p - mean normal stress tensor, T - absolute temperature, R – gas constant. To describe changes in the cohesion of the crack surfaces most researchers use the following dependence suggested by Serebrinsky based on the Langmuir-McLean isotherm [4]: θ=

C   H C + exp − g RT

(2)

where θ is parameter reflecting the degree of filling the surface of the crack with hydrogen atoms, gH is Gibbs free energy difference for hydrogen between the adsorbed inside the crystal lattice and bulk standard states of Fe. It is interesting to note that Serebrinsky deduced this expression based on the equation for the separation of impurity atoms on the surface of a monocrystal [5], according to which the equilibrium relative concentration of foreign substance at the grain boundaries fr (surface concentration) Cgb can be calculated using known fractional concentration of solute inside grains (internal concentration) C fr : fr

Cgb =

C fr    C fr + 1 − C fr exp − (E−e) RT 

(3)

where (E − e) is the difference in dissolution and sorption energies at grain boundaries. Serebrinsky applies C fr in this formula as a total concentration, meaning that all hydrogen moves on the free surface during decohesion. An important component of the HEDE model is the hydrogen degradation law. As a rule, almost all researches use the dependence, proposed by Serebrinsky, which is as

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follows. He suggested to determine the surface free energy using the known parameter θ value as   (4) γ (θ ) = 1 − 1.0467θ + 0.1687θ 2 γ (0) Here γ (θ ) and γ (0) are surface energy with and without hydrogen, respectively. This equation is obtained by approximating the graph presented in an unpublished article by co-authors of Serebrinsky. Moreover, co-authors obtained the graph for the case of deformation monocrystal in the crystallographic direction (110) of hydrogen saturated pure Fe (although it is unclear what to consider grain boundaries). After that Serebrinsky assumed that maximum value of the relative displacement of the crack edge, which will not lead to decohesion, weakly depends on the value of the parameter θ . Then using the energy ratio 2γ (θ ) = σz (θ )δ(θ ), there σz (θ ) is stress normal to crack surfaces and δ(θ ) – distance between them, the hydrogen degradation law can be determined as   σz (θ ) = 1 − 1.0467θ + 0.1687θ 2 σz (0) (5) The value of σz (0) should be used considering the recommendations set forth in the work [6]. The procedure for FE simulations of the HEDE model includes law for cohesive forces. It is assumed that the cohesive stress σ (δ) between two atomic planes passing into the crack surfaces depend on the distance δ between them arising due to deformations. With an undisturbed interatomic distance δ = a, the cohesion stress is zero, then, at a distance of the order of δ = 1.5a, it reaches a maximum and begin to decrease with a further increase in δ up to a break or a significant weakening of cohesion, characteristic of free crack surfaces. When this dependence σ (δ) is established, the surface energy γ can be calculated δ ∞ as γ = 1/2 a σ (δ)d δ ≈ 1/2 0 c σ (δ)d δ, where a – undisturbed interatomic distance,   and stress intensity factor [7] KIC = Eγ / 1 − υ 2 , then the standard criterion for crack growth is used. In FE simulations tension between the elements in the cohesive zone are given in the form of normal stress corresponding to the law for the cohesive forces σ (δ). In this case, as soon as the distance between them δ becomes greater than the critical δC (known for considered material and independent of hydrogen concentration), the cohesion breaks and a crack grows. Most researchers use the Needleman equation [8]: σ (δ) =

 δ 2 27 δ σz (θ ) 1− 4 δc δc

(6)

(θ)δc which after integration leads to the relation γ = 9σz16 . Needleman obtained this formula for a composite material with fibers having a rectangular cross-sectional shape, hence the fractional factor. And even though the grooves on all metal specimens for which the HEDE model was used are round, this formula is the most popular.

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Besides the Needleman equation, the following are used: • Linear dependence [9]:

 δ σ (δ) = σz (θ ) 1 − δc

(7)

The disadvantage of this dependence is the fundamental discrepancy with the cohesion law for small δ, therefore, a piecewise linear dependence is more often used [10, 11]: σz (θ ) δδ0 , 0 ≤ δ ≤ δ0   σ (δ) = (8) δ−δ0 σz (θ ) 1 − δc −δ0 , δ0 ≤ δ ≤ δc • Exponential dependence [12–14]: σ (δ) =

δ e1 δ σz (θ )e− δc δc

(9)

In the review paper [15], the authors note that the influence of different types of this law on the simulation results has not been completely studied, in some works it is characterized as strong, in several works, on the contrary, as insignificant. The procedure for accounting for cohesive forces (traction-separation law) is considered standard and “Cohesive analysis” is contained in the standard FE software packages, therefore, most authors do not pay due attention to this algorithm.

3 Numerical Simulations In the current work, we studied the features of applying the HEDE approach to the numerical description of the effect of hydrogen on the mechanical properties of materials in the following model task. We considered the uniaxial tension of a cylindrical bar made of high strength steel with a semicircular groove. The initial distribution of hydrogen concentration was assumed uniform throughout the specimen. We implemented the HEDE method in the integrated development environment Microsoft Visual Studio (VS) in C++. This choice is due to the ability to independently “assemble” the model, in other words, to add into the program absolutely all equations describing the process under study. As a result of this, we did not miss the slightest detail of the calculation and we learned all its subtleties, what could not be done by working with FE software. 3.1 Statement of the Problem Considering symmetry, the geometry of the studied model is a two-dimensional rectangular area with side 5a with a hole of radius a at the outer boundary. A rectilinear structured mesh consisting of triangles, used in all performed calculations, is shown in Fig. 1.

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Fig. 1. Mesh of the model.

Considering the condition of symmetry, we established a prohibition on movements at the nodes of the right edge along the horizontal axis Ox and at the nodes of the lower edge along the vertical axis Oy. A tensile load of magnitude p acting along the Oy axis was applied to the upper edge of the area. In the framework of the task, we can conditionally distinguish three of its components: • Static structural analysis • Hydrogen diffusion analysis • Cohesive analysis Since static and diffusion problems require different orders of magnitude of the iteration step in time, they must be calculated separately from each other. In turn, the cohesive analysis in our statement involves working with scalar quantities that continuously change in time and depend on the results of solving the diffusion problem. In this regard, the last two calculations should be carried out simultaneously. Thus, the algorithm of our work will consist of two stages: 1) solving the mechanical problem of the stress-strain state of the specimen and 2) diffusion analysis with the simultaneous calculation of cohesive stress.

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3.2 Numerical Procedure The calculation of the static structural problem is carried out as for non-stationary with a very small time step using the equation of motion. To obtain the characteristics of the stress-strain state of the specimen, we used Hooke’s law. The calculation of the static problem continues until the solution reach a conditionally stationary level. When this condition is satisfied, the calculation of the parameters of the mechanical stage is completed. The characteristics of the stress field obtained at this step are passed to the next stage – the diffusion problem. Hydrogen diffusion analysis is based on the Eq. (1). Using it, we calculate the redistribution of the hydrogen concentration, considering its initial content in nodes, which is determined by the value c0 , and parameters of the stress-strain state obtained in the course of solving the static problem. At the same time, cohesive stress analysis is launched. It was based on Eqs. (2) and (5) with which we calculated the current value of cohesive stress. Note that it is enough to calculate these values not over the entire mesh, but sequentially in the nodes lying on a straight line along which crack nucleation and propagation are assumed (on the lower edge of a rectangular area). The process of crack growth we have defined as follows. We established the criterion for breaking the cohesion between the atomic planes of a material as condition when the value of the elastic stress obtained in solving the static problem becomes higher than the level of cohesive stress. As soon as this relation is satisfied, the restriction on movement along the horizontal axis in the mesh node is removed, node gets the opportunity to break away from the horizontal face and to shift to the up under the action of a tensile load. Thus, the concept of the critical concentration of hydrogen in a node is introduced, at which, according to Eqs. (2) and (5), the value of cohesive stress becomes less than the value of the component of elastic stress acting along the vertical axis. As a result of this, a programmed initiation and propagation of a crack occurs at one internodal distance. After that, the recalculation of the static structural analysis is started again, and the calculation procedure is repeated. 3.3 Model Parameters The material of the test specimen is high-strength steel PSB1080, physical properties are shown in Table 1. We also set the following values: a = 200 μm, D = 2.5 · 10−11 m2 /s, VH = 2 · 10−6 m3 /mol, gH = 30 kJ /mol, T = 298 K, σz (0) = 4.5σT . During the calculations, it was found that, depending on the initial level of hydrogen concentration and the applied load, the following situations are possible: 1) the hydrogen concentration in the edge node, closest to the hole (in the «first» node), is such that the cohesive stress recalculated with it cannot reach a value less than elastic, as a result of which crack initiation does not occur; 2) the hydrogen concentration reaches a critical value, but only in the «first» node, thus, a crack is initiated and covers only one internodal distance, and its further growth stops; 3) the hydrogen concentration is sufficient to consistently exceed the critical value at all nodes of the area edge, which leads to an unstable growth of the main crack, up to the destruction of the whole area. And in our research, we were interested in the latter case. Such an outcome is guaranteed to us if

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Y. Sedova et al. Table 1. Physical properties of the material. Steel PSB1080 Tensile strength

σB = 1498 MPa

Yield strength

σT = 1276 MPa

Density

ρ = 7800 kg/m3

Bulk modulus

K = 160 GPa

Shear modulus

G = 79.3 GPa

the gap criterion in the «second» edge node is done. Based on this, we set the duration of the problem solution. This value corresponded to the time elapsed from the moment the diffusion calculation was started and until the gap criterion was done in the second edge node closest to the hole. 3.4 Results of the Numerical Simulations Here we present the detailed results of the calculation for level of the tensile load equal to p = 650 MPa and the initial hydrogen concentration of c0 = 0.23 ppm. Figure 2 shows the distribution of elastic stress acting along the vertical axis, obtained at the end of the solution of the static structural problem. Hereinafter, blue color in the image corresponds to the maximum value, red – the minimum.

Fig. 2. Distribution of elastic stress (max = 2150 MPa; min = −21.9 MPa).

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As can be seen from the presented results, the greatest tensile stress was near the groove on the lower edge of the area. The value at the interesting point for us on the first step of solving the problem – in the first node closest to the hole – was 2150 MPa. After the calculation of the static part was completed, the solution of the diffusion problem was launched. The gap criterion in the «first» node was completed almost instantly. The hydrogen content along the propagation line of the initiated crack at that moment is shown in Fig. 3. 0.2308

0.2307

Hydrogen concentration, ppm

0.2306

0.2305

0.2304

0.2303

0.2302

0.2301

0.23

0.2299

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Distance from groove, mm

Fig. 3. Distribution of hydrogen concentration along the lower edge.

It is seen that, as prescribed by Eq. (1), the hydrogen reached its maximum value at the groove surface in the region of maximum tensile pressure. The values of cohesive stress obtained at that time along the crack propagation line are shown in Fig. 4.

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2017.5

Cohessive stress, MPa

2017

2016.5

2016

2015.5

2015

2014.5

2014

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Distance from groove, mm

Fig. 4. Distribution of cohesive stress along the lower edge.

As expected, the minimum value of cohesive stress was also observed near the surface of the groove. Due to the presence in the specimen and further redistribution of hydrogen, this value dropped below the level of elastic stress, which lead to a loss of connection in the «first» node. This ended the calculations of the diffusion stage. The next step new cycle of solving the numerical simulation started, and we return to recalculating its static component. Figure 5 demonstrates the distribution of elastic stress normal to the crack surfaces, obtained at the end of the solution of the static structural problem for an area with a crack propagating by one internodal distance.

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Fig. 5. Distribution of elastic stress (max = 2342 MPa; min = −21.7 MPa).

We can see that the maximum values of elastic stress are now observed slightly to the right of the hole because the corner of the area is now free from fixing. And we are interested in the value that was achieved in the «second» node of the lower edge, it amounted to 1741 MPa. With knowledge of this value, we again proceed to the stage of diffusion analysis. The changes in the value of cohesive stress in the «second» node over time are illustrated in Fig. 6. As we can see, the value of cohesive stress decreased with time. On the graph, the horizontal dashed line corresponds to the stress normal to the crack surfaces in the «second» node obtained when solving the static problem (1741 MPa). Cohesive stress reached this value after 32.5 h, which is recognized as the initiation time of the cracking. At this moment, the next mesh node is torn off, and the task calculation is completed.

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2000

Cohessive stress, MPa

1950

1900

1850

1800

1750

1700

0

5

10

15

20

25

30

35

Time, h

Fig. 6. Time dependence of cohesive stress in the «second» node.

Changes in the hydrogen content in the «second» node over time are presented in Fig. 7. We can see that over time the hydrogen concentration in the node increased from the initial level of 0.23 ppm set by us to a certain value. The duration of its growth is limited by the time that cohesive stress reach values of elastic (32.5 h). At this moment, the hydrogen concentration in the «second» node reached the limit value of 0.312 ppm (horizontal dashed line), which in this case is the critical value. It should be noted that the critical concentration does not depend on the initial hydrogen content, it is determined only by the level of applied load.

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0.32

0.31

Hydrogen concentration, ppm

0.3

0.29

0.28

0.27

0.26

0.25

0.24

0.23

0

5

10

15

20

25

30

35

Time, h

Fig. 7. Time dependence of hydrogen concentration in the «second» node.

Figure 8 shows the distribution of hydrogen concentration along the crack propagation line at the time of separation of the «second» node. The horizontal dashed line on it marks the value of the critical hydrogen concentration. The vertical line is drawn along the abscissa value corresponding to the location of the «second» node on the edge. The intersection of the lines indicates that the calculation of the task was completed at that exact moment in time when the concentration value in the «second» node reached a critical value. At the subsequent points of the edge, the hydrogen content is even higher, which is caused by the redistribution of the elastic stress calculated in solving the static problem. Thus, with the successive advancement of the crack tip along the area, the concentration value will only increase. At the same time, the value of cohesive stress will decrease, gradually more and more approaching elastic stresses. This fact indicates the possibility of further unhindered crack growth, up to the destruction of the whole specimen.

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0.31

Hydrogen concentration, ppm

0.3

0.29

0.28

0.27

0.26

0.25

0.24

0.23

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Distance from groove, mm

Fig. 8. Distribution of hydrogen concentration along the lower edge after separation of the «second» node.

The described calculation procedure for a tensile load of p = 650 MPa was carried out for several values of the initial hydrogen concentration c0 . Based on this, we obtained the dependence of the initiation time of cracking on the level of the initial hydrogen concentration, shown in Fig. 9. According to the results obtained, the initiation time of cracking strongly depends on the value of the initial hydrogen concentration in the specimen. With an increase in the hydrogen concentration from 0.22 ppm to 0.245 ppm, the duration of the time during which the specimen could be exposed to an external load without loss of solidity decreased from almost 120 h to 20 min.

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120

Initiation time of the cracking, h

100

80

60

40

20

0

0.22

0.225

0.23

0.235

0.24

0.245

Initial hydrogen concentration, ppm

Fig. 9. Dependence of the initiation time of cracking on the level of the initial hydrogen concentration.

In addition, we compared our results with those obtained by other researchers during the simulation of the HEDE mechanism in the ABAQUS software suite. Figure 10 shows a comparison of the dependences of the initiation time of cracking on the level of the initial hydrogen concentration obtained in one of these works and the one that we established. A comparison was made with the results of Wang et al. [16] in which they study the destruction of a bar made of high-strength steel PSB1080 caused by the action of a tensile load of p = 650 MPa with a uniform initial distribution of hydrogen over the specimen. A significant difference in the formulation of this study from ours is that Wang considers the crack growth (with known parameters) already existing inside the specimen, its top is represented by a round hole in the center of the simulated rectangular area. To describe the effect of hydrogen on the mechanical properties of materials, the authors also apply the formula (1) and dependencies (2), (5). The values, they used in the equations, are set as follows: D = 4 · 10−11 m2 /s, VH = 2 · 10−6 m3 /mol, gH = 30 kJ/mol, T = 298 K, σz (0) = 4.5σT . In both cases, increasing the hydrogen content in the specimen leads to a decrease in the length of time before its destruction. However, it can be noted that even with the same level of external load and geometric parameters of the model, the critical level of substance concentration presented in the study [16] differs by an order of magnitude from that obtained by us.

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Present work Wang et al. 2016

1000

Initiation time of the cracking, h

900 800 700 600 500 400 300 200 100 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Initial hydrogen concentration, ppm

Fig. 10. Dependencies of the initiation time of cracking on the level of the initial hydrogen concentration.

Such a significant difference can be explained by the fact that Wang et al. during the calculating the parameter of the degree of filling the free surface of the crack with hydrogen atoms by the formula (2), use the expression for the mass hydrogen concentration. In our study, we recalculate it to the value of the partial concentration (by multiplying by the ratio of the atomic mass of iron to the atomic mass of hydrogen), as it should be done. This procedure is skipped in almost all works devoted to numerical modeling of HEDE, as a result, many articles have now been published, the quantitative estimates of which should be recognized as incorrect. It should also be noted that measurements of hydrogen concentration levels in most studies are not carried out. Even the initial concentration before loading is unknown. The only parameter that compares with the experiment is the incubation time of the crack. Under these conditions, the hydrogen concentration is a fitting parameter in the complex chain of dependencies used in the HEDE model. This issue requires additional research.

4 Discussion of the Results From a mechanical point of view, the effect of hydrogen on the properties of materials is a classic problem of the influence of a small parameter, since hydrogen concentrations critical for the strength and ductility of metals are usually small. One of the models

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describing this effect is the hydrogen enhanced decohesion (HEDE) approach, which in the framework of this work we implemented in the integrated development environment Microsoft Visual Studio (VS) in C++. Using the developed program, we performed a numerical solution of the model problem of the effect of hydrogen on the mechanical properties of materials. Our algorithm made it possible to obtain the dependence of the initiation time of cracking on the level of the initial hydrogen concentration. It was found that this value strongly depends on the content of the diffusing substance. According to the results obtained, with an increase in the level of initial hydrogen concentration in the specimen, the length of time until its destruction decreases significantly. One of the significant drawbacks of the HEDE model is the need for a crack or a concentrator with known parameters, that is required for calculations. This makes it impossible to simulate and study real structures interacting with internal and external hydrogen, because, as a rule, these parameters are a priori unknown. In addition, it should be noted that to date the concept of numerical modeling of the HEDE approach, includes dependences that are not related to the problems solved on its basis. For example, almost always researchers to describe the cracks growth in steels use the laws of hydrogen degradation of Serebrinsky, which he obtained for the case of deformation in the crystallographic direction (110) of pure Fe monocrystal hydrogen saturated. At the same time, the HEDE model is the generally recognized and currently most developed model of hydrogen embrittlement, and the performed work allows us to obtain a recognized standard calculation algorithm, based on which we can discuss specific details and create models more appropriate to experimental data. After comparing the results with the dependences presented earlier in other works, we can conclude that most authors do not pay due attention to strict adherence to the whole algorithm of the HEDE model and admit inaccuracies in their calculations, which leads to quantitatively incorrect results and depreciate performed research. In addition, it is important to pay attention to the fact that the lack of reliable data on the distribution of hydrogen concentration makes it impossible to unambiguously determine all the parameters of the model under consideration. On the one hand, this allows fitting to any experimental data, but on the other hand, it reduces the predictive engineering value of the model, since in the engineering calculation for strength, only qualitative compliance is not enough.

5 Conclusions In the framework of this work, we gave an overview of one of the basic models of mechanics, which is used to describe the effects of hydrogen embrittlement – the HEDE model. We studied the features of applying this approach to the description of the effect of hydrogen on the mechanical properties of materials on a model problem. For its implementation, we compiled a program in Microsoft Visual Studio (VS), written in C++, which allowed us to “assemble” a model ourselves, adding to it all the equations describing the process under study. We performed a numerical simulation of a standard

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crack growth problem based on the HEDE approach and established the dependences of the initiation time of cracking on the value of the initial hydrogen concentration. Fully immersed into the details of numerical implementation mechanism we can conclude that there are many fitting parameters in the model the main one of them is the local concentration of hydrogen at the crack tip. All methods for measuring local concentrations are indirect. Even in the case of applying subtle physical methods, mechanical surface preparation is required, during which it is impossible to maintain the natural, initial concentration of hydrogen. Therefore, the next stage of refinement and development of hydrogen embrittlement models will inevitably require new experimental results, development of methods for measuring the local concentration of hydrogen dissolved in metals, both in model specimens and in parts of real constructions. The combination of these factors allows us to conclude that it is impossible to apply the HEDE model in its present form to the numerical solution of real problems of engineering practice. Today, it cannot be fully used to describe and simulate the effect of hydrogen on the mechanical properties of materials. Therefore, the task of creating a model of such behavior remains relevant and requires further research. The algorithm, that we developed, can be used in the future, both for testing new models and for developing approaches to solving such problems of the stress-strain state of a solid saturated with hydrogen. Acknowledgments. This paper is based on research carried out with the financial support of the grant of the Russian Science Foundation (project no. 18-19-00160).

References 1. Kolaqev, B.A.: Bodopodna xpypkoct metallov. M.: Metallypgi, pp. 216 (1985) 2. Troiano, A.R.: The role of hydrogen and other interstitials in the mechanical behavior of metals. Trans. ASM 52, 54–80 (1960) 3. Shewmon, P.G.: Diffusion in Solids, pp. 200. Mc Graw-Hill Book Co, Inc., N. Y. (1963) 4. McLean, D.: Grain Boundaries in Metals, pp. 346. Clarendon Press, GB (1957) 5. Serebrinsky, S., Carter, E.A., Ortiz, M.: A quantum–mechanically informed continuum model of hydrogen embrittlement. J. Mech. Phys. Solids 52(10), 2403–2430 (2004) 6. Tvergaard, V., Hutchinson, J.W.: The relation between crack growth resistance and fracture process parameters in elastic–plastic solids. J. Mech. Phys. Solids 40(6), 1377–1397 (1992) 7. Olden, V., Alvaro, A., Akselsen, O.M.: Hydrogen diffusion and hydrogen influenced critical stress intensity in an API X70 pipeline steel welded joint–experiments and FE simulations. Int. J. Hydrogen Energy 37(15), 11474–11486 (2012) 8. Needleman, A.A.: Continuum model for void nucleation by inclusion debonding. Appl. Mech. ASME 54(3), 525–531 (1987) 9. Nguyen, O., Ortiz, M.: Coarse-graining and renormalization of atomistic binding relations and universal macroscopic cohesive behavior. J. Mech. Phys. Solids 50(8), 1727–1741 (2002) 10. Wang, Y., et al.: Prediction on initiation of hydrogen-induced delayed cracking in highstrength steel based on cohesive zone modeling. In: ASME Pressure Vessels and Piping Conference 6B (2014) 11. Guedes, D., et al.: The role of plasticity and hydrogen flux in the fracture of a tempered martensitic steel: a new design of mechanical test until fracture to separate the influence of mobile from deeply trapped hydrogen. Acta Mater. 186, 133–148 (2020)

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12. Rimoli, J.J., Ortiz, M.: A three-dimensional multiscale model of intergranular hydrogenassisted cracking. Phil. Mag. 90(21), 2939–2963 (2010) 13. Sun, Z., Benabou, L.: Modélisation de la fragilisation dynamique d’un matériau polycristallin sous traction monotone. CSMA2013 (2013) 14. Benabou, L.: Coupled stress-diffusion modelling of grain boundary segregation and dynamic embrittlement in a copper alloy. Model. Simul. Mat. Sci. Eng. 27(4), 045007 (2019). https:// doi.org/10.1088/1361-651X/ab1624 15. Olden, V., Thaulow, C., Johnsen, R.: Modelling of hydrogen diffusion and hydrogen induced cracking in supermartensitic and duplex stainless steels. Mater. Des. 29(10), 1934–1948 (2008) 16. Wang, Y.F., et al.: Failure analysis of pre-stressed high strength steel bars used in a wind turbine foundation: experimental and FE simulation. Mater. Corros. 67(4), 406–419 (2016)

Nonlinear and Multibody Dynamics, Chaos and Vibration

Approximating Unstable Operation Speeds of Automatic Ball Balancers Based on Design Parameters Lars Spannan(B)

and Elmar Woschke

Otto von Guericke University, 39106 Magdeburg, Germany {lars.spannan,elmar.woschke}@ovgu.de

Abstract. Automatic balancers present a modular possibility to counteract variable rotor unbalances during operation. Two or more balancing masses, usually spheres, can orbit in a fluid-filled annular cavity whose axis of symmetry coincides with the rotor axis. At supercritical speeds, the masses – driven by the rotor deflection – tend towards stationary positions inside the cavity opposing the primary rotor unbalance. Related to the phenomenon of rotating shafts being captured at resonances due to insufficient drive power, automatic ball balancers inhibit operation speed bands with non-synchronous vibrations where the rotor surpassed the resonance, but the balls continue to orbit with the eigenfrequency with respect to the inertial system. As a result, the balancing masses do not take stationary positions inside the cavity and the rotor is excited not only by the primary unbalance but also by the sub-synchronously orbiting balancing masses. The width of the operation speed band exhibiting non-synchronous behaviour depends on the balancing masses, the orbit radius, external damping of the rotor and viscous damping of the balls due to the fluid inside the cavity. For a planar oscillator in isotropic supports with a balancer containing two balancing balls, an explicit correlation between the stability border and the fluid damping is presented. In order to parameterize the fluid damping model, the drag on spheres in annular cavities is examined and a proposed relation based on the cavity geometry and the fluid properties is presented. Keywords: Automatic balancing Sub-synchronous vibration

1

· Stability · Drag model ·

Introduction

Rotating machinery can encounter variable mass distribution during operation, leading to unbalance excitation, which cannot be addressed by conventional balancing at the beginning of the product life cycle. Examples of aforementioned systems are washing machines, centrifuges for medical research, optical disc drives with varying content as well as grinding machines whose discs are worn unevenly. To reduce variable unbalance excitations, automatic ball balancers (ABB) can be equipped to the rotor, counteracting the uneven mass distribution with c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 223–233, 2022. https://doi.org/10.1007/978-3-030-92144-6_17

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balancing balls optical system

ABB unit

disc loading tray

Fig. 1. Partially disassembled DVD writer with ABB unit (pried open) to counterbalance variable disc unbalances.

freely movable masses (balls) inside an annular cavity whose symmetry axis coincides with the rotor axis, see Fig. 1 for an example application in an optical disc drive. Driven by inertial effects resulting from the common deflection of the rotor and the ABB, the balancing balls move towards the orbital position with maximum distance to the axis of rotation. Resulting from this passive functional principal, ABBs only achieve the desired effect of balancing at super-critical operation, where the rotor displacement and the unbalance excitation have undergone the typical phase shift when passing through resonance. In addition to the critical speed, which has to be surpassed for the intended operation of ABBs, a speed domain of unstable operation, which prevents the desired positioning of the balls just above the critical speed, has to be surpassed as well. The extent of this domain depends on the design parameters of the rotating system, in particular external damping and the selected fluid1 with which the cavity is filled and is environing the balancing balls. In the presented study, an explicit calculation of the unstable speed range for isotropically supported planar rotor systems with one ABB unit incorporating two balancing balls is derived. Section 2 describes the equations of motion of the planar system and Sect. 3 recapitulates the transformations used to gain a generalized eigenvalue problem whose frequency-dependent eigenvalues provide information about the stability of the operation, as presented in [1]. In Sect. 4 the Routh-Hurwitz criterion is applied to the eigenvalue problem and with the assumption of isotropic support of the rotor, an explicit relation between the viscous damping of the fluid, the external damping, the eigenfrequency of the rotor system and the balancing ball mass and orbit radius is deducted. Finally, a conclusion and outlook is given in Sect. 5.

1

Air is used in the most rudimentary design, which has low viscous damping.

Approximating Unstable Operation Speeds of Automatic Ball Balancers

2

225

Equations of Motion

Let m and J be the mass and mass moment of inertia of a planar rotor in the x–y plane, Fig. 2, excited by the primary unbalance mass m0 located ε0 away from the center of mass OM of the balanced rotor. The rotor is mounted orthotropically described by linear stiffnesses kx and ky and viscous damping coefficients bx and by , respectively. The rotation angle θ of the rotor is indicated by the position of the primary unbalance and describes the orientation of the rotor fixed coordinate system ξ–η. Due to manufacturing and assembly tolerances, the raceway of the balancing balls may encounter an eccentric geometric centre OG , described by angle γ and eccentricity E. The balancing balls of mass mb and radius r can move inside the cavity on an orbit with radius ε. The rotor-fixed orbital position of ball i with respect to the primary unbalance is described by ϕi . Considering an ABB with two identical balls sharing the same orbit, the potential and kinetic energy  1 kx x2 + ky y 2 2   2   2   mb x˙ 2i + y˙ i2 + Jb κ˙ 2i 1  2 2 2 2 ˙ m x˙ + y˙ + J θ + m0 x˙ 0 + y˙ 0 + T = 2 2 i=1

U=

(1) (2)

can be formulated. Inserting the no-slip rolling condition, the coordinates of the centres of mass can be expressed in terms of x, y, θ and ϕi x0 = x + ε0 cos(θ), y0 = y + ε0 sin(θ), ε κ i = θ − ϕi . r

xi = x + ε cos(θ + ϕi ) + E cos(θ + γ), yi = y + ε sin(θ + ϕi ) + E sin(θ + γ),

In accordance with [2–4] and others, the viscous effect on the balls due to the fluid are modelled in a linear manner by the torque MD = −β ϕ˙ i .

(3)

In combination with the translatoric damping forces and an external damping torque Fbx = −bx x, ˙

Fby = −by y, ˙

˙ Mβ0 = −β0 θ,

evaluation of the Euler-Lagrange equations yields, with M = m+m0 + a system of five differential equations

2 i=1

mi ,

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G

m0

ϕi

E γ

θ

M

y

ξ

mb , J b

ε0

m0

kx x

R

m, J by

ky

Fig. 2. Translational oscillator with primary unbalance and one counterbalancing mass of a balancer with eccentricity.

Mx ¨ + bx x˙ + kx x  = m0 ε0 sin (θ) +

2 

+ m0 ε0 cos (θ) θ˙ + +

i=1

Emb sin (γ + θ) + mb ε sin (θ + ϕi ) θ¨

i=1



2 







2 

  ˙ ˙ Emb cos (γ + θ) θ + mb ε θ + ϕ˙ i cos (θ + ϕi ) θ˙

i=1

mb ε θ˙ + ϕ˙ i cos (θ + ϕi ) ϕ˙ i + mb ε sin (θ + ϕi ) ϕ¨i

(4)

Approximating Unstable Operation Speeds of Automatic Ball Balancers

M y¨ + by y˙ + ky y  = − m0 ε0 cos (θ) +

2 

+ m0 ε0 sin (θ) θ˙ + +

Emb cos (γ + θ) + mb ε cos (θ + ϕi ) θ¨

i=1



2 





227

2 

  ˙ ˙ Emb sin (γ + θ) θ + mb ε θ + ϕ˙ i sin (θ + ϕi ) θ˙

i=1

mb ε θ˙ + ϕ˙ i sin (θ + ϕi ) ϕ˙ i − mb ε cos (θ + ϕi ) ϕ¨i

(5)

i=1

 J+

m0 ε20

+ β0 +

2 



2Emb ε sin (γ − ϕi ) ϕ˙ i

θ˙ = M0

2 

mb E sin (γ + θ) + mb ε sin (θ + ϕi ) x ¨

i=1

 − m0 ε0 cos (θ) + 2



Jb + mb E + mb ε + 2Emb ε cos (γ − ϕi ) θ¨ 2

i=1

+ m0 ε0 sin (θ) +

−

2

i=1

 

+

2 

2 

Emb cos (γ + θ) + mb ε cos (θ + ϕi ) y¨

i=1

Emb ε cos (γ − ϕi ) −

i=1

 Jb ε + mb ε2 ϕ¨i + Emb ε sin (γ − ϕi ) ϕ˙ 2i r

  Jb ε Jb ε2 2 2 ¨ − Emb ε cos (γ − ϕi ) − mb ε θ + mb ε ϕ¨i + β ϕ˙ i = r2 r   + mb ε E sin (γ − ϕi ) θ˙2 + sin (θ + ϕi ) x ¨ − cos (θ + ϕi ) y¨ ∀i ∈ {1, 2}.

(6)



2.1

(7)

Identification of the Viscous Drag Coefficient

Assuming a quasi steady state with the fluid flow velocity vflow equal to the speed of the rotor and small orbit velocities ϕ˙ i of the balls with respect to the rotor, a Stokes flow regime can be assumed, leading to a reciprocal relation between the Reynolds number Re =

2ϕ˙ i εr 2vflow r ≈ , νfluid νfluid

(8)

with νfluid being the kinematic viscosity of the fluid, and the resulting drag coefficient cd . Therefore, the product cd Re can be represented by a scalar, which

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depends on the geometry of the annular cavity and the relative ball size. Inserting the drag force FD =

1 2 ρfluid πr2 cd vflow sign(vflow ), 2

(9)

with ρfluid being the fluid density, into Eq. (3) leads to π β = cd Re ρfluid νfluid rε2 . 4

(10)

An appropriate value for cd Re can be determined empirically, see [5,6], or by CFD analysis for the specific geometry of the annular cavity and the balancing balls. One suitable estimation results from the widely studied drag forces on a sphere in contact with a flat wall, e.g. cd Re = 322, which was identified empirically in [7].

3

Condition for Stable Balancing

Following the procedure presented in [1], Eqs. (4), (5) and (7), evaluated for i = {1, 2}, can be transformed utilizing the Krylov-Bogoliubov-Mitropolsky method [8]. The coordinates x and y are substituted by x=x = Ax cos(θ) − Bx sin(θ), ˙x ˙ ˙ x cos(θ), = −θAx sin(θ) − θB ¨ = ∂ x x ˙ /∂t,

y = y = Ay sin(θ) + By cos(θ), ˙ y cos(θ) − θB ˙ y sin(θ), (11) y ˙ = θA y¨ = ∂ y ˙ /∂t,

yielding the additional conditions ˙ cos(θt), ˙ A˙ x = B˙ x sin(θt)/

˙ cos(θt). ˙ B˙ y = −A˙ y sin(θt)/

(12)

Considering the steady state, i.e. θ˙ = ω, θ¨ = 0, ϕ¨1 = 0 and ϕ¨2 = 0, a system of six ordinary differential equations (ODEs) is formed in terms of Ax , Bx , Ay , By , ϕ1 and ϕ2 . Approximation of the ODEs by their integral mean over one revolution removes the trigonometric terms in ωt. Following a linear perturbation analysis, i.e. substituting Ax = A0x + ΔAx , By = B0y + ΔBx ,

Bx = B0x + ΔBx , ϕ1 = ϕ01 + Δϕ1 ,

Ay = A0y + ΔAy , ϕ2 = ϕ02 + Δϕ2 ,

expanding the trigonometric terms in Δϕ1 and Δϕ2 and neglecting terms of higher order, the system of linear ODEs of first order A z˙ + B z = 0 with z = T [ΔAx , ΔBx , ΔAy , ΔBy , Δϕ1 , Δϕ2 ] , ⎡

A=

mb ε sin(ϕ01 ) M 01 ) ⎢ − mb ε cos(ϕ M ⎢ mb ε sin(ϕ01 ) ⎢ ⎢ M ⎢ 01 ) − mb ε cos(ϕ ⎢ M ⎢ ω cos(ϕ01 ) ω sin(ϕ01 ) ω cos(ϕ01 ) ω sin(ϕ01 ) β ⎣ 2 2 2 2 mb ε ω cos(ϕ02 ) ω sin(ϕ02 ) ω cos(ϕ02 ) ω sin(ϕ02 ) 0 2 2 2 2

−1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

mb ε sin(ϕ02 ) ⎤ M 02 ) ⎥ − mb ε cos(ϕ M ⎥ mb ε sin(ϕ02 ) ⎥ ⎥ M 02 ) ⎥ , − mb ε cos(ϕ ⎥ M ⎥

0

β mb ε

⎦

Approximating Unstable Operation Speeds of Automatic Ball Balancers

and

⎡

kx ω − bx − 2M 0 0 2 ω ⎢ ω 2Mkx bx − 2M 0 0 ⎢− 2 + 2M ω ⎢ by ky ω ⎢ 0 0 − 2M − 2 2M ω B=⎢ ky by ω ⎢ 0 − 2 + 2M ω − 2M ⎢ 2 0 ⎢ ω sin(ϕ01 ) ω 2 cos(ϕ01 ) ω 2 sin(ϕ01 ) ω 2 cos(ϕ01 ) − − ⎣ 2 2 2 2 2 2 ω 2 sin(ϕ02 ) ω 2 sin(ϕ02 ) 02 ) 02 ) − ω cos(ϕ − ω cos(ϕ 2 2 2 2

mb ωε cos(ϕ01 ) 2M mb ωε sin(ϕ01 ) 2M mb ωε cos(ϕ01 ) 2M mb ωε sin(ϕ01 ) 2M

0 0

229

⎤

mb ωε cos(ϕ02 ) 2M mb ωε sin(ϕ02 ) ⎥ ⎥ 2M mb ωε cos(ϕ02 ) ⎥ ⎥ 2M ⎥ mb ωε sin(ϕ02 ) ⎥ 2M ⎥

0 0

is obtained. By solving the eigenvalue problem of the system   B + λA v = 0

⎥ ⎦

(13)

of order six and determining if all eigenvalues λ have negative real parts (λ) < 0 ∀ λ ,

(14)

the stability of the system can be obtained for a given angular frequency ω of the rotor. However, numerical methods show inaccuracies when ϕ01 = −ϕ02 = π/2 or ϕ01 = −ϕ02 = π, i.e. when the ABB eccentricity is zero and the primary unbalance is very low or very high, respectively. This follows from the resulting poor condition of the matrices. Alternatively, the sixth order characteristic polynomial of Eq. (13) can be tested with the Routh-Hurwitz stability criterion [9]. With the variation of one design parameter of the ABB, a stability chart can be obtained, depicting the stable operating speeds, see Fig. 3. Since the choice of fluid is barely subject to boundary conditions, the variation of the fluid properties is well suited to influence the behaviour of the ABB unit. Hence, the stability maps are presented in the β–ω space. As can be seen from the graphs, stable balancing can only be achieved above the natural frequency if the viscous parameter of the liquid is sufficiently high. In the case of orthotropic supports, the stability area is divided in two regions, as only forward whirling motions of the rotor can be balanced [10].

a)

b)

Fig. 3. Stability chart examples for variable viscous parameter β for the case of (a) isotropic and (b) orthotropic (kx < ky ) supports.

230

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L. Spannan and E. Woschke

Explicit Stability Border for the Isotropic Eccentricity Free Case

For a given rotor and ABB configuration with two balls, the stability chart can be readily obtained as presented above. Assuming isotropic supports, i.e. kx = ky = k, bx = by = b and therefore k/M ≡ ω02 , and no eccentricity of the orbit raceway, the balancing balls will be positioned symmetrically in order to counterbalance the primary unbalance, resulting in ϕ02 = −ϕ01 ,

π/2 ≤ ϕ01 ≤ π .

(15)

Simplifying the matrices in Eq. (13) under the assumptions stated above, the characteristic equation can be written in a factorised form

  2 2 0 = 16M 4 (mb ε)2 ω 4 M 2 ω 2 − ω02 + ω 2 (2M λ + b)

 2  2  4(mb ε)4 λ2 ω 6 + 2λ2 ω 2 + ω 4 sin (ϕ01 ) cos2 (ϕ01 )      2  + βλ 2(mb ε)2 ω 4 M ω 2 − ω02 + 3λb + M 2 βλ ω 2 − ω02      (16) +4M λ2 ω 2 (mb ε)2 2ω 2 + ω02 + β (M λ + b) + βλω 2 b2 . From the first term in brackets follow the roots b ω2 − ω2 b ω2 + ω2 +i 0 ∨ λ=− +i 0 (17) 2M 2ω 2M 2ω with negative real parts. The second term in brackets is a fourth order polynomial a4 λ4 + a3 λ3 + a2 λ2 + a1 λ + a0 with the coefficients being rewritable as   a4 = 4ω 2 4(mb ε)4 ω 2 sin2 (ϕ01 ) cos2 (ϕ01 ) + M 2 β 2 ,   a3 = 4M βω 2 2(mb ε)2 ω 2 + (mb ε)2 ω02 + βb ,  2 a2 = M 2 β 2 ω 2 − ω02   (18) + ω 2 20(mb ε)4 ω 4 sin2 (ϕ01 ) cos2 (ϕ01 ) +6(mb ε)2 βω 2 b + β 2 b2 , λ=−

a1 = 2(mb ε)2 M βω 4 (ω − ω0 ) (ω + ω0 ) , a0 = 4(mb ε)4 ω 8 sin2 (ϕ01 ) cos2 (ϕ01 ). Since a4 > 0, it is necessary and sufficient [9] that a0 > 0, a1 > 0, a2 > 0, a3 > 0 and (19) 0 < a1 (a3 a2 − a4 a1 ) − a0 a23 . From a3 > 0 follows the familiar condition ω > ω0 for automatic balancing at super critical speeds. Differentiation of the right hand side of Eq. (19) with respect to ϕ01 yields d a1 (a3 a2 − a4 a1 ) − a0 a23 = − p sin(4ϕ01 ) dϕ01 d2 a1 (a3 a2 − a4 a1 ) − a0 a23 = − 4p cos(4ϕ01 ) dϕ201

(20) (21)

Approximating Unstable Operation Speeds of Automatic Ball Balancers

231

with    0 < p = 16(mb ε)4 M 2 β 2 ω 12 βb 2βb + 3(mb ε)2 ω 2   + 9(mb ε)2 ω02 βb + (mb ε)2 (ω 2 + ω02 ) . The roots of Eq. (20) deliver possible extrema of the stability criterion at ϕ01 = jπ/4, j ∈ {2, 3, 4}, depicted in Fig. 4. Evaluation of Eq. (21) shows that ϕ01 = 3π/4 is a minimum of the criterion, therefore the stability of the system is guaranteed if Eq. (19) holds for ϕ01 = 3π/4, which reads after factorization   0

(mb ε)2 ω02 ω 2 + ω02 . b ω 2 − ω02

(22)

From Eq. (22) the limit βmin = lim β = ω→∞

(mb ε)2 ω02 b

(23)

π π

Fig. 4. Balancing ball positions resulting in extrema of the stability criterion. Primary unbalance magnitude: (a) Medium (b) low (c) high.

can be derived, describing a minimum required value for the viscous drag parameter β, in order to be able to pass the unstable operation of the ABB, see Fig. 5. With Eq. (10) these requirements can be expressed as a minimum dynamic fluid viscosity. The assumptions of absence of raceway eccentricity and specific orbit positions of the counterbalancing masses are justified by the fact that these values are not subject to design, but are caused by manufacturing inaccuracies and the variable magnitude of the primary unbalance inherent to the application. Furthermore, sensitivity analyses of the stability considerations using Eq. (13) show only marginal influences on the stability border.

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Fig. 5. Stability chart for an ABB in isotropic supports showing the minimum viscous parameter value required to surpass unstable operation

5

Conclusion and Outlook

The problem of sub synchronous behaviour may be a minor issue in the design of ABB units. Since rotating machinery, which run above the first critical frequency, are operated with sufficient clearance to avoid substantial unbalance excitation. The frequency band of unstable behaviour, as shown in the stability maps, is often avoided due to already existing constraints concerning the operation speed. Nevertheless, the presented explicit determination of the stability border for ABB units in isotropic supports can simplify the estimation of stability problems significantly. An interesting result emerges from the minimum fluid viscosity required to surpass the sub synchronous behaviour at all. This is most relevant, when air filled ABB units are utilized in order to avoid the design of sealing and filling procedures. Due to the low viscosity, ABB designs with air filled cavities are most prone to unstable behaviour if external damping is insufficient. However, the presented model neglects the dissipative rolling friction between the balls and the raceway. This damping behaviour leads to stabilization and thus smaller areas of instability. In addition, no suitable factorization of the characteristic equation of the orthotropically supported ABB system was found, which would allow an analogous evaluation of the criterion to derive explicit representations for the stability limits. Notes. A software repository providing stability charts based on Eq. (13) is openly available in Zenodo at https://doi.org/10.5281/zenodo.3894373.

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References 1. Bykov, V.G., Kovachev, A.S.: Autobalancing of a rigid rotor in viscoelastic orthotropic supports considering eccentricity of the automatic ball balancer. AIP Conference Proceedings 1959(1):080,011, 10/gf3s79 (2018) 2. Sharp, R.: An analysis of a self-balancing system for rigid rotors. J. Mech. Eng. Sci. 17(4), 186–189 (1975). 10/bzcp8d 3. Lee J, Van Moorhem, W.: Analytical and experimental analysis of a selfcompensating dynamic balancer in a rotating mechanism. J. Dyn. Syst. Measur. Control 118(3), 468–475 (1996). 10/fg8sdj 4. Sperling, L., Merten, F., Duckstein, H.: Self-synchronization and automatic balancing in rotor dynamics. Int. J. Rotating Mach. 6(4), 275–285 (2000). 10/d25fj2 5. Spannan, L., Daniel, C., Woschke, E.: Experimental study on the velocity dependent drag coefficient and friction in an automatic ball balancer. Technische Mechanik 37(1), 62–68 (2017). 10/gf3s8h 6. Spannan, L., Daniel, C., Woschke, E.: Simulation of the ball kinetic in ball-type automatic balancing devices by solving the axisymmetric navier-stokes equations in annular cavities. In: Cavalca, K., Weber, H. (eds.) Proceedings of the 10th International Conference on Rotor Dynamics - IFToMM, vol. 63, pp. 109–118. Springer, Cham (2018). 10/gf3s8f 7. Jan, C.D., Chen, J.C.: Movements of a sphere rolling down an inclined plane. J. Hydraulic Res. 35(5), 689–706 (1997). 10/bp4x9v 8. Kryloff, N., Bogoliubov, N.: Introduction to non-linear mechanics. Annals of Mathematics Studies, Princeton University Press (1947). English translation 9. Popov, E.P.: The Dynamics of Automatic Control Systems. Pergamon Press (1962) 10. Ryzhik, B., Sperling, L., Duckstein, H.: Auto-balancing of anisotropically supported rigid rotors. Tech. Mech. 24(1), 37–50 (2004)

Shape Control and Modal Control Strategies for Active Vibration Suppression of a Cantilever Beam Aleksandr V. Fedotov(B) Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, V.O., Bolshoj pr. 61, 199178 Saint Petersburg, Russia [email protected]

Abstract. The study is devoted to the numerical simulation and comparison of two different control strategies, shape control and modal control, applied to the problem of suppression of forced bending vibrations of a thin metal cantilever beam at the first and the second resonance frequencies. The shape control strategy is based on the compensation of known distribution of the external disturbance in the static case, while the modal control strategy implies the correspondence between the control loops and the vibration modes of the object. The results show that the modal system can be efficient at both resonance frequencies. The shape control strategy provides efficient vibration suppression only at the first resonance, while at the second resonance frequency it is significantly less effective than the modal approach. Therefore, the modal method is preferable to the shape control method in the cases where it is necessary to suppress forced vibrations at several resonance frequencies of the object. Keywords: Active vibration suppression · Shape control · Modal control · Piezoelectric sensors and actuators

1 Introduction The present paper is devoted to the problem of vibration suppression of continuous systems, which is widespread in various fields of technology. These systems formally do not possess the properties of controllability and observability due to infinite number of degrees of freedom. They also tend to demonstrate resonance behavior, which in the case of low damping leads to high vibration amplitudes at the resonance frequencies and may cause the performance degradation and damage to the structure. Different passive or active systems can be used to protect the mechanical structure from the undesired vibrations [1–3]. Active control systems include feedback loops, which use sensors and actuators, and can provide the influence on the structure depending on its dynamics. There are known various strategies for organizing feedback control systems. We analyze three of them: local, modal and shape control strategies. The shape control method [4–6] is used to compensate the known distribution of the external excitation. It implies using only one feedback loop with collocated system of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 234–244, 2022. https://doi.org/10.1007/978-3-030-92144-6_18

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sensors and actuators. On the contrary, the modal system [7–9] allows one to control independently different vibration modes of the object regardless of the shape of the external excitation. This possibility is provided by using a separate feedback loop for each mode of the object to be controlled. More simple is a local method [8, 10], which implies using local connections sensor-actuator. In the local system multiple feedback loops could also be used. In order to compare the control strategies mentioned the problem of vibration suppression of a thin cantilever beam is considered. The control purpose is to suppress forced bending vibrations of the beam caused by the base excitation at the first and the second resonances. All control systems use the same number of piezoelectric sensors and actuators. Our previous investigations [8] have shown that under the considered conditions the modal method is more effective than the local one if it is needed to suppress several vibration modes of the object. Therefore, the aim of the present study is to compare the modal and the shape control methods for the above stated problem.

2 Theoretical Background 2.1 Shape Control Method The theoretical description of the shape control method for controlling the bending vibrations of Bernoulli-Euler beams using piezoelectric sensors and actuators is given in [6]. The main idea of this method is the compensation of the known distribution of the external excitation by the piezoelectric actuation: the actuation bending moment should be opposite to the statically admissible bending moment produced by the external load. The first problem is that in real cases control possibilities are usually limited: actuators used in the control system cannot fully compensate the shape of the external excitation. Therefore, it is needed to approximate the bending moment to be compensated by available actuators. For this purpose, it is suggested to use the equal-area-rule or to divide the beam into sections and compensate the deflection individually in each section by a single actuator. These variants of compensation are considered in [2]. In the present study two variants of compensation are compared: the “sections method” and the method based on the minimum deflection criterion, this methods are described below in Subsect. 4.1. Of course, all these methods are valid only if the distribution of the external excitation is constant. The second problem is that the time variation of the external load is often not known in advance, which makes necessary the use of feedback control systems. In this case, the collocated system of sensors and actuators is usually used. It means that the design of the sensor system repeats the one of the actuator system: sensors are located symmetrically with respect to actuators at the opposite side of the beam. All sensors and actuators are integrated in a single feedback loop. 2.2 Modal Control Method The modal control method, or IMSC (independent modal space control), implies the correspondence between the feedback loops and the vibration modes of the object. First formulated in 1966 [11], it was further developed in [12]. The application of this approach

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to active vibration control of continuous systems is considered in [7–9, 13]. Our previous study [8] has shown that the modal method is more effective than the local one if it is needed to suppress forced vibrations of the object at more than one eigenfrequency. The general scheme of the modal control system is shown in Fig. 1. Here y is the vector of sensor signals, which is transformed into the vector of the estimates of the generalized coordinates q˜ using the matrix T, which is called the mode analyzer. The length of the vector q˜ is equal to the number of feedback loops. In i-th feedback loop one ˜ i acting component of this vector, q˜ i , is transformed into the desired generalized force Q on the corresponding vibration mode of the beam, using the transfer function Ri (s). That means that R(s) is a diagonal matrix of the control laws, where the negative sign indicates ˜ is transformed into the vector of control the negative feedback. After that, the vector Q signals to the actuators u using the matrix F, which is called the mode synthesizer.

d u

A c t u a t o r s

S e n s o r s

Control object

F

-R(s)

y

T

Controller Fig. 1. Scheme of the modal control system

When creating the modal control system, one should specify the matrices T and F. To clarify this step, we need to introduce the matrices θ a and θ s . θ a is the excitation matrix, which shows, how strong is the influence of each actuator on each eigenmode of the object. θ s is the measurement matrix, which indicates, how strong is the influence of each eigenmode on each sensor. In order to provide the correspondence between the feedback loops and the vibration modes of the object, the following relations should be satisfied:  −1  −1 (1) F = θa , T = θs Unfortunately, the total amount of modes needed to describe the dynamics of the control object is usually greater than the number of controlled vibration modes. The problem of activation of higher, uncontrolled modes is known as a spillover effect, which can cause not only the mutual influence between the modal feedback loops, but also the instability of the closed-loop system. This effect can be minimized by enhancing the mode separation, for example, due to increasing the number of sensors and actuators.

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3 The Control Problem The scheme of the considered control problem is shown in Fig. 2. The control object is a thin cantilever beam made of aluminium with length of 50 cm and cross section of 3 × 35 mm. The beam is undergoing forced bending vibrations due to the base excitation. Small rectangular piezoelectric patches with dimensions 60 × 30 × 0.5 mm covered by electrodes on both sides are used as sensors and actuators. The sensors measure the bending deformation of the beam at specific locations, while the actuators cause this deformation.

actuators sensors

Controller

Fig. 2. Scheme of the control problem

The control purpose is to suppress forced bending vibrations of the beam in the frequency range containing the first and the second resonance frequencies. Each control system created includes two sensor-actuator pairs, where the elements of each pair are mounted to the beam symmetrically on both sides. For each control strategy the locations of sensor-actuator pairs on the beam are different. To evaluate the efficiency of the created control systems, the vibration amplitude of the upper endpoint of the beam is analyzed. This choice is caused by the fact that the vibration amplitude of this point is the biggest among all points of the beam for the vibration modes to be controlled. The solution of the control problem for each control system is computed from the frequency response functions (FRFs) of the beam obtained using finite element modeling.

4 Creating the Control Systems: Actuator and Sensor Placement 4.1 Shape Control Method The first stage of creating the control system is the placement of actuators and sensors on the beam. In this subsection we will consider the shape control method.

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In the shape control system, actuators should compensate the external excitation, which is equivalent to the distributed inertia load (Fig. 3). Here we consider the static case of loading with p0 = 1 N/m. Obviously, two discrete actuators cannot fully compensate this load. Therefore, we consider two variants of compensation (Fig. 4).

z

p1

p1

M1

M2

p0 x

Fig. 3. Cantilever beam with two sensor-actuator pairs

Beam deflection

0.02

0.01

w, mm

0

-0.01

-0.02

-0.03 "Sections method": x1 = 0.0645, x2 = 0.234, M1 = 0.52, M2 = 0.2935 -0.04

Minimum deflection: x1 = 0.05, x2 = 0.2045, M1 = 0.4351, M2 = 0.3184 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x, m

Fig. 4. Shape control: two variants of compensation of the external load

The first variant of compensation is obtained using the so-called “sections method”: the beam is divided into two sections, at the ends of which deflection and slope are zero. This method of compensation of the beam vibrations is described in [2]. For the case under consideration, the coordinates of the centers of piezopatches for this method are x 1 = 6.45 cm, x 2 = 23.4 cm, and the actuation moments are M 1 = 0.52 N·m, M 2 = 0.2935 N·m. The second method of compensation is based on the minimum deflection criterion: it means the minimization of the maximum deflection of the beam. At the free end of the beam, deflection is required to be zero, since the performance of the control system is analyzed from the vibration amplitude of this endpoint. For this method, the piezopatch coordinates and the actuation moments are the following: x 1 = 5 cm, x 2 = 20.45 cm, M 1 = 0.4351 N·m, M 2 = 0.3184 N·m. The maximum deflection of the beam for the first and the second methods of compensation is respectively 0.031 mm

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239

and 0.014 mm, while the maximum deflection for the uncontrolled case (corresponding to the tip of the free end) is 2.83 mm. The actuation moments obtained give the weighting factors for actuators and sensors used in the feedback loop of the control system. The weighting factors for the actuator and sensor systems are equal, since these systems are collocated. 4.2 Modal Control Method In the modal control system, two sensor-actuator pairs should be attached to the beam in those locations where they can most efficiently measure and affect the first and the second bending modes of the beam. These are the locations of maximum curvature of the considered eigenmodes. This curvature obtained numerically is shown in the Fig. 5. The first sensor-actuator pair is placed at the clamped end of the beam (x 1 = 3 cm), where the curvature of both modes gets maximum values, while the second pair is located approximately at the center of the beam (x 2 = 26.5 cm), where there is a local maximum of the curvature of the second bending mode. Curvature of the mode shapes

6

1st mode 2nd mode piezopatch boundaries

4

w''

2 0 -2 -4 -6

0

0.1

0.2

0.3

0.4

0.5

x, meter

Fig. 5. Curvature of the 1st and the 2nd bending modes of the beam with the piezopatch locations

After the placement of the piezoelectric patches, the matrices T and F for the modal control system (mode analyzer and synthesizer), which define the linear transformation of the measured and control signals, are specified. These matrices are calculated from the numerically obtained matrices θ a and θ s (excitation and measurement matrices) according to the Eqs. (1). The obtained values are the following:    −1 6.06 −0.41 = 100 · F = θa (2) 6.79 1.27    s −1 2.82 3.16 −5 = 10 · T= θ (3) −0.19 0.59

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5 Finite Element Modeling To compute the solution of the control problem, the frequency response functions of the beam are obtained using finite element (FE) modeling. The FE models of the beam are created using ANSYS software. The models of the beam are constructed from 3-node one-dimensional elements Beam189. The model without piezoelectric patches contains 100 elements, while the models with piezopatches include 120 elements. Harmonic analysis is performed for each model for different variants of excitation: the base vibration or the actuator excitation. For each variant, the measured values are the sensor signals and the vibration amplitude of the point at the free end of the beam. Figure 6 shows the FE model of the beam with sensors and actuators corresponding to the shape control system #2, which is created using the minimum deflection criterion, with actuator excitation.

Fig. 6. FE model of the beam with piezopatches for shape control system #2

The piezoelectric effect in this FE simulation is not modeled directly. Instead of this, for simplicity, the bending moments are applied to the end sections of the actuators, and the rotation of the end sections of the sensors is analyzed to compute the sensor signals.

6 Design of the Controller and Comparison of the Results The second stage of creating the control systems is the synthesis of the transfer functions for each feedback loop. They are designed using the loop shaping method. In order to model the delay in the feedback loop, in each transfer function the lowpass filter with the cut-off frequency of 200 Hz is included. Therefore, the gain values in each loop are limited by the occurring of instability at high resonance frequencies due to the phase shift. The transfer functions are designed to provide the best vibration suppression at the desired resonance frequencies. As an example, Eqs. (4, 5) and Figs. 7 and 8 show the transfer functions and the Bode diagrams for both loops of the modal control system. The black curves correspond to the control object, and the gray ones – to the open-loop system. Rm 1 (s) = Rm 2 (s) =

s4

2.74 · 108 s2 + 5.16 · 109 s + 1.08 · 1010 + 1513s3 + 3.35 · 105 s2 + 1.89 · 107 s + 3.6 · 109

4.73 · 1010 s2 + 5.94 · 108 s + 7.47 · 1012 s4 + 1861s3 + 1.26 · 106 s2 + 9.23 · 108 s + 3.72 · 1011

(4) (5)

magnitude, dB

Shape Control and Modal Control Strategies for Active Vibration 40 20 0 -20 -40 -60 -80

phase, °

10 180 120 60 0 -60 -120 -180 -240 10

1

1

2

10 frequency, Hz

2

10 frequency, Hz

10

10

241

3

3

magnitude, dB

Fig. 7. Bode diagram for the 1st loop of the modal control system 40 20 0 -20 -40 -60 -80 -100 -120

phase, °

10 180 120 60 0 -60 -120 -180 -240 10

1

1

2

10 frequency, Hz

2

10 frequency, Hz

10

10

3

3

Fig. 8. Bode diagram for the 2nd loop of the modal control system

The performance of all created control systems in the vicinity of the first and the second resonance frequencies is shown in Figs. 9 and 10. These figures show the vibration amplitude of the upper endpoint of the beam with and without control, which is calculated using the FRFs of the beam obtained by the FE modeling and the transfer functions in the feedback loops.

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1 resonance

0

no control shape control #1

-5

shape control #2, 1st res

-10

modal control, 1st loop modal control, both loops

w, dB

-15 -20 -25 -30 -35 -40

6

8

10

12

14

16

18

f, Hz

Fig. 9. Performance of the created control systems at the 1st resonance 2

-5

nd

resonance no control shape control #1

-10

shape control #2, 2nd res

-15

modal control, 2nd loop modal control, both loops

w, dB

-20 -25 -30 -35 -40 -45 30

40

50

60

70

80

90

100

110

120

f, Hz

Fig. 10. Performance of the created control systems at the 2nd resonance

For the modal method the results of three control systems are used: two systems with only one active feedback loop (either the first or the second), and a system with both feedback loops active. The performance of the latter system is not as efficient as in the case of using only one loop, since the gain factors in both loops of this system are reduced compared to the other cases to avoid instability at high frequencies. For the shape control system #2 two variants of transfer functions are designed: the first one is providing the most efficient performance of the system at the first resonance, and the second one – at the second resonance. For shape control system #1 only one transfer function is designed, because this system is unable to suppress the second vibration mode of the beam and does not work at the second resonance.

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It can be seen from the figures, that the modal systems efficiently suppress forced vibrations at both resonances. Shape control systems, especially the system #2, work slightly better than the modal system at the first resonance, but at the second resonance they in fact do not work at all. Therefore, the modal method is more efficient than the shape control method when it is needed to suppress several vibration modes of the object. The second conclusion is that in the framework of shape control approach the method of compensation based on the minimum deflection criterion is more effective than the “sections method”, which implies dividing the beam into sections. The numerical data on the performance of the designed control systems is summarized in Table 1. Here the gain factors in each control loop are presented as well as the difference in the vibration amplitude of the upper endpoint of the beam with and without control at both resonances. Table 1. Performance of the created control systems Control system

Gain

w1 , dB w2 , dB

Shape control #1

3.41

−22.09

1.00

Shape control #2, 1st resonance

3.08

−24.35

1.06

Shape control #2, 2nd resonance 0.763

0.10

−0.83

Modal control, 1st loop

3

−20.90

0.71

Modal control, 2nd loop

21

0.27

−20.04

Modal control, both loops

2.1 14.07 −16.85

−17.91

7 Conclusions The present study is devoted to realization and numerical comparison of shape control and modal control strategies for the problem of active suppression of forced bending vibrations of a thin cantilever beam. The purpose of the control systems created was to suppress forced vibrations of the beam at the first and the second resonances. Each control system includes two pairs of piezoelectric patches used as sensors and actuators, located in two different positions on the beam. For the shape control strategy, two variants of placement of the piezoelectric patches on the beam were considered, realizing two variants of compensation of the external excitation. The numerical modeling has shown that the modal control system designed can efficiently suppress vibration of the beam at both the first and the second resonances (the level of vibration amplitude is reduced by 17–20 dB). At the same time, the systems based on shape control demonstrate even better performance at the first resonance (reduction up to 24 dB), but at the second resonance they almost do not work at all (reduction less than 1 dB). In the framework of the shape control strategy, the variant of compensation based on minimum deflection criterion turned out to be more efficient at both resonances than the method based on dividing the beam into sections.

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Therefore, the modal control strategy is preferable to the shape control if it is needed to suppress several vibration modes of the object. This can be explained by the fact that the modal method implies using several feedback loops designed to suppress specific vibration modes of the object. On the contrary, the shape control systems include only one feedback loop, the design of which provides efficient suppression of only one mode (the first bending mode of the beam). In other words, the modal control system is more complicated than the shape control system, which determines its greater efficiency.

References 1. Fisco, N.R., Adeli, H.: Smart structures: part I—active and semi-active control. Sci. Iranica 18(3), 275–284 (2011). https://doi.org/10.1016/j.scient.2011.05.034 2. Nader, M.: Compensation of Vibrations in Smart Structures: Shape Control. Experimental Realization and Feedback Control. Trauner Verlag, Linz (2008) 3. Cruz Neto, H.J., Trindade, M.A.: On the noncollocated control of structures with optimal static output feedback: Initial conditions dependence, sensors placement, and sensitivity analysis. Struct. Control Health Monit. 26(10), e2407 (2019). https://doi.org/10.1002/stc.2407 4. Haftka, R.T., Adelman, H.M.: An analytical investigation of shape control of large space structures by applied temperatures. AIAA 23, 450–457 (1985). https://doi.org/10.2514/3. 8934 5. Irschik, H.: A review on static and dynamic shape control of structures by piezoelectric actuation. Eng. Struct. 24, 5–11 (2002). https://doi.org/10.1016/S0141-0296(01)00081-5 6. Irschik, H., Krommer, M., Pichler, U.: Dynamic shape control of beam-type structures by piezoelectric actuation and sensing. Int. J. of App. Electromag. Mech. 17, 251–258 (2003). https://doi.org/10.3233/JAE-2003-251 7. Zenz, G., Berger, W., Gerstmayr, J., Nader, M., Krommer, M.: Design of piezoelectric transducer arrays for passive and active modal control of thin plates. Smart Struct. Sys. 12(5), 547–577 (2013). https://doi.org/10.12989/sss.2013.12.5.547 8. Belyaev, A.K., Fedotov, A.V., Irschik, H., Nader, M., Polyanskiy, V.A., Smirnova, N.A.: Experimental study of local and modal approaches to active vibration control of elastic systems. Struct. Control Health Monit. 25(2), e2105 (2018). https://doi.org/10.1002/stc.2105 9. Peukert, C., Pöhlmann, P., Merx, M., Müller, J., Ihlenfeldt, S.: Investigation of local and modal based active vibration control strategies on the example of an elastic system. J. Machine Eng. 19(2), 32–45 (2019). https://doi.org/10.5604/01.3001.0013.2222 10. Udwadia, F.E., Hosseini, M., Wada, B.: Distributed control of large-scale structural systems. Comput. Aided Civil Inf. Eng. 13(6), 377–387 (1998). https://doi.org/10.1111/0885-9507. 00116 11. Gould, L.A., Murray-Lasso, M.A.: On the modal control of distributed parameter systems with distributed feedback. IEEE Trans. Autom. Control 11(4), 729–737 (1966). https://doi. org/10.1109/TAC.1966.1098463 12. Meirovitch, L.: Dynamics and Control of Structures. Wiley, New York (1990) 13. Fedotov, A.V.: Active vibration suppression of Bernoulli-Euler beam: experiment and numerical simulation. Cyber. Phys. 8(4), 228–234 (2019). https://doi.org/10.35470/2226-4116-20198-4-228-234

Axially Symmetric Oscillations of Circular Cylindrical Shell with Localized Mass on Winkler Foundation George V. Filippenko1,2(B) and Tatiana V. Zinovieva1 1 Institute for Problems in Mechanical Engineering, V.O., Bolshoj pr., 61, St. Petersburg

199178, Russia 2 St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg 199034, Russia

Abstract. The oscillations of a circular cylindrical shell of finite length with additional inertia mass are explored. The shell contacts with an elastic medium, which reaction is represented by Winkler model. Such systems can model various elements of buildings, hydro technical constructions, bridges, oil rigs, various pipes etc. The problem of shell vibrations is investigated using the theory of Kirchhoff-Love shells. The shell is loaded with concentrated mass in the form of zero width “mass belt”. The axisymmetric free harmonic oscillations of the shell are explored. In addition, a computer model using ANSYS software has been developed and the shell oscillations have been analyzed by the finite element method. At some values of the system’s parameters, its anomalous behavior occurs. The nature of the system in the vicinity of dispersion curves special points has been investigated. The influence of system parameters on these processes is analyzed using both approaches. Keywords: Cylindrical shell · Shell vibrations · Winkler foundation

1 Introduction Cylindrical shells have long been studied in connection with rich applications in engineering and construction. This is one of the most important elements in modeling acoustic waveguides, various pipelines, supports of offshore drilling rigs and other hydraulic structures [1–4]. The model of a cylindrical shell based on the classical Kirchhoff-Love theory often serves as a mathematical model describing real structures. The variant of this theory based on Lagrangian analytical mechanics was used in the work [5–12, 14]. This theory was successfully used in [9, 10] for statics calculations, in [13, 14] and [11, 12] – for investigation of wave processes and oscillations of rotation shells with arbitrary meridian. A number of authors studied the interaction between shell and the medium, which was the acoustic fluid in [13–16], and the soil, which was modeled by Winkler foundation [17–19]. Often, in the mathematical model it is necessary to take into account the inertial load on the shell in the form of massive “belts”. They can be, for example, massive stiffening ribs or ice layer around the support [20]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 245–257, 2022. https://doi.org/10.1007/978-3-030-92144-6_19

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This work is devoted to free axially symmetric vibrations of a cylindrical shell under the influence of both Winkler foundation and massive “belt” of zero width (Fig. 1a). The system oscillations at different combinations of belt mass and base rigidity were studied. The statement of the problem is similar to considered in [17, 18]. The second part of the paper considers the statement of the problem for KirchhoffLove shell theory. In the third part, the scheme of constructing the exact analytical solution (the vibration field) of this problem and, on its basis, the dispersion equation is represented. On the basis of the formulas in the fourth part, the corresponding vibrational fields and dispersion curves are investigated numerically. In order to verify the obtained results, alternative calculations of the natural frequencies and shapes of the shell are used in the work by means of the finite element method in commercial software product ANSYS. A good agreement of both approaches is obtained. Practically useful conclusions on the strength of the structure are made; in particular, it is shown that at some parameters of the system it is possible to create the oscillations mode, which can be dangerous for the structure strength.

2 Statement of the Problem Let us consider the free vibrations of an infinitely extended circular cylindrical shell of Kirchhoff – Love type. The cylindrical coordinate system (r, ϕ, z) and local coordinate system (t, k, n) are introduced (Fig. 1b). Here the axis z coincides with the cylinder axis, the vectors t, n are respectively the tangent and normal unit vectors to the shell and the vector k is the unit vector along the axis z. The displacement vector u = (ut , uz , un )T (T – is the sign of the transpose operation) describes the vibrational field in the shell. Only axisymmetric oscillations will be considered below, therefore ϕ will be omitted in the function arguments.

Fig. 1. Physical model (a) and coordinate systems (b)

Similar to Winkler foundation for the plates, Winkler force f w can acts on the boundary of the shell (reaction of Winkler foundation – “Winkler springs” Fig. 1a) f w = (Kt ut , Kz uz , Kn un );

−∞ < z < +∞,

(1)

where Kt,z,n are the coefficients of elasticity of Winkler foundation in the corresponding direction.

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    The following notations are introduced: cs = Eh/ 1 − ν 2 ρ is the velocity of deformation waves propagation of the middle surface of the cylindrical shell; E, ν and ρs , is Young’s modulus, Poisson’s ratio and volumetric density of the shell material, respectively, h is the thickness of the shell, ρ = ρs h is the surface density, R is the radius of the cylindrical shell. The dependence of all processes on time is assumed to be harmonic with frequency ω and the time factor exp{−iω t} is omitted everywhere. In addition, the dimensionless parameters are introduced: the dimensionless frequency w = ωR/cs and the parameter α 2 = (1/12)(h/R)2 characterizing the relative thickness of the shell. In the axisymmetric case, the balance of the forces acting on the cylinder (taking into account Winkler force as in (1)) can be written in the form (−∞ < z < +∞)   L − K + w2 (I + l Mδ(z)) u = 0, −l/2 < z < l/2; (2) where ⎛

⎞ w2 + d 2 ∂z2 0 0 ⎠, L=⎝ ν ∂z 0 w2 + ∂z2 2 2 2 4 0 −ν ∂z w − α (1 − 2ν∂z + ∂z ) − 1

I = diag{1, 1, 1},

 1 − ν2 R diag{Kt , Kz , Kn }, M = diag{Mt , Mz , Mn }, K = diag Kt0 , Kz0 , Kn0 = h/R E   ∂ μt,z,n d 2 = 1 + 4α 2 ν− , ∂z = R , Mt,z,n = . ∂z ρlR μ

2π Rμ

t,z,n t,z,n Here Mt,z,n = ρlR = 2π R2 lhρ is the ratio of the corresponding inertial mass of the s whole “belt” to the mass of the shell without it.

3 Getting the View for the Vibration Field The method of solution is similar to the one considered in [20]. The solution of Eq. (1) on the Left and on the Right of the line z = 0 (the place of application of concentrated mass) is searched in the form: (ut , uz , un )T = Aeiλz (ς, ξ, γ )T .

(3)

Here A, ς, ξ, γ are the arbitrary constants; z and λ are the longitudinal coordinate and the required wave number which are considered to be dimensionless to the radius R of the shell (λ := λR, z := z/R). As a result we get two homogeneous algebraic systems of the species: Lλ x = 0,

(4)

where x is the eigenvector (it is normalized to the unit length) and matrix Lλ is the Fourier image of the operator L − K + w2 I in (2).

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The condition of existence of a nontrivial solution of the system (3) leads to the equation of the species det Lλ (w, λ) = 0, which roots can be written in the form λj = λj (w), j = 1, . . . , 6. After these roots substitution in the system (3) eighen vectors x j of this system can be found and, with the accuracy to six undefined constants, the solutions on the left and right of the line z = 0 can be found after it. Then we satisfy the conditions at both ends of the shell and sew the solutions on the line z = 0. The condition of the existence of a nontrivial solution obtained in this way leads to the equation of the species det A(w, λ(w), M, K) = 0,

(5)

whose solution is dependence w = w(M, K). The corresponding curves, in the future, will be called “branches of the dispersion curve” or simply “branches”. After substitution of the corresponding roots wm , m = 1, 2, . . . in the system, we find the solution decomposition coefficients Aj , j = 1, . . . , 6 (3) and, consequently, the type of vibration field in the system. Let us restrict ourselves to consideration of the shell motions that do not contain a rotational component ut . In this case, the problem (1) is reduced to finding the vectors of displacements (uz , un ) and generalized forces (fz , fn ) [20].

4 Numerical Results To calculate the presented graphical dependencies in FORTRAN [21], the nondimensional parameters of the shell are taken as follows: ν = 0.3, h/R = 0.01, l = 4. In addition, for calculations in ANSYS package, the following dimensional shell parameters are taken: R = 1 m, E = 2.1 · 1011 N/m2 , ρs = 7850 kg/m3 . Taking into account the fact that only axially symmetric oscillations are considered, and they do not contain a rotational component, a reduced mass matrix M = diag{Mz , Mn } is introduced, where either the component Mn ≡ M and Mz = 0 or Mz ≡ M and Mn = 0. The corresponding oscillations of the shell will be denoted by the letters “N” and “Z”. If the both components are present, this type of shell vibration will be denoted by the abbreviation  “ZN”. Similarly, a reduced stiffness matrix of Winkler

foundation K = diag Kz0 , Kn0 is introduced. Three value sets of this matrix components are considered: {Kz = 0, Kn ≡ K}, {Kz ≡ K, Kn = 0} and {Kz = 0.7K, Kn ≡ K}. The ratio between the stiffness matrix components shown in the last example is often used by engineers when designing specific structures. For illustration, two types of dependencies will be considered. The first type is the dependencies of dimensional frequency f (Hz) both on Winkler foundation stiffness K (kN/m3 ) (dispersion curves in Figs. 3 and 4) and on dimensionless concentrated mass M (dispersion curves in Fig. 6). For the dependencies of the first type, the branches starting from M = 0 with sequentially increasing frequencies are marked with numbers 1, 2,… respectively, and the branches themselves are called “first”, “second”, etc. for convenient. The second type is the dependence of components of the dimensionless displacement vectors and forces from the longitudinal coordinate z (profiles of real parts of these components are demonstrated in Figs. 5, 7, 8, 9 and 10). Note, that the displacement

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vector (uz , un ) is normalized to the largest module of vector components in the given range −l/2 < z < l/2. The vector of forces (fz , fn ) is normalized in the same way. Calculations of the dispersion curves were controlled by calculations in ANSYS package (finite element method) in the reference points M = 0.0, 0.1, 1.0, 10.0. The axially symmetric element SHELL209 was used for the shell modeling. Concentrated mass was determined by the specification of inertial characteristics of element MASS21. At first, let us consider the influence of Winkler foundation stiffness on the eighenfrequencies of the shell. In this case, Eq. (1) is reduced to the form    −l/2 < z < l/2 (6) L + w2 I − K/w2 u = 0, Note that the increase of Winkler foundation rigidity leads to a general reduction of   the attached mass I − K/w2 of the system shell – Winkler foundation (6) and, thus, to an increase in its eighenfrequencies. In this case, the influence of Winkler foundation decreases with increasing branch number (which corresponds to higher frequencies: lim K/w2 = 0). w→∞ In the following, we will consider the first two eigen frequencies, which feel the influence of Winkler foundation rigidity most acutely. The first one corresponds mainly to longitudinal (Fig. 2a) and the second one to bending oscillations (Fig. 2b).

(a) First mode K Z = 100.0 E 6, K N = 0

(b) Second mode K Z = 0, K N = 100.0 E 6

Fig. 2. Modes of the shell for M = 0

Fig. 3. Dispersion curves for the first branch (a) and for the second branch (b)

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Note that the “Winkler springs” acting normally to the shell (curves 2 in Fig. 3a and 3b) have a stronger influence on the character of those dispersion curves, which respond primarily to bending vibrations, and “Winkler springs” acting along the axis of the shell (curves 1 in Fig. 3a and 3b) have a stronger influence on the character of those dispersion curves, which respond primarily to longitudinal vibrations. If both types of “springs” are present, all dispersion curves are affected (curves 3 in Fig. 3a and 3b). This will happen, for example, in the case KT = 0.7K, KN = 0.7K that is often used in the calculation of real structures. The calculations accuracy can be estimated by comparing the eigenfrequencies obtained by FORTRAN program with ones obtained by package ANSYS in the reference points (Table 1). This table relative error does not exceed 0,01%.

(a) Normal “springs”

(b) Longitudinal “springs”

Fig. 4. Dispersion curves

(a) K Z

2000 E 6, K N

0, w 821.21

(b) K Z

2320 E 6, K N

0, w 827.53

Fig. 5. Displacements uz (1), un (2) for the modes from points A (a) and B (b) (see Fig. 4)

Let’s consider the dispersion curves behavior on a larger range of rigidy (Fig. 4a,b). In the case of longitudinal “springs”, the intersection of these curves is possible (Fig. 4b). The greatest influence is on the odd branches, which correspond mainly to longitudinal vibrations. Their curves form a pseudo-dispersion branch (circles on the Fig. 4b), where the forms of oscillations that correspond to the first form are dominated. Figure 5a and 5b show the modes calculated for the points of dispersion curves A and B in Fig. 4b, respectively.

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Table 1. Reference frequencies KN = 0;

KN = 100.0E6;

KN = 0;

KN = 100.0E6;

KZ = 0

KZ = 0

KZ = 100.0E6

KZ = 70.0E6

Fortran

Ansys

Fortran

Ansys

Fortran

Ansys

Fortran

Ansys

617.51

617.48

621.81

621.86

638.40

638.45

636.85

636.89

804.58

804.54

822.97

823.05

805.97

806.06

823.97

824.06

Now let’s consider the influence of the mass belt on eigenfrequencies in the case of absence of Winkler foundation. In this case, Eq. (1) is reduced to the species   L + w2 (I + l Mδ(z)) u = 0, −l/2 < z < l/2 (7) Note that the growth of belt mass leads to the growth of the inertial term (I + l Mδ(z)) and thus to an increase of the shell inertial properties and, consequently, to a decrease of the system “shell – mass belt” own frequencies. Figure 6a shows the branches 1, 2 of the dispersion curve for the case Mn = 0. This corresponds to the interaction of the mass belt with the shell along the axis z only. The characteristic feature of this case is the absence of intersection of these branches (note, that there are no intersections for the branches with large numbers). The dispersion curves of solid lines represent the results of the calculations in FORTRAN program, and the circles are the control calculations in ANSYS. Calculations, carried out by both methods for the mass M = 10.0 for the first five frequencies, show that the relative error does not exceed 0.4%. Figure 6b shows the branches 1, 2 for the case Mz = 0, Mn ≥ 0. Letters a, b, c indicate the branches calculated for KN = 0.0, 1.0E8, 1.0E9 kN/m3 at Kz = 0.0. It corresponds to the interaction of the mass belt with the shell only by the normal. In contrast to the previous case, there is a visible intersection of branches. The branches are crossed, or it is just a quasi-intersection (similar to the “veering” of the dispersion curves in the plane (ω, λ)) – this question remains open. However, from the point of view of the calculator, the curves can be considered really intersecting and then the intersection point is the point where the both types (Z- and N) of oscillations are possible. A characteristic feature of the graphs with dispersion curves is alternation of branches that “feel” the inertial mass increase, with branches that do not “feel” it. It is is due to the presence of a node or a standing wave beam at the point of mass attachment for the corresponding component of the displacement vector. For example, vibration beams of the component uz at the point z = 0 are observed on the branches with odd numbers, and nodes are observed on the branches with even numbers. Therefore, in the case of Z-vibrations on odd-numbered branches, the frequency naturally drops with mass growth, infinitely draw upon the next eighenfrequency, where the system can implement the node at the point of mass location. Note that in the case of Z-vibrations, the mass interacts with the shell like a concentrated mass with longitudinal vibrations of the rod [18].

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(a) Z - type masses for K n = 0

(b) N - type masses for K n = 0, 1.0 E 8, 1.0E 9 (curves a,b,c)

Fig. 6. Dispersion curves for Kz = 0

For the component un the situation is reverse: the vibrations at the point z = 0 are observed on the branches with even numbers, and the nodes are observed on the branches with odd numbers. Therefore, in the case of N-oscillations on even-numbered branches, the frequency naturally decreases with the growth of mass, but unlike the previous case the intersection of branches is observed. If both components in the vector are non-zero (ZN-type of the oscillations), this case inherits properties of both Z- and N-type of oscillations. Note that in the case of N-vibrations, the concentrated mass interacts with the shell like a pointed mass with the beam during the bending vibrations [18]. Let us consider more detailed the vibration field in the intersection point vicinity of the branches 1 and 2 for the case of N-vibrations (Fig. 6b).

(а) displacements u z (1) , un (2)

(b) forces f z (1) , f n (2)

Fig. 7. N - type masses first branch, M = 10.0, K = 0.0

Components of displacement and force vectors (Fig. 7a and 7b) keep their shape for the first branch (calculated at K = 0) throughout its entire length, in accordance with the analysis (the component un implements a node at the point z = 0, so it does not “feel” the mass change at this point). The longitudinal displacement component is dominated. Note that the longitudinal force component dominates too (Fig. 7b). The relative contribution of the normal force component is almost equal to zero. For the second branch, calculated at KN = 1.0E9, the relative contribution of the component uz decreases with increasing mass from M = 0.0 to M = 0.1 and vice versa,

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the contribution of the component un increases, it quickly becomes highly localized in the vicinity of mass belt (Fig. 8a) [22]. At the same time the break of the normal force component increases, the relative contribution of the amplitude of the longitudinal force component decreases sharply to some constant value (Fig. 8b). Further, as the mass increases to M = 10.0, the graphs for displacements and forces practically do not change. Note that the localization of vibrations is observed only for oscillations, which has component Mn > 0 (N- and ZN-type). The magnitude of Mz component does not change the character of localization qualitatively. The calculations show that the position of the dispersion curves intersection point significantly depends on the stiffness of the Winkler foundation (Fig. 6b). This stiffness can vary within a wide enough range [23], even during operation of a particular pipeline throughout the year. Thus, the values of this coefficient for the clay from the state of “wet, softened” to the state of “hard clay” vary from 1e6 to 200e6. In case of clay freezing, this coefficient reaches 1000e6 (Table 2). The maximum Winkler hardness value is reached for rocks (15000e6). Table 2. Stiffness of Winkler foundation for clay Type of soil

K, kN/m3 Min

Max

Wet, softened clay

1 000

5 000

Damp clay

5 000

50 000

Low humidity clay

50 000

100 000

Solid clay

100 000

200 000

Frozen soil

200 000

1 000 000

At the same time, the degree of vibrations localization is relatively weakly affected by the rigidity of the foundation (Fig. 8a and 8b). Qualitatively, the character of oscillations does not change. Increasing the rigidity of Winkler base leads to increased rigidity of the whole system. It leads, on the one hand, to the eighen frequencies increase (Fig. 4b), and on the other hand – to a decrease of the of mass “belt” relative influence. Therefore, the vibrations become less localized (Fig. 9a and 9b). The growth of belt mass naturally leads to the increase of localization (compare Figs. 9a and 10a with Figs. 9b and 10b). Note that in this figures the curves are normalized per unit under different conditions, so it makes sense to compare exactly the degree of localization, not the amplitude. The location of the intersection point of curves with rigidity increasing can be traced by analyzing the system of Eqs. (1), rewriting the third equation of the system as follows   (8) L3,1 ut + L3,2 uz + L3,3 un + w2 1 + l Mn δ(z) − Kn /w2 un = 0. Note that the curve at a certain ratio of Kn and Mn (which   intersection is observed determines the value l Mn δ(z) − Kn /w2 in (8)). It follows that the intersection point of curves 1 and 2 shifts with growth Kn into the region of large masses Mn (Fig. 6b). When

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(a) displacements u z (1) , un (2)

(b) forces f z (1) , f n (2)

Fig. 8. N - type masses, second branch, M = 0, 0.004, 0.1 (curves A, B, C); Kz = 0; Kn = 1000.0E6

(a) displacements u z (1) , un (2)

(b) forces f z (1) , f n (2)

Fig. 9. N - type masses, second branch, M = 0.004, Kn = 0.0, 1.0E9 (curves A, C); Kz = 0.

(a) displacements u z (1) , un (2)

(b) forces f z (1) , f n (2)

Fig. 10. N - type masses, second branch, M = 0.01, K = 0.0, 1.0E9 (curves A, C); Kz = 0

we consider the intersection points of branches with large numbers (corresponding to high frequencies w), this shift becomes smaller, which follows from a decrease of the summand Kn /w2 with increasing frequency.

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5 Conclusions The work investigated the character of cylindrical oscillations of Kirchhoff-Love shell on Winkler foundation with additional inertia in the form of “mass belt”. At a certain ratio of parameters, the effect of intersection or quasi-intersection of dispersion curves is possible, which corresponds to the change of the oscillation pattern in the vicinity of the intersection point (quasi-intersection). The paper shows that such intersections can occur both due to the changes in the nature and rigidity of the foundation, and due to the magnitude and nature of the solid belt attachment. In the latter case, the situation is aggravated by localization of oscillations in the vicinity of the massive belt. The mutual influence of the base and the belt has been studied. If the concentrated mass has a normal component in a vector describing inertia properties of the mass belt, then there are intersection points of dispersion branches (the coordinates of which significantly depend on the relative contribution of this normal component). In the vicinity of these points, there is a sharp change in the oscillations type from predominantly longitudinal to predominantly bending type, the latter being localized in the area of this “belt”. This effect is occurred even with small relative masses of the “belt”, which constitute a percentage of the shell total mass. With the growth of normal stiffness component of Winkler foundation, intersection point of dispersion branches shifts to the area of large relative masses of the “belt”. It can be noted that effect of oscillations type changing which can greatly depend on the condition of the surrounding soil during the year can be dangerous for the integrity of real constructions. On the other hand, by varying of the load fixing, it is possible to achieve a more stable operation mode of the structure within a given range of parameters. The calculations fulfilled according to the above theory are in good agreement with the finite element method. It should be noted that asymmetry of boundary conditions at the ends of the shell and not the central location of the massive belt can significantly change the quantitative parameters of the phenomenon. Note that Kirchhoff-Love theory allows consideration of multilayer shells also, provided that the characteristics of several layers are reduced to those of a single, averaged layer [24]. This makes it possible to analyze the effects under study for such important applications as oscillations of cylindrical supports and pipelines weakened by hydrogen corrosion.

References 1. Novak, P., Moffat, A.I.B., Nalluri, C., Narayanan, R.: Hydraulic Structures, 4th edn. CRC Press, New York (2007) 2. El-Reedy, M.: Offshore Structures: Design, Construction and Maintenance. Gulf Professional Publishing, Oxford (2012) 3. Palmer, A.C., King, R.A.: Subsea Pipeline Engineering, 2nd edn. PennWell Corp, Oklahoma (2008) 4. Gerwick, B.C.: Construction of Marine and Offshore Structures, 3rd edn. CRC Press, New York (2007)

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5. Eliseev, V.V.: Mechanics of Deformable Solids. Polytechnic University Press, St. Petersburg (2003).In Russian 6. Eliseev, V., Vetyukov, Y.: Finite deformation of thin shells in the context of analytical mechanics of material surfaces. Acta Mech. 209(1–2), 43–57 (2010). https://doi.org/10.1007/s00707009-0154-7 7. Eliseev, V., Vetyukov, Y.: Theory of shells as a product of analytical technologies in elastic body mechanics. Shell Struct. Theory Appl. 3, 81–85 (2014) 8. Eliseev, V.V., Zinovieva, T.V.: Lagrangian mechanics of classical shells: theory and calculation of shells of revolution. Shell Struct. Theory Appl. 4, 73–76 (2018) 9. Evgrafov, A.N. (ed.): Advances in Mechanical Engineering. LNME, Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11981-2 10. Zinovieva, T.V., Smirnov, K.K., Belyaev, A.K.: Stability of corrugated expansion bellows: shell and rod models. Acta Mech. 230(11), 4125–4135 (2019). https://doi.org/10.1007/s00 707-019-02497-6 11. Zinovieva, T.V.: Wave dispersion in cylindrical shell. St. Petersburg State Polytechnical University Journal 4–1(52), 53–58 (2007). (In Russian) 12. Zinovieva, T.V.: Calculation of Shells of Revolution with Arbitrary Meridian Oscillations. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 165–176. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-53363-6_17 13. Evgrafov, A.N. (ed.): Advances in Mechanical Engineering. LNME, Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72929-9 14. Evgrafov, A. (ed.): Advances in Mechanical Engineering. LNME, Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29579-4 15. Filippenko, G.V., Wilde, M.V.: Backwards waves in a fluid-filled cylindrical shell: comparison of 2D shell theories with 3D theory of elasticity. In: Proc. of the Int. Conf. “Days on Diffraction 2018”, pp. 112–117. IEEE, St. Petersburg, Russia, (2018). https://doi.org/10.1109/DD.2018. 8553487 16. Evgrafov, A.N. (ed.): Advances in Mechanical Engineering. LNME, Springer, Cham (2017). https://doi.org/10.1007/978-3-319-53363-6 17. Filippenko, G.V., Wilde, M.V.: Backwards waves in a cylindrical shell: comparison of 2D shell theories with 3D theory of elasticity. In: Proc. of the XLVI Summer School_Conference Advanced problems in mechanics (APM-2018), pp. 79–86. Institute for Problems in Mechanical Engineering, St. Petersburg, Russia, (2018). http://apm-conf.spb.ru/proceedings2018 18. Latifov, F.S., Mekhtiev, M.F., Bergman, R.M.: Asymptotic analysis of free vibrations of a spherical shell in contact with elastic surrounding media. Izvestia AN SSSR, Mekhanika Tverdogo Tela 5, 174–176 (1985). (In Russian) 19. Erba¸s, B., Kaplunov, J., Nobili, A., Kılıç, G.: Dispersion of elastic waves in a layer interacting with a Winkler foundation. J. Acoust. Soc. Am. 144(5), 2918–2925 (2018). https://doi.org/ 10.1121/1.5079640 20. Filippenko, G.V.: Wave Processes in the Periodically Loaded Infinite Shell. In: Evgrafov, A.N. (ed.) Advances in Mechanical Engineering. LNME, pp. 11–20. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11981-2_2 21. Chapra, S.C., Canale, R.P.: Numerical Methods for Engineers, 7th edn. McGraw-Hill Education, New York (2015)

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22. Mikhasev, G.P., Tovstik, P.E.: Localized Oscillations and Waves in Thin Shells. Asymptotic Methods. PHYZMATLITE, Moscow (2009).In Russian 23. Umansky, A.A.: Designer’s Handbook, vol. 1. Book of Requirements, Moscow (2013).In Russian 24. Kossovich, L.Y., Wilde, M.V., Shevzova, Y.V.: Asymptotic integration of dynamic elasticity theory equations in the case of multilayered thin shell. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform. 12(2), 56–64 (2012) (In Russian). https://doi.org/10.18500/1816-9791-201212-2-56-64

Mechanical and Civil Engineering Applications

Statistical Quality Analysis of Bag-in-Box Packaging for Food Products S. A. Atroshenko(B) Institute for Problems of Mechanical Engineering Russian Academy of Sciences, Saint Petersburg 199178, Russia [email protected]

Abstract. The work is devoted to statistical analysis of the quality of food bags, consisting of flexible bag, box and self-closing crane. Two types of packages with barrier layer EVON and MetPET are compared. At the analysis, the following statistical methods are used: histograms, quality cobweb, horizontal histograms and desirability function. It is shown that, depending on the indicators used, packages with different barrier layers have a different level of quality, but exceed the requirements of the standard. Keywords: EVON barrier layer · MetPET barrier layer · Quality level

1 Introduction Bag-in-Box packaging combines a flexible bag, a box and a self-closing crane. Bag-inBox packaging is ideally designed for the following purposes: 1. 2. 3. 4. 5.

extend the shelf life of liquid or semi-liquid food [1] or industrial products; save the quality of the product, even after opening the package; be able to consume as much as necessary at any time; optimize the packaging capacity required for the product; when the product is removed, the flexible bag is compressed by gravity, while pouring the product from the filled bag, air does not penetrate the package; 6. bottling process can be clean or aseptic. In the production of Bag-in-Box packaging, films with barrier layers of two types are used: a copolymer of ethylene and vinyl alcohol EVOH (Ethylene Vinyl Alcohol) and metallized laminated polyester MetPET. 1.1 Barrier Layer EVOH EVOH is a copolymer of ethylene and vinyl alcohol. The EVOH barrier film is a crystallizing material, which is characterized by significant resistance to oxygen and moisture,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 261–277, 2022. https://doi.org/10.1007/978-3-030-92144-6_20

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as well as a good combination of strength, flexibility and transparency. The barrier properties of EVOH depend on the content of polyvinyl alcohol, and the ease of thermal processing depends on polyethylene. The alcohol groups of the polymer are moisture sensitive and therefore, as a rule, EVOH is made the central part of the coextruded structure with several outer layers of polyester, polyethylene or polypropylene (Fig. 1).

polyethylene

EVON

polyethylene

Fig. 1. Film structure using an EVOH barrier layer.

The EVOH film production process is embedded in just one process using an extruder. The coextrusion process is a method of obtaining products from polymeric materials by forcing a melt of material through a forming hole in an extruder, which consists in combining two or more liquid polymers into a multilayer single structure. The result of this transformation is a single multilayer structure with excellent barrier properties. A film using an EVOH barrier layer has a low oxygen permeability compared to peers. For example, at a temperature of 23 °C and a humidity of 65%, the permeability of oxygen for EVOH is approximately 0.5 cm3 /m2 /day, which is not comparable to the permeability of polyethylene terephthalate (PET) – it is approximately 80 cm3 /m2 /day or polyamide (RA) – 30 cm3 /m2 /day at the same conditions. 1.2 Barrier Layer MetPET MetPET – is metallized laminated polyester (Fig. 2).

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polyethylene Met PET polyethylene Fig. 2. Film structure using a MetPET barrier layer

The process of obtaining MetPET is divided into two technological processes: metallization and lamination. Metallization occurs by vacuum deposition of sprayed aluminum onto the surface of polymer films. The process of lamination of films consists in adhesion (bonding) of two or more layers using adhesive - a substance capable of bonding materials by surface adhesion. The result of this conversion is a single multilayer structure with excellent barrier properties, but less flexible than EVOH. The main parameters of the film include: 1. oxygen penetration rate is an important parameter when creating products for storing substances with a limited shelf life; 2. film elasticity is a parameter that can affect the resistance to cracking under the action of bending; 3. recommended shelf life of the film is the period from the date of manufacture of the film to the manufacture of products from it should not exceed a certain time. Shelf life is determined based on the type of film, storage conditions and transportation of the film; 4. resistance to serious deformations (cracking under the action of bending) is a parameter determined during the test period. The measure is the number of micro-holes per square meter after a certain number of tests; 5. film tensile strength is a parameter measured by the load on the film by stretching to rupture or residual deformation; 6. stretching is increasing the length of the film in percent to break during stretching, both in the direction of action of the machine and in the transverse direction; 7. delamination resistance is an important parameter, measured by the force required to separate the two layers of the laminated film.

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Cranes. Many types of cranes are used for Bag-in-Box packages. It should be borne in mind that not all cranes provide absolute tightness of the package. Company uses self-closing cranes in the production, which are the most reliable. In addition, they have a large number of adapters, which allows the use of cranes with almost any package design. When choosing a crane, the following basic parameters must be considered: 1. type of crane and seal (neck): the color division of cranes and covers is accepted, depending on their application. For example, lids come in two colors: yellow is non-heat resistant, white is heat-resistant. In addition, each valve is designed for a specific seal (neck); 2. the force required to remove and insert the crane into the neck: usually the crane and the neck are made of different polymers, therefore, when the temperature changes, these materials can behave differently: expand, soften, deform. To determine the best combination of crane and neck, it is important to know the force of removing and / or inserting the tap into the neck; 3. neck resistance to cracking under stress is a parameter, the determination of which avoids cracks when the neck and crane come into contact; 4. storage temperature of cranes and seals are their properties depend on the storage conditions of cranes and seals, both during production and during operation of the finished product. Recommended storage temperature from 4 to 30 °C; 5. shelf life of cranes and necks is the recommended shelf life between the date of manufacture of cranes and necks and their installation in bags of Bag-in-Box packaging should not exceed two and a half years; 6. air volume in the crane is a parameter measured in the volume of water required to fill the crane to the edge; 7. crane integrity: each crane in the production process is subjected to leak tests; 8. integrity of the tap-neck joint: the tap-neck joint must remain airtight at different temperature levels; 9. crane valve strength: to measure the parameter, the strength test of the crane valve mechanism is carried out. To prevent the formation of irregularities on the film surface, an electrostatic field is created at the entire production stage. Soldering the neck and installing the lid takes place before the films are joined, which avoids damage to the lower layer of the film. One machine for production is designed for two sleeve bags, therefore, with longitudinal soldering, the central “iron” is double to ensure better quality seams. With transverse soldering, the “iron” is also double. After longitudinal soldering, the bags are marked. The marking shall indicate: 1. 2. 3. 4. 5. 6.

bag volume unit name sleeve designation package number order number release time

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On the marking, the sleeve is designated as CC and COC, which makes it possible to distinguish the location of the sleeve with the bags on the machine and simplifies quality control: the right sleeve is marked CC, the left one is marked COC. Next, the packages undergo visual inspection by the operators on the installation and are stacked in a corrugated box. In the future, the finished product is selectively checked for compliance with the requirements of the standard. The main parameters for the finished package are as follows: 1. The safekeeping temperature and humidity: packages should be stored in covered dry warehouses at a distance of not less than one meter from heating and calefactory appliances, away from strongly smelling substances at temperatures from minus 5 to plus 40 °C. Direct sunlight is not allowed on finished products. 2. Shelf life of empty packages: - the warranty period of storage should not exceed 24 months from the date of manufacture; 3. Package sizes; 4. The orientation of the crane: the crane should be oriented in the right direction for the end user. The correct orientation is displayed in the form of clock hands indicating where the crane should be turned, for example, for six hours; 5. The tests for resistance of welds are carried out to peel and tear the bag with compressed air. 6. Packing empty bags: it is necessary to control the number of bags in a box and on a pallet; 7. The pollution prevention: filled bags should not be stored or transported near potential sources of pollution. 1.3 Quality Control of the Bag The most important parameters that determine the quality of the film are the rate of penetration of oxygen and resistance to serious deformations. Oxygen permeability (OP) is a very important and sometimes defining barrier property of a material. When developing a package that provides specified storage periods, it is important to know the methods for determining gas permeability and have the appropriate devices for this purpose. By permeability of a material is meant the intensity of the transfer of molecules of another substance through this material. Moreover, for the same material it can be different in relation to different substances (gases) and is determined in relation to any particular one. Permeability depends on a large number of factors, such as the chemical structure of the polymer used, its degree of crystallization, ambient humidity and others. In this regard, a material with good barrier properties under normal conditions may not be suitable for specific conditions of use. All this makes urgent the need to measure and control the permeability of the used polymer material at various stages of its production and in real conditions of use. For example, in a film with an EVOH barrier layer at a temperature of 23 °C and a humidity of 65%, the oxygen permeability is about 0.5 cm3 /m2 /day, which is not comparable to the permeability of polyethylene terephthalate (PET) – about 80 cm3 /m2 /day or polyamide (PA) – 30 cm3 /m2 /day under the same conditions. Therefore, a film with an EVOH barrier layer is most often used

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for storage of products, the quality of which oxygen can have a negative effect and be a consequence of oxidation and deterioration. There are many methods for determining the rate of penetration of oxygen. The most popular is the method for determining the oxygen permeability of polymer films using the ASTM D3985-02 coulometric sensor [2], which was developed in 1973 and officially published after eight years of testing. This method is known in Germany as DIN 53380, and in Japan as JIS K-7126. The basis of this method is the use of a coulometric sensor operating according to the principle of the Faraday law. Each oxygen molecule passing through the sensor causes the appearance of 4 electrons, which in turn create a weak electric current detected by the device. ASTM D 398502 defines that, because of its effectiveness, a coulometric sensor can be considered an internal standard that does not require calibration. The sample is clamped into a measuring cell, inside which oxygen-free gas is supplied to remove oxygen residues. After the sensor detects a stable zero, pure (99.9%) oxygen is supplied to the outer part of the cell. Molecular oxygen, penetrated through the film, is transferred together with the gas to the sensor, which determines its amount. Resistance to severe deformation (cracking due to bending) can also be measured in several ways. The most popular in European countries and practically not widespread in Russia is GelboTest. GelboTest is used to compare various flexible barrier materials, such as polymer bags. As you know, a key feature of a flexible barrier material is its ability to maintain its properties after repeated deformation – GelboFlex.

Fig. 3. Principle of operation of the Gelbo apparatus

The essence of the test lies in the fact that the test material is subjected to repeated twisting on a special apparatus. The Gelbo apparatus, as a rule, consists of two frames to which the sample is attached (Fig. 3). One of the frames is static, the other is mobile. The test sample is subjected to repeated bending and compression due to the creation of a movable frame of rotational and translational motion. Typically, the material bends at a frequency of forty-five cycles per minute for a certain time. The severity and number of compressions may vary depending on the instance being tested. The quality of the material being tested is determined by the presence or absence of tiny holes in the sample, which are detected using colored turpentine (turpentine), which stains a white background through a bent film. If we compare two different films (one with the EVOH barrier layer, the other with MetPET), then after the GelboTest, the oxygen permeability of the first film remains almost at the same level, and the oxygen permeability of the second film increases. From this it can be concluded that the ability

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267

OP, cm3/m2/day

Number of bends

Fig. 4. Change in oxygen penetration rate after GelboTest

of a film with a MetPET barrier layer to withstand multiple bending is lower than that of a film with an EVOH barrier layer. The oxygen permeation test results after GelboTest are shown in Fig. 4. Quality Control of Finished Products. Control of finished products is an integral part of production. The company carries out quality control at the beginning of the order, fifteen minutes after the start of the order and at the end of the order. A visual quality control is also carried out during the entire time the order is completed and periodic quality control is performed by the senior operator. During periodic quality control by the senior operator, the following data is reflected in a special form: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

machine side (CC or COC); control time; box number; package number; seam overlap; perforation appearance (if any); readability marking; presence of foreign particles; seams appearance; plug quality; tube orientation; packet format (length and width);

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13. the position of the neck (the distance from the center of the neck to the edge of the bag or to the perforation and the distance from the center of the neck to the left edge of the bag); 14. the presence of air; 15. rupture (pressure at which the rupture occurred and the location of the rupture); 16. extension of peripheral joints; 17. stretching the seams of the neck. Parameters: overlap of joints, appearance of perforation, readability of marking, presence of foreign particles, appearance of joints, presence of air, extension of peripheral joints, extension of neck joints are assessed by presence (plus/minus). The remaining parameters are characterized by specific values. Sampling for quality control occurs according to the instructions. For example, in the production of bags of three liters with the number of bags in a box of five hundred pieces, quality control is carried out for every fourth box on each side of the machine. Three packages are selected from each box and checked for compliance with all parameters indicated in the specification for a concrete product. Tensile Tests. The tensile test is most interesting when testing the finished product. The tensile test is carried out on a tensile testing machine by filling the bag with compressed air. For each type of package, depending on the volume, combination of films and film thickness, a minimum discontinuity value and a break location are provided. Bursting of the package can occur in the center or at the weld. The break at the weld is often accompanied by delamination of the film. Based on the data on the break, we can make a conclusion about the quality of the initial material – the film, the quality of the production process, the strength and quality of the finished product.

2 Statistical Analysis Methods The following statistical methods were used to analyze product quality: 1. 2. 3. 4.

histogram; quality cobweb; horizontal histograms; desirability function.

2.1 Histogram Histogram is a bar graph based on data obtained over a certain period, which are divided into several intervals. The number of data falling into each of the intervals (frequency) is expressed by the height of the column. 2.2 The Differential Method (Quality Cobweb) The differential method of assessing product quality is carried out by comparing the indicators of individual properties of one product group with the corresponding indicators

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of another product group. At the same time, it is determined whether the desired quality has been achieved as a whole, as well as how similar property indicators differ from each other. For a more accurate and more informative assessment characterizing the quality of the product, it is usually built a chart comparing quality indicators (quality cobweb). The values of the characteristics indicate on the scales. The points connect with each other and get several polygons. It can be seen from the quality cobweb that the area occupied by the polygon of properties of one product group differs from the area occupied by the polygon of properties of a similar group. This indicates that the quality of similar product groups is different. The distance from the center to the angle of the polygon on the axes (scales) corresponds to the limit value of the parameter – an indicator of the property. 2.3 Horizontal histograms Horizontal histogram is the operational method of visual analysis of quality and competitiveness [3]. The essence of the method is to present product options in the form of histograms. To begin with, selection criteria are determined, their significance in percent is established and the obtained values are plotted on the ordinate axis. Perpendiculars to the axis are restored from each point, the width of the obtained horizontal fields corresponds to the significance of each criterion. There are two ways to build. 1. The values of the criteria are laid off to the right and left of the ordinate axis - this method allows you to compare only two options, but with a higher degree of visual clarity - the right and left parts of the resulting image are compared. 2. Values of criteria are plotted only to the right. In this case, you can compare a larger number of options, but the visibility is significantly reduced. On the abscissa axis, the values of the parameters are plotted, for example, in points. Perpendiculars to the abscissa axis are laid off through the obtained points. The resulting figures are compared in area and then conclusions are drawn about which of the presented samples is better. 2.4 Desirability Function If no connection is found between several optimization parameters, then they can be combined into one generalizing parameter using the desirability function [4]. Desirability indicators – dimensionless non-discrete quality characteristics varying from zero to one for any range of dimensional quality indicators x i . The construction of a generalized function is based on the idea of converting the natural values of particular parameters into a dimensionless scale of desirability.

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Desirability indicators are calculated using the equation:   1 1 = d = exp − , 1/ y y e

(1)

where 0 < y < ∞ – supporting index (partial optimization parameter). Dimensional values x i of natural quality indicators are converted into dimensionless supporting index y according to the equation y = a0 + a1 xi .

(2)

To find the coefficients a0 , a1 , it is necessary to have the standard values of the indicators of desirability d, the values of dimensionless indicators y, as well as the values of dimensional indicators x i for two levels of quality gradations. Table 1 shows the gradation of quality depending on the values of indicators of desirability. Table 1. Values of desirability indicators and corresponding dimensionless supporting index. Quality gradation

Desirability indicator d

Supporting index y

«Excellent»

≥0,80

≥4,50

«Good»

≥0,63

≥2,18

«Satisfactorily»

≥0,37

≥1,00

«Bad»

T σ ); 1/T ε is the coefficient of retardation. Plane strain is considered. Let us use the approach introduced in [9]. The state of the viscoelastic medium is steady with respect to the moving coordinate system (0, x, z), where x = x  – Vt, z = z . The displacements ui and stresses σ ij depend on the coordinates (x, z) and are independent of time. Introducing the notations for strains, and stresses below, we summaries that εij∗ , and σij∗ satisfy the equations equivalent to the equilibrium, strain compatibility and Hooke’s law equations for an isotropic elastic body. ∂εij ∂εij ∗ ∂t = εij − Tε V ∂x = εij ,  ∂σ ∂σ σij + Tσ ∂tij = σij − Tσ V ∂xij = σij∗ , i = ui∗ , p(x) − Tσ V ∂p(x) ui − Tε V ∂u ∂x ∂x  uz∗ (x) = A p∗ (x)

εij + Tε

= p∗ (x),

(5)

  An operator A p ∗ (x) corresponds to an elastic problem solution.

4 The Solution for the Steady-State Regime The contact pressure is distributed uniformly p(x,0) = P(0)/l, x ∈ (-∞, ∞) initially. The function dD/dt which is the rate of surface approach possesses an asymptote, that is ∂ω∗ (x, t) = D∞ . t→∞ ∂t lim

(6)

It means that the contact pressure tends to the constant distribution while t → ∞ p∞ (x) = lim p(x, t). t→∞

(7)

Equations (6), (7) corresponds to the steady-state regime, the sliding velocity V, the contact area , and the surface approach D are not depend on time in this regime [3]. So as K ω (x) is periodic function, contact pressure p∞ (x), displacements on the surface u∞ z (x) are periodic functions of coordinate x as well while t → ∞.

308

A. Lyubicheva

The solution of the following equation gives the displacements on the surface u∞ z (x): uz(∞) (x) − Tε V

∂uz∞ (x) ∂p∞ (x) = A[p∞ (x) − Tσ V ] ∂x ∂x

(8)

Let us introduce a notation:   ϕ(x) = A p∞ (x) =



l 0

K(ξ − x)p∞ (ξ )d ξ .

(9)

  If the kernel K(ξ − x) of the operator A p∞ (x) is a periodic function than: l ∂K(ξ −x) d ϕ(x) p∞ (ξ )d ξ dx  = 0 ∂x

=−

l 0

∂K(ξ −x) p∞ (ξ )d ξ ∂ξ 

= − K(ξ − x)p∞ (ξ )|ξ =l − K(ξ − x)p∞ (ξ )|ξ =0 + 

l (ξ ) ∂p(ξ ) . d ξ = A = 0 K(ξ − x) ∂p∞ ∂ξ ∂ξ

= l 0

(ξ ) K(ξ − x) ∂p∞ ∂ξ d ξ =

(10)

The solution of the Eq. (8) can be found as: l

uz(∞) (x)

=

e Tε V l

Tε V (e Tε V

 l ∂ϕ(x + χ) − TχV ϕ(x + χ) − Tσ V e ε dχ. ∂x − 1) 0

(11)

The last term under the integral in (11) is 

l 0

  l ∂ϕ(x + χ ) − TχV 1 e ε d χ = e− Tε V − 1 ϕ(x) + ∂x Tε V



l

χ

ϕ(x + χ )e− Tε V d χ .

0

Finally, the displacements on the surface can be written in the following form l

uz(∞) (x)

= ϕ(x) +

(e

l Tε V



Tσ 1− Tε − 1)

e Tε V

 l  0

0

l

K(ξ − (x + χ ))

∂p∞ (ξ ) − TχV e ε dξdχ. ∂ξ (12)

Fig. 1. The contact scheme of the viscoelastic body and the composite material

Wear of Composite Materials in Full Contact with a Viscoelastic Body

309

The last equation shows that the viscoelastic problem solution is a sum of elastic problem solution ϕ(x) and an additional part. Calculations above are obtained with respect to the periodicity of the kernel K(ξ − x) and pressure distribution function p∞ (x) only. Now we assume that the wear rate coefficient K ω (x) is a step function:  Kω1 , x ∈ [nl, a + nl], Kω (x) = , (13) / [nl, a + nl] Kω2 , x ∈ were K ω1 i K ω2 are the wear rate coefficients of the composite components (K ω1 > K ω2 ) which has structure shown in Fig. 1. The contact pressure distribution at t → ∞ is derived from the wear law (1) and Eqs. (6), (13): ⎧ 1/α  ⎪ ⎨ p1 = p˜ D∞ , x ∈ [nl, a + nl], K (14) p∞ (x) =  ω1 1/α ⎪ ⎩ p2 = p˜ D∞ , x ∈ / a + nl] [nl, Kω2 The kernel of the operator A[p(x)] for the 2-D periodic contact problem [10] is    π(ξ − x)  2(1 − ν 2 ) . (15) ln 2sin K(ξ − x) = −  πE l We can establish the displacements on the viscoelastic surface for the periodic wear contact problem from Eqs. (9), (14), (15) and (12):      l 2) a  π(ξ −x)   π(ξ −x)  (∞) uz (x) = − 2(1−ν ξ + p ln 2 p ln 2 d d ξ + sin sin 1 2 0 a πE l l      l  

    l − TχV π(a−(x+χ )) π(−(x+χ )) Tσ e Tε V + l 1 − Tε p 0 e ε ln 2sin  − ln 2sin  dχ , l l (e Tε V −1)

(16) here p is equal to p2 –p1 . Since the displacements of the boundary due to wear ω∗ (x, t) are equidistant in time, and we consider the complete contact of the surfaces, the dimensionless shape of the composite at t → ∞ will be determined as follow: f (x) = uz(∞) (x)

πE 2(1 − ν 2 )P∞

(17)

Dimensionless parameters are presented below: xˆ = x/l; χˆ = χ /l; ξˆ = ξ/l; aˆ = a/l; m =

Kω2 Tσ ; ζ = l/Tε V ; γ = . Kω1 Tε (18)

The pressure p1 and p2 can be expressed through the constant load P∞ and ratio of the wear coefficients of components m: p1 =

P∞ m1/α P∞ ; p2 = , (1 − aˆ m1 )l (1 − aˆ m1 )l

(19)

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A. Lyubicheva

here m1 = (1 − m1/α ). The expression for the worn surface of the rigid inhomogeneous half-space in dimensionless form is      1/α 1   aˆ m 1 ˆ − xˆ )d ξˆ + ˆ − xˆ )d ξˆ + π( ξ π( ξ ln 2 ln 2 f (ˆx) = − (1−ˆ sin sin am1 ) 0 (1−ˆam1 ) aˆ     1 −χˆ ζ   ζ m1 sin π(ˆa − (ˆx + χˆ )) − ln 2sin π(ˆx + χ) ˆ  d χˆ . + (eζe−1) (1 − γ ) (1−ˆ ln 2 e am ) 0 1

(20) So, the shape of the inhomogeneous body surface in steady-state regime depends on the following parameters: â, m – triboparameters of the composite; γ – the quotient of the relaxation T σ and retardation T ε times of the viscoelastic material; ζ = (l/V )/T ε – the period divided to the sliding velocity and retardation time of viscoelastic material T ε.

5 Results and Discussion The curves in Fig. 2 illustrate the shape of the composite surface in the steady-state regime of wear, calculated according to the Eq. (20) for different values of the parameter ζ. This worn surface becomes wavy due to the wear process. The shape of the worn surface is not symmetrical in the range [10–2 ÷ 102 ] of parameter ζ, contrary to the shape which forms in contact with elastic counter body. Parameter ζ is analog of the Deborah number for this problem, the solution tends to elastic one with longitudinal Young’s modulus while ζ is relatively large.

Fig. 2. The shape of the worn surface of the composite on a period in steady-state regime; m1 = 0.3, â = 0.2, γ = 10–3 ; ζ = 1 (line); ζ = 5 (dot); ζ = 10 (long dash); ζ = 1000 (close to elastic counterbody, dash).

An amplitude Amp of the wavy surface vary with respect to parameter ζ = l/T ε V. It can be defined as a sum of minimum and maximum absolute values of the function f (x).

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The locations of minimum and maximum of the function are the roots of the following equation π 2E ∂f (ˆx) = 0. ∂ xˆ 2(1 − ν 2 )P∞ The calculations of amplitude Amp provided numerically for different parameters, dependences on parameter ζ are presented in Fig. 3 (left). The curves have horizontal asymptotes, asymptotic values are characterized by solution of wear contact problem for elastic counterbody longitudinal Young’s modulus. Analysis of the curves shows that the amplitude does not monotonically depend on parameter â = a/l. The dependence of the amplitude on â is demonstrated in Fig. 3 (right) for different ζ values. Maximum of the amplitude achieves when the lengths of components of the composite are approximately equal. For example, the maximum value of the amplitude is located at â = 0.55, while the other parameters are ζ = 1, m1 = 0.3, γ = 10–3 .

Fig. 3. The dependence of the amplitude of the worn surface waves on ζ (left), and on a/l (right) for m1 = 0.3, γ = 10–3 .

The amplitude depends on the wear coefficients of the composite components. The curves in Fig. 4 shows that amplitude is as higher as lower the ratio of the wear coefficients of components m. It is clear, that for the homogeneous material with m = 1 the surface stays flat Amp = 0. The wear rate in the steady-state stage is characterized by the effective wear coefficient D∞ , which can be represented in the form: ⎧ ⎫α ⎨ ⎬ P∞  .

D∞ = ⎩ p˜ l aˆ K −1/α + (1 − aˆ )K −1/α ⎭ ω1

ω2

This characteristic does not depend on the properties of the viscoelastic counter body for this problem formulation but depends only on the tribological characteristics of the composite, on the wear law, and on the load.

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Fig. 4. The dependence of the amplitude of the worn surface waves on m for a/l = 0.5, γ = 10–3 , and various ζ.

6 Conclusions The wear contact problem is considered for the periodic composite material and the viscoelastic body. Wear of inhomogeneous materials is characterized by the waviness arising on its surface. The shape of the worn surface and contact characteristics are determined for the steady-state regime of the wear process. The amplitude of the waviness depends on the triboparameters of the composite, period of its structure, the viscoelastic properties of the counterbody material, and the sliding velocity. The viscoelastic properties of the material have a significant influence on the operational waviness. The shape of the worn surface is not symmetrical in the [10–2 ÷ 102 ] range of parameter ζ (the analog of the Deborah number); contrary to the shape which forms in contact with elastic counter body. The amplitude of waviness of the surface increases for higher values of parameter ζ and lower values of the ratio of the wear coefficients of components. So, the analysis of the problem permits to control the operational waviness choosing optimal parameters. Acknowledgments. The present work was supported by Russian scientific foundation grant 19– 19-00548.

References 1. Goryacheva, I., Dobychin, M.: The wear of non-uniformly hardened surfaces. Treniye Iznos 7(6), 985–992 (1986). In Russian 2. Goryacheva, I., Torskaya, E.: Contact problems for wear of bodies with a variable surface wear resistance coefficient. Treniye Iznos 13(1), 185–94 (1992). In Russian 3. Goryacheva, I.G.: Contact Mechanics in Tribology. Springer Netherlands, Dordrecht (1998)

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4. Chekina, O., Keer, L.: Wear-contact problems and modeling of chemical-mechanical polishing. J. Electrochem. Soc. 145(6), 2100–2106 (1998) 5. Makarova, N., Polonik, M.: Optimization of structural parameters of cement composites with high wear resistance. IOP Conf. Ser.: Mater. Sci. Eng. 365, 032068 (2018). https://doi.org/ 10.1088/1757-899X/365/3/032068 6. Aleksandrov, V., Kudrova, F.: An exact solution of the periodic contact problem for an elastic layer taking wear into account. J. Appl. Math. Mech. 66(4), 647–654 (2002) 7. Shpenev, A.: Friction and wear of fiber composites with abrasive particles on contact surface. J. Frict. Wear 39(3), 188–194 (2018) 8. Kragelskii, I., Dobychin, M., Kombalov, V.: Friction and Wear Calculation Methods. Oxford (1982) 9. Goryacheva, I.: Contact problem of rolling of a viscoelastic cylinder over a base of the same material. J. Appl. Math. Mech. 37(5), 877–885 (1973) 10. Shtayerman, I.: Contact problem of the theory of elasticity. Leningrad (in an English Translation by Foreign Technology Div., FTD-MT-24-61-70) (1970)

Pressure Concentration in 2D Elastic Regular Rough Contacts: The Effect of Asperity Interaction Ivan Y. Tsukanov(B) Ishlinsky Institute for Problems in Mechanics RAS, Prospekt Vernadskogo, 101-1, Moscow 119526, Russia

Abstract. The analysis of the effect of elastic interaction between asperities of 2D regular rough surface on the peak pressure is performed on the basis of contact problems for sinusoidal geometry with single, double and infinite contact zones. The new asymptotic expression for pressure distribution in the contact of two sinusoidal asperities with an elastic half-plane was obtained using Gladwell’s approach. The results of the analysis show, that the effect of elastic interaction leads to increase of maximum pressure value and reduction of the contact zone size. Despite significant discrepancy in pressure distributions for different types of contact (single, double and periodic), elastic interaction produces only slight effect on the peak pressure dependence at light and moderate loads. Keywords: Regular rough surface · Contact problem · Asperity interaction

1 Introduction Local pressures at contact interaction of elastic bodies can be high enough to cause unallowable plastic deformation or fracture of materials. The calculation of peak pressure is also actual for joints loaded periodically, which can undergo fatigue damage of contacting surfaces. The surface microgeometry plays a major role in forming pressure peaks, which make the stress state more severe. The peak pressure value depends on asperities elastic interaction, which effect can be stronger due to the multiscale nature of surface roughness. 2D regular rough surfaces are the surfaces, the roughness amplitude of which is significantly different in longitudinal and transverse directions. They usually appear at turning or milling technological operations, which have one basic direction of a tool path. The height distribution of a such type of surfaces is often close to uniform [1, 2]. The first simple approach to calculate contact characteristics of a 2D regular rough surface, including peak pressure, is to represent it as a sine wave, having an equivalent amplitude and a frequency (period). However, that approach doesn’t take into account small scale components of roughness, which strongly influence on the stress-strain state of contacting bodies, especially at small loads. In the more recent studies, a quasiregular rough surface is represented as a Fourier series, having harmonics of different amplitudes and frequencies [3, 4]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 314–319, 2022. https://doi.org/10.1007/978-3-030-92144-6_25

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The influence of roughness on the peak pressure value for the contact of the Hertzian type was studied [5] on the basis of the complete contact solution [2], approximately taking into account continuous or discontinuous type of contact. Semi-analytical and numerical studies [6–8], dedicated to investigation of the influence of elastic interaction of 2D sinusoidal roughness asperities on the contact area and peak pressure behavior, show considerable discrepancy in comparison with the Hertz theory. However, comparison of the contact problem for sinusoidal asperities with Hertz theory is not quite correct, as the shape of asperities significantly influences on the pressure distribution [9]. In the present study the influence of asperities interaction is analyzed for the case of sinusoidal asperities. The considered analysis is based on the analytical solutions of the problems for single, double and periodic contacts.

2 Contact Problems for 2D Sinusoidal Geometry with Different Type of Interaction 2.1 Periodic Contact Problem Periodic contact problem is the basic contact problem for sinusoidal geometry, as trigonometric functions are periodic. Considering only the first harmonic of the rough surface, the periodic contact problem would reflect the long-range interaction between asperities of the longest wavelength. the scheme of the problem is shown in Fig. 1.

Fig. 1. Scheme of the periodic contact problem for a 2D sinusoidal wavy surface indenting into an elastic half-plane.

The profile of a wavy surface is described by the function   2π x f (x) =  1 − cos , λ

(1)

where  is an amplitude and λ is a period of a wavy surface. The close form solution of the problem (see Fig. 1) was obtained in early 1939 by Westergaard [10]. Contact pressure distribution is given by  √ πx 2π a 2πE ∗  2π x p(x) = cos − cos , (2) cos λ λ λ λ

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where a is a contact half-width; E * is a reduced modulus of elasticity: 1 − ν12 1 − ν22 1 = + . E∗ E1 E2

(3)

In Eq. (3) E 1 , E 2 , ν1 , ν2 are Young’s moduli and Poisson’s ratios of the materials of a wavy surface and a half-plane respectively. The relation between the applied (mean) pressure p and the maximum value of pressure p0 for partial contact was given in [11]  π E ∗ p p0 = 2 . (4) λ It is interesting to notice, that Eq. (4) is exactly the same as in the Hertz theory, considering curvature radius of a sine wave at the origin R = λ2 /4π 2  [2]. For the case of complete contact the following relation holds true p0 =

2π E ∗  . λ

(5)

2.2 Contact of a Single Sinusoidal Asperity with an Elastic Half-Plane In the case of a single sinusoidal asperity, indenting into an elastic half-plane (no elastic interaction), the expression for the contact pressure distribution was obtained in [12] in the form of series: p(x) =

 ∞  2π E ∗ 1 − (x/a)2 (−1)k J2k+1 (2π a/λ)U2k (x/a), λ

(6)

k=0

where U i (x) is the Chebyshev polynomial of the second kind of degree i; Jk (x) is the Bessel function of the first kind of integer order k. The relation between the maximum and the applied pressure can be determined from the system of equations π E ∗ p0 = λ p=

2π  a/λ

J0 (t)dt,

(7)

π 2 E ∗ aJ1 (2π a/λ), λ2

(8)

0

2.3 Contact of Two Single Sinusoidal Asperities with an Elastic Half-Plane At light loads only several asperities are in contact, and with increasing of the load the more asperities come into contact. This situation can be modeled by contact problem for the finite system of punches. However the close-form analytical solution is possible only

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Fig. 2. Scheme of the contact problem for two interacting sinusoidal asperities, indenting into an elastic half-plane.

for two symmetrical interacting punches because otherwise the coefficients of additional polynomial in integral equation should be determined numerically to fulfill side (rigid interconnection and consistency) conditions. [13, 14]. Consider the contact problem for the two sinusoidal asperities, indenting into an elastic half-plane. The scheme of the problem is shown in Fig. 2. Approximating the shifted cosine function (see Fig. 2) as a fourth degree polynomial, the asymptotic expression for contact pressure distribution can be obtained by using Gladwell’s solution [14]:    E∗ (9) x2 x2 − b2 d 2 − x2 , p(x) = 2 Rλ where b and d are the coordinates of contact zone boundaries and λ is a distance between asperities, corresponding to period of a wavy surface (see Fig. 2). The relation between values of b, d and λ is [14]:

λ = 2(b2 + d 2 ). (10) Shifting the origin to the contact zone center of one asperity, the following expression holds true: 2a = b − d .

(11)

Considering curvature radius of sinusoidal asperity as R = λ2 /4π 2 , Eq. (9) can be transformed as follows:  π 2 E ∗ 2  (12) x 16λ2 a2 − 64a4 − λ4 + 8λ2 x2 − 16x4 , p(x) = 4 λ where a is a contact half-width (see Fig. 2). The nominal (mean) pressure is given by evoking the equilibrium equation 2 p= λ

b d

  4a2 π 3 E ∗ a2 1− 2 . p(x)dx = λ3 λ

(12)

Maximum pressure value, which can be obtained by using Eq. (12), is not located at the center of the contact zone, so it should be determined numerically.

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3 Results and Discussion Contact pressure distributions and dependences of the maximum pressure calculated from the solutions of corresponding contact problems (see. Sects 2.1, 2.2 and 2.3) for the specific values of dimensionless nominal pressure p/p∗ are presented in Fig. 3 (p∗ = π E ∗ /λ).

Fig. 3. Contact pressure distributions for sinusoidal asperities, indenting into an elastic half-plane at p/p∗ = 0.4 (a) and dependences of maximum pressure on p/p∗ (b): single asperity contact (dash-dot line); double asperity contact (solid line); periodic contact (dashed line); Hertz theory (dash-dot line with ▲).

Figure 3 (a) shows that the single asperity contact problem and the periodic contact problem for moderate value of load give similar pressure distributions. However, the effect of elastic interaction leads to moderate increase of the maximum pressure value and reduction of the contact zone size. Comparison with Hertz theory shows, that single sinusoidal asperity gives significantly larger contact zone size and smaller peak pressure value. This discrepancy is produced by change in profile slope of sine curve. Meanwhile, periodic contact solution gives closer results to Hertz theory due to combined effect of slope and asperities elastic interaction. A decrease in the size of the contact zone is observed also for the double asperity contact. At the same time the maximum pressure value grows significantly in comparison with single asperity contact, and the pressure distribution becomes asymmetric, because elastic coupling is stronger near the valley between asperities (see Fig. 2). In comparison with periodic problem rise in the peak pressure for the double asperity contact provided by asymmetry of pressure distribution due to the non-symmetric side conditions at the ends of contact zones [14]. Graphs of maximum pressure vs mean pressure, presented in Fig. 3 (b) show, that significant discrepancy between results appears at p/p∗ ≈ 0.25. This statement agrees with analysis of multiscale wavy contact problem with continuous contact configuration [12]. The graph of peak pressure for the periodic contact problem completely matches the result for the Hertz theory in accordance with Eq. (4). It is interesting to note, that dependence, obtained by periodic contact solution gives approximately average results between single and double contacts in the rage of moderate applied pressures p/p∗ = 0.25 . . . 0.5. This result allows to use close form expressions of the periodic contact

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problem in approximate calculation of peak pressure for multiscale surfaces contact in the case of light and moderate loads. Unlike the real contact area [8], the maximum pressure value is less sensitive to asperity interaction for sinusoidal geometry in plane elasticity theory. This contact characteristic mainly depends on geometric parameters (amplitude, frequency) of the highest asperities.

4 Conclusion The considered analysis of the pressure concentration (peak pressure dependence on load) in single, double and periodic sinusoidal contacts show, that the effect of elastic interaction leads to increase of the maximum pressure value and reduction of the contact zone size. Despite significant discrepancy in pressure distributions, elastic interaction produces only slight effect on the maximum pressure dependence at light and moderate loads. It can be concluded, that the solution of the periodic problem, which leads to simple expressions for contact characteristics can be used for approximate calculation of maximum pressure in multiscale contact problems, concerning 2D regular rough (wavy) surfaces. Funding. This research was funded by the Russian Foundation for Basic Research, grant 20-0100400.

References 1. Khusu, A.P., Vitenberg, Y.R., Palmov, V.A.: Roughness of Surfaces: Theoretical Probabilistic Approach. Nauka, Moscow (1975).(In Russian) 2. Johnson, K.L.: Contact Mechanics. Cambridge University Press (1985). https://doi.org/10. 1017/CBO9781139171731 3. Manners, W.: Partial contact between elastic surfaces with periodic profiles. Proc. R. Soc. Lond. 454(1980), 3203–3221 (1998) 4. Manners, W.: Methods for analyzing partial contact between surfaces. Int. J. Mech. Sci. 45(6–7), 1181–1199 (2003) 5. Paulin, C., Volle, F., Sainsot, P., Coulon, S., Lubrecht, T.: Effect of rough surfaces on rolling contact fatigue theoretical and experimental analysis. Tribology series 43, 611–617 (2003) 6. Vergne, P., Villechaise, B., Berthe, D.: Elastic behavior of multiple contacts: asperity interaction. J. Tribol-T ASME 107(2), 224–228 (1985) 7. Berthe, D., Vergne, P.: An elastic approach to rough contact with asperity interactions. Wear 117(2), 211–222 (1985) 8. Seabra, J., Berthe, D.: Influence of surface waviness and roughness on the normal pressure distribution in the Hertzian contact. J. Tribol-T ASME 109(3), 462–469 (1987) 9. Tsukanov, I.Y.: Effects of shape and scale in mechanics of elastic interaction of regular wavy surfaces. P. I. Mech. Eng. J. J. Eng. 231(3), 332–340 (2017) 10. Westergaard, H.M.: Bearing pressures and cracks. J. Appl. Mech. T-ASME 6, 49–52 (1939) 11. Dundurs, J., Tsai, K.C., Keer, L.M.: Contact between elastic bodies with wavy surfaces. J. Elast. 3, 109–115 (1973) 12. Tsukanov, I.Y.: Partial contact of a rigid multisinusoidal wavy surface with an elastic halfplane. Adv. Tribol. 2018, 1–8 (2018) 13. Shtayerman, I.Ya.: Contact Problem of the Theory of Elasticity. Gostekhteoretizdat, Leningrad (in an English translation by Foreign Technology Div., FTD-MT-24-61-70) (1970) 14. Gladwell, G.M.L.: Contact Problems in the Classical Theory of Elasticity. Springer, Netherlands, Dordrecht (1980)

Effect of Friction in Sliding Contact of Layered Viscoelastic Solids Elena V. Torskaya(B) and Fedor I. Stepanov Ishlinsky Institute for Problems in Mechanics RAS, Vernadskogo prosp. 101-1, 119526 Moscow, Russia

Abstract. Sliding contact of a smooth indenter and a two-layered half-space is considered taking into account rheological properties of materials. Tangential stresses at the surface due to friction are also taken into account in problem formulation. The case of a viscoelastic layer bonded with a rigid half-space is analyzed as well as an opposite one, which is viscoelastic half-space covered by rigid layer. The problem is formulated as quasi-static. Numerical-analytical method of solution is based on boundary element method and iteration procedure. The solution, which is similar for both cases, is developed based on double integral Fourier transforms. New analytical solution is used to calculate influence coefficients for the computation procedure. Contact pressure, energy dissipation and internal stresses are analyzed in dependence on friction coefficient, sliding velocity, layer thickness and Poisson’s ratio. The analysis of stresses inside the layer or in the viscoelastic substrate can be used to predict the coating fracture in friction interaction. Keywords: Sliding friction · Coating · Viscoelasticity

1 Introduction Relatively compliant polymer coatings are often used in friction units to provide damping, anti-noise and other effects. Usually such materials are much more compliant than the substrates and have rheological properties. A friction interaction of the coatings can be considered using the model of viscoelastic layer bonded to a rigid half-space. An opposite case is relatively hard coatings on polymers, which are used to decrease wear or adhesion in friction contact [1], or to protect the materials from chemical degradation. For both cases contact problem solution and calculation of internal stresses can be used to find optimal coating thickness and other changeable parameters, which provide reliability of the coating-substrate system. 2-D contact problems for a layer with rheological properties and constant Poisson ratio are formulated and solved in [2, 3]; contact pressure and sliding resistance due to energy dissipation are analyzed based on the solutions. 3-D quasi-static contact problems for indenters sliding over a viscoelastic half-space were developed in [4–6]. The same quasi-static contact problems for two-layered structures, in which a coating or a substrate is viscoelastic, were solved in [7, 8]. In these papers no boundary friction was considered; only sliding resistance due to viscoelasticity. The effect of adhesive friction component © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 320–330, 2022. https://doi.org/10.1007/978-3-030-92144-6_26

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on energy dissipation was analyzed in [9, 10] for viscoelastic half-space. Friction contact of viscoelastic layers was considered only for simplified model of material, for example in [11]. In this study, a quasi-static contact problem for a viscoelastic coating (or for a viscoelastic substrate with a rigid coating) is formulated and solved taking into account friction forces.

2 Problem Formulation Let’s consider a contact problem for a rigid smooth slider and a layer with thickness h bonded with a half-space. Indenter slides with constant velocity V along the Ox axis; and it is loaded with vertical force Q (Fig. 1). Origin of coordinate system (x,y,z) is placed at the center of indenter, the Oz axis is directed normally to the unloaded surface of the layer. The origin of the coordinate system is located at the point of initial layer-indenter contact.

Fig. 1. Scheme of contact

The following boundary conditions are considered at the surface (z = 0): w(x,y) = f (x,y) + D, (x,y) ∈ ; τxz = μσz (z,y), (x,y) ∈ ; σz = τxz = 0, (x,y) ∈ / ; τyz = 0

(1)

Here  is the contact zone, w(x,y) is the vertical displacement of the upper layer boundary, D – is the indentation depth of indenter, σz ,τxz ,τyz are normal and tangential stresses. The shape of indenter is specified by a smooth function f(x,y). Friction in this formulation of the problem is represented in the form of the Amonton-Coulomb law with μ as a coefficient of friction.

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Contact pressure p(x, y) = −σz (x, y) and contact area are to be found. Equilibrium condition is the following: ¨ p(x, y)dxdy (2) Q= 

We use also the condition of zero normal stresses at the boundary of the contact zones. Conditions at the layer-substrate interface (z = −h) satisfy the case of perfect adhesion. For the case of rigid plate, which is bending on the viscoelastic substrate we have equal boundary displacements: w(1) = w(2) , ux(1) = ux(2) , uy(1) = uy(2)

(3)

Here ux and uy are tangential displacements. Indexes (1) and (2) corresponds to the layer and the substrate respectively. For the case of rigid substrate conditions (3) transform to the following w = 0, ux = 0, uy = 0

(4)

Viscoelastic material is used as a coating or as a substrate in combination with essentially more hard material, which can be modeled as a rigid half-space or as a plate with flexural stiffness. Mechanical properties of linear viscoelastic material are defined by the following stress-strain relations [9]:  1 t 1 t(t) + t(t)K(t − t)dt; G G −∞   1 t   1 ex (t) = σx (t) − ν(σy (t) + σz (t)) + σx (t) − ν(σy (t) + σz (t)) K(t − τ)d τ; E E −∞   1 t   1 σy (t) − ν(σx (t) + σz (t)) + σy (t) − ν(σx (t) + σz (t)) K(t − τ)d τ; ey (t) = E E −∞   1 t   1 ez (t) = σz (t) − ν(σy (t) + σx (t)) + σz (t) − ν(σy (t) + σx (t)) K(t − τ)d τ; E E −∞   t K(t) = k exp − ω γ(t) =

(5)

Here ν is Poisson’s ratio, E and G are Young’s modulus and shear modulus, respectively. The creep kernel is an exponential function which depends on relaxation time 1/k and retardation time ω.

3 Method of Solution First let’s consider normal and tangential loads distributed uniformly within the square with side 2a, which moves over the viscoelastic layer. Conditions at the surface are the following: (1) = −μq, |x| ≤ a,|y| ≤ a σz(1) = −q, τxz (1) σz(1) = 0, τxz = 0, |x| > a,|y| > a (1) τyz

=0

(6)

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In the case of a layered elastic half-space, the problem of determining stresses and displacements is solved using methods based on double integral Fourier transforms [12]. It is shown that the normal displacements of the upper layer boundary are determined by the following expression:  π/2  ∞ q (γ ,ϕ,λ,χ ) cos(y γ sin ϕ) cos(x γ cos ϕ)d γ d ϕ w (x ,y ,0) = − 2G 0 0   μq 2π ∞ − τ (γ ,ϕ,λ,χ )[sin(y γ sin ϕ) × cos(x γ cos ϕ) (7) 2G 0 0 





+ cos(y γ sin ϕ) sin(x γ cos ϕ)]d γ d ϕ Here x ,y ,w are dimensionless coordinates and normal displacement of the surface related to the half-side a of the square, χ = E1 (1 + ν2 )/E2 (1 + ν1 ), γ ,ϕ are the coordinates in the space of double integral Fourier transforms, λ = h/a is dimensionless thickness of the layer, (γ ,ϕ,λ,χ ) and τ (γ ,ϕ,λ,χ ) are obtained during solution of the system of linear functional equations derived from boundary conditions (1–3) by using the biharmonic functions for definition of the stresses and displacements. Expressions defining (γ ,ϕ,λ,χ ) and τ (γ ,ϕ,λ,χ ) are very complex [12], but it should be mentioned that in both cases we have linear dependence on q: q=q

4 sin(γ cos ϕ) sin(γ sin ϕ) π2 γ 2 sin ϕ cos ϕ

(8)

Here q is the result of application of the double integral Fourier transforms to the pressure distribution given by Eq. (6). The case of distributed load moving with constant velocity over a viscoelastic halfspace was considered in [13] in assumption, that shear modulus is time-dependent operator and the Poisson ratio is a constant. Following [13] we obtain normal displacements of the surface of viscoelastic layer from (7) in the coordinate system related to the center of the square containing the distributed load:  π/2  ∞ 1 (γ ,ϕ,λ) cos(y γ sin ϕ) w (x ,y ,0) = − 2G 0 0    0    × cos(x γ cos ϕ) + K(−τ ) cos((x + V τ )γ cos ϕ)d τ d γ d ϕ 







π/2  ∞

−∞

μ τ (γ ,ϕ,λ) sin(y γ sin ϕ) 2G 0 0    0  × cos(x γ cos ϕ) + K(−τ ) sin((x + V  τ )γ cos ϕ)d τ d γ d ϕ −

−∞

 π/2  ∞ μ τ (γ ,ϕ,λ) cos(y γ sin ϕ) − 2G 0 0    0    × sin(x γ cos ϕ) + K(−τ ) cos((x + V τ )γ cos ϕ)d τ d γ d ϕ −∞

(9)

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here K(t) is the creep kernel (5), V  is the sliding velocity divided by a (V  = V /a), G is the instantaneous shear modulus. For the exponential creep kernel introduced in Eq. (5), the time integrals in Eq. (9) can be analytically calculated. As a result, we have the following relation:  π/2  ∞ 1 (γ ,ϕ,λ) cos(y γ sin ϕ) w (x ,y ,0) = − 2G 0 0   ωV  γ cos ϕ sin(x γ cos ϕ) + cos(x γ cos ϕ) × cos(x γ cos ϕ) + c dγ dϕ 1 + (V  ωγ cos ϕ)2  2π  ∞ μ − τ (γ ,ϕ,λ)[sin(y γ sin ϕ) 2G 0 0   ωV  γ cos ϕ sin(x γ cos ϕ) + cos(x γ cos ϕ)  × cos(x γ cos ϕ) + c 1 + (V  ωγ cos ϕ)2  + cos(y γ sin ϕ)×   ωV  γ cos ϕ cos(x γ cos ϕ) + sin(x γ cos ϕ)  ]d γ d ϕ sin(x γ cos ϕ) + c 1 + (V  ωγ cos ϕ)2 





(10)

Here c = k · ω. Due to constant pressure q appearing linearly in functions , (10) may be used for calculation of influence coefficients in boundary elements method, when contact pressure p(x,y) is obtained as a piecewise function. Expressions (1) and (2) lead to the following system of linear equations: ⎛

4a2 ⎜ κ1 ⎜ 1 ⎜ . ⎝ ..

· · · 4a2 · · · κN1 . . .. . .

⎞ ⎛ p1 0 ⎜ .. −1 ⎟ ⎟ ⎜ .. ⎟ × ⎜ . . ⎠ ⎝p

κ1N · · · κNN −1

N

D

⎞ Q ⎟ ⎜ f1 ⎟ ⎟ ⎜ ⎟ ⎟=⎜ . ⎟ ⎠ ⎝ .. ⎠ ⎞

⎛

(11)

fN

where p1 . . . pN are unknown constant pressures in each of N elements, values f1 . . . fN j are defined by the indenter shape. Coefficients ki are obtained from (10):  π/2  ∞ 1 j  (γ ,ϕ,λ) cos(yij γ sin ϕ) κi = − 2G 0 0   ωV  γ cos ϕ sin(xij γ cos ϕ) + cos(xij γ cos ϕ) dγ dϕ × cos(xij γ cos ϕ) + c 1 + (V  ωγ cos ϕ)2  2π  ∞ μ − τ (γ ,ϕ,λ)[sin(yij γ sin ϕ) 2G 0 0   ωV  γ cos ϕ sin(xij γ cos ϕ) + cos(xij γ cos ϕ) × cos(xij γ cos ϕ) + c 1 + (V  ωγ cos ϕ)2 + cos(yij γ sin ϕ)   ωV  γ cos ϕ cos(xij γ cos ϕ) + sin(xij γ cos ϕ) ]d γ d ϕ × sin(xij γ cos ϕ) + c 1 + (V  ωγ cos ϕ)2

(12)

Effect of Friction in Sliding Contact

325

Here (xij2 +yij2 )1/2 is a distance between square elements,  (γ ,ϕ,λ) = (γ ,ϕ,λ)/q. For the first step of iteration procedure we choose a contact area, which is known to be larger than the actual one. For this area solution of Eq. (11) contains some elements with negative pressure. In the next step of iteration, the values of negative elements are set to null and therefore the value of N decreases. As the result of iteration procedure we obtain the contact pressure distribution p(x,y) and the contact area . Contact pressure distributions are used to calculate internal stresses in the viscoelastic layer or substrate. Expressions for such calculations were obtained for elastic layered half-space in [12]. The stresses have no dependence on elastic modulus, so the same expressions can be used both for elastic and viscoelastic layered structures.

4 Results The method, described above, was used to consider a sliding contact of a spherical indenter (with radius R) and a layered half-space with viscoelastic material used as a layer or as a substrate. The following dimensionless parameters were used for analysis: dimensionless coordinates (x∗ ,y∗ ) = (x,y)/R, velocity V ∗ = V ω/R = V  ω · a/R, layer thickness h∗ = h/R, load Q∗ = Q/R2 Gl (here Gl is longitudinal shear modulus), Poisson ratio ν and friction coefficient μ, and contact pressure p∗ (x,y) = p(x,y)/Gl . Dimensionless stresses are also related to longitudinal shear modulus Gl . First purpose of the analysis was to evaluate the influence of friction on contact pressure distribution. It is important to make this evaluation with relatively large friction coefficient, which occurs, for example, in dry friction of rubbers [14]. For μ = 2 at the first step we estimated the contribution of frictional forces to the total influence coefficients calculated according to (12). It is obtained that in the case of a viscoelastic layer, this contribution is maximal for relatively low velocities (up to 7 percent). For the problem of a rigid bending plate bonded with a viscoelastic half-space, this contribution is negligible. When calculating the contact pressure, the difference in pressure distribution does not exceed the calculation error. Therefore, frictional forces need to be taken into account only when determining the stresses arising from sliding. Based on the analysis of the influence of the input parameters of the problem on the contact pressures made in [8], for the study of stresses, cases were selected in which a significant effect of the rheological properties of the material on the contact pressure was noted. Examples of such distributions are shown in Fig. 2 for a viscoelastic layer (a) and a viscoelastic half-space with a rigid layer on the surface (b). The asymmetric type of the pressure distributions is caused by rheological properties of the half-space. Stress analysis is important for material fracture prediction. Below are the results of calculations of tensile-compressive stresses in a viscoelastic material, since tension is one of the main causes of its fracture. The reduced stresses influencing the rate of damage accumulation process [15] were also evaluated. For the case of rigid plate bonded to viscoelastic half-space the maximal values of tensile stresses are localized in the viscoelastic material at it’s bound with the plate. The results for different rigid layer thickness and two friction coefficients are presented in Fig. 3. Tensile stresses are negative, and compressive stresses are positive. The influence of the friction coefficient is predictable; the larger this coefficient, the greater the

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Fig. 2. Contact pressure distribution within the contact area: V ∗ = 1/3, h∗ = 0.1, c = 5, ν = 0.3, Q∗ = 0.15 (a); c = 6, h∗ = 0.013, ν = 0.4, Q∗ = 2.0, V ∗ = 0.05 (b)

maximum stress values. An increase in the layer thickness on the one hand leads to an increase in the maximum pressures, on the other hand the interface moves away from the surface. Also, with increasing layer thickness, the influence of the rheological properties of the substrate on the contact pressure decreases. These tendencies lead to the fact that the maximum tension at the given calculation parameters takes place under the thinnest layer; the maximum compressive stresses first increase with increasing layer thickness and then decrease. Distributions of reduced stresses under the rigid layer are presented in Fig. 4. The stresses are concentrated at the layer-substrate interface independently on the value of friction coefficient, but the coefficient effects essentially on the stress maximum. For the case of friction contact with viscoelastic layer bonded to a rigid half-space tension-compression inside the layer should be analyzed. Such stress distribution is presented in Fig. 5 for the case of relatively thin layer. The tensile and compressive maxima are at the surface. The zones of tension and compression extend from the surface to the interface with the half-space. This means that if a crack has formed on the surface, there are conditions for its propagation deep into the layer. An analysis was made of the effect of friction on tensile-compressive stresses on the layer surface (see Fig. 6). Curves 1–3, obtained for different values of the coefficient of friction, allow us to conclude that there is a significant effect of friction on these stresses.

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327

Fig. 3. Tensile-compressive stresses at the interface between rigid bending plate and viscoelastic half-space for μ = 1.0,0.4 (curves 1 and 2), c = 6, ν = 0.4, Q∗ = 3.0, V ∗ = 0.05, h∗ = 0.0066 (a), h∗ = 0.0165 (b), h∗ = 0.05 (c)

Fig. 4. Reduced stresses in viscoelastic half-space under the rigid bending plate for μ = 1.0(a), 0.4(b), c = 6, ν = 0.4, Q∗ = 3.0, V ∗ = 0.05, h∗ = 0.0066

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Fig. 5. Tensile-compressive stresses σx∗ inside viscoelastic layer: V ∗ = 1/3, h∗ = 0.1, c = 5, ν = 0.3, Q∗ = 0.15, μ = 0.4

Fig. 6. Tensile-compressive stresses σx∗ at the surface of viscoelastic layer μ 0.2,0.4,1.0(curves 1–3 respectively); V ∗ = 1/3, h∗ = 0.1, c = 5, ν = 0.3, Q∗ = 0.15

=

5 Conclusions In this study a quasistatic sliding of a smooth indenter over a viscoelastic layer bonded with a rigid half-space is considered as well as the same contact for a rigid plate bonded with viscoelastic half-space. Tangential forces at the surface caused by friction are taken into account in the contact problem formulation. New numerical-analytical method of

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solution, which is similar for both cases, is developed based on double integral Fourier transforms. The influence of friction on contact pressure distribution was evaluated. It is obtained that in the case of a viscoelastic layer, this contribution is small; and for the problem of a rigid bending plate bonded with a viscoelastic half-space, this contribution is negligible. The analysis of stresses inside the layer or in the viscoelastic substrate can be used to predict the coating fracture in friction interaction. The maximal tangential stresses in viscoelastic coatings depend essentially on the value of friction coefficient. In the case of a viscoelastic half-space with a wear-resistant hard coating, fracture of the viscoelastic substrate by the fatigue mechanism may occur. If the material is rubber, for which the criterion of reduced stresses is relevant, then the probable type of failure is delamination of the coating. Funding. This research was funded by Russian Science Foundation, grant 18-19-00574.

References 1. Bai, C., Liang, A., Cao, Z., Qiang, L., Zhang, J.: Achieving a high adhesion and excellent wear resistance diamond-like carbon film coated on NBR rubber by Ar plasma pretreatment. Diam. Relat. Mater. 89, 84–93 (2018) 2. Mark, A.V.: The uniform motion of rectangular and parabolic punches in a viscoelastic layer. J. Appl. Math. Mec. 72(4), 492–498 (2008) 3. Aleksandrov, V.M., Mark, A.V.: Quasistatic periodic contact problem for a viscoelastic layer, a cylinder, and a space with a cylindrical cavity. J. Appl. Mech. Tech. Ph. 50(5), 866–871 (2009) 4. Aleksandrov, V.M., Goryacheva, I.G., Torskaya, E.V.: Sliding contact of a smooth indenter and a viscoelastic half-space (3D problem). Dokl. Phys. 55(2), 77–80 (2010) 5. Koumi, K.E., Chaise, T., Nelias, D.: Rolling contact of a rigid sphere/sliding of a spherical indenter upon a viscoelastic half-space containing an ellipsoidal inhomogeneity. J. Mech. Phys. Solids 80, 1–25 (2015) 6. Kusche, S.: Frictional force between a rotationally symmetric indenter and a viscoelastic half-space. ZAMM J. Appl. Math. Mech. 97(2), 226–239 (2016) 7. Stepanov, F.I., Torskaya, E.V.: Modeling of sliding of a smooth indenter over a viscoelastic layer coupled with a rigid base. Mech. Sol. 53(1), 60–67 (2018) 8. Torskaya, E.V., Stepanov, F.I.: Effect of surface layers in sliding contact of viscoelastic solids (3-D model of material). Front. Mech. Eng. 5, 26 (2019) 9. Goryacheva, I.G., Stepanov, F.I., Torskaya, E.V.: Sliding of a smooth indentor over a viscoelastic half-space when there is friction. J. Appl. Math. Mec. 79(6), 596–603 (2015) 10. Goryacheva, I., Stepanov, F., Torskaya, E.: Effect of friction in sliding contact of a sphere over a viscoelastic half-space. In: Neittaanmäki, P., Repin, S., Tuovinen, T. (eds.) Mathematical Modeling and Optimization of Complex Structures. CMAS, vol. 40, pp. 93–103. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-23564-6_6 11. Goryacheva, I., Miftakhova, A.: Modelling of the viscoelastic layer effect in rolling contact. Wear 430–431, 256–262 (2019) 12. Nikishin, V.S., Shapiro, G.S.: Space Problems of Elasticity Theory for Multilayered Media. Vych. Tsentr Akad Nauk SSSR, Moscow (1970).in Russian

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13. Alexandrov, V.M., Goryacheva, I.G.: Moving with constant velocity of distributed load over a viscoelastic half-space. In: Proceedings of the 5th Russian conference “Mixed problems of deformable bodies mechanics”, pp. 23–25. Saratov, Russia (2005), in Russian 14. Morozov, A.V., Petrova, N.N.: Method of evaluating the coefficient of friction of frostresistant sealing rubbers. J. Frict. Wear 37(2), 124–128 (2016) 15. Goryacheva, I.G., Stepanov, F.I., Torskaya, E.V.: Fatigue wear modeling of elastomers. Phys. Mesomech. 22(1), 65–72 (2019)

The Simulation the Contact Interaction of the Needle and Brain Tissue Tatiana Lycheva and Sergey Lychev(B) Ishlinsky Institute for Problems in Mechanics RAS, Prospekt Vernadskogo 101-1, Moscow 119526, Russia http://www.ipmnet.ru/∼lychev/

Abstract. The needle is modeled as an elastic hollow cylindrical flexible rod, partially immersed into a viscoelastic material that simulates brain tissue. The controlled force and moment are applied at the end of the rod. The insertion of the needle is modeled as sliding with friction along a channel whose walls compress the needle. The compression force varies along the axis of the embedded part of the needle and changes in time. The magnitude of the compression forces is determined from the solution of the initial-boundary-value problem. The compression stiffness of the rod is assumed to be infinite, i.e. its deformation is reduced only to bending. Along the axis, the rod moves like an absolutely rigid body. The interaction of a viscoelastic material and a needle is modeled in the linear Winkler approximation as a dynamic system “beam-viscoelastic base” with a time-variable interaction zone length. Keywords: Contact interaction · Brain tissue · Immersion · Elastic needle · Viscoelastic media · Analytical solution · Non self-conjugate operators

1

Introduction

Needle immersing in brain soft tissue has attracted substantial attention in last decade in view of its applications in biopsies and brachytherapy. There is an extensive literature on this subject. While no attempt has been made to provide an exhaustive survey, mention only a few of publications. General observations on how one can model the mechanical behavior of soft brain tissue one can find in [1– 4]. The experimental methods for identification proposed mathematical models are discussed in [5–7]. In sight of significant challenge in nonlinear modelling for soft biological tissues, especially brain tissue, specific finite element algorithms are developed in [8]. Some of them are applied for particular problem for needle insertion [9,10]. Experimental data on the process of insertion is given in [11]. On particular importance is the work [12] which deals with optimization problem c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 331–349, 2022. https://doi.org/10.1007/978-3-030-92144-6_27

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for the shape of needle tip, aimed at reducing resistance during immersion of the needle. Note also the paper [13] in which the robotically steering of the needle is discussed. Despite extensive research, the problem for adequate mathematical modelling of the needle immersion into the brain tissue is still open. The present paper is aimed at developing such models. In rough approach the needle penetration into the brain tissue might be considered as motion of cylindrical hollow rod along the passage, resulting of local tissue damage at the tip of the needle. Apart from controlled force P and moment M that are applied on free end of the needle, it is subjected to numerous resistance forces. The most dominant are friction component T , which arises along the contact zone between the immersed part of the needle and biological tissue, and an inelastic resistant force R at the needle’s tip, related with the energy expended to the damage. In a slow motion the friction traction T along the immersed part is estimated with Amontons–Coulomb law T = κS, where κ is the coefficient of friction, and S is a normal component of interaction between the needle and brain tissue in the passage. While cognizant of the complexity of spatial problem for S we suppose to simplify it with the following approximation of Winkler type S = ku + η

∂u . ∂t

Here k is an resultant elastic modulus, related with elastic characteristics for the tissue, η denotes resultant viscosity modulus, related with tissue viscous properties, while u denotes the deflection of the needle’s axis. The characteristics for brain tissue is estimated according to experimental data, given in [5–7]. The calculations provided in the present paper used as an input the following values: k = 106 Pa/m, η = 106 Pa s/m. A needle is modelled as a hollow thin-walled long cylinder made of stainless steel. The following parameters are used in modelling. Length of the needle is supposed to be l = 0.1 m. The outer diameter of the annular cross section is taken to be equal to D = 0.002 m while the inner diameter is d = 0.0015 m. We assume that the needle is manufactured from stainless steel AISI 304, which elastic modulus E and Poisson’s ratio ν are the follows: E = 200 GPa, ν = 0.29. Figure 1a shows a cross section of a needle. The bending of the needle in most simple approach can be mathematically formalized in the framework of the Euler-Bernoulli beam theory. In so doing one can estimate completely the elastic properties of the needle with just one parameter, which is bending rigidity B B=

 Eπ  4 D − d4 . 64

Calculation for above parameters gives B = 0.107379 H/m2 .

Interaction of the Needle and Brain Tissue

(a) A cross section of a needle

333

(b) Principle diagram of cantilever testing

Fig. 1. Cantilever testing for a needle

The background unknown function is the displacement of the beam axis u = u(x) which should be found from the solution of corresponding boundary value problem. Such problems in several statements will be considered below. Within the framework of adopted beam model the stress resultants, i.e. the bending moment M and the shear force Q are related with displacement function u as follows M = −Bu , Q = M  = −Bu . Hereinafter the prime stands for the derivative with respect to axial beam coordinate x. The above discussion makes it clear that the determination of a “immersion law”, which relates the time depend immersion of a needle with corresponding resistive resultant force, is based on the modelling of time dependent displacements of the needle’s axis. The main part of present paper addresses this issue.

2

The First Problem

We start with very elementary problem for cantilever elastic beam, fixed on its left tip and loaded with some force P and moment M on the right tip (see Fig. 1b). Note that on this stage no elastic or viscoelastic foundation is taken into account. The boundary value problem in this case can be stated as follows

u |x=l

u(iv) = 0, u|x=0 = u |x=0 = 0, M P , u |x=l = . = B B

This problem has elementary solution u=

  x2   x −l −M . P 2B 3

The displacements for unit P and M are shown on Fig. 2. With obtained solution one can determine the region of permissible values for P and S from the following requirement: maximal needle’s deflection has to be no more then some predefined

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value, say, umax = αl, where α is a quotient of permissible deflection and needle’s length. In this context the region of permissible loading can be found from the condition:  l2 2 Pl + M . umax ≤ 2B 3 After simplification we get the inequality M≤

2Bα 2 − P l. l 3

In calculations we take α = 0.3. It results in the region of permissible loading, shown on Fig. 3.

(a) Unit force loading

(b) Unit moment loading

Fig. 2. Deflections of cantilever needle

Fig. 3. Region of permissible loading

Interaction of the Needle and Brain Tissue

3

335

The Second Problem

Now make the problem a little harder. Suppose that the beam is statically immersed into elastic media up to depth a. Under the assumptions adopted before, the affect of the elastic media can be taken into account by resultant elastic Winkler foundation with modulus k. This results in the following statement: Bu(iv) + H(a − x)ku = 0, u |x=l

u|x=0 = u |x=0 = 0, M P , u |x=l = , = B B

⎧ ⎪ ⎨0, H(z) = 1/2, ⎪ ⎩ 1,

where

z0

is a Heaviside step function. This problem can also be solved with elementary but slightly more cumbersome reasoning. The solution may be represented by the following expressions  u1 , 0 ≤ x ≤ a u= , u2 , a < x ≤ l where u1 defines the deflections of immersed part of the needle, √   2 cosh η sin η  cosh μ 2p cos μ + ν sin μ + ν cos μ sinh μ u1 = ς  √  sinh η  cosh μ 2 2p cos(η−μ) + ν (3 sin(η−μ)+sin(η+μ)) − ς  √ + 2 cos μ sinh μ 2p sin η − ν cos η , (1) while u2 determines the deflections of it’s free part, 1 u2 = (ζ − σ) cos(2η) + (ζ + σ) cosh(2η) − 2ζ ς  √ + 2 [(p − α(a − x)ν) sin(2η) − (p + α(a − x)ν) sinh(2η)] . (2) Above formulas use the designations aα η=√ , 2 ζ=

xα μ= √ , 2

ν = α [m + p(a − l)] ,

α3 (a − x)2 [3m + p(2a − 3l + x)] , σ = α [m + p(2a − l − x)] , 6 ς = α3 (cos(2η) + cosh(2η) − 2) .

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For the sake of brevity we also introduce the relative force p and the relative moment m as follows M P . p= , m= B B The distributions for displacements, bending moments and shear forces for different depth of penetration and parameters, specified at the beginning of the paper, is shown in Table 1 below. Table 1. Bending moment and shear force in partially immersed needle

4

The Third Problem

Make now the problem even harder with account of viscous behaviour for Winkler foundation. The corresponding boundary value problem can be written as: Bu(iv) + H (a − x) (ku + η u) ˙ = 0,  u|x=0 = u |x=0 = 0, u |x=l =

M (t) , B

P (t) , B u(0) = 0.

u |x=l =

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337

Here and throughout dot over a variable denotes the derivative with respect to time variable t. Note that the problem becomes evolutionary, since it includes derivatives with respect to both variables, spatial, x, and time t. That is what makes it necessary to set initial data for u. For the purpose of present paper we can set it equal to zero. We are seeking the solution of the form u = u0 + u1 , where u0 is an auxiliary polynomial function which satisfies given boundary conditions but does not meet neither differential equation nor initial conditions: x4 (a(3x − 5a)p + 3(5a − 2x)m) . (3) 60a3 The second item of the solution, u1 , has to satisfy the following boundary value problem  4     ∂ 1 ∂   + = q , u = −u (4) k + η u   , 1 0 1 0 ∂x4 B ∂t t=0 t=0 u0 =

where q0 denotes dummy RHS, which can be calculated with account of (3) as follows  4   ∂ 1 ∂ q0 = − + k + η u0 ∂x4 B ∂t  1   4 ap kx (3x − 5a) − 120B(a − 3x) =− 60a3 B  + 3m kx4 (5a − 2x) + 120B(a − 2x)  ˙ . (5) + ηx4 [a(3x − 5a)p˙ + 3(5a − 2x)m] The solution for (4) can be represent in the form of expansion u1 =

∞ 

Un φn ,

Un = Un (x),

φn = φn (t).

n=1

Here Un are nontrivial normalized solutions for eigenvalue problem  4     λ U, U  x=0 = U  x=0 = U  x=a = U  x=a = 0. U (iv) = a They are 1 U1 = √ , a

  3 x 1−2 , U2 = a a

 x  x  1 − e−2λn 1  −λn cos λn Un = √ sin λn − λn e − υn a 2 a  x  e−λn xa − e−2λn +λ xa 1 + e−2λn sin λn − + 2 a 2 x −λ x x λn x −λ −λ  n n n a +e eλn a −λn − e−λn a −λn e a sin λn + cos λn , + 2 2

(6)

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where υn denotes the squares of normalization factors: υn =

ae−4λn  2λn + e4λn (2λn + 3) − 3(e2λn + 1)2 sin(2λn ) 8λn − 12eλn (e2λn − 1) cos λn + (3e4λn − 3 − 4λn e2λn ) cos(2λn )

 + 8e2λn (3 cosh λn − 2λn sinh λn ) − 3 . (7)

The corresponding eigenvalues can be calculated as follows λ1 = λ2 = 0,

n = 3, 4, . . . , ∞,

λn = n−2 ,

where k are the roots of the transcendental equation cos  =

1 . cosh 

It is appropriate to say something about their search, because direct searching entailed some numerical algorithm, which generally does not assure lack of omissions in the sequence for k . To avoid this we suggest first and second asymptotic approaches k ≈ 1k ≈ 0k , i.e. 0k = π (1/2 + k) ,

1k = 0k +

1 . (−1)1+k cosh 0k + tanh 0k

They give the values very close to k , as can be seen in Table 2 below. Table 2. The first several eigenvalues and relative inaccuracy for their asymptotic representations     k − 0k  /k k − 1k  /k

k k

1 4.730040744862704 3.7 × 10−3 −5

7.7 10− 7

2 7.853204624095838 9.9 × 10

4.0 10−11

3 10.99560783800167 3.110−6

2.3 × 10−15

Thus, in order to obtain standard machine precision, one have to tabulate only first three values for k (they are given in the table) and use second asymptotic approach for all the others. First few normalized eigenfunctions are shown on the Fig. 4. To obtain coordinate functions of the expansion, φn it is necessary first to calculate Fourier coefficients An , n = 1, 2, . . . for auxiliary function u0 at t = 0 with the following integral l Un (u0 |t=0 ) dx.

An = 0

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Fig. 4. First five normalized eigenfunctions

The first two of them have the form A1 = −

a5/2 (ap − 4m)   , 120 t=0

A2 = a5/2

13ap − 54m  √  . t=0 840 3

All the other can be represented by single formula: An =

    a3 eλn  √ 2ap 120 − λ4n + (λ4n − 120) cos λn − 30λn sin λn cosh λn 5 30λn υ n    + 3m 20λ2n sin λn sinh λn − 3(λ4n − 40)(cos λn cosh λn − 1)  . (8) t=0

The Fourier coefficients for dummy RHS l An =

Un q0 dx 0

can also be obtained analytically. Calculations for the first two of them gives Φ1 =

Φ2 =

 4  1 √ p(a k − 120B) + a3 (aη p˙ − 4km − 4η m) ˙ , 120 aB

 1 √ p(2520aB − 13a5 k) + 18m(3a4 k − 280B) a3/2 840 3B

 ˙ − 13ap) ˙ . (9) + a4 η(54m

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For all the other one can obtain the following formula Φn = −

  e−2λn λ 4 4 4 4ae n p cos λn cosh λn (a k(λn − 120) − 120Bλn ) √ 60aBλ5 υ n

 4 4 4 4 − a k(λn − 120) + 30a kλn (cos λn sinh λn − sin λn cosh λn ) + 120Bλn    λ 4 4 4 2λ 4 4 4 + m 18e n a k(λn − 40) − 40Bλn − 9(e n + 1) cos λn (a k(λ − 40) − 40Bλ )    4 2λ 2 4 λ 4 + 60a (e n − 1)kλn sin λn + a η 4ae n p˙ 120 − λn   4 + cosh λn (λn − 120) cos λn − 30λn sin λn + 30λn cos λn sinh λn    λ 4 2λ 4 2λ 2 . + 3m ˙ 6e n (λn − 40) − 3(e n + 1)(λn − 40) cos λn + 20(e n − 1)λn sin λn

(10)

Now everything is ready to formulate Cauchy problems for φn :  4   λ 1 ∂ + k + η φn = Φn , φn (0) = −An . a4 B ∂t The solution for this problem can be obtained in form: φn =

φ0n

+

φ∗n ,

φ0n

−

= −An e



λ4 B +k a4



t η

,

φ∗n

t =



−

e

λ4 B +k a4



t−τ η

Φ0 (τ ) dτ.

0

An important special case of this solution is for the constant p, m:       ∞ λ4 B λ4 t n B +k  − an4 +k ηt − Ba4 4 η a u = u0 + Un Φn 4 e 1 − e − A . n λn B + ka4 n=1 The computations with material parameters specified at the beginning of the paper was provided for several values of penetration depth. The results are shown on Fig. 5, 6 and 7. The surfaces on these figures illustrate the values for displacement function with respect to spatial coordinate, x and time t. Thus they can be viewed as the representation of evolution in time the transverse motion and bending of the needle. Note that with the increase of time the profiles of the needle’s axis tends to what were obtained in previous section from elementary solution for elastic foundation. This observation correlates with physical nature of the modelling process, because the influence of viscosity expected to reduce with an increasing of time. Also note that the model at this stage of reasoning still neglect the motion along the needle’s axis, so the immersion depth should be viewed as fixed parameter.

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Fig. 5. Evolution in time of the needle bending with viscoelastic anchoring. Penetration depth is 10% of total length

Fig. 6. Evolution in time of the needle bending with viscoelastic anchoring. Penetration depth is 50% of total length

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Fig. 7. Evolution in time of the needle bending with viscoelastic anchoring. Penetration depth is 100% of total length

5

The Forth Problem

Now we can proceed to solve complete problem with account of immersion, varying in time. To this end consider the problem ˙ = 0, Bu(iv) + H (a(t) − x) (ku + η u) u|x=0 = u |x=0 = 0, u |x=l =

M (t) , B

(11)

P (t) , B u(0) = 0.

u |x=l =

Difference between this and previous statements is in the fact that parameter a = a(t) varies in time. It’s minor at first glance specific feature significantly changes the way in which one can obtain analytical solution, because the conventional separation of variables in no more valid [14]. Actually in general case it is not possible to apply separation of variables to this problem, but for specific kinds of “immersion law” a(t) it is still possible. To show this, introduce new independent variables ξ, τ that relates with x, t as follows x . τ = t, ξ = a(t) Denote by v new dependent variable v(ξ, τ ) that relates with u(x, t) as composition u(x, t) = v(ξ(x, t), t).

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In new variables equation (11) takes the form  ∂ 4 v a4 a˙ ∂v ∂v − + kv + η = 0. ∂ξ 4 B ∂τ a2 ∂ξ It will be possible to separate variables ξ and τ if a2 a˙ = β/3 = Const. So we arrive at the differential equation that has the following solution  a = 3 βt + γ. Here γ is a constant of integration. With this dependence for a(t) the equation can be resolved to the following form  ∂4v β ∂v a4 ∂v + − kv + η = 0, ∂ξ 4 3B ∂ξ B ∂τ which allows for separation of variables. Thus one can seek the solution v(ξ, τ ) in series ∞  Vn (ξ)φn (τ ), v(ξ, τ ) = n=1

and, consequently, the solution for u(x, t) as: u(x, t) =

∞ 

Vn (x/a(t)) φn (t),

n=1

where Vn (ξ) are the eigenfunction of the following differential operator L[V ] =

∂4V β ∂V , − 4 ∂ξ 3B ∂ξ

    V  ξ=0 = V  ξ=0 = V  ξ=1 = V  ξ=1 = 0.

This operator is conjugated with the following one L∗ [V ] =

∂4V β ∂V , + ∂ξ 4 3B ∂ξ

    V  ξ=0 = V  ξ=0 = V  ξ=1 = V  ξ=1 = 0.

Together they constitute mutually conjugate pair and generate bi-orthogonal complete system in L2 (0, 1) space of complex valued functions with scalar product 1 V, W  = V W dx, 0

where (.) denotes complex conjugation. It might be asked whether such specific kind for a(t) be useful for applications. The graphs of a(t) for several parameters β with γ = 1 is shown on Fig. 8. The

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Fig. 8. Immersion rate dependency that allow for separation of variables

corresponded values for β are shown in rendered circles. One can see that such “immersion laws” are rather realistic and for small values of β tends to linear dependence. To obtain expressions for coordinate functions φn (t) we will take advantage of the bi-orthogonal system, which elements can be obtained as non-trivial solutions of mutually conjugate eigenvector problems β  V = χV, 3B

V (iv) −

    V  ξ=0 = V  ξ=0 = V  ξ=1 = V  ξ=1 = 0,

    β  Λ = χΛ, Λ ξ=0 = Λ ξ=0 = Λ ξ=1 = Λ ξ=1 = 0. 3B The solutions of differential equations can be written as Λ(iv) +

V =

4 

k ξ

Ck e

,

k=1

Λ=

4 

∗

Ck∗ ek ξ ,

k=1

where ’s are the roots of corresponded algebraic characteristic equations: ⎧

⎪

 h 3 3 1⎨  3 2χ ± 4 − − = 2⎪ 3h 3 2 ⎩

∗ 1,2

∗ 3,4

⎫ √ ⎪ 2h2 − 8 3 3χ ⎬ √ , 1,2 √ 3 ⎪ 6 h ⎭ ⎧ ⎫ √ √ ⎪

√ √ ⎪

3  2h2 − 8 3 3χ ⎬ 1⎨  3 2χ h 3 3 2 3 6 hβ √ √ 3,4 = − + + ± 4 , √ 3 √ ⎪ 2⎪ 3h 3 2 6 h ⎩ ⎭ 3B 3 2h2 − 8 3 3χ ⎫ ⎧ √ √ ⎪

√ √ ⎪

3  2(h∗ )2 − 8 3 3χ ⎬ 2 3 6 h∗ β 1⎨  3 2 χ h∗ 3 3 √ √ + − = − , ± 4 √ 3 √ ⎪ 2⎪ 3 h∗ 3 2 6 h∗ ⎭ ⎩ 3B 3 2(h∗ )2 − 8 3 3χ ⎧ ⎫ √

√

√ √ ⎪ 3  ∗ )2 − 8 3 3χ ⎪ 3 ⎨ ⎬ ∗ ∗ 2(h 2 6 h β 1 h 3 3  3 2 χ √ √ ± 4 − + = − . √ 3 √ ∗ ∗ ⎪ 2⎪ 3 h 3 2 6 h ⎩ ⎭ 3B 3 2(h∗ )2 − 8 3 3χ √ √ 2 3 6 hβ √ − √ 3B 3 2h2 − 8 3 3χ

√ 3

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Here

 3 h= (β/B)4 + 768χ3 + 9 (β/3B)2 ,

 3 h = (β/B)4 + 768χ3 + 9 (β/3B)2 . ∗

Substitution of obtained above solutions into the boundary conditions results in  ∗ = 0  = 0, M∗ .C M.C where M, M∗ are the following matrices   12   3 M =  2 11  1 e  3 e1 1

2 2 3 2

2 3 3 3

2 4 3 4

2 2 2 3 2 4 2 e 3 e 4 e 3 2 3 3 3 4 2 e 3 e 4 e

    ,  

  (1∗ )2   (1∗ )3 M∗ =   (1∗ )2 e1∗   (∗ )3 e1∗ 1

∗ 2 (2 ) ∗ 3 (2 )

∗ 2 (3 ) ∗

∗ 3 (3 )

∗ 2 (4 ) ∗

∗ 3 (4 )

∗

∗ 2 2 ∗ 2 3 ∗ 2 4 (2 ) e (3 ) e (4 ) e ∗

∗

∗

∗ 3 2 ∗ 3 3 ∗ 3 4 (2 ) e (3 ) e (4 ) e

    ,   

 C  ∗ are formal vectors, composed with constants of integration: and C,  = (C1 , C2 , C3 , C4 ), C

 ∗ = (C1∗ , C2∗ , C3∗ , C4∗ ). C

The eigenvalues can be found as the roots of spectral equation, which results from substitution of obtained solutions into boundary conditions: det M = e2 +3 (2 − 3 )(1 − 4 ) + e1 +4 (2 − 3 )(1 − 4 ) − e1 +3 (1 − 3 )(2 − 4 ) − e2 +4 (1 − 3 )(2 − 4 ) + e1 +2 (1 − 2 )(3 − 4 ) + e3 +4 (1 − 2 )(3 − 4 ) = 0.(12) Due to the fact that the eigenvector problem is not self-conjugate, the roots occupy general positions complex plane. Meanwhile the peculiarity of the problem cause some specificity on their distribution. It is worth making some comments on it. Firstly, it should be noted that the multiple root placed at zero in the case of self-conjugate counterpart (for β = 0) is split into pair of mutually conjugated roots that are shifting from the origin to increase with the value for β. This is illustrated on Fig. 9. Secondly, note that all the others roots are

(a) Slow penetration, β = (b) Faster penetration, (c) More faster penetra0.3 β = 0.8 tion, β = 1.5

Fig. 9. Position of complex zeros

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situated on real line, approximately near positions for similar eigenvalues of self-conjugate problem. The distance between them decreases sharply with their sequence number. This feature is illustrated on Fig. 10, where relative deviation of zeros, obtained from non self-conjugate problem, from their self-conjugate prototypes are shown. In that regard one can conclude that the usage of eigenvalues, obtained from asymptotic approach for Problem 3, wouldn’t be a big mistake, at least for small values for β.

(a) First regular zero in (b) First regular zero in (c) First regular zero in the vicinity of χ λ43 the vicinity of χ λ44 the vicinity of χ λ45

Fig. 10. Relative deviations of regular zeros

To obtain complex-valued eigenfunctions one can calculate the constants of integration in following form  2 4 [e4 (3 − 2 ) + e3 (2 − 4 ) + e2 (4 − 3 )] , C1 = 1  2 4 C2 = [e4 (1 − 3 ) + e1 (3 − 4 ) + e3 (4 − 1 )] , 2  2 4 C3 = [e4 (1 − 3 ) + e2 (3 − 4 ) + e3 (4 − 1 )] , 3 C4 = e3 (2 − 1 ) + e2 (1 − 3 ) + e1 (3 − 2 ). Similarly, C1∗

 =

C2∗ = C3∗

 

=

4∗ 1∗ 4∗ 2∗ 4∗ 3∗

2   ∗ ∗ ∗ e4 (3∗ − 2∗ ) + e3 (2∗ − 4∗ ) + e2 (4∗ − 3∗ ) , 2   ∗ ∗ ∗ e4 (1∗ − 3∗ ) + e1 (3∗ − 4∗ ) + e3 (4∗ − 1∗ ) , 2   ∗ ∗ ∗ e4 (1∗ − 3∗ ) + e2 (3∗ − 4∗ ) + e3 (4∗ − 1∗ ) , ∗

∗

∗

C4∗ = e3 (2∗ − 1∗ ) + e2 (1∗ − 3∗ ) + e1 (3∗ − 2∗ ). Taking into account the bi-orthogonality property of obtained conjugate eigensystems, we arrive at the countable sequence of uncoupled Cauchy problems with respect to φn (τ ):

Interaction of the Needle and Brain Tissue

∂φn + ∂τ



χn B k + η η(βτ + γ)4/3

φn =

347

Bn (τ )B . η(βτ + γ)4/3

They can be solved in standard way: φn = An φ0n +

B Nn η

τ φ0 (s)(βs + γ)3/4 Bn (s) ds,

k −η t−

φ0n = e

3Bχ √ βη 3 γ+βt

,

s=0

where Nn are the normalization factors: Nn = Vn , Λn  . The account for the force and moment, applied on the free tip of the needle can be done with auxiliary function in such a manner as in Problem 3.

6

Conclusion

The analytical solutions, obtained in the paper, provide a qualitative assessment for the process of needle interaction with brain tissue. Obviously, the detailed modelling of such complicate problem may be realized numerically, for example, with finite element method. Nevertheless, proposed analytical expansions illustrate the specificities of interaction process and can give greater insight into its nature. These include: 1. The total friction force between the boundary surface of the needle and the channel, formed during interaction, varies in time and depends on the rate of the penetration. Moreover, due to viscoelastic properties of biological tissue, considered mechanical system exhibits the properties of the system with memory. 2. Deceleration of the dive speed and temporary stops cause the relaxation of resistance force. 3. It is possible to reduce the dive resistance by variation of horizontal force and momentum applied to the end of the needle. 4. Mathematically, considered problem has much in common with the problem for growing solid [15,16] due to the fact, that during the immersion process mechanical couplings are continually evolving. All of these features correspond to experimental facts and can be taken into account for designing robotic devices that delivers the needle to a given point of a brain tissue [17]. Acknowledgements. The study was partially supported by the Russian Government program (contract # AAAA–A20–120011690132–4) and partially supported by RFBR (grant No. 19–58–52004 and grant No. 18–29–03228).

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References 1. Fallenstein, G.T., Hulce, V.D., Melvin, J.W.: Dynamic mechanical properties of human brain tissue. J. Biomech. 2(3), 217–226 (1969). https://doi.org/10.1016/ 0021-9290(69)90079-7 2. van Dommelen, J.A.W., Hrapko, M., Peters, G.W.M.: Mechanical properties of brain tissue: characterisation and constitutive modelling. In: Kamkim, A., Kiseleva, I. (eds.) Mechanosensitivity of the Nervous System. MECT, vol. 2, pp. 249–279. Springer, Dordrecht (2009). https://doi.org/10.1007/978-1-4020-8716-5 12 3. Darvish, K.K., Crandall, J.R.: Nonlinear viscoelastic effects in oscillatory shear deformation of brain tissue. Med. Eng. Phys. 23(9), 633–645 (2001). https://doi. org/10.1016/S1350-4533(01)00101-1 4. Rashida, B., Destradeb, M., Gilchrista, M.: Mechanical characterization of brain tissue in compression at dynamic strain rates. J. Mech. Behav. Biomed. Mater. 10, 23–38 (2012). https://doi.org/10.1016/j.jmbbm.2012.01.022 5. Millera, K., Chinzeib, K.: Constitutive modelling of brain tissue: experiment and theory. J. Biomech. 30(11–12), 1115–1121 (1997). https://doi.org/10.1016/S00219290(97)00092-4 6. Millera, K., Chinzeib, K.: Mechanical properties of brain tissue in tension. J. Biomech. 35, 483–490 (2002). https://doi.org/10.1016/s0021-9290(01)00234-2 7. Budday, S., et al.: Mechanical characterization of human brain tissue. Acta Biomaterialia 48, 319–340 (2017). https://doi.org/10.1016/j.actbio.2016.10.036 8. Lehocky, C.A., Shi, Y., Riviere, C.N.: Hyper-and viscoelastic modeling of needle and brain tissue interaction. In: Computer Science, Medicine, 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp. 6530–6533 (2014). https://doi.org/10.1109/EMBC.2014.6945124 9. Wittek, A., et al.: Subject-specific non-linear biomechanical model of needle insertion into brain. Comput. Methods Biomech. Biomed. Eng. 11(2), 135–146 (2008). https://doi.org/10.1080/10255840701688095 10. Abolhassaniab, N., Patelab, R., Moallem, M.: Needle insertion into soft tissue: a survey. Med. Eng. Phys. 29(4), 413–431 (2007). https://doi.org/10.1016/ j.medengphy.2006.07.003 11. Jiang, S., Li, P., Yu, Y., Liu, J., Yang, Z.: Experimental study of needle-tissue interaction forces: effect of needle geometries, insertion methods and tissue characteristics. J. Biomech. 47(13), 3344–3353 (2014). https://doi.org/10.1016/j.jbiomech. 2014.08.007 12. Oldfield, M.J., Leibinger, A., Seah, T.E.T., Rodriguez y Baena, F.: Method to reduce target motion through needle–tissue interactions. Ann. Biomed. Eng. 43(11), 2794–2803 (2015). https://doi.org/10.1007/s10439-015-1329-0 13. Misra, S., Reed, K.B., Schafer, B.W., Ramesh, K.T., Okamura, A.M.: Mechanics of flexible needles robotically steered through soft tissue. Int. J. Robot. Res. 29, 1640–1660 (2010). https://doi.org/10.1177/0278364910369714 14. Polyanin, A.D., Lychev, S.A.: Decomposition methods for coupled 3D equations of applied mathematics and continuum mechanics: partial survey, classification, new results, and generalizations. Appl. Math. Model. 40(4), 3298–3324 (2016). https:// doi.org/10.1016/j.apm.2015.10.016 15. Lychev, S.A., Koifman, K.G.: Geometric aspects of the theory of incompatible deformations. Part I. Uniform configurations. Int. J. Nanomech. Sci. Technol. 7(3), 177–233 (2016). https://doi.org/10.1615/NanomechanicsSciTechnolIntJ.v7.i3.10

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16. Lychev, S.A., Koifman, K.G.: Geometric aspects of the theory of incompatible deformations. Part II. Strain and stress measures. Nanosci. Technol. 10(2), 97–121 (2019). https://doi.org/10.1615/NanoSciTechnolIntJ.2018024573 17. Goryacheva, I.G., et al.: Control of insertion of indenter into viscoelastic tissue using a piezoelectric drive. Mekhatronika Avtomatizatsiya Upravlenie 21(5), 304– 311 (2020). https://doi.org/10.17587/mau.21.304-311. (in Russian)

Modeling of Fatigue Wear in Rolling Contact of Elastic Bodies Almira Meshcheryakova1,2(B)

and Irina Goryacheva1,2

1 Ishlinsky Institute for Problems in Mechanics RAS, Prospekt Vernadskogo, 101-1,

Moscow 119526, Russia 2 Moscow Institute of Physics and Technology (National Research University), 9 Institutskiy

Per, Moscow Region, Dolgoprudny 141701, Russia

Abstract. Fatigue wear and damage accumulation take place in the wheel and rail interaction, rolling bearings and so on. In this study the three-dimensional contact problem for a cyclic rolling of an elastic sphere over an elastic half-space is considered which allows us to study the effect of relative slip velocity on the damage accumulation and kinetics of fatigue wear. The contact region consists of a stick and slip subregions, which configuration is unknown in advance. The relation between normal and shear stresses in the slip subregions is described by the Coulomb law. It is assumed that the rolling body and the base have the same elastic properties, and the Hertz solution is used to calculate the normal stress distribution in contact region. Determination of the shear stress is based on the variational method. The results show that the wear process consists of the incubation stage and then the detachment of the layers of definite thickness takes place. The thickness of the detached layers decreases with time. The influence of the relative slip velocity, sliding friction coefficient in contact region and parameters of the damage accumulation function on the fatigue wear process was also studied. The developed model can be used to predict the fatigue wear features in rolling contact of elastic bodies. Keywords: Rolling contact · Fatigue wear · Wear kinetics

1 Introduction During the operation of machinery elements in rolling contact conditions, after number of cycles their contacting surfaces begin to damage. It was shown that the damage of the material accumulates in the subsurface layer of the contacting bodies [1]. The modeling of contact fatigue wear is based on the analysis of the stress state of subsurface layer including the loading history and material properties. The model to calculate the contact fatigue wear in two-dimensional rolling contact taking into account the residual stresses is presented in [2], where for the contact shear stress calculation the Carter solution [3] is used. The effect of relative slip velocity, sliding friction coefficient, and residual stresses on the distribution of the maximum shear stresses is studied. In [4] the influence of relative slip velocity and viscoelastic layer properties on the distribution of the contact and internal stresses in two-dimensional © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 350–359, 2022. https://doi.org/10.1007/978-3-030-92144-6_28

Modeling of Fatigue Wear in Rolling Contact of Elastic Bodies

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rolling contact of elastic bodies is analyzed. The stress state analysis in rolling with slip in application to the wheels of freight cars is given in [5]. The numerical method to calculate the rolling contact fatigue wear based on the representation of the threedimensional contact stress distribution as the number of sections is used in [6]. The effect of the sliding friction coefficient, relative slip velocity and the load on the contact fatigue damage accumulation in rolling contact of a steel wheel and a rail is studied numerically in [7]. The contact fatigue wear modeling using the finite element method in rolling contact with slip is considered in [8–10]. In this study the results of modeling of contact fatigue wear in rolling contact of elastic sphere over the elastic half-space is presented. The effect of longitudinal, lateral and spin components of slip velocity on the subsurface stress state and wear process is studied.

2 Problem Formulation Let us consider the steady rolling with slip of an elastic sphere over an elastic half-space made of the same material. A constant normal force P, longitudinal and lateral tangential forces T x , T y and rotational moment M are applied to the sphere. The contact scheme is shown in Fig. 1.

Fig. 1. The scheme of the elastic sphere (1) rolling over the elastic half-space (2)

The problem is considered in a moving system of coordinates (O, x, y, z). For normal stresses distribution in the contact area the Hertz solution is used.   x 2  y 2 p(x,y) = p0 1 − − , (x,y) ∈  a b

(1)

where p0 – maximum value of the normal stress in contact region. In rolling, the contact region of the sphere and the half-space is divided to the stick and slip subregions, which boundaries are determined in solving. In the slip subregion S the relation between the normal and shear stresses is described by the Coulomb law: | τ (x, y)| = μp(x, y), (x, y) ∈ S

(2)

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where | τ (x,y)| =



2 + τ2 . τxz yz

(3)

The direction of shear stresses τ(x,y) is opposite to the direction of slip velocity vector s(x,y) [11]: τ(x,y) = μp(x,y)

s(x,y) , (x,y) ∈ S |s(x,y)|

(4)

where |s(x,y)| =



sx2 + sy2 .

(5)

In rolling contact the slip velocity between the roller and the half-space with same elastic properties is described with the following relation [12]:   

V x − ωz y B x − x ,y − y τ(x,y)dx dy , (6) s(x,y) = −V V y + ωz x 

where ω  ωx ,ωy ,ωz – the angular velocity of spinning, Δx and Δy – the longitudinal and lateral relative slip velocity: x =

V − ωy R ωx R , y = V V

where R – curvature radius of rolling body surface in point O. Components of matrix B(x – x  , y – y ) are defined with the formulas [11]:



cos  3ν sin2  − 1   B11 x − x ,y − y = − , π Gr 2





ν sin  1 − 3 cos2      B12 x − x ,y − y = B21 x − x ,y − y = − , π Gr 2



cos  ν − 3ν sin2  − 1   B22 x − x ,y − y = − π Gr 2

(7)

(8)

(9)

(10)

where ν is Poisson ratio, G

is shear modulus of the material of the sphere and the half-space. where u uix ,uiy , i = 1, 2 – the shear displacement of the sphere and the half-space. At the boundaries of the stick and slip subregions the conditions of the continuity for shear stresses and displacements functions must be satisfied.

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3 Method of Solution 3.1 Shear Stresses Calculation The problem of shear stress calculation is solved using the variational method, where the minimized functional is built taking into account the boundary conditions for stresses and displacements in the contact interaction region:   τ (x,y), s(x,y)))dxdy (11) F τ,s( τ) = (μp|s(x,y)| − ( 

In this case, the existence of stick and slip subregions is taken into account. The proof of the equivalence of the problem of minimizing functional (11) to the problem of calculation of shear stresses function that satisfies the boundary conditions is given in [11]. The following dimensionless parameters are considered: x1 =

x y p(x, y) τ(x,y) s(x,y) , y1 = , p1 (x1 ,y1 ) = , , τ1 (x1 ,y1 ) = , s1 (x1 ,y1 ) = a a p0 μp0 x V (12) x =

V − ωy R y ωz a ωx R μp0 , y = ,W = ,H = ,γ = , V V V x x Gx

(13)

where a is the radius of the contact region. The numerical solution of the variational problem was obtained using the gradient projection method. A detailed description of the numerical method for solving the rolling problem of two elastic bodies in the variational formulation is given in [2, 6]. 3.2 Damage Accumulation Model To study the process of fatigue wear of an elastic base, a linear damage summation model [12] is used, where the damage at the point of elastic half-space is associated with the maximum values of principal shear stresses and the number of cycles. Material is considered as damaged when the value of the damage function becomes equal to the threshold value. In accordance with the linear damage summation model [12] the following equation is used to calculate the damage accumulation rate:

 τmax (x,y,z,t) m ∂Q(x,y,z,t) =c (14) q(x,y,z,t) = ∂t E where E is the modulus of elasticity of the half-space and the sphere material, τ max (x, y, z) is the maximum value of the principal shear stress at the point (x, y, z), c and m are the constants, which are determined from the experimental data. The principal shear stresses are calculated from the following equation [13]: τmax (x,y,z) =

1 (σ1 (x,y,z) − σ3 (x,y,z)) 2

(15)

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where {σ1 , σ2 , σ3 } are the solutions of the characteristic equation:    σx − σ τxy τxz    τxy σy − σ τzy  = 0,    τ τzy σz − σ  xz

(16)

if at each point the condition σ3 < σ2 < σ1 is satisfied. The internal stress state of the elastic foundation is calculated using the Boussinesque and Cerruti solutions for normal and tangential forces applied to the elastic half-plane [13].

4 Results and Discussion The contact and internal stresses distribution in rolling contact of elastic bodies made of the same elastic material depend on the sliding friction coefficient, longitudinal, lateral slip velocities, spin angular velocity, linear rolling velocity and material properties. In calculation we fix the material properties (E, ν), the linear rolling velocity (V) and sliding friction coefficient (μ). The effect of longitudinal, lateral slip velocities and angular velocity of spinning is studied using the dimensionless parameters (12–13) γ , H and W respectively. 4.1 Influence of Longitudinal Slip Velocity The shear stress distribution calculated for different values of longitudinal slip velocity when the lateral and spin components of slip velocity are absent (W = H = 0) is shown in Fig. 2.

Fig. 2. Shear stresses distribution for rolling contact of elastic sphere and half-space when W = 0, H = 0 and 1 – γ = 16, 2 – γ = 4, 3 – γ = 1

The principal shear stress distribution calculated for the same values of longitudinal slip velocity W = H = 0 is illustrated in Fig. 3. The results show that with the growth of longitudinal slip velocity, corresponding to the decrease of parameter γ , the maximum value of principal shear stress close to the surface of half-space increases. Figure 4 demonstrates the compressive-tensile stresses distribution at the surface of half-space in the cross-section y = 0.

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Fig. 3. Principal shear stresses distribution for rolling contact of elastic sphere and half-space when W = 0, H = 0 and 1 – γ = 16, 2 – γ = 4, 3 – γ = 1

Fig. 4. Compressive-tensile stresses distribution for rolling contact at the surface of half-space (y = 0, z/a = 0.1) when W = 0, H = 0 and 1 – = 16, 2 – γ = 4, 3 – γ = 1

The contact fatigue damage distribution in the elastic half-plane under rolling friction conditions is calculated for different number of cycles. Figure 5 illustrates the wear process for different values of longitudinal slip velocities and parameter m.

Fig. 5. Wear kinetics for rolling contact of elastic sphere and half-space when W = 0, H = 0 and 1 – γ = 16, 2 – γ = 4, 3 – γ = 1

The results indicate the dependence of the dimensionless coordinate z/a on the number of cycles N. The case with full slippage (γ = 1) corresponds to the steady surface wear after the first subsurface fracture occurs.

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4.2 Influence of Lateral Slip Velocity To analyze the influence of lateral slip velocity on fatigue wear in rolling contact the shear stress distribution is calculated for different values of parameter H when the angular velocity of spinning is equal to 0 (W = 0). The location of stick and slip subregions for rolling with slippage in longitudinal and lateral directions is shown in Fig. 6.

Fig. 6. Stick (yellow color) and slip (blue color) subregions configuration for rolling contact of elastic sphere and half-space when W = 0, γ = 10 and 1 – H = 0, 2 – H = 1, 3 – H = 2

The isolines of principal shear stresses at the cross-section y = 0 in presence of longitudinal and lateral slip velocities are shown in Fig. 7.

Fig. 7. Isolines of principal shear stresses in rolling contact at the cross-section y = 0 when W = 0, γ = 10 and 1 – = 0, 2 – H = 1, 3 – H = 2

The results illustrate that with the growth of lateral slip velocity the maximum value of principal shear stresses slightly shifts in the rolling direction and to the surface of half-space. To study the effect of lateral slip velocity on the wear kinetics the dependence of the dimensionless coordinate z/a on the number of cycles N is calculated for different values of parameter H and power of principal shear stresses in the damage function m. The results are shown in Fig. 8. The numerical analysis illustrates that with the growth of lateral spin velocity the detached layer thickness decreases and the subsurface wear rate tends to the constant value.

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Fig. 8. Wear kinetics for rolling contact of elastic sphere and half-space when W = 0, γ = 10 and 1 – H = 0, 2 – H = 1, 3 – H = 2

4.3 Influence of Angular Velocity of Spinning The shear stress distributions calculated in the presence of longitudinal slip velocity and angular velocity of spinning is presented in Fig. 9.

Fig. 9. Shear stresses distribution for rolling contact of elastic sphere and half-space when γ = 10, H = 0 and 1 – W = 0, 2 – W = 1, 3 – W = 2

The results show that in case of rolling with spin the shear stress distribution is asymmetrical about the axis Ox passing through the center of the contact region. The dependence of the dimensionless coordinate z/a on the number of cycles N calculated for different values of parameters H and m is shown in Fig. 10.

Fig. 10. Wear kinetics for rolling contact of elastic sphere and half-space when γ = 10, H = 0 and 1 – W = 0, 2 – W = 1, 3 – W = 2

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The results illustrate that the parameter W, which refers to the angular velocity of spinning, slightly affects the detached layer thickness during fatigue wear for all considered values of parameter m.

5 Conclusion The kinetics of wear process in rolling with slippage under a constant load applied to the roller is studied. The contact normal and shear stresses distribution was used to calculate the internal stresses in the elastic half-space based on the boundary element method. The rate of damage accumulation caused by contact fatigue is assumed to be a function of the maximum values of the principal shear stresses. The influence of longitudinal and lateral slip velocities and angular velocity of spinning on the kinetics of the wear process was studied numerically. It is shown that the wear process has a delamination character. The results illustrate that with the growth of spin velocity the detached layer thickness decreases and the subsurface wear rate tends to the constant value. Acknowledgments. This work was partially supported by the Ministry of Science and Higher Education within the framework of the Russian State Assignment under contract No. AAAAA20–120011690132-4 and partially supported by RFBR according to the research project № 19–31-90015.

References 1. Sadeghi, F., et al.: A review of rolling contact fatigue. J. Tribol. 131(4), 1–15 (2009) 2. Goryacheva, I.G., Torskaya, E.V.: Modeling the accumulation of contact fatigue damage in materials with residual stresses under rolling friction. J. Frict. Wear 40(1), 33–38 (2019) 3. Carter, F.W.: On the action of a locomotive driving wheel. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 112(760), 151–157 (1926) 4. Goryacheva, I.G., Torskaya, E.V., Zakharov, S.M.: The effect of relative slippage and properties of the surface layer on the stress-strain state of elastic bodies in rolling friction. Frict. Wear 24(1), 5–15 (2003) 5. Zakharov, S.M., Goryacheva, I.G.: Rolling contact fatigue defects in freight car wheels. Wear 258(7–8), 1142–1147 (2005) 6. Walvekar, A.A., et al.: A novel modeling approach to simulate rolling contact fatigue and three-dimensional spalls. J. Tribol. 140(3), 031101 (2018) 7. Ghodrati, M., Ahmadian, M., Mirzaeifar, R.: Three-dimensional study of rolling contact fatigue using crystal plasticity and cohesive zone method. Int. J. Fatigue. Elsevier Ltd. 128, 105208 (2019) 8. Zhao, X.J., et al.: Effect of spherical and ballast dents on rolling contact fatigue of rail materials. Wear Elsevier Ltd. 450–451, 203254 (2020) 9. Sakalo, V., et al.: Computer modeling of processes of wear and accumulation of rolling contact fatigue damage in railway wheels using combined criterion. Wear Elsevier Ltd. 432–433, 102900 (2019) 10. Li, Y., et al.: Study on the Effect of Residual Stresses on Fatigue Crack Initiation in Rails. Int. J. Fatigue. Elsevier BV 139, 105750 (2020)

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11. Goldshtein, R.V., Zazovskii, A.F., Spektor, A.A., Fedorenko, R.P.: Solution of threedimensional contact problems of rolling with slip and adhesion by variational methods. Adv. Mech. 5(3/4), 61–102 (1982) 12. Goryacheva, I.G.: Contact Mechanics in Tribology. Springer Netherlands, Dordrecht (1998) 13. Johnson, K.L.: Contact Mechanics. Cambridge University Press (1985)

Minisymposium “Scientific and Technical Creativity in Experimental Mechanics”

Dynamics of Hollow Cylindrical Magnets Wilhelm Rickert(B) and Wolfgang H. M¨ uller Institute of Mechanics, Chair of Continuum Mechanics and Constitutive Theory, Technische Universit¨ at Berlin, Einsteinufer 5, 10587 Berlin, Germany {rickert,wolfgang.h.mueller}@tu-berlin.de

Abstract. A system of rigid permanent magnets of cylindrical shape is studied. In particular their motion due to gravity and their mutual magnetic interaction is investigated. Additionally, if the magnets move inside a copper tube an electric current is induced which in turn leads to another interaction that influences the motion of the magnets. The magnets are hollow cylinders centered around a rigid rod, which confines their movement to one dimension. Therefore, a coupled system of one-dimensional equations of motion arises. This system contains non-linear force terms as well as damping terms, which do not only depend on the positions and velocities of the magnets, but also on the orientation of their axial magnetization. The orientation determines if repulsion or attraction occurs. In contrast to approaches in literature, an exact closed-form solution for the magnetic field of a hollow cylindrical magnet in terms of elliptical integrals is used. This field is known for each magnet in its respective rest frame. However, the magnetic fields need to be known in the laboratory inertial frame in which the equations of motion are solved. Therefore, a transformation from the magnet’s rest frame to the laboratory frame is required. It is customary to simplify the Lorentz boost transformation using a low velocity approximation, which is reasonable for most engineering applications. Finally, the motion of the magnets is calculated for different numbers of magnets and different initial configurations.

Keywords: Dynamics of permanent magnets oscillations · Lorentz transformation

1

· Non-linear

Introduction and Problem Statement

The combination of electrodynamics and mechanics is a technically relevant topic that is also theoretically challenging. The effects that are relevant for this paper are magnetic braking and magnetic oscillation in general. These phenomena are exploited in several technical applications such as magnetic braking systems [2,4,21], energy harvesting systems [20] as well as vibration suppression of beams [18]. In these systems the interactions between electrodynamics and mechanics are not only complex, but to some extend uncertain. This manifests in the so called Abraham–Minkowski controversy, which raises the question of the correct electromagnetic momentum and force expression if magnetizable c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 363–380, 2022. https://doi.org/10.1007/978-3-030-92144-6_29

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or polarizable matter is present. The force expression itself enters the mechanical balances laws and thus influences the motion of a body. In order to study this interaction it is beneficial to analyze simple experiments involving both mechanics and electrodynamics. From a mechanical point of view the rigid body is the simplest extensive object and permanent magnets are the simplest examples of extensive bodies with electromagnetic properties. Therefore, experiments involving these objects are considered. Several problems that are closely related are analyzed. The dynamic interaction between cylindrical magnets is investigated by confining them to a onedimensional and frictionless motion on a non-conducting rod, which is schematically depicted in Fig. 1a. By adding a copper tube to the assembly, see Fig. 1b, one could first study the classical induction experiment of a slowly falling magnet through this tube as the gravitational force is compensated by an induction force. Second, additional magnets can be introduced and the resulting dynamics can be analyzed. It should be noted that in order for the middle magnet in Fig. 1b to be repelled by the other two fixed magnets, the magnetizations are opposing such that M 1 = −M 2 = M 3 . In all problems azimuthal symmetry is assumed and air resistance is neglected. Due to the linearity of Maxwell’s equations together with the assumption of permanent magnetizations, an uncoupled analysis of the electromagnetic fields is possible. Furthermore, since the problems are mechanically constrained by the rod, the motion of the magnets is one-dimensional, which renders the numerical integration of Newton’s equation of motion an easy task. The scenario in Fig. 1b is the more complicated. Therefore the following derivations are with respect to the damped magnetic oscillator. However, in the result section also the falling magnet inside a copper tube without additional magnets and the double magnetic oscillator from Fig. 1a are studied. It should be noted that most problems presented in the following were analyzed in literature before, see for example [1–3,7,9,10]. However, several simplifying assumptions were made, such as the two disk approximation or even the dipole approximation for the magnetic field of a cylindrical magnet. Other simplifications involve neglecting the edges of the copper tube. In this work none of these assumptions are made and the problems are solved numerically with no attempt of analytical solutions. Additionally the induction force is usually given without any preliminary considerations. However, in context of moving magnets, field transformations must be considered. In contrast to the cited literature, this work uses the Lorentz transformation in order to obtain the induction law in a rational manner, see for example [5] or [8]. Another conspicuity in the above cited literature is that the Abraham–Minkowski controversy seems to be completely ignored. Therefore, the controversy is briefly discussed here and its significance is investigated. Finally note that there are many different variants and notations in electrodynamics. In this work we follow the electromagnetic theory as presented in [8] and the notation from [12,22]. For a brief but complete summary of the used equations and corresponding symbols the reader is referred to [15].

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Fig. 1. Schematic depiction of a double magnetic oscillator assembly in which one magnet is fully clamped and the others are free to move vertically, because their motion is confined by a vertical rod. The magnets next to each other have opposing magnetizations, such that they are repelling. In 1b a copper tube is added and the top magnet is also fully clamped allowing only the magnet in the middle to move vertically.

2

Magnetic Field and Transformations

The analytical solution to the magnetic field of a potentially hollow cylindrical magnet with constant axial magnetization M = M0 ez is given in the cylindrical coordinate system of the magnet {ρ , ϕ , z  } by, see [16],  1  ˜n R M0 μ0  1  2 (k ρ, z˜) = (−1)n+m −2)K(k )+2E(k ) eρ B α (˜ n,m n,m n,m 2 π n,m=0 Hn,m kn,m

  Z˜m 2˜ ρ 2˜ ρ ˜ n + ρ˜ − R + 2 Π(h2n , kn,m ) + 2 K(kn,m ) ez , (1) Dn h2n hn for ρ˜ =

ρ >0, rm

z˜ =

z , rm

˜ = ρ˜eρ + z˜ez x

(2)

and

ro + ri ˜ m, , 2δ = ro − ri , (·) = (·)r 2 see Fig. 2. Furthermore, the auxiliary functions are defined as rm =

˜ n = 1 + (−1)n δ˜ , R 2 ˜ 2 + Z˜ 2 + 2˜ ˜n , = ρ˜2 + R ρR Hn,m n m

˜ , Z˜m = z˜ + (−1)m H 2 kn,m = 4ρ˜R˜ n/H2n,m ,

(3)

˜n , Dn = ρ˜ + R

h2n = 4ρ˜R˜ n/Dn2 . (4) The functions K(k), E(k) and Π(p2 , k) are the complete elliptic integrals of first, second and third kind, respectively, as defined in [17].

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The cylindrical coordinates used in the definitions above are in the corresponding eigensystem {ρα , ϕα , zα } of a magnet, see Fig. 2. In order to evaluate the magnetic field B α of the α-th magnet in the laboratory system it is required, first, to plug in the coordinate transformation z  = z − zα to Eq. (1) and, second, the transformation laws for the electromagnetic fields need to be considered.

Fig. 2. Depiction of the coordinate systems and geometric quantities.

It is customary to use the so called Lorentz boost transformation, which is essentially a relativistic Galileiean transformation and as such only applicable if both frames move at a constant speed relative to each other. This is not the case, because the magnet’s frame is accelerated with respect to the so-called inertial laboratory frame in which the resting copper tube is situated. However, as pointed out in [13, pg. 2379], the accelerated frames of the magnets can be connected to the laboratory frame by an “infinite sequence of comoving inertial frames.” In particular, since the considered frames in this work always have parallel axes, no Thomas precession needs to be considered, so that the application of the Lorentz boost is correct and not only an approximation. The low velocity approximations of the Lorentz transformations of some selected electromagnetic fields are given by, see [5]: E  = E + wr × B , 

D =D+ 

P =P −

1 c2 w r × H 1 c2 w r × M

B = B − , ,



1 c2 w r

×E ,

H = H − wr × D , 

M = M + wr × P ,

(5a) (5b) (5c)

where the dashed quantities refer to the moving frame and quantities without a dash are measured in the laboratory frame. The relative velocity between the frames, wr , is required to be small, i.e., wr · wr /c2  1, where c is the speed of light. These equations represent the fact that, e.g., a moving magnetic field leads

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to an electric field that is observed in a stationary frame. In a similar manner a moving magnetization causes polarization. Most authors simply ignore terms with a leading factor of c−2 , which results in statements such as P  = P and as a result: If there is no polarization in one frame then there is also no polarization in another frame. However, mathematical precaution forbids such immediate conclusions, because the fields in Eq. (5) can actually be of the order of the speed of light or c−2 . This is the same as claiming that the limit of x2 /x2 for x → ∞ is zero, which is obviously not true. Furthermore, even if M  is known, the magnitude of M is not known a priori. Therefore, it is worth investigating the transformations more rigorously. For example by substituting Eq. (5c)2 into Eq. (5c)1 one finds  r P  = (1 − w rc·w )1 + c12 wr ⊗ wr · P − c12 wr × M  2 (6) ≈ P − c12 wr × M  , where the dashed M  is used in contrast to Eq. (5c)1 . Therein, the approximation 1 − wr · wr /c2 ≈ 0 is in perfect agreement with the low velocity assumption. In Eq. (6) M  is the (most often known) magnetization in the moving frame with a known magnitude. Then it can safely be concluded that P ≈ 0 if P  = 0 as it is the case for a moving magnet. Analogously it can be shown that E  ≈ E + wr × B  ,

(7)

where the dashed magnetic flux B  is used in contrast to Eq. (5a)1 . From this it can be concluded that for a moving permanent magnet with E  = 0 the electric field observed in the laboratory frame is given by E ≈ wr × B  .

(8)

Hence, the magnetic field from the moving frame does not need to be transformed in order to obtain the corresponding electric field. Similar statements can be made for charges and electric currents, which are not shown here. In summary, a moving permanent magnet produces an electric field in the laboratory frame which is given by Eq. (8) and the polarization created due to the moving magnetization can be neglected. These statements are important for the force evaluation in Eq. (13) in the next section.

3

Total Force Analysis

The total force on the α-th magnet occupying the volumetric domain Ωα is given by

f (EM) dV +

Fα = Ωα

∂Ωα

(EM)

fI

dA ,

(9)

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where f (EM) and f I are the volumetric and surface force density, respectively. Volumetric forces are due to charges as well as currents and gradients in electromagnetic fields. Surface forces arise from steep gradients of electromagnetic fields across, e.g., the boundary of a magnet, where the magnetization suddenly jumps on a macroscopic scale. However, both forces are unknown if magnetizable or polarizable matter is present. This conundrum is referred to as Abraham–Minkowski controversy and is investigated in [15]. Unfortunately, there is no resolution of this problem yet. Rather there are several force model candidates that can be discussed. Starting with the introduction of the electromagnetic momentum densities, named after Lorentz, Abraham and Minkowski, respectively, gL = D × B ,

g A = D × μ0 H ,

gM = D × B ,

(10)

these are inserted into the so-called electromagnetic momentum balance ∂g (EM) = ∇ · σ (EM) − f (EM) , ∂t

(11)

from which the volumetric force density f (EM) can be obtained by using Maxwell’s equations and identities from vector calculus. This procedure does not yield the force densities in a unique manner, because the electromagnetic stress tensor, σ (EM) , may absorb certain terms. For an in-depth discussion see [15]. However, once a volumetric force density is obtained, the surface force density arises from (EM) = n · [[w ⊗ g (EM) + σ (EM) ]] , (12) fI where n is the outward surface normal, w is the surface velocity and [[·]] is the jump operator. As a reference model the so-called generalized Lorentz model is used, which is given by:   ∂P L f f +∇×M ×B , f = (q − ∇ · P )E + q v + j + ∂t (13) σ L = − 12 (0 E · E + μ10 B · B)1 + 0 E ⊗ E + μ10 B ⊗ B ,     f LI = qIf − n · [[P ]] E + J fI − [[P ]]w⊥ + n × [[M ]] × B , where · is the mean value across a surface and the various quantities are explained in [8] or [15]. In [14] it was shown that difference in the total force between the Lorentz model and any force model compatible with Eq. (11) is given by

d (g (EM) − g L ) dV . (14) F (EM) − F L = − dt Ω By analyzing the momentum differences one finds in view of Eq. (10) the following differences are obtained

d d A L M L E × M dV , F − F = B × P dV . (15) F − F = 0 μ0 dt Ω dt Ω

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Therefore, in order to see a model dependent difference in the total force one has to construct an experiment in which both electric and magnetic effects are present. Furthermore, by noting that 0 μ0 = c−2 one could easily argue that for real experiments the integral as well as its temporal change together with the leading factor c−2 are insignificant when compared to other forces such as gravitation or friction. Hence, there is no difference between the total forces of the different models that needs to be considered for the Abraham model. For the Minkowski model the difference to the Lorentz force vanished identically in the experiments described here, because P ≡ 0. Therefore, the force model choice does not matter in this scenario and the Lorentz model is used. From the generalized Lorentz model two contributions arise in the analyzed settings: A magnetic force, F mag , due to the mutual interaction between the permanent magnets and an induction force, F ind , due to the induced current inside the copper tube. Since Maxwell’s equations are linear and the magnets are assumed to be permanently magnetized, one can analyze the electric and magnetic effects separately. Afterwards the force contributions may be added such that the total force is given by F = F mag + F ind .

(16)

Both contributions are derived in the next sections. 3.1

Magnetic Interaction

In a purely magnetic setting with several permanent magnets the force densities in Eq. (13) reduce to   f L = 0 , f LI = n × [[M ]] × B , (17) because the magnetizations are constant, such that ∇ × M vanishes. Furthermore, no charges or currents as well as no polarization are present in the moving frame. Furthermore, the polarization due to the Lorentz transformation in Eq. (6) can be neglected by means of the arguments in Sect. 2. In the scenario depicted in Fig. 1b without the copper tube the total magnetic field is simply given by a superposition of the eigenfields B=

3 

Bα .

α=1

Hence, the total force on the second (middle) magnet with M = M 2 is given by 3  F mag = (n × [[M 2 ]]) × B α dA . (18) α=1∂Ω 2

However, since the a magnet cannot accelerate on its own, the total force must vanish if B 1 and B 3 are not present, i.e.,

(n × [[M 2 ]]) × B 2 dA = 0 . (19) ∂Ω2

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Furthermore, since the magnetic flux densities are continuous in the exterior domain of a respective magnet, the total magnetic force acting on the second magnet is given by

(20) F mag = (n × [[M 2 ]]) × (B 1 + B 3 ) dA . ∂Ω2

This integral can be simplified by noting that [[M 2 ]] = −κ2 M0 ez and hence

F mag = κ2 M0 eϕ × (B 1 + B 3 ) dA − κ2 M0 eϕ × (B 1 + B 3 ) dA . (21) Γ2o

Γ2i

where Γ2o and Γ2i are the inner and outer mantle of second magnet, respectively. Since the components of the magnetic fields do not depend upon ϕ one has, e.g.,

2π eϕ × B 1 =

Bz1 eρ

−

Bρ1 ez

,

eρ dϕ = 0

(22)

ϕ=0

and the corresponding integral reduces to

eϕ × B 1 dA = −2πez ro Bρ1 (ρ = ro , z) dz , Γ2o

I2 (z2 ) := [z2 − H, z2 + H] .

z∈I2

(23) Interestingly the resulting force expression only has a component in z-direction, which is compatible to the assumption of a one dimensional setting. Hence, one may define fmag (z2 ) := ez · F mag and find

 ρ=ro ρBρ1 + ρBρ3 fmag (z2 ) = −2πκ2 M0 dz . (24) z∈I2 (z2 )

3.2

ρ=ri

Electric Interaction

Similarly to the magnetic analysis, a single magnet inside a copper tube is considered. However, instead of calculating the force on the moving magnet, the total force acting on the copper tube is determined. Due to the moving magnetic field of the permanent magnet, an electric field is observed in the laboratory frame of the tube, which then results in a diffusive electric current. In order to see this, both the magnetic and electric field are decomposed into magnet’s and tube’s contribution in the respective frames: B = B magnet + B tube , 

B =

B magnet

+

B tube

,

E = E magnet + E tube , E  = E magnet + E tube ,

(25)

where the dash denotes the fields in the moving frame, in which magnet has no electric field, i.e., E magnet = 0. Additionally, the tube has no magnetic field

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in the laboratory frame, such that B tube = 0. In total, the electric field in the laboratory frame is calculated by using Eq. (8) and is given by E = E magnet = E magnet − wr × B magnet = −v(t)ez × B magnet .

(26)

Since the copper tube is conducting, Ohm’s law is used j f = σE = −σv(t)ez × B magnet ,

(27)

where σ is the conductivity of copper. Note that the fields B tube and E tube are not considered since they are of second order. Therefore, the force expressions in Eq. (13) reduce to: (28) f L = j f × B , f LI = 0 , where B = B magnet . The total force due to induction acting on the magnet is equal but opposite to the force acting on the copper tube and hence

  f ez × B × B dV F ind = − j × B dV = σv(t) ∂Ωtube

= σv(t)

∂Ωtube



 Bρ Bz eρ − Bρ2 ez dV = −σv(t)ez

∂Ωtube

(29) Bρ2 dV ,

∂Ωtube

where the Bρ Bz eρ term vanishes in the ϕ-integration. Note that this total force has also only a z-component and that the direction of F ind is always opposite to the magnets movement wr . Hence one may define ez · F ind =: −v(t)find (z) with ρ=R

o z=L−z2

ρBρ2 (ρ, z) dρ dz .

find (z) = 2πσ ρ=Ri

4

(30)

z=−z2

Equation of Motion and Numerical Analysis

The equation of motion for the moving magnet is obtained from Newton’s law m

d2 x = F (EM) + mg dt2

⇒

m

d2 z = ez · F (EM) − mg dt2

(31)

and with Eqs. (24) and (30) one has m

dz d2 z find (z) − mg . = fmag (z) − dt2 dt

(32)

In order to solve this equation for each movable magnet numerically, material parameters are required, a non-dimensionalization is essential and the right hand side must be investigated. In view of the expressions in Eqs. (21) and (29) the evaluation of the right hand side is probably numerically costly. Even if the force expressions are simplified by using the symmetries of the problem, the integration

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Table 1. Selected properties of the magnet and the copper tube from [19] and [11], respectively. Property

Symbol Value

Total axial length of the magnets

2H

31.50 mm

Outer diameter of the magnets

2ro

62.00 mm

Inner diameter of the magnets

2ri

10.00 mm

Remanence

μ0 M0

1.7 T

Mass density of copper

ρ0

7.00 g cm−3

Length of the copper tube

L

315.00 mm

Outer diameter of the copper tube 2Ro

98.00 mm

Inner diameter of the copper tube 2Ri σ conductivity of copper

160.00 mm (1.25–6) 107 Ω−1 m−1

of complicated compositions of elliptical functions is inefficient. Bearing in mind that for the integration of Eq. (31) many evaluations of the right hand side are required, the force expression are simplified and the method of pre-sampling is applied. In order to normalize the differential equation, the following non dimensional quantities are introduced  m ˜ ˜ = rm (˜ ρeρ + z˜ez ) , t = tref t = t˜ , x = rm x μ0 M02 rm (33) 2 2 ˜ ˜ ˜ B = μ0 M0 B , F = Fref F = μ0 M0 rm F . Additionally dimensionless damping and gravity factors are defined D=

2 σμ0 rm , tref

γ=

m0 g0 2 μ0 M02 rm

(34)

such that the equation of motions becomes d˜ z d2 z˜ + D f˜ind (˜ z ) − f˜mag (˜ z ) = −γ 2 ˜ dt dt˜

(35)

ind ˜ find . The numerical integration is perwith fmag = Fref f˜mag and find = Fref formed using Mathematica [6] Furthermore one has with the values from Tab. 1 m ≈ 650 g and also:

D ≈ 6.17 ,

γ ≈ 8.54 × 10−3 ,

Fref ≈ 745 N ,

ind Fref ≈ 1011

Ns . m

(36)

In Fig. 3 the pre-sampled functions are depicted for the reference values from Tab. 1 for the magnetic oscillator assembly from Fig. 1b. The numerical integration was performed by using Mathematica [6]. It can be seen that the induction

Dynamics of Hollow Cylindrical Magnets

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force function is roughly of the same order as the magnetic interaction. However, in view of Eq. (35) and the factor D roughly about a value of six, the induction force dominates the magnetic interaction for non-vanishing velocities such that strong damping is to expected. From the plots it is obvious that the magnet dipole model (dashed line) cannot represent the magnetic interaction correctly. In particular the contact force of the interaction between the magnets is vastly underestimated. However, at least the order of magnitude of this dipole force is similar for both the correct magnetic field and the dipole field. This is achieved by putting the magnetic dipole moment equal, instead of the magnetizations themselves: mdip = Vcyl M cyl , where Vcyl is the volume of the cylindrical magnet. However, this does not lead to satisfactory results for the induction force. The dipole model yields approximately 20% less than the true force value predicted by the analytical magnetic field of the cylindrical magnet. This in turn will lead to a different apparent damping coefficient.

Fig. 3. Pre-sampled dimensionless force functions from numerical integration for the case of the damped magnetic oscillator assembly depicted in Fig. 1b. The dashed lines ˜ with corresponds to the dipole model. The forces are functions of κ = z˜ − (˜ z1 + 2H) ˜ z˜3 − 2H], ˜ which is the dimensionless edge to edge distance between the z ∈ [˜ z1 + 2H, magnets, i.e., κ = 0 represents the contact of two magnets.

5

Results

In order to visualize the results the phase space is investigated for different initial conditions of the magnets. Three different scenarios are analyzed in the subsequent sections: (I) double magnet oscillator: two movable magnets and one fixed magnet on a rod;

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(II) falling magnet: a single magnet falling through a copper tube; (III) damped magnetic oscillator: a movable magnet between two fixed magnets on a rod inside a copper tube. 5.1

Double Magnet Oscillator

In Fig. 4 several initial conditions for the double magnetic oscillator are depicted and the corresponding initial conditions are given in Table 2 in terms of numbers. In the first two Figs. 4a and 4b both magnets are initially at rest, but the third magnet starts at different heights. Then, with the same elevated position of the third magnet as in Fig. 4a, in Fig. 4c both magnets start with an initial velocity v0 in different directions. Table 2. Initial conditions for the double magnet oscillator for different scenarios shown in Fig. 4. Case z2 (t = 0) z˙2 (t = 0) z3 (t = 0) z˙3 (t = 0) (a)

4H

0

20H

0

(b) (c)

4H

0

8H

0

18H

− 2trm ref

22H

rm 2tref

Fig. 4. Schematic view of different initial conditions for the double magnet oscillator.

In Fig. 5 two-dimensional subsets of the four-dimensional phase space zi /dt˜. For this scenario, where the third (˜ z2 , v˜2 , z˜3 , v˜3 ) are shown, where v˜i = d˜ magnet starts at some distance to the others, its motion remains regular and periodic in the considered time interval. Note that the amplitude of position becomes very high when compared to the initial position and to the position of

Dynamics of Hollow Cylindrical Magnets

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the second magnet. This explains why the phase space looks so regular: far away from the other magnets only the gravitational force has a significant influence and thus the classical parabolic curve becomes apparent. The situation is different for the second (middle) magnet, which executes a complicated motion, such that its phase space curve follows through two intersecting elliptic paths that constitute a cycle. However, apart from the mere resemblance of different cycles no limiting cycle is visible as the part of the path corresponding to the bigger ellipse bends and never overlaps completely with a previous one. This behavior is apparent even if the initial transient behavior is excluded and also continues to exist for further time integration. In view of the magnetic force function in Fig. 3a this system could be compared to a two-point-oscillator with non-linear springs, where the non-linearity triggers chaotic behavior. Of course, the term “chaotic” depends upon its definition and is often used laxly.

Fig. 5. Phase diagrams for two magnets of the double magnetic oscillator with initial conditions from Table 2 (a) for the time interval t˜ ∈ [0, 6000]. The squares represent the initial data and the triangles the static equilibrium positions.

In contrast to the previous initial conditions, the phase space for both magnets with small distances in shown in Fig. 6. In this scenario the motion of the third magnet is vastly different from before in the sense that there is no clear periodic behavior for large distances z3 /H > 100 recognizable. This reason for

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this can be found in the interaction with the other magnets. Since the initial magnetic energy is larger when compared to case (a), the second magnet travels greater distances and is thus able to interact with the third magnet more vividly, leading to disturbances in the quadratic profile. Furthermore note that the two elliptic shapes look very different in case (b) when compared to (a). Both path segments are stretched out and one even resembles a quadratic path.

Fig. 6. Phase diagrams for two magnets of the double magnetic oscillator with initial conditions from Table 2 (b) for the time interval t˜ ∈ [0, 6000]. The squares represent the initial data and the triangles the static equilibrium positions.

In Fig. 7 the phase space of the situation (c) is shown, in which both magnets start at an elevated position close to each other and with additional kinetic energy form their initial velocity. Here the chaotic behavior also becomes apparent. Interestingly, for both phase space subsets one could identify elliptical shaped paths.

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Fig. 7. Phase diagrams for two magnets of the double magnetic oscillator with initial conditions from Table 2 (c) for the time interval t˜ ∈ [0, 6000]. The squares represent the initial data and the triangles the static equilibrium positions.

5.2

Falling Magnet

The classical textbook and physics lecture example of the decelerated fall of a magnet through a copper tube is shown in Fig. 8. The magnet enters the copper tube with a vanishing initial velocity. The quadratic profile starts to develop since the magnet is not fully inside the copper tube and thus the induction force, which slows down the free fall, has not grown to its full extend, compare Fig. 3b. However, after a short time this transient behavior vanishes and stationary falling begins since gravitation and inductions forces compensate each other. After t˜ = 2 × 103 , which corresponds to t ≈ 7.9 s, the magnet has traveled through the copper tube and continues its free fall. It should be noted that the transition between stationary falling and free fall takes only a small amount of time. Thus, one may conclude that the “edge effects” of entering and leaving the tube are negligible, at least for the given set of parameters. The motion predicted by the dipole model (dashed line) shows overall the same behavior although the velocity is higher due to the weaker damping. Therefore, a travel time of roughly 6 s underestimates the true value given above significantly.

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Fig. 8. Position over time for a falling magnet inside a copper tube. The dashed line results from the dipole model. The solid black lines represent the points where the magnet enters and exits the copper tube fully.

5.3

Damped Magnetic Oscillator

Finally, a combination of the above experiments is considered in Fig. 9: A magnet moving in between two fixed magnets inside a copper tube. As was mentioned before, the damping due to induction dominates the magnetic interaction for the real material parameters in Tab. 1. Thus, the motion of the magnet quickly comes to a stop at the midpoint between the two fixed magnets, which is the point of static equilibrium. The initial position of the magnet does not matter, the result is the same. This demonstrates the clear separation of order of magnitudes (from high to low): induction force, magnetic force, and the gravitational force.

Fig. 9. Position over time magnet between two fixed magnets inside a copper tube. The solid black lines represent the positions where the magnet has face to face contact with the fixed magnets and the dashed line the static equilibrium position. Two motions of are shown for two different initial heights with zero initial velocity.

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Conclusion and outlook

Several variations of the electromagnetic interaction between cylindrical permanent magnets and a copper tube were analyzed in a rational manner. To this end the corresponding electromagnetic fields and the resulting total force on the magnets were computed in a mechanically one dimensional setting. It was shown that the Abraham–Minkowski controversy has no impact on the translational motion of rigid magnets, which may be the reason why the pertinent literature does not even mention the controversy at all. Furthermore, the induction force is obtained via the Lorentz boost transformation rather than just stating the final result. From a theoretical standpoint the assumption that the magnetic field information propagates instantly is in contradiction with special relativity. However, since the magnetic field information propagates at the speed of light, the retardation may be neglected on the time scale of the mechanical motion of the magnets. Another point of criticism is that the equations of electrodynamics are treated relativistically, while no change of observer is discussed for the mechanical movement. Newton’s law holds true in inertial frames such as the laboratory frame, but the magnetic force interaction is calculated in the moving frame. From the analysis of specific initial value problems we may conclude: – the non-stationary disturbances in the motion of a falling magnet into a copper tube at entering and exiting the tube are negligible for strong magnets, – Abraham–Minkowski controversy has a low impact on the translational motion of rigid bodies, – the dipole approximation is not capable of predicting the force to distance characteristics properly as near field solutions are important. In future work the four-dimensional phase space could be analyzed more systematically.

References 1. Donoso, G., Ladera, C.L., Mart´ın, P.: Magnet fall inside a conductive pipe: motion and the role of the pipe wall thickness. Eur. J. Phys. 30(4), 855–869 (2009) 2. Ebrahimi, B., Khamesee, M.B., Golnaraghi, F.: Permanent magnet configuration in design of an eddy current damper. Microsyst. Technol. 16(1–2), 19–24 (2008) 3. Hahn, K.D., Johnson, E.M., Brokken, A., Baldwin, S.: Eddy current damping of a magnet moving through a pipe. Am. J. Phys. 66(12), 1066–1076 (1998) 4. Heald, M.A.: Magnetic braking: improved theory. Am. J. Phys. 56(6), 521–522 (1988) 5. Hutter, K., Ven, A.A.F., Ursescu, A.: Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids. Springer, Heidelberg (2006). https:// doi.org/10.1007/3-540-37240-7 6. Wolfram Research, Inc., Mathematica, Version 10.2. Champaign, IL (2015) 7. Irvine, B., Kemnetz, M., Gangopadhyaya, A., Ruubel, T.: Magnet traveling through a conducting pipe: a variation on the analytical approach. Am. J. Phys. 82(4), 273–279 (2014)

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8. Kovetz, A.: Electromagnetic Theory. Oxford University Press, Oxford (2000) 9. Levin, Y., Silveira, F.L., Rizzato, F.B.: Electromagnetic braking: a simple quantitative model. Am. J. Phys. 74(9), 815–817 (2006) 10. MacLatchy, C.S., Backman, P., Bogan, L.: A quantitative magnetic braking experiment. Am. J. Phys. 61(12), 1096–1101 (1993) 11. Matula, R.A.: Electrical resistivity of copper, gold, palladium, and silver. J. Phys. Chem. Ref. Data 8(4), 1147–1298 (1979) 12. M¨ uller, I.: Thermodynamics. In: Interaction of Mechanics and Mathematics Series. Pitman (1985). ISBN 9780273085775 13. Nelson, R.A.: Generalized Lorentz transformation for an accelerated, rotating frame of reference. J. Math. Phys. 28(10), 2379–2383 (1987) 14. Reich, F.A.: Coupling of continuum mechanics and electrodynamics: an investigation of electromagnetic force models by means of experiments and selected problems. Doctoral thesis, Technische Universit¨ at Berlin, Berlin (2017) 15. Reich, F.A., Rickert, W., Stahn, O., M¨ uller, W.H.: Magnetostriction of a sphere: stress development during magnetization and residual stresses due to the remanent field. Contin. Mech. Thermodyn. 29(2), 535–557 (2016). https://doi.org/10.1007/ s00161-016-0544-8 16. Reich, F.A., Stahn, O., M¨ uller, W.H.: The magnetic field of a permanent hollow cylindrical magnet. Contin. Mech. Thermodyn. 28(5), 1435–1444 (2015). https:// doi.org/10.1007/s00161-015-0485-7 17. Reich, F.A., Rickert, W., M¨ uller, W.H.: An investigation into electromagnetic force models: differences in global and local effects demonstrated with selected problems. Contin. Mech. Thermodyn. 30(2), 233–266 (2018) 18. Sodano, H.A., Bae, J.-S., Inman, D.J., Belvin, W.K.: Concept and model of eddy current damper for vibration suppression of a beam. J. Sound Vibr. 288(4–5), 1177–1196 (2005) 19. Stainless Steel Encased N42 Neodymium Magnet. Magnet Expert Ltd. (2020). https://www.first4magnets.com/circular-disc-rod-c34/63mm-dia-x-31-5mmthick-stainless-steel-encased-n42-neodymium-magnet-80kg-pull-p3463#ps 0 3533|ps 1 1329 20. Tang, L., Yang, Y.: A nonlinear piezoelectric energy harvester with magnetic oscillator. Appl. Phys. Lett. 101(9), 094102 (2012) 21. Tossman, B.E.: Variable parameter nutation damper for sas-a. J. Spacecraft Rock. 8(7), 743–746 (1971) 22. Truesdell, C.A., Toupin, R.: The classical field theories. In: Handbuch der Physik, Bd. III/1, pp. 226–793; appendix, pp. 794–858. Springer, Berlin (1960). https:// doi.org/10.1007/978-3-642-45943-6 2

Experimental Study on Hydraulically-Driven Fracture Initialization and Propagation in the Gelatin Mixture M. B. Babenkov1,2 , E. A. Belousova1 , B. A. Islamov1 , S. F. Lebedev1 , V. A. Timoshenko1(B) , and A. U. Vasilyeva1 1 Higher School of Theoretical Mechanics, Peter the Great St. Petersburg Polytechnic

University, Polytechnicheskaya, 29, 195251 Saint-Petersburg, Russia [email protected] 2 Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bolshoy pr. 61, V.O., 199178 Saint Petersburg, Russia

Abstract. The main purpose of our study is to observe the crack propagation in real time. When pursuing this goal, the researchers usually provide full-size tests using the concrete mixtures and real proppant. However, such experiments are quite expensive. Another disadvantage is the inability to fully control and record the crack propagation due to non-transparency of the rock modelling material. This is the reason why we decided to do simpler, cheaper and more obvious series of experiments to achieve our purpose. In our experimental setup the transparent gelatin mixture and the ink-dyed water model the rock formation and the hydraulic fracturing fluid respectively. We also consider the analytical PKN model and compare the results of the calculations with the experimental results. In the work we directly observe the fracture growth and the petals formation. The process was recorded on video. Further, we also obtain analytical and experimental dependencies of the flow rate, pressure and the fracture growth over time. Keywords: Hydraulic fracturing · PKN Model · The gelatin mixture · Hydraulic fracture propagation

1 Introduction One of the most common oil recovery technologies is hydraulic fracturing. The technology consists in the injection of hydraulic fracturing fluid into the perforated well under high pressure. As a result, a fracture appears near the well and propagates into the rock formation. After a certain time, the proppant is added to the hydraulic fracturing fluid to fix the crack opening. Flow capacity of fracture depends significantly both on its shape and proppant distribution. In article [1] a triaxial hydraulic fracturing apparatus and 3D scanning device for a specimen of coal layer were applied to monitor of pressure inside of fracture and its geometry in time. As a result, it has been found that the pressure is inversely proportional to the difference in horizontal stresses and relates to the rate of fluid injection. The authors © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 381–396, 2022. https://doi.org/10.1007/978-3-030-92144-6_30

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also noticed that constant vertical and increasing horizontal stresses lead to decreasing in pressure and crack width. Moreover, they stated that the propagation of natural fractures perpendicularly to the direction of minimal horizontal stress gives the connection with the main fracture. Another series of experiments [2] considers the hydraulic fracture propagation in the sandstone, shale and granite formations. The authors use the X-ray computer scanner to obtain spatial distribution of cracks. They stated that the length of the cracks in the cores of sandstone, slate and granite varies greatly. Initial heterogeneities in the materials, such as the layered structure of slate, as well as grain boundaries and pores in granite, played an important role in the propagation and tortuosity of cracks [2]. In addition, the authors observed microcracks around the injection well in the sandstone core. In paper [3] the authors considered the hydraulic fracturing in an elastic gelatin matrix. As a result, they obtained a disc-like shape crack and found that some parameters like the flow rate, Young’s modulus of the matrix and fluid viscosity can influence the crack formation. They stated that the crack radius increases with time as t a , where a = 0.48 ± 0.04. In addition, the authors measured the crack evolution over time. In comparison with the previous works, the direct observation of the fracture growth is possible in the experiment setup given bellow. It allows to measure directly the following quantities: flow rate, pressure and the resulting fracture geometry. The gelatin mixture was chosen as a convenient material, transparent enough to observe the crack propagation in real time. In turn, adding the ink to the fracturing fluid helps to significantly improve the crack visibility.

2 The Experimental Installation In this experiment the gelatin mixture models the rock formation, and the colored water models the hydraulic fracturing fluid. The scheme of the installation is represented in Fig. 1.

Fig. 1. The scheme of the installation

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Table 1. The components of the installation and their properties Components

Properties

1

Injection syringe

2 th., volume 60 ml

2

Tube from the drip bulb

Inner diameter 6 mm

3

Brass triple adapter

Outer diameter 5 mm

4

Water manometer

Maximum pressure 0.1 MPa

5

Polyvinylchloride pipe

Inner diameter 12 mm

6

Medical needle

Diameter 1 mm

7

Transparent container

138x104x69 mm

8

Brass tube

Inner diameter 2 mm

9

Materials for connection and leakproofness

Hot glue, threads, jointing material

All necessary components of the installation are in Table 1. The assembly of the installation is as follows. Brass tube with hole 8 is frozen into container with gelatin mixture 7. The water injection system is connected to tube 8 with medicine needle 6. The water injection system consists of two connected injection syringes with volume of 60 ml each one. Both syringes, manometer 4 and container with tube are connected with tubes system (2, 3, 5). The installation is represented in Fig. 2.

Fig. 2. The installation

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3 The Experiment and Discussion In this experiment a 1.1 L container fully filled with the gelatin mixture is used (see Fig. 3). The brass tube having 1.5 mm in diameter represents the well bore. The tube has one perforation.

Fig. 3. The container with the gelatin mixture and frozen brass tube (the perforation is on the left picture)

The hydraulic fracturing fluid is injected with the system of two syringes. The ink-dyed water models the hydraulic fracturing fluid. The development of the crack is represented in Fig. 4 and Fig. 5. The crack formed in horizontal direction mainly at an angle to vertical, forming two petals lying in different planes. Both petals spread radially from the tube, which means that the fracture stress is constant in petals plane. Such positions of the petals supposed to be caused by residual temperature stresses during the hardening process. Mentioned stresses supposed to appear due to two reasons. The first one is hardening the gelatin mixture in the fridge, and the second one is difference between thermal conductivities of the gelatin mixture and the brass tube, which has the greater value of this parameter. Consequently, the brass tube cools faster than the gelatin mixture, and this causes the appearance of temperature gradient. It means that the residual stresses appear near the tube during the hardening process. The crack opening occurs in the direction of minimum stresses. The direction of minimum stresses deviates from the vertical by an angle approximately equal to 30 degrees. It is caused by the inclination of the container edges. The petals spread in mutually perpendicular planes, which can also be explained by the container configuration. During the experiment, we have observed that one of the crack petals spread significantly faster and reached the gelatin mixture surface. Supposedly it can be caused by the following reasons: – the decrease of the depth leads to the decrease of the lateral pressure; The vertical pressure decreases due to hydrostatic law, and the lateral pressure is directly proportional to the vertical ones. It makes it possible to the fracture to propagate in mentioned direction because the rupture pressure decreases.

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– the increase the area of the container with simultaneous decrease of the depth leads to decrease of the lateral pressure; If the stresses are the same and the linear dimensions grows, the compression ratio grows too. We consider the crack opening to be constant along its half-length, that means the area near the crack plane has the same compression ratio along its total length. Consequentially, when the area grows, the stresses decrease, and this causes the decreasing in rupture pressure. – there is an impact of rigid boundaries. The first two reasons can be neglected due to the small depth and small changes in the area of the container. However, the third reason makes a significant contribution to the development of the crack. The boundary is made from another material with its own elastic properties. The gelatin mixture glues to this boundary, and it causes the boundary stresses.

Fig. 4. The crack growth. Frontal view, see video [4] (the perforation is on the other side of the tube)

It is quite important to note that even in scale of this experiment there is a partial collapse of the crack after the removing the pressure from the syringe pistons, as in the real hydraulic fracturing. That means that there is excess pressure in the crack. During the experiment there were several technical difficulties, which affected the accuracy of the obtained results. They include the appearance of residual stresses and the impact of the container configuration. Moreover, the perforation hole was comparable with the diameter of the tube, which could greatly affect the course of the experiment. Certain difficulties were also caused by the achievement of acceptable transparency of

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Fig. 5. The crack growth. Side view, see video [5]

the gelatin mass. The accuracy of the experiment could be decreased by the significant impact of the boundaries, which can be neglected by conducting an experiment on larger containers. Our team tried two water injection techniques, the first one is applying the sudden dynamic load and the second one is applying the quasistatic load. It was impossible to finish the experiment with the first kind of injection because the pressure was too large. Such pressure led to leakage in the connections between the components and made it impossible to rupture the gelatin mixture. In the second case the injection was gradual. The liquid always follows the path of the least resistance. Consequently, the sudden pressure increase has damaged the components connections in the first instance. In turn, it causes the pressure decrease, and as a result, the liquid doesn’t rupture the gelatin mixture. In another case, the pressure grows gradually both in the components’ connection places and the brass tube. Such a uniform pressure growth has let us to observe the hydraulic fracturing process.

4 The Results of the Experiment The fluid pressure, the fluid rate and the growth of the crack were measured during the experiment. All values are presented in following Table 2 and Table 3: As a result of the data processing, the dependencies of the fluid rate q, the pressure P, the length L and the height h of the crack over time were obtained. Approximation functions were selected based on the formulae presented in the mathematical modelling part. According to these formulae, the pressure and crack length have a power relationship. The dependencies are shown on Fig. 6, Fig. 7, Fig. 8 and Fig. 9: The formulae used for approximation the graphs presented below (Fig. 6, Fig. 7, Fig. 8 and Fig. 9) are as follows: q(t) = 1.3 t 0.7

(1)

P(t) = 7 · 103 t 0.5

(2)

L(t) = 7.7 t 0.23

(3)

h(t) = 3.2 t 0.5 + 1.3

(4)

Experimental Study on Hydraulically-Driven Fracture Table 2. The fluid rate and the hydraulic fracturing pressure Fluid rate,cm3 /s

Time, s

Time, s

Pressure, kPa

0.00

0.0

0.92

6.7

0.74

0.5

1.23

7.8

0.99

1.0

1.39

8.2

1.24

1.5

2.46

10.9

1.75

2.0

3.39

12.8

2.23

2.5

3.70

13.5

2.98

3.0

4.62

15.0

3.73

3.5

5.54

16.5

4.73

4.0

7.24

18.8

5.48

4.5

8.47

20.4

6.73

5.0

9.39

21.5

7.73

5.5

9.70

21.8

8.98

6.0

11.09

23.3

10.48

6.5

12.17

24.4

11.48

7.0

12.63

24.9

12.73

7.5

13.71

25.9

13.98

8.0

13.86

26.1

14.73

8.5

14.79

26.9

15.98

9.0

16.94

28.8

16.98

9.5

17.71

29.5

18.24

10.0

19.25

30.7

Table 3. The length and the height of the crack Time, s

The length of the crack, mm

Time, s

The height of the crack, mm

0.00

0.0

0.00

0

0.24

6.0

0.24

3.3

2.73

8.7

0.48

4.6

3.48

10.0

2.76

5.9

5.74

11.3

3.48

7.8

9.48

12.0

5.16

9.2

12.49

12.7

7.68

9.8

15.48

14.0

9.24

11.1

17.48

15.3

11.16

11.8

19.24

16.7

19.08

15.0

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Fig. 6. The dependency of the fluid rate over time. The dots represent the experimental data and the solid curve represents the approximation function

Fig. 7. The dependency of the fluid pressure over time. The dots represent the experimental data and the solid curve represents the approximation function

Fig. 8. The dependency of the length of the crack over time. The dots represent the experimental data and the solid curve represents the approximation function

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Fig. 9. The dependency of the height of the crack over time. The dots represent the experimental data and the solid curve represents the approximation function

The flow rate as a function of time is found by differentiating the approximate function q(t), as follows (Fig. 10): Q(t) = 0.9 t −0.3

(5)

Fig. 10. The dependency of the flow rate over time

After that our group made the experiment of determination of the elastic characteristics of the gelatin mixture used. As a specimen we took the gelatin mixture of cylindrical form with the radius R = 5.1 cm and the height h = 3.5 cm. The specimen was loaded with mass, and after that the deformation in the longitudinal and transverse directions was measured. The elastic characteristics are calculated according the following formulae [6]: d · l (6) d · l where ν is Poisson ratio; l, d are longitudinal and transverse dimensions of the specimen before the deformation; l, d is the deformation of the specimen in the longitudinal ν=

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and transverse directions; F ·l (7) S · l where E is Young modulus, F is the force applied normal to the square; S is the specimen section area to which the force is applied. The specimen before and after the load is shown on Fig. 11 and Fig. 12: E=

Fig. 11. The gelatin mixture before the deformation

Fig. 12. The gelatin mixture under the load

For calculations the following data is used (Table 4): Table 4. Experimental data m,g

l, mm

d ,mm

l  ,mm

d  ,mm

l,mm

d ,mm

586

35

101

27

106

8

5

The letters l and d without a prime represent the geometrical characteristics of the specimen before the loading and these letters followed with the prime represent the same characteristics after the loading.

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As the result, according to the formulae (6) and (7) the elastic characteristics of the gelatin mixture are the follows: E = 3200Pa, ν = 0.22 The instruments uncertainties are presented in Table 5: Table 5. The instruments uncertainties The ruler, mm

The medical syringe, ml

The manometer, kPa

Scales, g

0.5

0.5

1

0.01

5 Mathematical Modelling The height, the length of the fracture and the fluid pressure can also be calculated analytically. For such calculations mathematical models of spreading the hydraulic fracturing crack are used [7]. One of them is the Perkins-Kern-Nordgren (PKN) model. This model has several assumptions. The liquid flow supposed to be one-dimensional, and the liquid is Newtonian. This way the mechanical task becomes two-dimensional in homogeneous isotropic medium, and the growth of the fracture spreads in one direction. Additionally, the model doesn’t include the gravitational effects. The PKN model assumes that the fracture has a constant height significantly less than the total crack length (see Fig. 13). This assumption makes possible to consider that the change in the parameters of the crack along its length is not significant, and the deformation of the rock can be considered in each vertical section as a plane strain condition. It is assumed that the excess pressure in each section is constant, then the fracture opening profile is elliptical. Due to the assumption that the pressure is constant, we can write the coupling equation between the excess pressure and the maximum fracture opening in each section [8]: Wmax (x) =

2H pnet (x) E

(8)

The solution of the problem of stationary flow of Newtonian liquid in a cylindrical pipe with elliptical section, considering the small width of the crack compared to its height, gives [6]: 64μ ∂pnet =− Q(x, t) 3 H ∂x π Wmax

(9)

where μ is dynamic viscosity of hydraulic fracturing fluid, H is the height of the fracture, Q(x,t) is local fluid flow through x section of unit fracture height.

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Fig. 13. Geometrical concept of the model

After the substitution of Eq. (8) into Eq. (9) and further integration, we get the expression for excess pressure pnet in dependence on coordinate x: 2 pnet (x) = H



 41  3 2μQ(x,t) E  (L − x) π

(10)

Then considering the Eq. (10), the maximum crack opening Wmax in dependence on coordinate x is determined as: 1  4 2μQ(x,t) Wmax (x) = 4 (11) − x) (L  πE It is also necessary to use boundary condition at the tip of the fracture L to determine the crack closure at the fracture tip: pnet (L) = 0

(12)

Considering the assumptions that the fluid flow is constant along the fracture, the leakage of fluid is absent, and the time of the fracture development is small enough, we can use approximation formulae modelling the hydraulic fracturing crack [9]:   3  15 4 EQ t5 L(t) = 0, 68 4 2μH

(13)

1  2μQ2 5 1 t5 Wmax (0, t) = 2, 5 EH

(14)

  4 2  15 1 (E ) Q t5 pnet (0, t) = 2, 5 6 16H

(15)

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6 The Calculation Results It is necessary to use some information (Table 6), which was obtained during the experiment, to evaluate the PKN model. Table 6. Information obtained during the experiment Experimental information needed

The PKN model

Local fluid flow The height of the crack

The length of the crack Fluid pressure The width of the crack

The results are represented in Fig. 14, Fig. 15 and Fig. 16:

Fig. 14. The dependency of the length of the crack over time. The solid curve represents the experimental data and the dashed curve represents the analytical solution

The graph of the length dependency (Fig. 14) shows the smooth increase in the length of the crack over time. It is worthwhile noting that the PKN model gives more accurate results some time after the crack appearance. While considering the graph of the width dependency (Fig. 15), it can be noticed that the PKN model shows the growth of the crack over time at early times. Further, the graph shows the slight decline, which can be explained with the partial collapse of the crack and redistribution of the injected liquid. In Fig. 16 we can see the divergence between the pressure curves for the experimental data and the analytical solution. It is caused by the fact that the model applicability conditions don’t completely match the experiment. The PKN model supposes the height and the flow rate to be constant, but in the experiment they are not as shown in Fig. 9 and Fig. 10. If we consider the constant values for the height H = 10 cm and the flow rate Q = 1 cm3 /s, the graph shape for the analytical solution (Fig. 16, the dot-dash curve) will be similar to the experimental one (Fig. 16, the solid curve).

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Fig. 15. The dependency of the width of the crack over time. The vertical line represents the end of injection

Fig. 16. The dependency of the pressure over time. The solid curve represents the experimental data, the dashed curve represents the analytical solution and the dot-dash curve represents the corrected solution

In Fig. 16 we also see that the PKN model overestimates the pressure in the beginning. The possible explanation of this fact is that the PKN model is designed for calculations in rock formation, and we use the gelatin mixture, which is more malleable. In addition, the pressure slowly increases in the real experiment. Further, we notice that the model underestimates the pressure after the certain moment. It is connected with the fact that in the real experiment we reach the pressure value, which is enough to further crack propagation on its own, but the model does not take it into account.

7 Conclusion The performed experiment allows one to observe the crack propagation in real time. As a result, the fluid pressure, the fluid rate and the growth of the crack were measured, and the empirical expressions were obtained.

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In addition, the PKN model was considered in this experiment. This analytical model supposes the liquid flow to be one-dimensional; the liquid supposed to be Newtonian; the propagation of the crack is in vertical direction and the medium assumed to be homogeneous and isotropic. Moreover, there is no gravitational effects in this model. The time evolution of the pressure and the crack length calculated based on the PKN model were compared with the experimental results. The comparison between analytical solution and experiment data shows that the PKN model quite accurately describes the crack growth. Moreover, the best accuracy has been achieved some time after the appearance of the crack. However, the analytical model doesn’t describe the fracture grows after the rupture, although the fracture continued to grow in the experiment. However, for the pressure the PKN model results significantly differ from the experimental values. It is explained by the fact that the PKN model suppose the height and the flow rate to be constant while they change during the experiment. For the constant values of these parameters, the experimental data agrees well with the analytical solution. The time evolution of the crack width was also studied. According to the calculations, the crack width is quite small, as expected. In addition, there is a slight decrease of the width, which can be explained by the redistribution of the injected fluid and, as a consequence, by the partial collapse of the crack. In addition, during the experiment two crack petals with different dimensions lying in different planes were obtained. It is important to notice that the PKN model doesn’t describe the appearance, amount and position of crack petals. The installation can be modified to alter the experiment. For example, it is possible to add the contrast proppant to the transparent fracturing fluid to monitor its propagation in crack plane. Another option is the injection of the fracturing fluid with the lower temperature in comparison with the gelatin mixture temperature. It can give a possibility to observe the temperature field evolution in the gelatin mixture with a thermal camera and allow one to examine the suggested models [10, 11] and their equivalents. The PKN model still gives appropriate results, which supposes the idea of the scale invariance of the process. Acknowledgements. The authors are thankful to Kovalev O. for useful discussions. We thank the anonymous reviewer for their constructive comments, which helped us to improve the manuscript.

References 1. Zhang, F., Ma, G., Tao, Y., Liu, X., Feng, D., Li, R.: Experimental analysis of multiple factors on hydraulic fracturing in coalbed methane reservoirs. https://doi.org/10.1371/journal.pone. 0195363 (2018). Accessed 5 April 2018 2. He, J., Lin, C., Li, X., Wan, X.: Experimental investigation of crack extension patterns in hydraulic fracturing with shale, sandstone and granite cores (2016). Accessed 1 December 2016 3. Lai, C.-Y., Zheng, Z., Dressaire, E., Wexler, J.S., Stone, H.A.: Experimental study on pennyshaped fluid-driven cracks in an elastic matrix. Proc. R. Soc. A 471, 20150255 (2015). https:// doi.org/10.1098/rspa.2015.0255 4. www.youtube.com/watch?v=G6Mq9GrVcGA

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5. www.youtube.com/watch?v=uGTkHDRUprc 6. Putilov, K.A.: The course of Physics. In: Mechanics, Acoustics, Molecular Physics, Thermodynamics, vol. 1, 560 p. (1963) (in Russian) 7. Yew, C.H.: Mechanics of Hydraulic Fracturing. Gulf Publishing Company, Houston, USA (1997) 8. Esipov, D.V., Kuranakov, D.S., Lapin, V.N., Cherny, S.G.: Mathematical models of hydraulic fracturing. Comput. Technol. 19(2), 33–61 (2014). (in Russian) 9. Yildizdag, K., Weber, F., Konietzky, H.: Hydraulic fracturing. TU Bergakademie Freiberg, Geotechnical Institute 10. Babenkov, M.B.: Temperature of rock formation and fracturing fluid during the hydraulic fracturing process. In: IOP Conference Series: Earth and Environmental Science, vol. 193, issue 1. IOP Publishing (2018). https://doi.org/10.1088/1755-1315/193/1/012076 11. Babenkov, M.B., Starobinskii, E.B.; The non-stationary temperature field of the fracturing fluid during the propagation of planar 3D hydraulic crack. In: AIP Conference Proceedings, vol. 2216, issue 1. AIP Publishing (2020). https://doi.org/10.1063/5.0003420

Correlation Model of Fracturing Fluid Viscosity with Regard to Proppant Concentration D. A. Nikiforov, A. A. Rybakovskaya(B) , I. S. Senkin, and O. I. Tsykunov Peter the Great St. Petersburg Polytechnic University, St Petersburg, Russia

Abstract. The aim of this work is to study the behavior of the viscosity of a fracturing fluid with different concentrations of suspended particles, using the example of a cross-linked solution of guar gum with the addition of quartz sand with a fraction of 0,63 – 1,25 mm. The purpose is to obtain an empirical equation for modeling the behavior of fracturing fluid, which moving in capillaries and rock cracks, with different particle contents. In addition, by analyzing the obtained dependences, it is possible to determine the critical content of particles in the polymer, which will optimize the progress of the polymer in cracks when modeling the hydraulic fracturing process. The viscosity values of the mixture were obtained using a flow viscometer based on the patent formula for calculating the viscosity of non-Newtonian fluids.

1 Introduction Polymers are widely used in oil production. They are used at many stages of field development. During drilling, polymers act as cooling, cleaning and protective agents as drilling fluids. They are also used as fluids that reduce resistance, which allows increasing the drilling speed. Polymers are used in the treatment of well bottom zones to improve the performance of injection wells or flooded production wells by blocking high permeability zones. They can act as agents that reduce the mobility of water or reduce the ratio of mobilities of water and oil in tertiary methods to increase oil recovery. All polymers used in the above processes have common characteristics: high molecular weight and high solution viscosity at low polymer concentrations. In the production of oil and gas, three types of polymers are used: polysaccharides (biopolymers), modified and synthetic polymers (polyacrylamides and hydrolyzed polyacrylamides). Due to its low cost, some of the most common polymers are the synthetic polymer hydroxypropyl guar and the biopolymer guar gum. The Fig. 1 shows the structural units of the polymers. Hydroxypropyl guar and guar gum are used to increase oil production through hydraulic fracturing. In this process, a proppant, such as sand, suspended in a solution thickened with guar or hydroxypropyl guar solution is pumped into the well under pressure in order to create and expand cracks in the rocks and to allow oil to leak into the well. Due to the cross-linking with borate, transition metal ions (Zr and Ti) or with ethylene glycol dinitrate (EGDN), gelation or crosslinking of the polymer solution often occurs (Fig. 2). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 397–406, 2022. https://doi.org/10.1007/978-3-030-92144-6_31

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Fig. 1. Structural formula of (a) guar gum unit and (b) Hydroxypropyl Guar (HPG) unit

Fig. 2. The crosslinked polymer

This study will be based on a solution of guar cross-linked with sodium tetra borate. Guar at low concentrations has high low shear rate viscosity at low concentrations, excellent suspending properties, pseudo plastic (shear thinning) rheological properties, excellent stability in a wide pH range (>12), and excellent retention of viscosity at temperatures up to 150 °C. One of the most important characteristics of a fracturing fluid is viscosity. It is required to assess proppant movement in the rock fracture and fracture opening time. As is known, viscosity refers to the property of a liquid to resist the shear or sliding of some fluid layers relative to others, resulting in the conversion of the kinetic energy of the ordered motion of the layers into the energy of the thermal motion of the molecules. A power law describes the shear stress τ for non-Newtonian fluids:  n ∂υ , (1) τ =K ∂y Where: K is the flow consistency index (SI units Pa·sn ),

Correlation Model of Fracturing Fluid Viscosity

399

∂υ/∂y is the shear rate or the velocity gradient perpendicular to the plane of shear (SI unit s−1 ), n is the flow behavior index (dimensionless). The quantity represents an apparent or effective viscosity as a function of the shear rate (SI unit Pa·s):  n−1 ∂υ μeff = K (2) ∂y Guar gel exhibits non-Newtonian properties. It belongs to pseudo plastic fluids for which the dimensionless coefficient n < 1. In this case, the viscosity of the polymer varies depending on the shear rate; the effective viscosity should decrease with increasing velocity gradient. This shear sensitivity implies that the guar properties observed under static fluid conditions are different from those under fluid flow conditions. The guar viscosity should therefore describe the fluid capability to transport proppant while the fluid is flowing during a fracturing treatment, and to suspend proppant particles while it is static after the fracturing treatment. Changes in the effective viscosity of a fracturing fluid are influenced not only by the environmental conditions (shear rate, pressure and temperature), but also by the size, shape and concentration of solid particles. The mechanical friction of the particles between themselves and with the layers of liquid increases the resistance of flow. This work is focused specifically on studying the effect of the solid phase on the viscosity of a polymer solution during the flow. The measurement of viscosity in dynamics is possible based on Poiseuille’s law, which determines the flow rate of a fluid with a steady flow of a viscous incompressible fluid in a thin cylindrical tube of circular cross section:  R  υ(r)rdr (3) Q = υ(r)dS =2π S

r

Poiseuille’s law is applicable only in case of laminar flow, when length of the tube exceeds the length of the initial section, which is necessary for the development of a laminar flow in the tube with a parabolic velocity profile. The velocity profile in the cross section of a cylindrical tube for a Newtonian fluid (n = 1), where η is the coefficient of dynamic viscosity, P1 – P2 is the pressure drop at the ends of the tube, and L is the cylindrical tube length: υ(r) =

P1 − P2 2 (R − r 2 ) 4ηL

(4)

Poiseuille’s equation will take the classical form after integrating from zero to R: Q=

π R4 (P1 − P2 ) 8ηL

(5)

Now the equation for a non-Newtonian fluid should be received, taking into account the fact that based on the power law, the velocity profile in the section is:   1  n+1 n+1 n dP 1 n n n υ(r) = R (6) −r n + 1 dz 2K

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For a power law, the Poiseuille equation takes the form: Q=2

− 1n

3+ 1n

πR



P1 − P2 LK

1 n

1 1 + 3n

(7)

Determining the coefficients characterizing the viscosity of a non-Newtonian fluid based on the received Eq. (7) is a complex and time-consuming task. There is an express method for determining the viscosity of non-Newtonian fluids in patent [5]. This method was described by a simplified formula received based on the above laws. The measurement method includes pumping water and non-Newtonian fluid through the channel, calculating the second flow rate and average flow rate based on the measured data and the subsequent calculation of the viscosity by the formula:  w 4 υav g w (8) η =η · g υav Where: ηw is water viscosity (Newtonian fluid), mPa·s; ηg is viscosity of non-Newtonian fluid, mPa·s; w is the average flow rate of water (Newtonian fluid), based on the measurements, m/s; υav g υav is the average flow rate of non-Newtonian fluid, based on the measurements, m/s. In this case, the viscosity of water (Newtonian fluid) is taken according to the reference data. With the known diameters of the channel cross sections and the viscosity of the water, the solution of the problem of determining the output parameters included in the right hand side of the presented function (8) and the determination of the viscosity of a non-Newtonian fluid are simplified significantly.

2 Experimental Setup The purpose of this work is to study the behavior of the viscosity of a fracturing fluid with different particle contents when moving in channels and rock fractures. A guar gum solution cross-linked with sodium tetra borate was used in the experiment. The viscosity of the obtained gel with various concentrations of proppant was studied in a special installation (Fig. 3). The installation is a flowing viscometer, which consists of a tank with the original liquid (1), a pump (2), a pressure indicator (3), a stand (4), a tube of radius R (5) and a drain tank (6). The gel containing sand is pumped into the tube by a pressure P1 , determined by a pressure indicator installed immediately after the pump. Further flowing through a tube of length L, the liquid flows into a tank. The pressure at the end of the tube P2 is atmospheric. A timer is also needed to determine the outflow time of a known volume of liquid. For the preparation of a cross-linked polymer, scales, reagents (guar gum, sodium tetra borate, distilled water) and sand are required.

Correlation Model of Fracturing Fluid Viscosity

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Fig. 3. The experimental setup

3 Experimental Procedure 1) Preparation of fracturing fluid (Fig. 4).

a.

b.

Fig. 4. The experimental liquids (a) guar gel and (b) crosslink guar gel

Guar gel was prepared in several steps. The first step was preparing a solution of guar gum with a concentration of 2,4 g/l. After mixing, the solution was left for 1 min in a tank. The resulting solution appeared to be cloudy due to the presence of insoluble endosperm particles. Next, a cross linker with a volume of 15 ml. per 4,5 l. was added to the resulting guar gum solution. In this experiment, sodium tetra borate solution of the maximum concentration (3,2 g / 100 ml. at 25 °C) is used as a cross linker. After adding a cross linker, a cross-linked guar gum gel is formed within 10 min.

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2) Then, the required amount of sand is added to the cross-linked gel and mixed thoroughly. 3) At the experimental setup (Fig. 5), the necessary parameters are measured for the gel with particles.

Fig. 5. Photo of the experimental setup

4 Results Measurements were done for water, gel without sand and gel with six different concentrations of sand. Each experiment was repeated 4 times. The viscosity of water for a temperature of 25 °C was taken from the reference data equal to 0,89 mPa·s. The tube diameter is 0,012 m. In each experiment, the time was measured, during which 2 L of fluid was pumped. After calculations according to the patent formula (8), the following results were obtained, they presented in Table 1.

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Table 1. Results Exp. No

Concentration of sand in gel, kg/m3

Water

0 time, s

υ, m/s

100

200

time, s

η, mPa·s

time, s

η, mPa·s

time, s

η, mPa·s

1

9,8

1,804

13

241,871

13,56

286,318

15,78

525,095

2

10

1,768

12,3

193,834

14,31

355,115

14,67

392,221

9,7

1,823

12

175,604

15,07

436,78

15,39

475,077

3

300

400

time, s

η, mPa·s

1

16,06

563,368

17

2

15,4

476,313

16,8

time, s

500 η, mPa·s

600

time, s

η, mPa·s

time, s

η, mPa·s

707,303

17,22

674,601

17,65

744,633

17,87

863,591

821,843

17,69

829,318

5 Discussion of Result As is known, the viscosity of mixtures depends on the average diameter of the molecules. Therefore, the viscosity of a liquid varies greatly in the presence of small amounts of colloids, the particles of which are significantly larger than the molecules of the solvent. Thus, in dilute suspensions, emulsions, and colloidal solutions, the viscosity increases linearly with increasing volumetric filling V2 / V1 = ω of the medium and does not depend on particle sizes (V1 is the volume of the dispersion medium, V2 is the volume of the dispersed phase). In order to find out what kind of dependence of the viscosity versus volume content of quartz sand, the results were plotted (6) Fig. 6. The graph shows that this dependence has a linear character, despite the errors of some measurements due to the human factor. These errors correspond to acceptable experimental uncertainties. In order to obtain an equation characterizing the relationship between the viscosity of the mixture and the concentration of particles, a trend was drawn. Ultimately, we have a correlation dependence: ηg = 222, 9 + CV · 1, 1194 = η0 + CV · 1, 1194 Where: ηg is mixture viscosity, mPa·s; η0 is mixture viscosity without proppant, mPa·s; C V is volume content of proppant, kg/m3 .

(9)

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1000

Dynamic viscosity (η),mPa·s

900 800 700

The trend equation: η = 1,1194·Cv + 222,9

600 500

R² = 0.9559 400 300 200 100 0 0

100

200

300

400

500

600

700

Volume content of sand (Cv), kg/m3

Fig. 6. The dependence of the viscosity of the fracturing fluid on the volumetric content of sand (proppant)

A couple of additional experiments must be performed with the concentration of sand in the gel 350 and 700 kg/m3 to confirm the obtained empirical formula (Table 2). Table 2. The additional experiments Exp. No

Concentration of sand in gel, kg/m3 350

700

time, s

η, mPa·s

time, s

η, mPa·s

1

16,15

599,156

19,27

1096,456

2

16,33

723,589

19,51

1033,999

3

16,39

639,256

19,29

993,753

4

16,25

556,636

19,83

969,456

average η

-

629,6593

-

1023,416

formula η

-

614,69

-

1006,48

deviation %

2,44%

1,68%

It can be seen that the values obtained experimentally have a small deviation from the values calculated by the empirical formula (Fig. 7). This confirms the possibility of using the formula to model the behavior of a fracturing fluid.

Correlation Model of Fracturing Fluid Viscosity

405

η(C v)

Dynamic viscosity (η),mPa·s

1000

800

600 original trend 700

400

350 average

200

0 0

100

200 300 400 500 Volume content of sand (Cv), kg/m3

600

700

800

Fig. 7. The comparison of the prediction and experimental results

6 Conclusion 1. In the future, the results of the work should be clarified by increasing the number of experiments in which the concentration of guar, cross-linker, and size of the sand fraction will vary. It is also possible to use other proppant. Thus, the correlation model will be improved. 2. It was experimentally confirmed that fracturing fluid actually behaves as nonNewtonian and its viscosity linearly depends on the volume concentration of particles. Because of a series of experiments, a correlation model was obtained for a gel with an average guar concentration with added sand with a fraction of 0,63 – 1,25 mm. 3. This dependence can be used in modeling the hydraulic fracturing process to assess the viscosity of the mixture and determine the most effective injection scenario.

References 1. Fomin, S., Watterson, J., Raghunathan, S., Harkin-Jones, E.: The run-off condition and hydraulic jump in rimming flow of a non-Newtonian fluid. paper FEDSM 2001–18186 (2001) 2. Gardner, D.C., Eikerts, J.V.: Rheological Characterization of Cross-linked and Delayed Crosslinked Fracturing Fluids Using a Closed-Loop Pipe Viscometer. paper SPE 12028 (2011) 3. Hall, C.D., Dollarhide, F.E.: Effects of fracturing fluid velocity on fluid loss agent performance. Truns. AIME 231, 555–560 (2001) (1964) 4. Ihejirika, B., Dosunmu, A., Eme, C.: Performance Evaluation of Guar Gum as a Carrier Fluid for Hydraulic Fracturing, paper SPE-178297-MS (2015) 5. Kovalev, B.I., Erofeeva, A.A., Alashkevich, U.D.: Method for Measuring the Viscosity of Non-Newtonian Liquids, patent 6. Nielsen, L.: Mechanical properties of polymers and polymer compositions, a training manual

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7. Nurullaev, A.S., Ermilov, A.V., Darovskikh: Rheological properties of polymer composite materials filled with dispersed particles, paper - UDC 678.6/.7, 541.64/.68 8. Osovskaya, I.I., Antonova, V.S.: The Viscosity of Polymer Solutions, a Training Manual, 2nd ed. /62 s, UDC 532.13 (2016) 9. Sedelkin, V.M., Potekhin, L.N., Oleinikova, E.V.: Rheological properties of polymer solutions with solid fillers based on plant waste for the manufacture of nanostructured filtration membranes, paper - UDC: 66.081.6–278E.M

Minisymposium “Nonlinear Waves in Continuous Media”

Solution of Equations for Plane Deformation of Nonlinear Model of Complex Crystal Lattice Anatolii N. Bulygin(B) and Yurii V. Pavlov Institute of Problems in Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russia

Abstract. Mathematical methods of solving equations of plane deformation for nonlinear model of deformation of crystalline medium with cubic symmetry are developed. In the nonlinear model, the plane deformation describes by macro-shift vector (Ux , Uy ) and micro-shift vector (ux , uy ). A general solution to the macrofield equations was found. Stress tensor (σxx , σxy , σyy ) and the macro-shift vector (Ux , Uy ) are expressed through two arbitrary analytic functions. The two related nonlinear microfield equations are reduced to two independent equations: the Laplace equation and the double sine-Gordon equation with variable amplitude. The second equation is transformed under simplifying assumptions, to simpler form (the usual sine-Gordon equation, double sine-Gordon equation, sine-Gordon equation with variable amplitude). New methods of solving these equations have been proposed. The found solutions are illustrated by graphs. The received results are discussed.

Keywords: Plane deformation equation

1

· Nonlinear model · Sine-Gordon

Introduction

In recent years, huge experimental material has been accumulated in research at the atomic level of the relief of solid surface, large deformations of the crystal lattice, the occurrence of defects, the formation of new phases, fragmentation of the crystal lattice and phase transformations of the martensitic type under the action of intense external forces, temperature or electromagnetic fields. The success of experimental research and the development of modern technologies for the production of new materials with internal nanostructure stand new problems for the mechanics of continuous media. Classical continual mechanics are no longer adequate for the new problems that arise from deep penetration into the area of nanoscopic scales. The main disadvantage of the classical model is that in it displacements are assumed to be small, not bringing atoms outside the cell of the crystal lattice. This limitation does not allow to describe the fundamental restructuring of the cell and the change in the structure of the crystal c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 409–425, 2022. https://doi.org/10.1007/978-3-030-92144-6_32

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medium under intense external influences. These processes are not described by linear models. Nonlinear theory of deformation of crystalline bodies with complex lattice is proposed in works [1,2]. In the nonlinear model, the displacements of atoms can be extremely large. The nonlinear model describes the fundamental restructuring of the lattice, phase transformations, superlattice formation, occurrence of defects in the initially ideal lattice, reduction of activation energy of structural restructuring processes depending on external influences and lattice parameters. Development of nonlinear model of solid media mechanics corresponds to modern stage of nanotechnologies development.

2

Nonlinear Model of Plane Deformation

In the nonlinear model [1,2], plane deformation is described by macro-shift vector U and a micro-shift vector u. They can be found from the system of equations σij = λijmn emn − sij Φ(us ), ux + uy Um,n + Un,m Φ(us ) = 1 − cos us , , emn = , us = b 2

σij,j = 0,

χij,j = (p − slj elj )

∂Φ(us ) , ∂ui

χij = kijmn εmn ,

εmn =

um,n + un,m . 2

(1)

(2)

Here σij , χij , sij are tensors of stress, micro-stress and nonlinear restriction, respectively, λijmn , kijmn are elasticity tensor and micro-elasticity tensor, emn , εmn are tensors of deformation and microdeformation, p is half of effective interatomic potential barrier, b is size of a primitive cell. For crystal media with cubic symmetry sij = sδij , and tensors λijmn and kijmn have form of Huang tensor Cijmn = C12 δij δmn + C44 (δim δjn + δin δjm ) + (C11 − C12 − 2C44 )δijmn .

(3)

Here δij is the Kronecker delta and δijmn = 1, if all indexes are the same and δijmn = 0 otherwise. The nonlinear model of plane deformation is the system of four connected nonlinear equations (1), (2).

3

Mathematical Methods of Implementation for Nonlinear Model of Plane Deformation

Exact analytical solutions of the equation system (1), (2) can be found if some constraints have place for the medium properties or types of its deformation.

Solution of Equations for Plane Deformation of Nonlinear Model

3.1

411

Solution of Macrofield Equations

We will assume that the crystal medium of cubic symmetry is weakly anisotropic (λ11 − λ12 − 2λ44 = 0). Then the macrofield equations take the form σxx,x + σxy,y = 0,

σyx,x + σyy,y = 0,

(4)

σyy = λθ + 2μUy,y − sΦ(us ), σxx = λθ + 2μUx,x − sΦ(us ), σxy = μ(Ux,y + Uy,x ), θ = Ux,x + Uy,y .

(5)

One must add the Beltrami-Michell condition to relations (4) σxx,yy + σyy,xx − 2σxy,xy −

λ μΔ [sΦ(us )] Δ(σxx + σyy ) + = 0, 2(λ + μ) (λ + μ)

(6)

where Δ = ∂ 2 /∂x2 + ∂ 2 /∂y 2 . A general solution to the macrofield equations (4)–(6) was obtained in [3]. It was shown that the stress tensor σij and the macro vector U can be expressed through two functions ϕ(z) and ψ(z) of complex variable z = x + iy and the function Φ(us ), which has sense of the potential of volume sources for macrostresses   1 μs   Φ[u (ξ, η)] s dξdη, (7) σxx − σyy + 2iσxy = 2 zϕ (z) + ψ  (z) + π λ + 2μ (ζ − z)2 



2μW = κ2 zϕ (z) + ψ  (z) +

1 μs π λ + 2μ

 A

A

Φ[us (ξ, η)] dξdη, (ζ − z)2

(8)

Here κ = (λ + 3μ)(λ + μ), ζ = ξ + iη, and the line over denotes complex conjugation. The formulas (7), (8) coincides to Mushelishvili’s formulas [4] for the case of volumetric forces with potential Φ(us ).

4

Solutions of Microfield Equations

Microfield equations (2), (3) can be written as two equations, from which we will find a vector of micro-shifts u R sin us , b R = sin us . (9) b

k44 Δux + (k12 + k44 )(ux,xx + uy,xy ) + (k11 − k12 − 2k44 )ux,xx = k44 Δuy + (k12 + k44 )(ux,xy + uy,yy ) + (k11 − k12 − 2k44 )uy,yy

Instead of the components (ux , uy ) we enter values us and um = (ux − uy )/b. Then the sum and difference of Eqs. (9) are written as 4R sin us , b2 (10) (k11 + k44 )Δum + 2(k12 + k44 )um,xy + (k11 − k44 )(us,xx − us,yy ) = 0.

(k11 + k44 )Δus + 2(k12 + k44 )us,xy + (k11 − k44 )(um,xx − um,yy ) =

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Instead variables (x, y) we introduce new variables (ξ, η) x+y ξ= √ , 2l1

x−y η= √ , 2l2

(l1 , l2 ) − const.

(11)

In the new variables, the Eqs. (10) take the form 2

1 (k11 l12

2

− k12 ) ∂∂ξu2m

2

+ k12 + 2k44 ) ∂∂ξu2s + l12 (k11 − k12 ) ∂∂ηu2s 2 + l12l2 (k11 − k44 )um,ξη = 4R (12) b2 sin us , ∂ 2 um 1 + l2 (k11 + k12 + 2k44 ) ∂η2 + l12l2 (k11 − k44 )us,ξη = 0.

1 (k11 l12

2

One can see that the Eqs. (12) are related. They are split if k11 = k44 . Under this condition the system of Eqs. (12) splits into two independent equations ∂ 2 um ∂ 2 um k44 − k12 + = 0, k1 = , 2 ∂ξ ∂η 2 k12 + 3k44 ∂ 2 us ∂ 2 us R + = sin us . 2 2 ∂ξ ∂η p

k12

In Eqs. (13), (14) we taking into account that   b k12 + 3k44 b k44 − k12 , l2 = . l1 = 2 p 2 p

(13) (14)

(15)

The solution to Eqs. (13) is an arbitrary analytic function of a special argument. In particular, um = Re g(ξ + ik1 η),

um = Im g(ξ + ik1 η).

(16)

The solution of Eqs. (14) defines the amplitude R. From the macrofield equations we find  1 1   div U = sΦ(us ) + ϕ (z) + ϕ (z) . (17) λ + 2μ λ+μ Equation (17) allows to write the amplitude R in the form R = P1 + 2P2 cos us , p

(18)

where P1 = 1 − 2P2 −

  s ϕ (z) + ϕ (z) , p(λ + μ)

P2 =

s2 . 2p(λ + 2μ)

(19)

Taking into account (17), the Eq. (14) takes the form of double sine-Gordon equation ∂ 2 us ∂ 2 us + = P1 sin us + P2 sin 2us . (20) ∂ξ 2 ∂η 2

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413

It differs from the classic double sine-Gordon equation in that the amplitude P1 is not a constant magnitude, but an arbitrary harmonic function. There are no analytical methods in the literature to solve such an equation. However, if we accept some conditions, the Eq. (20) can result in a simpler form. If the function P1 changes smoothly in the study domain, we can accept P1 ≈ const and Eq. (20) can be considered as classical double sine-Gordon equation. For the case s2  2p(λ2μ) and s  p(λμ)/(σxx σyy ) the Eq. (20) takes the form of classic sine-Gordon (SG) equation ∂ 2 us ∂ 2 us + = P1 sin us . 2 ∂ξ ∂η 2

(21)

If s2  2p(λ + 2μ) and s ≈ p(λ + 2μ)/(σxx + σyy ), then the Eq. (20) takes the form of SG equation with variable amplitude ∂ 2 us ∂ 2 us + = p(x, y) sin us , 2 ∂ξ ∂η 2 where p(x, y) = 1 − 2P2 +

  s ϕ (z) + ϕ (z) , p(λ + μ)

(22)

Δp(x, y) = 0,

is an harmonic function.

5

Modified Lamb Method of Solving Sine-Gordon Equation

Different methods of solving the SG equation with constant amplitude  2  ∂ u ∂2u k + 2 = p0 sin u ∂x2 ∂y

(23)

are described in the literature. Here (k, p0 ) are constants. Widely known method based on substitution u = 4 arctg (Φ1 (x)Φ2 (y)),  

dΦ1 dx dΦ2 dy

2

(24)

=

1 a + bΦ21 + cΦ41 = P1 (Φ1 ), k k

(25)

=

1 c + dΦ22 + aΦ42 = P2 (Φ2 ). k k

(26)

2

Here (a, b, c, d) are constants and b + d = p0 . The solution (24)–(26) in the literature is associated with Lamb [5], although Steuerwald [6] received it first.

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The functions Φ1 (x), Φ2 (y) can be found from (25), (26) by inversion of the corresponding elliptic integrals. In literature, the canonical algorithm of the inversion of elliptic integrals is considered Legendre method. The Legendre approach is quite complex and begins with finding the roots of the corresponding polynomials P1 (Φ), P2 (Φ). Depending on the kind of roots (real, complex, purely imaginary, different coincidental), a corresponding substitution is made and after rather cumbersome calculations, the integrals (25) or (26) are transformed to the canonical kind and their inversion is performed. The authors of the article propose another way to find functions Φ1 (x) and Φ2 (y). First, we modify the Eqs. (25), (26) by introducing new constants Φ1 (x) = AΨ1 (ξ), Φ2 (y) = Ψ2 (η), ξ =

x y K(ν1 ), η = K(ν2 ). L1 L2

(27)

Here (A, L1 , L2 , ν1 , ν2 ) are constants and K(ν) is complete elliptic integral of the first kind. Before the function Ψ2 (η), we do not write a constant multiplier because the Φ1 (x) and Φ2 (y) functions in the solution (24) include multiplicatively. Taking into account (27), the Eqs. (25), (26) take the form 

dΨ1 dξ

2 = 

L21 b L21 a c A2 L21 4 2 + Ψ Ψ . + 1 2 2 k A2 K (ν1 ) k K (ν1 ) k K2 (ν1 ) 1

dΨ2 dη

2 =

 L22 c + dΨ2 + aΨ24 . 2 kK (ν2 )

(28)

(29)

Comparing the Eqs. (28), (29) to differential equations for Jacobi functions, we find Ψ1 (ξ)and Ψ2 (η) by identifying the corresponding coefficients. So by matching the Eq. (28) to equation

2 d sn(ξ, ν1 ) = 1 − (1 + ν12 ) sn2 (ξ, ν1 ) + ν12 sn4 (ξ, ν1 ), dξ

(30)

we find that Ψ1 (ξ) = sn(ξ, ν1 ), if

L21 a = 1, k A2 K2 (ν1 )

c A2 L21 b L21 = −(1 + ν12 ), = ν12 . 2 k K (ν1 ) k K2 (ν1 )

(31) (32)

Here sn(η, ν) and further cn(η, ν), dn(η, ν), tn(η, ν) = sn(η, ν)/cn(η, ν) are Jacobi elliptic functions. Similarly we find Ψ2 (η). Comparing the Eq. (29) to

2 d tn(η, ν1 ) = 1 + (2 − ν22 )tn2 (η, ν1 ) + (1 − ν12 ) tn4 (η, ν1 ), dη

(33)

we get Ψ2 (η) = tn(η, ν1 ),

(34)

Solution of Equations for Plane Deformation of Nonlinear Model

if

a L22 = 1 − ν22 , k K2 (ν1 )

d L22 c L22 = 2 − ν22 , = 1. 2 k K (ν2 ) k K2 (ν2 )

415

(35)

If we exclude constant (a, b, c, d) from Eqs. (32), (35), then we find the dependence of constant (A, L1 , L2 ) on modules (ν1 , ν2 ). 1 (1 + A2 )(A2 + 1 − ν12 )K2 (ν1 ), A2 1 B 2 = 2 (1 + A2 )(A2 + 1 − ν22 )K2 (ν2 ), A H2 =

(36)

A4 = (1 − ν12 )(1 − ν22 ), L1 L2 H= , B= , l = k/p0 . l l The equalities (36) are similar to variance ratios. Considering that solutions of the SG equation are demanded in modern mathematical physics, the authors considered it appropriate to give a complete list of found solutions  H 2 = A12 (1 + A2 )  1 + (1 − ν12 )A2 K2 (ν1 ), 2 1 2 2 2 2 1) A dn (ξ, ν1 )dn(η, ν2 ), B = A2 (1 + A ) 1 + (1 − ν2 )A K (ν2 ), 1 , A4 = (1 − ν12 )(1 − ν22 ) 2  2 H 2 = A12 (1 + A2 ) 1 + (1 − ν12 )A 2 K (ν1 ), 2 2 2 2 dn(ξ, ν1 ) B = (1 + A ) 1 + (1 − ν1 )A K (ν2 ), , 2) A 1 − ν22 dn(η, ν2 ) A4 = , 1 − ν12 sn(ξ, ν1 ) , 3) A tn(η, ν2 )

H 2 = A12 (A2 − 1)(A2 − ν12 )K2 (ν1 ), B 2 = A14 (A2 − 1)(A2 − ν12 )K2 (ν2 ), ν12 A4 = , ν12 + ν22 > 1, 1 − ν22

H 2 = A12 (1 + A2 )(A2 + 1 − ν12 )K2 (ν1 ), 4) A tn(ξ, ν1 ) tn(η, ν2 ), B 2 = A12 (1 + A2 )(A2 + 1 − ν22 )K2 (ν2 ), A4 = (1 − ν12 )(1 − ν22 ), 2 1 2 2 2 2 5) A sn(ξ, ν1 ) tn(η, ν2 ), H = A2 (A − 1)(A − ν1 )K (ν1 ), 2 1 2 2 2 2 B = ν 2 (A − 1)(A − ν1 )K (ν2 ), cn(ξ, ν1 ) 1 tn(η, ν2 ), 6) A dn(ξ, ν1 ) A4 = ν12 (1 − ν22 ), ν12 + ν22 > 1,

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 H 2 = (1 + A2 ) 1 + (1 − ν22 )A2 K 2 (ν1 ), tn(ξ, ν1 ) B 2 = A12 (1 + A2 ) 1 + (1 − ν22 )A2 K2 (ν2 ), , 7) A 1 − ν12 tn(η, ν2 ) A4 = , ν12 + ν22 > 1, 1 − ν22  ν2 H 2 = A41ν 2 (1 + A2 ) (1 + A2 )ν22 − 1 K2 (ν1 ), 2  cn(ξ, ν1 ) B 2 = A12 (1 + A2 ) (1 + A2 )ν22 − 1 K2 (ν2 ), , 8) A cn(η, ν2 ) ν 2 (1 − ν22 ) , A4 = 12 ν2 (1 − ν12 )  H 2 = (1 − ν12 )(A2 ν22− 1) 1 + (1 − ν 22 )A2 K2 (ν1 ), dn(η, ν2 ) B 2 = A12 (A2 ν22 − 1) 1 + (1 − ν22 )A2 K2 (ν2 ), , 9) A cn(ξ, ν1 ) ν12 sn(η, ν2 ) , A4 = 2 ν2 (1 − ν12 )(1 − ν22 ) H 2 = A12 (A2 − 1)(A2 ν12 − 1)K2 (ν1 ), 2 1 2 2 2 2 1 dn(ξ, ν1 ) , B = A2 (A − 1)(1 − (1 − ν2 )A )K (ν2 ), 10) A 1 cn(ξ, ν1 ) tn(η, ν2 ) A4 = , ν12 + ν22 > 1, ν12 (1 − ν22 )

(37)

H 2 = A12 (A2 − 1)(A2 ν12 − 1)K2 (ν1 ), dn(ξ, ν1 ) B 2 = (A2 − 1)(A2 ν12 − 1)K2 (ν2 ), tn(η, ν2 ), 11) A 1 − ν22 cn(ξ, ν1 ) A4 = , ν12 + ν22 > 1, ν12  H 2 = (1 − ν12 )(A2 ν22− 1) 1 + (1 − ν 22 )A2 K2 (ν1 ), dn(η, ν2 ) B 2 = A12 (A2 ν22 − 1) 1 + (1 − ν22 )A2 K2 (ν2 ), , 12) A cn(ξ, ν1 ) ν12 sn(η, ν2 ) , A4 = 2 ν2 (1 − ν12 )(1 − ν22 )  H 2 = A12 (2ν12 − 1)A2 + 2ν22 − 1 K2 (ν1 ), dn(ξ, ν1 ) sn(η, ν2 ) B 2 = (2ν12 − 1)A2 + 2ν22 − 1 K2 (ν2 ), , 13) A ν 2 (1 − ν22 ) sn(ξ, ν1 ) dn(η, ν2 ) , ν12 + ν22 > 1, A4 = 22 ν1 (1 − ν12 ) H 2 = A14 (1 − ν12 )(ν22 − A2 )(A2 + 1 − ν22 )K2 (ν1 ), sn(η, ν2 ) B 2 = A12 (ν22 − A2 )(A2 + 1 − ν22 )K2 (ν2 ), 1 , 14) A ν 2 (1 − ν12 )(1 − ν22 ) cn(ξ, ν1 ) dn(η, ν2 ) A4 = 2 , ν12 + ν22 > 1, ν12

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2 1 2 2 2 2 15) A tn(ξ, ν1 ) sn(η, ν2 ), H = ν22 (A − 1)(A − ν2 )K (ν1 ), cn(η, ν2 ) B 2 = 12 (A2 − 1)(A2 − ν22 )K2 (ν2 ), A , 16) A tn(ξ, ν1 ) dn(η, ν2 ) A4 = ν22 (1 − ν12 ), ν12 + ν22 > 1,

tn(ξ, ν1 ) H 2 = (A2 − 1)(A2 ν22 − 1)K2 (ν1 ), , B 2 = A12 (A2 − 1)(A2 ν22 − 1)K2 (ν2 ), sn(η, ν2 ) dn(η, ν2 ) 1 − ν12 18) A tn(ξ, ν1 ) , A4 = , ν12 + ν22 > 1, cn(η, ν2 ) ν22  − ν12 )A2 K2 (ν H 2 = A12 (A2 ν12 − 1) 1 + (1 1 ),  dn(ξ, ν1 ) B 2 = (1 − ν22 )(A2 ν12 − 1) 1 + (1 − ν12 )A2 K2 (ν2 ), cn(η, ν2 ), 19) A ν22 sn(ξ, ν1 ) , ν12 + ν22 > 1, A4 = 2 ν1 (1 − ν12 )(1 − ν22 ) 17) A

In the list of solutions (37): 1 > ν1 , ν2 > 0. The last line of some solutions contains an additional condition for ν1 , ν2 , when the values of H and B are real. The number of SG equation solutions can be increased by noting that solutions (23) will 1−G 1 u = 4 arctg , (38) u = 4 arctg , G 1+G if u = 4 arctgG is a solution of SG equation (23). This statement is also true for solutions of the one-dimensional SG equation. Note that the solution (24) assumes that p0 > 0. If p0 < 0, then the solution is (39) u = π + 4 arctg(AΦ1 (x)Φ2 (y)). With respect to solid medium mechanics, for single-component twodimensional solutions can be given different interpretations. It is possible to consider that u(x, y) are the shifts perpendicular to the plane (XOY ). Then z = u(x, y) can be thought of as the deformation relief of the originally planar surface. Note that the surface z = u(x, y) has a negative Gaussian curvature and has the following property. If in the solution (1) put that v = ±u + (2m + 1)K(ν2 ),

m = 0, ±1,

(40)

and ν1 = ν2 , then on the lines y = ±x + (2m + 1)B,

(41)

forming square grid we have tg

u = 1, 4

u = π.

(42)

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For a crystal lattice, this fact means that on the lines (41) the crystal lattice shifts by half the cell size. Other solutions have such property, for example, for solutions (37) the grid forms by lines y = ±x + 2mB.

(43)

The second interpretation option assumes that u(x, y) describes the displacements of atoms in the (XOY ) plane parallel to the OX or OY axis. Special programs are used to visualize microdeformations. Figure 1 shows an undeformed crystal medium. The following figures show the images of the deformed medium.

Fig. 1. Not deformed crystal.

In figures with index a) shifts u(x, y) lie in plane OXY in direction of axis OY , and with index b) offsets u(x, y) are perpendicular to plane OXY . Figure 2 shows the deformed medium according to the solution (1) with ν1 = ν2 = 0.999.

u 20 0 0 y

20 0 x

(a)

20 20

(b)

Fig. 2. Crystal is deformed according to the solution (1).

The function dn(x, ν) is doubly periodic. It has no zeros or poles. For this reason, the solution (1) describes deformation when the displacements of atoms are smaller than the cell size. In this case, the lattice is not fundamentally rebuilt (its topology does not change). The intersection of the plane u = π with graph of the solution, as it is shown in Fig. 2 (b), is a square grid corresponding to lines (41).

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Figure 3 shows the deformed medium according to the solution (8) of the list (37) with ν1 = 0.75, ν2 = 0.985. The function cn(x, ν) is doubly periodic also, but it has zeros and poles. In this case, the shifts of atoms can exceed the cell size, and the crystal structure is fundamentally rebuilt. Micropore-type defects occur in the medium.

20 0 u

10 x 0

20 10

10

0 10

20

y

20

(a)

(b)

Fig. 3. Microdeformation according to the solution (8).

Figure 4 shows the microdeformation described by the solution (15) of list (37) with ν1 = ν2 = 0.99. It can be seen that the deformations of different domains of the fragment vary greatly.

2 u 0 10 2 0 y

10 0 x

(a)

10 10

(b)

Fig. 4. Microdeformation according to the solution (15).

Figure 5 shows the microdeformation described by the solution (19) of the list (37) with ν1 = ν2 = 0.995. Deformation field has the form of oval. The deformations inside and outside the ovals are different, but the crystal medium is continuous.

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2 20

u 0 10 2 20

0 y 10 10

0 x

10 20

(a)

20

(b)

Fig. 5. Microdeformation according to the solution (19).

An undeniable advantage of the modified method is the fact that it allows from solutions that are expressed through Jacobi elliptic functions, to obtain solutions that are expressed through circular and hyperbolic functions. This can be done by using known limit relations 1. ν → 0, 2. ν → 1,

sn(u, ν) → sin u, sn(u, ν) → th u,

cn(u, ν) → cos u, dn(u, ν) → 1, cn(u, ν) → ch1 u , dn(u, ν) → ch1 u .

Using (44), we find solutions to the SG equation ⎧ sin √x2l sin √x2l cos √x2l ⎪ ⎪ ⎪ , , ⎪ ⎪ sin √y2l cos √y2l cos √y2l ⎪ ⎪ ⎨ x tg 2l y x u = 4 arctgG(x,y), G = tg tg , y , ⎪ tg 2l 2l 2l ⎪ ⎪ x ⎪ ⎪ th 2l y x ⎪ ⎪ th th . ⎩ y , th 2l 2l 2l

(44)

(45)

For the top row, the solution is (39), which contains π. The second and third string expressions are the solution without π. New solutions need to be added to them ⎧     sh yl ch ψ sh yl ch ψ ⎪ ⎪ ⎪     ⎪ ⎪ th ψ sin x sin ψ , th ψ cos x sh ψ , ⎪ ⎪ l l     ⎨ sh y cos ψ ch yl cos ψ G = tg ψ  l (46)   , , tg ψ ⎪ sh xl sin ψ ch xl sin ψ ⎪ ⎪ ⎪  x  sn(y/b, ν)  x  cn(y/b, ν) ⎪ 1 4 ⎪ ⎪ √ , , tg ⎩ 1 − ν 2 tg 4 a cn(y/b, ν) a sn(y/b, ν) 1 − ν2 where ψ is an arbitrary constant, √ 1 + 1 − ν2 a= √ , 4 1 − ν2

b=1+



1 − ν2.

(47)

Solution of Equations for Plane Deformation of Nonlinear Model

421

Figure 6 shows the medium whose deformation is describes by equation

 cos( xl shψ) u = 4 arctg cth ψ , ψ = 0.15. (48) sh( yl chψ) In this case, the micropore defects are arranged along the axis OX. Away from line OX in the direction of axis OY the medium does not deformed practically.

2 u 0 10 2 0 y

50

10

0 x

50

(a)

(b)

Fig. 6. Crystal is deformed according to the Eq. (48).

Figure 7 shows the medium whose micro-shifts describes the equation

 sh( xl cos ψ) u = 4 arctg tg ψ , ψ = π/4. sh( yl sin ψ)

(49)

The half space x < 0 is compressed and x > 0 is expanded. As a result, defect of the main crack type occurs on the beam y = 0, x > 0.

2 u 0

50 25

2

0 y

50 25

25

0 x

25

50 50

(a)

(b)

Fig. 7. Crystal is deformed according to the Eq. (49).

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The modified Lamb method allows finding solutions to the one-dimensional SG equation k

d2 u = p0 sin u. dx2

(50)

in the form

⎧ x √ H ⎪ ⎪ , = 1 + ν, ⎨ ν sn l  xH u = π + 4 arctg G(x), G = √ cn H H ⎪ ⎪ = 1 + ν, ⎩ ν  x , l dn H ⎧ x H 1 ⎪ ⎪ √ , = 1 + 1−ν 2 , dn ⎨ 4 l 1−ν 2  H u = 4 arctg G(x), G = x sn H 4 ⎪ H ⎪ = 1 − 1−ν 2 . ⎩ 1−ν 2  x , l cn H

5.1

(51)

(52)

Solution of Sine-Gordon Equation with Variable Amplitude

There are no methods in the literature to solve the SG equation with an arbitrary variable amplitude. In the works [7–13] are built functionally invariant solutions of the (2+1) and (3+1)-dimensional SG equations for constant and wide but particular kind of variable amplitudes. For equation ∂2u ∂2u + 2 = p(x, y) sin u ∂x2 ∂y

(53)

one can find a solution if the amplitude p(x, y) is the modulus or square of the modulus of some holomorphic function from the variable z = x + iy. So if we accept that u = 4 arctg eϕ(x,y) , (54) then the Eq. (53) becomes

 ϕxx + ϕyy = ϕ2x + ϕ2y − p(x, y) th ϕ.

(55)

One can see from (55) that (54) will be solution (53), if ϕxx + ϕyy = 0,

ϕ2x + ϕ2y = p(x, y).

(56)

This result allows to build a solution (53) based on the function of the complex variable z. So, if f (z) = fr + ifi , the Laplace equation will satisfy fr ,

fi ,

fr fi ,

ln |f (z)|,

arg f (z).

(57)

The number of harmonic functions can be increased by taking two holomorphic functions f (z) and ψ(z) = ψr + iψi . Then harmonic pairs will be u1r = fr ± ψr , u1i = fi ± ψi , u2r = fr ψr − fi ψi , u2i = fr ψi + fi ψr fr ψr + fi ψi fi ψr − fr ψi u3r = , u3i = . ψr2 + ψi2 ψr2 + ψi2

(58)

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423

New harmonic functions can be constructed from each such pair, by the rule (57). Any one of these can be taken as a function of ϕ(x, y) and then (54) will be the solution to the SG equation with the corresponding amplitude. Figure 8 shows a microdeformation when in a formula (54) ϕ(x, y) = Re (th z). The corresponding solution has the form of a kink, the transition domain of which is periodically perturbed.

2 u 0 10

10

x

0

0

10

(a)

y

10

(b)

Fig. 8. Crystal is deformed according to the Eq. (54).

A new class of solutions to Eq. (53) can be obtained by using the Laplace operator transformation formula in conformal transformations Δξη = |f  (z)|2 Δxy ,

ξ + iη = f (z).

(59)

On the basis (59) u[fr (x, y), fi (x, y)] will be solution (53) with amplitude p(x, y) = |f  (z)|2 , if u(ξ, η) is a solution (53) with the amplitude p0 = 1. Consequently, all solutions of the SG equation with constant amplitude found above by conformal transformation can be converted into solutions of the SG equation with amplitude p(x, y) = |f  (z)|2 . Using harmonic functions, other solutions can be found to the SG equation with variable amplitude. Direct calculations can verify that the Eq. (53) satisfies ⎧ π + 2 am(f, ν), ⎪ ⎪

 ⎪ ⎪ cn(f, ν) ⎪ ⎪ , ⎨ π + 2 arcsin 

dn(f, ν) u= (60) sn(f, ν) ⎪ 2 ⎪ 1−ν , ⎪ 2 arcsin ⎪ ⎪ dn(f, ν) ⎪ ⎩ 2 arcsin dn(f, ν). Here f (x, y) is a harmonic function. For the first three solutions p(x, y) = ν 2 (fx2 + fy2 ) and for the last solution p(x, y) = fx2 + fy2 . Figure 9 shows the microdeformation of the crystal medium according to the first solution (60) for the case f = −arctg(y/x), ν = 0.4 and p = ν 2 /(x2 + y 2 ). Defects occur in the plane x = 0 and in the vicinity of the coordinates origin.

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2 u 10 0 0 y

10 0 x

(a)

10 10

(b)

Fig. 9. Crystal is deformed according to first Eq. (60).

6

Conclusion

General solutions to the plane deformation equations of the nonlinear model are given. The obtained solutions make it possible to set and solve specific boundary problems, which model physical and mechanical phenomena implemented in modern technologies of obtaining solid materials with internal nanostructure, as well as describe processes, which take place during their target application.

References 1. Aero, E.L.: Microscale deformations in a two-dimensional lattice: structural transitions and bifurcations at critical shear. Phys. Solid State 42, 1147–1153 (2000). https://doi.org/10.1134/1.1131331 2. Aero, E.L.: Significantly nonlinear micromechanics of medium with variable periodic structure [in Russian]. Uspekhi Mekhaniki 1, 130–176 (2002) 3. Aero, E.L., Bulygin, A.N., Pavlov, Y.V.: The solutions of nonlinear equations of flat deformation of the crystal media allowing martensitic transformations: complex representation for macrofield equations. Mater. Phys. Mech. 35, 1–9 (2018). https://doi.org/10.18720/MPM.3512018 1 4. Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Springer, Dordrecht (1977). https://doi.org/10.1007/978-94-017-3034-1 5. Lamb, G.L., Jr.: Analytical descriptions of ultrashort optical pulse propagation in a resonant medium. Rev. Mod. Phys. 43, 99–124 (1971). https://doi.org/10.1103/ RevModPhys.43.99 ¨ 6. Steuerwald, R.: Uber Enneper’sche Fl¨ achen und B¨ acklund’sche Transformationen. Abh. Bayer. Akad. Wiss., N.F.40, 1–105 (1936) 7. Aero, E.L., Bulygin, A.N., Pavlov, Y.V.: Solutions of the three-dimensional sineGordon equation. Theor. Math. Phys. 158, 313–319 (2009). https://doi.org/10. 1007/s11232-009-0025-3 8. Aero, E.L., Bulygin, A.N., Pavlov, Y.V.: Solutions of generalized (3+1) sineGordon equation [in Russian]. Nelineinyi Mir 7, 513–517 (2009)

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9. Aero, E.L., Bulygin, A.N., Pavlov, Y.V.: New approach to the solution of the classical sine-Gordon equation and its generalizations. Differ. Equ. 47, 1442–1452 (2011). https://doi.org/10.1134/S0012266111100077 10. Aero, E.L., Bulygin, A.N., Pavlov, Y.V.: Functionally invariant solutions of nonlinear Klein-Fock-Gordon equation. Appl. Math. Comput. 223, 160–166 (2013). https://doi.org/10.1016/j.amc.2013.07.088 11. Aero, E.L., Bulygin, A.N., Pavlov, Y.V.: Solutions of the sine-Gordon equation with a variable amplitude. Theor. Math. Phys. 184(1), 961–972 (2015). https:// doi.org/10.1007/s11232-015-0309-8 12. Aero, E.L., Bulygin, A.N., Pavlov, Y.V.: Exact analytical solutions for nonautonomic nonlinear Klein-Fock-Gordon equation. In: dell’Isola, F., Eremeyev, V.A., Porubov, A. (eds.) Advances in Mechanics of Microstructured Media and Structures. ASM, vol. 87, pp. 21–33. Springer, Cham (2018). https://doi.org/10.1007/ 978-3-319-73694-5 2 13. Bulygin, A.N., Pavlov, Y.V.: Methods of finding of exact analytical solutions of nonautonomous nonlinear Klein-Fock-Gordon equation. In: Altenbach, H., Belyaev, A., Eremeyev, V.A., Krivtsov, A., Porubov, A.V. (eds.) Dynamical Processes in Generalized Continua and Structures. ASM, vol. 103, pp. 147–161. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11665-1 8

Solitary Acoustic Pulses Propagating at the Tip of an Elastic Wedge Pavel D. Pupyrev1,3(B) , Alexey M. Lomonosov2 , and Andreas P. Mayer3 1

2

Prokhorov General Physics Institute of the Russian Academy of Sciences, Vavilov Street 38, 119991 Moscow, Russia [email protected] Scientific and Technological Center of Unique Instrumentation of the Russian Academy of Sciences, Butlerova street 15, 117342 Moscow, Russia 3 HS Offenburg – University of Applied Sciences, Klosterstrasse 14, 77723 Gengenbach, Germany

Abstract. Nonlinear acoustic waves are considered that have displacements localized at the tip of an elastic wedge. The evolution equation governing their propagation is discussed and compared with its analogues pertaining to nonlinear acoustic surface and bulk waves. Solitary wave solutions of the evolution equation have been determined numerically for the cases of two rectangular edges which may be viewed as generated by splitting a half-space, consisting of crystalline silicon, into two quarterspaces. For these two geometries, the kernel in the nonlinear terms of the evolution equation has been calculated from the second-order and third-order elastic constants of silicon, and weak dispersion due to tip truncation has been considered. Solitary pulse shapes have been computed and collisions of solitary pulses have been simulated for various relative speeds of the two collision partners. Collision scenarios for the two wedge geometries were found to differ considerably. Special attention is paid to the peculiar interaction of two initially identical solitary pulses.

Keywords: Wedge waves

1

· Nonlinear waves · Solitary waves

Introduction

An elastic wedge is a waveguide for acoustic waves with associated displacements localized at its apex line [1,2] (see also the review of one-dimensional waveguides [3]). The localization along a one-dimensional structure like the apex line precludes any diffraction losses. Moreover, an ideal wedge does not define any length scale, which makes the system non-dispersive. On the other hand, it still has a geometric parameter, the opening angle of the wedge, allowing the system to be adjustable. In fact, this is a very critical parameter with respect to the existence of wedge waves in isotropic elastic media [4,5], where these waves can be classified as symmetric or antisymmetric with respect to the midplane of the c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 426–437, 2022. https://doi.org/10.1007/978-3-030-92144-6_33

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wedge. For a positive Poisson ratio, we expect that at least one antisymmetric wedge mode exists in rectangular and sharp-angle wedges, and the smaller the opening angle of the wedge, the more modes exist. If obtuse-angle wedges are considered, the existence of wedge waves is not guaranteed anymore. However, one symmetric mode may appear when the wedge angle is close to 180◦ . There are also obtuse wedge angles, where no wedge mode exists. For the case of anisotropic elastic media, the situation is much more complicated. Here, the existence and mode properties strongly depend on the wedge orientation. This has important implications on the nonlinear properties of wedge waves as will be discussed below. Another characteristic quantity of wedge waves that strongly depends on the wedge angle is the phase velocity. The wedge modes are slower than the surface acoustic waves (SAW) on both wedge faces and their velocities monotonically tend to zero with wedge angle decreasing to zero [2,3]. Vice versa, with increasing wedge angle the mode velocity comes closer to the velocity of surface waves, the wedge wave loses its tip-localization and finally degenerates into a SAW. From the experimental point of view, the fabrication of an ideal wedge geometry is not trivial. Because of the wave localization, it is extremely sensitive to the tip quality [6] which is hard to maintain when applying mechanical treatments. This is the reason why in first experimental investigations of wedge waves these modes were found to be dispersive [7]. The proper way to generate a nondispersive wedge is to use the cleavage method in crystals, successfully applied in [8–10]. In other words, it is much easier to introduce dispersion than to avoid it. However, in connection with the topic considered in this contribution, namely the propagation of nonlinear waves at elastic wedges, dispersion plays a significant role. In this paper, two geometrical modifications are considered: wedge truncation and deposition of a film on one of the two wedge faces. For these cases, the dispersion relations were obtained within perturbation theory to lowest order in a parameter characterizing the tip modification [11], which will be made use of in the following. The dispersion laws resulting from the two different geometrical modifications of an ideal wedge occur as linear terms in the integro-differential equation that governs waveform evolution of wedge waves influenced by second-order nonlinearity and weak linear dispersion. As in the case of surface acoustic waves, the effective nonlinearity in this non-integrable evolution equation is non-local. Nevertheless, it allows for solutions corresponding to solitary pulses. For two examples of anisotropic rectangular wedges, results of a numerical study are presented that reveal peculiar interaction scenarios of such solitary pulses. The theory of nonlinear wedge waves and the evolution equation in the presence of dispersion will be discussed in the following section. For a more detailed review of acoustic wedge waves see [5,12].

2

Theory

An ideal wedge has one-dimensional translational invariance along the tip direction x1 , which means that for a plane wedge wave corresponding to a wavenumber

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q the dependence of the displacement field on the coordinate x1 can be separated from that on the other coordinates. On the other hand, the fact that this system does not define any length scale implies that the displacement field depends on the other coordinates x2 and x3 via the dimensionless variables qx2 and qx3 . The general Ansatz for the displacement field has the form uα (x1 , x2 , x3 , t) = Uα (qx2 , qx3 )ei(qx1 −vt) .

(1)

In addition to a number of analytical approaches [13–16] which mainly apply to sharp-angle wedges, wedge modes, in general, have been described with the help of two numerical methods, the widely used finite element method (FEM) [1] and the Laguerre function method (LFM) [2]. Both methods are based on Hamilton’s principle, the displacements Uα (y, z) are expanded in a series of special functions with unknown coefficients, and the variational approach leads to an eigenvalue problem. The only difference between these methods is in the structure of expansion (or test) functions: for the former method, they are localized in each element of the mesh, and for the latter method, they extend over the whole volume of the wedge. A detailed comparison of these two methods and convergence tests have been provided in a recent publication for a more complex system consisting of two wedges and a layer between them [17]. In our calculations of the kernel in the nonlinear evolution equation presented below, the LFM was used not only because of its efficiency but also because of its scalability. The latter property is important when dealing with nonlinearity, which requires the calculation of overlap integrals of displacement gradients for different harmonics, i.e. for different values of wavenumber q in Eq. (1). Because of this property, recalculation of the linear solution for different wavenumber ratios can be avoided. The derivation of the nonlinear evolution equation is based on an asymptotic expansion with multiple scales. The displacement field is presented as an expansion in powers of a small parameter ε having the magnitude of typical strains 2 (2) 3 (2) uα = εu(1) α + ε uα + O(ε ). At first order of ε, a homogeneous linear boundary-value problem is obtained. Its time-harmonic solution contains the modal function Uα and the wave velocity v. In addition, the multiple scales method suggests the introduction of a stretched coordinate X = εx1 describing “slow” variations along the propagation direction. A wedge wave pulse is presented in the following form ∞ u(1) α (x1 , x2 , x3 , t)

= 0

dq A(q, X)Uα (qx2 , qx3 )ei(qx1 −vt) + c.c. 2π

(3)

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At second order of ε the nonlinear evolution equation describes the change of the pulse shape. It can be presented in terms of strain amplitudes B(q) = iqA(q)  q   k dk ∂B =q G i B(k)B(q − k) ∂X 2π q 0

 +∞  dk ∗  q   q n ∗ G B(k)B (k − q) , (4) +2 2π k k q

where the exponent n is equal to 2. The right-hand side of Eq. (4) consists of two integrals over the wavenumber k, which parameterize possible generation of the harmonic q via nonlinear mixing of the harmonics k and q−k. The kernel function G (or weight function) describes the efficiency of the nonlinear interaction. It has the structure of a spatial overlap integral of the three displacement gradients, corresponding to the harmonics q, k, and q − k, normalized in an appropriate way [18]  1 ∗ [Dβ (1)Uα (y, z)] [Dδ (ζ)Uγ (ζy, ζz)] G(ζ) ∝ ζ(1 − ζ) Sαβγδμν [Dν (1 − ζ)Uμ ((1 − ζ)y, (1 − ζ)z)] dydz,(5) where the operator Dβ (ζ) is defined as (iζ, ∂/∂y, ∂/∂z)β . The components of the sixth-rank tensor S are linear combinations of second-order and third-order elastic moduli [19]. The kernel function has the symmetry G(ζ) = G(1 − ζ) and its values in the interval [0, 0.5] fully define the nonlinear behavior. For the case of bulk longitudinal or surface acoustic waves, analogous nonlinear evolution equations can be derived which differ from Eq. (4) only in the second term corresponding to the down-conversion processes. This term describes the generation of the harmonic q from the harmonics k > q and −(k − q) (see Fig. 1b). The efficiency of this process is related to the corresponding efficiency of the up-conversion process described by the first term in Eq. (4), however, reduced by the factor (q/k)2 . The exponent n of the ratio q/k in Eq. (4) reflects the different localization properties of the three types of waves, longitudinal bulk, surface, and wedge waves, for which it is equal to 0, 1, and 2, respectively [20]. The similarity of the evolution equations for the three wave types allows the application of the same methods for the computation of solitary pulses and for the simulation of their evolution and interaction. In first experimental studies of linear wedge waves, a dispersive behavior was found, caused either by tip imperfections or by the finite height of the wedge. The former cause of dispersion could be made use of in nondestructive testing and structural health monitoring of the tip quality [21]. The latter dispersion effect is not relevant here at sufficient wedge height, since it only appears at large wavelengths where nonlinear effects are less important. Another simple modification of the wedge geometry allowing for dispersion is the introduction of a film deposited at one of the wedge faces. This wedge configuration was considered with potential applications to sensors [22].

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Fig. 1. Up-conversion (a) and down-conversion (b) processes corresponding to the different terms in the right-hand side of Eq. (4).

The following simple modifications were considered: surface coating and tip truncation, which are characterized by parameters like the film thickness h and the cross-section area S of the truncated part of the wedge. Assuming these parameters to be small, the dispersion law can be described with the help of perturbation theory [11], resulting in the following relations between frequency ω and wavenumber k: ω 2 = (vk)2 (1 − 2khF1 ),

(6a)

ω = (vk) (1 − 2k SF2 ).

(6b)

2

2

2

In these formulas, v is the velocity of the wave in an ideal wedge, and the parameters F1/2 depend on the wedge angle and the linear elastic moduli of the wedge material. The dispersion provides additional terms in the evolution Eq. (4), which now reads as ⎡ q    k dk ∂B μ ⎣ = aμ q B(q) + q G i B(k)B(q − k) ∂X 2π q 0

⎤ +∞  dk ∗  q   q n G B(k)B ∗ (k − q)⎦ , (7) +2 2π k k q

where parameter n is equal to 2 for wedge waves and the parameter μ is equal to 2 or 3 for the coating or tip truncation dispersion type, respectively. For a specific linear dispersion law corresponding to parameter μ = 4 and artificial nonlinear properties corresponding to a constant kernel function, analytic pulse solutions have been found [23]. For the cases mentioned above, Eq. (7) is not integrable and it was investigated numerically with focus on the existence of solitary pulses. Subsequently, solutions of the initial value problem were simulated to study pulse interaction and evolution.

3

Results

The symmetry of the linear solution plays a critical role in the efficiency of second-order nonlinearity. As mentioned in the introduction, in rectangular or

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sharp angle wedges made of isotropic media, the modes are antisymmetric. For these cases, the kernel functions in the integrals in Eq. (5) vanish, and hence nonlinearity is absent in the evolution equation. However, if the wedge is made of a homogeneous anisotropic material and the wedge lacks reflection symmetry with respect to its midplane, the modes cannot be classified as pure symmetric and pure antisymmetric and their propagation is affected by cumulative secondorder nonlinearity. In this paper, two rectangular wedges being adjacent silicon configurations are investigated, which could be considered as two opposite parts of a cleaved silicon sample (Fig. 2a). The properties of linear solutions were previously investigated in [5,10], the fundamental solutions in systems I and II (Fig. 2a) differ substantially. The wedge mode in the system I (II) propagates with phase velocity 4531 m/s (4461 m/s) and the displacement pattern is almost symmetric (antisymmetric).

Fig. 2. Adjacent wedge configurations I and II in silicon (a). The corresponding kernel functions (b): solid line – configuration I, dashed line – configuration II. Real parts vanish and are not presented. Inset: enlarged curve corresponding to the configuration II.

For these configurations, the corresponding kernel functions are presented in Fig. 2b. The normalization of the linear solution Uα was chosen such that |Un (0, 0)| = 1, where label n denotes the normal direction to the top plane in Fig. 2a for both configurations. This also allows us to interpret the strain amplitude B as a Fourier amplitude of the surface slope on the top face at the tip. For both systems, the kernel functions are purely imaginary, but they differ substantially in shape and absolute value. This quantitative difference is related to the relative magnitudes of symmetric and antisymmetric parts of the mode displacements. The nonlinearity in configuration I is small due to the smallness of the symmetric component, and it is larger in configuration II because here it is the irrelevant antisymmetric part which is small. A numerical study of the existence of solitary solutions in the considered systems was performed in analogy to [5,24]. In these configurations, for both

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dispersion types, corresponding to film deposition on one of two wedge faces or tip truncation, solitary pulses were found. However, in the case of dispersion due to coating, further simulations of pulse evolution in the presence of small perturbations (like small unavoidable dispersive radiation or other purely numerical effects) were found to become unstable. In the following, we confine our investigation to the case of dispersion due to tip truncation. Solitary pulse shapes for the two considered configurations are presented in Fig. 3.

Fig. 3. Solitary pulse shapes for configurations I – case a) and II – case b).

It can be easily shown [24] that for the dispersion law considered here a solitary wedge wave solution has scaling property described by the following equation (and similar for longitudinal bulk and surface waves, corresponding to the exponents n = 0, 1 in Eq. (7))

 (1) (8) uα,1 (x1 , 0, 0, t) = κSα κ1/2 (xR ± κvt − xR 0 ) . The parameter κ scales the amplitude (or height), width, and velocity of the pulse simultaneously. In this equation the retarded coordinate xR is equal to x − vt, and the effective pulse velocity becomes (1 ∓ κ)v. The pulse height scales linearly with the parameter κ, whereas the pulse width is proportional to κ−1/2 . When simulating profiles shown in Fig. 3, dispersion parameters in Eq. (7) were chosen in such a way that the pulse widths were approximately equal, while the parameter κ = 0.007 and κ = 0.02 for configurations I and II respectively. Once the parameter a3 was adjusted to −30 · 10−12 and −18 · 10−12 for systems I and II, it was used with the same values for further simulations. For negative dispersion parameters, only slow solitary pulses were found having velocities (1 − κ)v. The solitary pulses presented in Fig. 3 for configurations I/II have almost opposite polarities and substantially different amplitudes (or heights), which is also related to the different magnitudes of the kernel function. Simulations of pulse propagation have been carried out by integration of the evolution Eq. (7). The evolution of two identical solitary pulses located at a certain distance and traveling initially with equal velocities reveals another remarkable difference between systems I and II. In Fig. 4 the surface slope at

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the tip is shown in the coordinate system tracking the pulses with the linear wedge wave velocity. The ordinate axis x (corresponding to X in the evolution Eq. (7)) is the distance along the wedge tip, traveled by this coordinate system with speed v, and the abscissa axis xR is the retarded coordinate x − vt along the tip introduced in Eq. (8). A single solitary pulse would have a life-line tilted to the vertical line, indicating that the pulse traveling velocity (1 − κ)v differs from the linear one. For a wide range of initial distances between solitary pulses, in configuration I, the pulses behave as being coupled together and oscillating (Fig. 4a), whereas for the configuration II, they attract each other, which leads to a collision (Fig. 4b).

Fig. 4. Interaction of identical solitary pulses in the tracking coordinate system. Configuration I – a) and configuration II – b).

We now discuss collision simulations of solitary pulses having initially different amplitudes. For configuration I and amplitude ratios close to one, the solitary pulses collide without visible radiation (the elastic scenario in Fig. 5a). In the collision process, the two pulses “exchange their identities” without penetrating each other. This behavior is very similar to the collision scenario of two Korteweg – de Vries solitons with initial velocities that do not differ very much [25]. In the light of this collision scenario, where the two pulses do not overlap, the oscillatory behavior of two identical solitary pulses discussed above (Fig. 3a) may be interpreted as a sequence of repeated attractions and “noncontact collisions”. For high amplitude ratios, the solitary pulses collide with partial destruction (an inelastic scenario in Fig. 5b). For configuration II only the latter scenario is observed in the simulations, namely collision and partial destruction (similar behavior is shown in Fig. 4b). However, the remaining solitary pulse substantially increases its amplitude.

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Fig. 5. Collision of two solitary pulses in configuration I. Initial amplitude ratio: 3/10 – a) and 1/7 – b).

Finally, we investigated how solitary pulses emerge out of initial conditions corresponding to pulses that differ from solitary ones. For the initial profile, we chose a solitary pulse scaled with a variable factor. The following results are shown only for configuration II because corresponding simulations for configuration I differ substantially only for high scaling factors. This difference will be discussed below. In the case of the scaling factor being equal to 0.5 (Fig. 6a), a high level of radiation (i.e. quasi-linear waves) is generated which looks like a train of pulses that flatten and slow down as they propagate. In addition, a dominant pulse is formed which seems to slowly broaden, but its evolution is difficult to assess because of interference with linear waves appearing due to periodic boundary conditions. For scaling factors greater than one, the pulse transforms to several solitary pulses almost without radiation, two solitary pulses for the factor 2.5 are shown in Fig. 6b. These findings evidence a clear tendency of solitary pulse formation: the higher the scaling factor, the more solitary pulses emerge, as may be expected when comparing to integrable systems. The distribution of their amplitudes is not trivial. As shown in Fig. 7, starting from the value of the scaling factor approximately equal to 2.7, the height of the second solitary pulse (cross markers) grows faster with increasing scaling factor than the height of the first pulse (circle markers). At the value of scaling factor equal to 3.6 their amplitudes and velocities become almost equal and the attraction, specific for this configuration, leads to pulse collision (Fig. 6c), after which a new fast solitary pulse is formed. For the case of configuration I, which exhibits the characteristic oscillatory behavior of identical solitary pulses (Fig. 4a), the highest solitary pulses combine into an oscillating couple. Further increase of the scaling factor leads to the formation of three coupled solitary pulses. Their subsequent evolution may lead to their separation.

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Fig. 6. Solitary pulse formation in configuration II from an initial pulse being a scaled solitary pulse with factors 0.5 – case a), 2.5 – case b), and 3.6 – case c). In configuration I the factor is 4 – case d).

relative amplitudes

40 30 20 10 0 1

1.5

2

2.5

3

3.5

4

scaling factor of initial solitary pulse

Fig. 7. Relative amplitude of emerged solitary pulses as function of scaling factor (amplitude relative to height of initial solitary pulse for scaling factor equal to 1).

4

Conclusions

The symmetry of acoustic modes plays a decisive role in the efficiency of nonlinear effects on guided acoustic waves. For wedge waves in crystals, effective

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second-order nonlinearity strongly depends on the wedge orientation and vanishes for antisymmetric modes in certain symmetric configurations (as well as in isotropic media). The second-order nonlinearity effects can be reduced to a scalar kernel function describing the interaction of harmonics. Two adjacent rectangular configurations in silicon were considered which have strongly different linear and nonlinear properties. In addition, two dispersion models based on perturbation theory were applied and solitary wave solutions were found. For the dispersion due to tip truncation, the solitary pulses in the two considered configurations have different polarities and interaction scenarios: formation of an oscillating system or attraction with partial destruction. The results of collisions of solitary pulses may depend on the initial amplitude or velocity ratios and initial distance. In addition, the formation of solitary pulses out of an initial pulse was analyzed. The initial pulse was chosen as a solitary pulse, multiplied by a scaling factor. The amplitudes of the emerging solitary pulses were determined numerically as functions of the scaling factor. Qualitatively similar results were obtained with a scaled Gaussian pulse chosen as the initial condition for the simulations. In conclusion, the numerical simulations, carried out for collisions of wedge acoustic solitary pulses and for generation of such solitary pulses out of certain prescribed initial conditions, reveal interesting behavior, not expected from integrable systems like the Korteweg - de Vries or Benjamin - Ono equations, especially oscillating pulse speeds. Acknowledgements. Financial support by Deutsche Forschungsgemeinschaft (Grant No. MA1074/11-3) and the Russian Foundation for Basic Research (Grant No. 19-0200682) is gratefully acknowledged.

References 1. Lagasse, P.E.: Higher-order finite-element analysis of topographic guides supporting elastic surface waves. J. Acoust. Soc. Am. 53(4), 1116 (1973) 2. Moss, S.L., Maradudin, A.A., Cunningham, S.L.: Vibrational edge modes for wedges with arbitrary interior angles. Phys. Rev. B 8, 2999 (1973) 3. Lagasse, P.E, Mason, I.M., Ash E.A.: Acoustic surface waveguides-Analysis and assessment. IEEE Trans. Microw. Theory Tech. MTT-21, 225 (1973) 4. Zavorokhin, G.L., Nazarov, A.I.: On elastic waves in a wedge. J. Math. Sci. 175, 646 (2011) 5. Pupyrev, P.D.: Linear and nonlinear wedge waves in solids. Ph.D. thesis, Prokhorov General Physics Institute of the Russian Academy of Sciences (2017) 6. de Billy, M., Hladky-Hennion, A.C.: The effect of imperfections on acoustic wave propagation along a wedge waveguide. Ultrasonics 37, 413 (1999) 7. Jia, X., de Billy, M.: Observation of the dispersion behavior of surface acoustic waves in a wedge waveguide by laser ultrasonics. Appl. Phys. Lett. 61, 2970 (1992) 8. Adler, R., Hoskins, M., Datta, S., Hunsinger, B.J.: Unusual parametric effects on line acoustic waves. IEEE Trans. Sonics Ultrason. SU-26, 345 (1979) 9. Lomonosov, A.M., Hess, P., Mayer, A.P.: Silicon edges as one-dimensional wave guides for dispersion-free and supersonic leaky wedge waves. Appl. Phys. Lett. 101, 031904 (2012)

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10. Pupyrev, P.D., Lomonosov, A.M., Hess, P., Mayer, A.P.: Symmetry effects on elastic wedge waves at anisotropic edges. J. Appl. Phys. 115, 243504 (2014) 11. Sokolova, E.S., Kovalev, A.S., Timler, R., Mayer, A.P.: On the dispersion of wedge acoustic waves. Wave Motion 50, 233 (2013) 12. Hess, P., Lomonosov, A.M., Mayer, A.P.: Laser-based linear and nonlinear guided elastic waves at surfaces (2D) and wedges (1D). Ultrasonics 54, 39 (2014) 13. McKenna, J., Boyd, G.D., Thurston, R.N.: Plate theory solution for guided flexural acoustic waves along the tip of a wedge. IEEE Trans. Sonics Ultrason. SU-21, 178 (1974) 14. Krylov, V.V.: Geometrical-acoustics approach to the description of localized vibrational modes of an elastic solid wedge. Sov. Phys. Tech. Phys. 35, 137 (1990) 15. Mozhaev, V.G.: Ray theory of wedge acoustic waves. Moscow Univ. Phys. Bull. 30, 38 (1989) 16. Parker, D.F.: Elastic wedge waves. J. Mech. Phys. Solids 40(7), 1583 (1992) 17. Pupyrev, P.D., Nedospasov, I.A., Mayer, A.P.: Guided acoustic waves at the intersection of interfaces and surfaces. Ultrasonics 95, 52 (2019) 18. Sokolova, E.S., Kovalev, A.S., Mayer, A.P.: Second-order nonlinearity of wedge acoustic waves in anisotropic media. Wave Motion 50, 246 (2013) 19. Mayer, A.P.: Surface acoustic waves in nonlinear elastic media. Phys. Rep. 256, 237 (1995) 20. Lomonosov, A.M., Pupyrev, P.D., Hess, P., Mayer, A.P.: Nonlinear one-dimensional guided wedge waves. Phys. Rev. B 92, 014112 (2015) 21. Yang, C.H., Hsu, C.H., Du, S.N.: A new method for the inspection of tool wear based on the dispersion of ASF modes. In: Proceedings of IEEE Ultrasonics Symposium, pp. 2061–2063 (2007) 22. Tung, P.H., Yang, C.H.: Anti-symmetric flexural modes for the detection of humidity variation. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60, 771 (2013) 23. Pupyrev, P.D., Lomonosov, A.M., Sokolova, E.S., Kovalev, A.S., Mayer, A.P.: Nonlinear acoustic wedge waves. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds.) Generalized Models and Non-classical Approaches in Complex Materials 2. ASM, vol. 90, pp. 161–184. Springer, Cham (2018). https:// doi.org/10.1007/978-3-319-77504-3 8 24. Eckl, C., Kovalev, A.S., Mayer, A.P., Lomonosov, A.M., Hess, P.: Solitary surface acoustic waves. Phys. Rev. E 70, 046604 (2004) 25. Drazin, P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge University Press, Cambridge (1989)

Minisymposium “Extreme Loading on Structures”

Free Vibrations of Thin Elastic Orthotropic Cantilever Cylindrical Panel G. R. Ghulghazaryan1(B) and L. G. Ghulghazaryan1,2 1 Armenian State Pedagogical University after Khachatur Abovyan, Tigran Mets 17,

0010 Yerevan, Armenia {ghulghazaryangurgen08,ghulghazaryanlusine08}@aspu.am 2 Institute of Mechanics of NAS Armenia, Marshal Baghramyan ave. 24/2, 0019 Yerevan, Armenia

Abstract. Using the system of equations corresponding to the classical theory of orthotropic cylindrical shells, the free vibrations of thin elastic orthotropic cantilever cylindrical panel are investigated. In order to calculate the natural frequencies and to identify the respective natural modes, the generalized KantorovichVlasov method of reduction to ordinary differential equations is used. Dispersion equations for finding the natural frequencies of possible types of vibrations are derived. An asymptotic relation between the dispersion equations of the problem at hand and the analogous problem for a cantilever rectangular plate is established. A relation between the dispersion equations of the problem and the boundary-value problem of a semi-infinite orthotropic cantilever cylindrical panel is derived. As an example, the values of dimensionless characteristics of natural frequencies are derived for an orthotropic cantilever cylindrical panel. Keywords: Boundary vibrations · Eigenfrequencies · Cylindrical panel

1 Introduction It is known that, at the free edge of an orthotropic plate planar and flexural vibrations can occur independently of each other [1–6]. When the plate is bent these vibrations become coupled and giving raise to two new types of vibrations localized at the free edge: predominantly tangential and predominantly bending vibrations. The transformation of the one type of vibration into the other occurs at the free end of a thin cylindrical elastic panel. For these vibrations a complex distribution of frequencies of natural vibrations occurs depending on the geometrical and mechanical parameters of finite and infinite cylindrical panels [4–11]. With the increase of the number of free edges of a cylindrical panel the distribution becomes increasingly complex [5–11]. Recently free vibrations of a thin elastic orthotropic cylindrical panel with free edges were investigated [20]. Using generalized Kantorovich-Vlasov method the corresponding dispersion equations are derived to find the natural frequencies of possible types of vibrations. In general, the form of dispersion equations depends on the boundary conditions. Therefore, the frequency distributions will be different for different boundary conditions. It would be © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 441–462, 2022. https://doi.org/10.1007/978-3-030-92144-6_34

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interesting to investigate the change in distribution of natural frequencies with the change in boundary conditions by considering cantilever plates (orthotropic rectangular plates with one rigid clamped edge and the other three edges free) and cantilever cylindrical panels (orthotropic cylindrical panels with one rigid clamped end and the other three edges free). The investigation of the edge resonance of cantilever plates and cantilever cylindrical panels is of practical importance since such elements are important components of modern structures and constructions. Therefore, the question of free vibrations of these elements is of vital importance and it demands special attention. Meanwhile, it is one of the most difficult problems in the theory of vibrations of plates and shells [5]. In practice these difficulties are resolved by using a combination of analytical and asymptotic theories, as well as by numerical methods. In the present work, for the first time, free vibrations of cantilever plate and cantilever cylindrical panel are investigated. It is shown that due to difference in boundary conditions the dispersion equations of the considered problem is different from the one derived in [20]. It is proved that the problem prevents separation of variables for the given boundary conditions. It can be proved that such problems for cylindrical shells of orthotropic materials with simple boundary conditions are self-conjugate and nonnegative definite. Therefore, the generalized Kantorovich-Vlasov method can be applied to them [12–16]. As the basic functions the following eigenfunctions of the problem are used:    wVIII = θ 8 w, w|β=0,s = w β=0,s = w β=0,s = w β=0,s = 0, 0 ≤ β ≤ s, (1) The problem (1) is a self-conjugate and has a positive simple discrete spectrum with a limit point at the infinity. The eigenfunctions corresponding to the eigenvalues θ 8m , m = 1, ∞, of the problem (1) have the form: wm (θm β) =

1 2 3 x1 (θm β) + x2 (θm β) + x3 (θm β) + x4 (θm β), 0 ≤ β ≤ s, m = 1, +∞,    θm β θm β θm β θm β x1 (θm β) = chθm β − ch √ cos √ − sh √ sin √ , 2 2 2 2 √ θm β θm β x2 (θm β) = shθm β − 2 ch √ sin √ , 2 2 √ θm β θm β x3 (θm β) = sin θm β − 2sh √ cos √ , 2 2 θm β θm β θm β θm β x4 (θm β) = cos θm β − ch √ cos √ + sh √ sin √ , 2 2 2 2

     x1 (θm s) x2 (θm s) x3 (θm s)   x4 (θm s) x2 (θm s) x3 (θm s)           =  x1 (θm s) x2 (θm s) x3 (θm s)  , 1 = − x4 (θm s) x2 (θm s) x3 (θm s)  ,       x (θm s) x (θm s) x (θm s)  x (θm s) x (θm s) x (θm s) 1 2 3 4 2 3      x1 (θm s) x4 (θm s) x3 (θm s)   x1 (θm s) x2 (θm s) x4 (θm s)          2 = − x1 (θm s) x4 (θm s) x3 (θm s)  , 3 = − x1 (θm s) x2 (θm s) x4 (θm s)  ,       x (θm s) x (θm s) x (θm s)  x (θm s) x (θm s) x (θm s) 1

4

3

1

2

4

(2)

(3)

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This eigenfunctions with their first and second derivatives define an orthogonal basis in a Hilbert space L2 [0, s] [16]. Here θm , m = 1, +∞, are the positive zeros of the determinant of Vronsky for functions (3) at the point β = s. Let us define βm

s =

2   wm (θm β) d β/

0

βm

s (wm (θm β))2 d β, 0

s = 0

  2 wm (θm β) d β/

s

  2 wm (θm β) d β .

(4)

0

Notice that, the derivatives in formulas (3) and (4) are taken with respect to θm β and βm → 1, βm → 1 at m → +∞.

2 The Statement of the Problem and the Basic Equations It is assumed that the generatrices of the cylindrical panel are orthogonal to the ends of the panel. The curvilinear coordinates (α, β) are defined on the median surface of the shell where α(0 ≤ α ≤ l) and β(0 ≤ β ≤ s) are the lengths of the generatrix and the directing circumference, respectively; l – is the length of the panel; and s – is the length of the directing circumference. As the initial equations describing vibrations of the panel, we will use the equations corresponding to the classical theory of orthotropic cylindrical shells written in the selected curvilinear coordinates α and β (Fig. 1): ∂ 2 u2 ∂ 2 u1 ∂ 2 u1 B12 ∂u3 − B − (B + B ) + = λu1 , 66 12 66 ∂α 2 ∂β 2 ∂α∂β R ∂α  ∂ 2 u2 ∂ 2 u2 ∂ 2 u2 ∂ 2 u1 B22 ∂u3 μ4 4B66 − (B12 + B66 ) − B + − B66 − 22 ∂α∂β ∂α 2 ∂β 2 R ∂β R2 ∂α 2    2 4 3 3 ∂ u2 ∂ u3 ∂ u3 μ +B22 2 − = λu2 , B22 3 + (B12 + 4B66 ) ∂β R ∂β ∂β∂α 2 − B11

(5)

  ∂ 4 u3 ∂ 4 u3 ∂ 4 u3 μ4 B11 4 + 2(B12 + 2B66 ) 2 2 + B22 4 ∂α ∂α ∂β ∂β   4 3 3 B12 ∂ u1 μ B22 ∂ u2 B22 ∂ u2 ∂ u2 − + B22 3 + (B12 + 4B66 ) − + 2 u3 = λu3 R ∂β ∂β∂α 2 R ∂α R ∂β R Here, u1 , u2 and u3 are projections of the displacement vector on the directions α β, and on the normal to the median surface of the shell, respectively; R is the radius and of the directing circumference of the median surface; μ4 = h2 /12 (h is the thickness of the shell); λ = ω2 ρ, where ω is the angular frequency, ρ is the density of the material; Bij are the elasticity coefficients. The boundary conditions has the form [17]:    u3  ∂u1 4μ4 ∂ 2 u3 1 ∂u2  ∂u2 ∂u2 − + + + = 0, = 0, ∂β R α=0 ∂α ∂β R ∂α∂β R ∂α α=0     B12 ∂ 2 u3 1 ∂u2  ∂ 3 u3 B12 + 4B66 ∂ 3 u3 1 ∂ 2 u2  ∂ 2 u3 + + = 0, + + = 0. ∂α 2 B11 ∂β 2 R ∂β α=0 ∂α 3 B11 ∂α∂β 2 R ∂α∂β α=0

∂u1 B12 + ∂α B11



(6)

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Fig. 1. Middle surface of a cylindrical panel

u1 |α=l = u2 |α=l = u3 |α=l

 ∂u3  = =0 ∂α α=l

 ∂u2 u3  B12 ∂u1 + −  = 0, B22 ∂α ∂β R β=0,s  B12 ∂ 2 u3 ∂ 2 u3 1 ∂u2  + + = 0, B22 ∂α 2 ∂β 2 R ∂β β=0,s

 ∂ 3 u3 B12 + 4B66 ∂ 3 u3 1 ∂ 2 u2 4B66 1 ∂ 3 u2  + + + = 0, ∂β 3 B22 ∂β∂α 2 R ∂β 2 B22 R ∂α 2 β=0,s  ∂u2 ∂u1  + = 0. ∂α ∂β 

(7)

(8)

α=0,s

Relations (6) and (8) are the conditions of free edges for α = 0 and β = 0, s, respectively, while conditions (7) indicate that the edge α = l is rigid-clamped.

3 The Derivation and Analysis of the Characteristic Equations Let’s formally replace the spectral parameter λ by λ1 , λ2 and λ3 in the first, second, and third equations of the system (5), respectively. The solution of system (5) is searched in the form  (θm β), wm (θm β)} exp(θm χ α), m = 1, +∞. (u1 , u2 , u3 ) = {um wm (θm β), vm wm (9)

Here,wm (θm β), m = 1, ∞, are determined from (2) and um , vm and χ are unknown constants. In this case, the conditions (8) are obeyed automatically. Let us insert Eq. equations are multiplied by the vector functions   (9) into Eq. (5). The obtained wm (θm β), wm (θm β), wm (θm β) in a scalar way and then integrated in the limits from 0 to s. From the first two equations we have  B22 (B12 + B66 )  2 2 2 2 B22 B12 a gm dm )um = εm χ am − a2 βm lm + εm a dm , (cm + εm B11 B66 B11 B66 (10)

Free Vibrations of Thin Elastic Orthotropic Cantilever

 2 2 (cm + εm a gm dm )vcm = εm bm − a2 gm lm ,

445

(11)

From the third equation, by taking into account the relations (10) and (11), the characteristic equation is obtained  B12 2 2 Rmm cm + εm cm + bm βm − χ am + a2 [Rmm gm dm − 2 lm bm βm ] B22 (12) B12 2 2 2 a dm (bm + χ ) + a4 gm lm2 βm = 0, m = 1, ∞ . +εm B11 B12 2 B22  B12 2 B12 B22 χ + β + η + (βm − βm ) ; B11 B11 m B11 2m B11 B66

B22  2 β − η1m ; bm = B1 χ 2 − B11 m     B66 2 B22  B66 2 4 2 2 2  2 cm = χ − B2 χ + ; (13) η + η2m χ + (βm − η1m ) β − η B11 1m B11 m B11 2m am =

2 −B B B11 B22 − B12 12 66 ; B11 B66 2 β  − 2B B β  B11 B22 βm − B12 12 66 m m B2 = , B11 B66 4B66 2 dm = χ − βm ; B11

B1 =

B22 2 B22  B22 2 χ − βm + η , B66 B11 B11 1m B12 + 4B66 2 λi 2 lm = χ − βm , ηim = , i = 1, 3 , B22 B66 θm2   B11 4 2(B12 + 2B66 )  2 B66 2 = a2 χ − βm χ + βm βm − η , B22 B22 B22 3m gm =

Rmm

a2 =

h2 2 1 θ , εm = . 12 m Rθm

Let χj , j = 1, 4, be pairwise different roots of Eq. (12) with non-positive real parts and (j) (j) (j) χ4+j = −χj , j = 1, 4 . Let (u1 , u2 , u3 ), j = 1, 8, be nontrivial solutions of type (9) of the system (5) at χ = χj , j = 1, 8, respectively. The solution of the problem (5–8) is searched in the form 8 (j) ui = ui wj , i = 1, 3. (14) j=1

Let us insert Eq. (14) into the boundary conditions (6) and (7). Each of the obtained equation is multiplied by w(θm β), except of the second one, which is multiplied by

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G. R. Ghulghazaryan and L. G. Ghulghazaryan

w (θm β), and then integrated in the limits from 0 to s. As a result, we obtain the system of equations. (m)

8 

Mij wj

j=1

2 a2 g d cm + εm m m

(j)

(j) (j)

= 0 , i = 1, 4

8 M (m) exp(θ χ α)w  m j j ij (j)

j=1

(j) (j)

2 a2 g d cm + εm m m

= 0 , i = 5, 8 ,

m = 1, ∞

(15)

B12 (j)  B12 (j) 2 2 B12 B22 (j)  2 bm βm − cm + εm a dm (βm − η1m ) 2 B11 B11 B11   B12  B12 2 B22 (j)  −a2 lm βm χj2 + βm − η1m , B11 B11 B11     B12 B22  B22 2 (j) (j) (j) (j) B22 2 (m) χj + βm + η1m M2j = βm χj am + bm + a2 4cm − lm B66 B11 B66 B11   B B B 12 22 12 (j) (j) (j) 2 +a2 εm dm − 4a2 gm χ2 , 4bm + B11 B66 B22 j       2 β B11 B22 βm − B12 B12 (j)  B12  (j) (m) m 2 2 2 2 (j) 4B66 4 β cm + εm a gm χ − χj − bm βm , M3j = χj − B11 m B22 j B11 B22 B11    B12 + 4B66  (j) (j) 2 a 2 gm M4j(m) = χj χj2 − βm cm + εm B11    2 β  − 4B B β  B12 + 4B66 (j)  4B66 4 B11 B22 βm − B12 12 66 m 2 m × χj − χj − bm βm , B22 B11 B22 B11

(j) (m) 2 (j) (j)  M5j = bm + a gm lm βm ,  B22 (B12 + B66 )  (j) (j) (m) 2 2 B22 B12 (j) M6j = βm χj am − a2 βm lm + εm a dm , B11 B66 B11 B66

(j) (j) (m) (m) 2 2 (j) (j) 2 2 (j) (j) M7j = cm + εm a gm dm , M8j = χj cm + εm a gm dm , j = 1, 8. (m)

M1j

(j)

= χj2 am −

(16) The superscript j in parentheses means that the corresponding function is taken at χ = χj . In order to the system (15) has a nontrivial solution, it is necessary and sufficient that ⎞ ⎛ 4     (m) 2 zj ⎠Det Tij  = 0, m = 1, ∞ , (17)  = exp⎝− j=1

i,j=1

  4   (m) 4   (m) T11 = Mij  , T12 = (−1)i−1 Mij exp(zj ) , zj = θm χj l, i,j=1 i,j=1   8,4   (m)   (m) 8,4 , T22 = (−1)i−1 Mij  . T21 = Mij exp(zj ) i=5,j=1

i=5,j=1

(18)

It is shown numerically that the left side of this equality becomes small when any two roots of Eq. (12) become close to each other. This highly complicates calculations

Free Vibrations of Thin Elastic Orthotropic Cantilever

447

and can lead to false solutions. It turns out that from the left side of Eq. (17) a multiplier that tends to zero can be separated when the roots approach each other. Let us introduce the following notations:   [zi zj ] = θm l(exp(zi ) − exp(zj ))/(zi − zj ), [zi zj zk ] = θm l [zi zj ] − [zi zk ] /(zj − zk ), [z1 z2 z3 z4 ] = θm l([z1 z2 z3 ] − [z1 z2 z4 ])/(z3 − z4 ), σ1 = σ1 (χ1 , χ2 , χ3 , χ4 ) = χ1 + χ2 + χ3 + χ4 , σ2 = σ2 (χ1 , χ2 , χ3 , χ4 ) = χ1 χ2 + χ1 χ3 + χ1 χ4 + χ2 χ3 + χ2 χ4 + χ3 χ4 , σ3 = σ3 (χ1 , χ2 , χ3 , χ4 ) = χ1 χ2 χ3 + χ1 χ2 χ4 + χ1 χ3 χ4 + χ2 χ3 χ4 , σ4 = σ4 (χ1 , χ2 , χ3 , χ4 ) = χ1 χ2 χ3 χ4 , σ k = σk (χ1 , χ2 , χ3 , 0), σ k = σk (χ1 , χ2 , 0, 0), k = 1, 4.

(19)

In this case, σ 4 = σ 4 = σ 3 = 0. Let fn , n = 1, 6, be a symmetric polynomial of nth order in variables χ1 , χ2 , χ3 , χ4 . It is known that it can be uniquely expressed in terms of elementary symmetric polynomials. By introducing the notations. fn = fn (σ1 , σ2 , σ3 , σ4 ); f n = fn (σ 1 , σ 2 , σ 3 , 0); f n = fn (σ 1 , σ 2 , 0, 0), n = 1, 6; (20) f1 = σ1 , f2 = σ12 − σ2 ; f3 = σ13 − 2σ1 σ2 + σ3 ; f4 = σ14 − 3σ12 σ2 + σ22 + 2σ1 σ3 − σ4 ; 6

4

2 2

3

f 5 = σ 51 − 4σ 31 σ 2 + 3σ 1 σ 22 + 3σ 21 σ 3 − 2σ 2 σ 3 ; f 6 = σ 1 − 5σ 1 σ 2 + 6σ 1 σ 2 − σ 2 ,

(21)

and performing elementary operations with columns of determinant (17), we obtain    8  (m) 2 = K 2 Det mij i,j=1 , m = 1, ∞ , (22) Det Tij  i,j=1

K = (χ1 − χ2 )(χ1 − χ3 )(χ1 − χ4 )(χ2 − χ3 )(χ2 − χ4 )(χ3 − χ4 ) .

(23)

The expressions for mij are given in Appendix 1. The Eq. (17) are equivalent to the equations  8 Det mij i,j=1 = 0 , m = 1, ∞ . (24) By taking into account the possible relations between λ1 , λ2 and λ3 we conclude that Eq. (24) determine frequencies of the corresponding types of vibrations. For λ1 = λ2 = λ3 = λ, the Eq. (12) are the characteristic equations of the system (5), and the Eq. (24) are the dispersion equations of the problem (5–8). In Sect. 6, the asymptotics of the dispersion Eq. (24) for εm = 1/θm R → 0 (transition to a cantilever rectangular plate or to vibrations localized at the free edges of the cantilever cylindrical panel) and for θm l → ∞ (transition to a semi-infinite cantilever cylindrical panel or to vibrations localized at the free edges of the cantilever cylindrical panel) are investigated. For checking the reliability of the asymptotic relations found in Sect. 6, the free planar and bending vibrations of a cantilever rectangular plate are investigated in the next two sections.

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G. R. Ghulghazaryan and L. G. Ghulghazaryan

4 Planar Vibrations of an Orthotropic Cantilever Rectangular Plate Let an orthotropic rectangular plate is defined in a triorthogonal system of rectilinear coordinates (α, β, γ ) with the origin on the free face plane such that the coordinate plane αβ coincides with the midsurface of the plate and the principal axes of symmetry of the material are aligned with the coordinate lines (Fig. 2). Let s and l be the width and the length of the plate, respectively. The problem of the existence of free planar vibrations of a cantilever rectangular plate is investigated. As the initial equations consider the equations of low-amplitude planar vibrations of the classical theory of orthotropic plates [17] ∂ 2 u1 ∂ 2 u1 ∂ 2 u2 = λu1 ; − B − (B + B ) 66 12 66 ∂α 2 ∂β 2 ∂α∂β ∂ 2 u2 ∂ 2 u2 ∂ 2 u1 − (B12 + B66 ) − B66 2 − B22 2 = λu2 , ∂α∂β ∂α ∂β

− B11

(25)

Fig. 2. Rectangular plate with rigid clamped edge

Here α (0 ≤ α ≤ l) and β (0 < β < s) are the orthogonal rectilinear coordinates of a point on the middle surface; u1 and u2 are the displacements in α and β directions, respectively; Bik , i, k = 1, 2, 6, are the coefficients of elasticity; λ = ω2 ρ, where ω is the natural frequency; and ρ is the density of the material. The boundary conditions have the form [17]   B12 ∂u2  ∂u1  ∂u2 ∂u1 + + = = 0, (26) ∂α B11 ∂β α=0 ∂α ∂β α=0 u1 |α=l = u2 |α=l = 0,

(27)

  ∂ u2  ∂ u1  B12 ∂ u1 ∂ u2 + + = = 0. B11 ∂α ∂β β=0,s ∂α ∂β β=0,s

(28)

Here conditions (26) and (28) mean that the edges α = 0 and β = 0, s are free, while conditions (27) indicate that the edge α = l is rigid-clamped. The problem (25–28) does not allow separation of variables. The differential operator corresponding to this problem is self-conjugate and nonnegative definite. Therefore, the generalized KantorovichVlasov method of the reduction to ordinary differential equations can be used to find

Free Vibrations of Thin Elastic Orthotropic Cantilever

449

vibration eigenfrequencies and eigenmodes [12–16]. The solution of the system (25) is searched in the form  (θm β)} exp(θm yα), (u1 , u2 ) = {um wm (θm β), vm wm

m = 1, +∞ .

(29)

In this case, the conditions (28) are satisfied automatically. Let us insert (29) into Eq. (25). Then,  the obtained equations are multiplied by vector functions   (θ β) in a scalar way and integrated in the limits from 0 to s. As a wm (θm β), wm m result, the system of equations is obtained B66  B12 + B66  2 (β − ηm ))um − yβm vm = 0, B11 m B11 (30) B12 + B66 B22  2 yum + (y2 − βm + ηm )vm = 0, B66 B66   2 = λ/ θ 2 B   where ηm m 66 , θm , and βm , βm , are determined in Eqs. (2) and (4), respectively. By equating the determinant of system (30) to zero, the following characteristic equation of the system of Eq. (25) is found: (y2 −

cm = y4 − B2 y2 +

  B11 + B66 2 2 B22  B66 2 ηm y + (βm − η2m ) βm − ηm = 0 , m = 1, +∞ . (31) B11 B11 B11

Let y1 and y2 be various roots of Eq. (31) with non-positive real parts and y2+j = −yj , j = 1, 2. As the solution of system (30) for y = yj , j = 1, 4, we take (j)

um =

B12 + B66  B66  (j) 2 β yj , vm = yj2 − (β − ηm ) , j = 1, 4 . B11 B11 m

(32)

The solution of the problem (25–28) can be presented in the form u1 =

4 j=1

(j)

um wm (θm β) exp(θm yj α)wj , u2 =

4 j=1

(j)

 vm wm (θm β) exp(θm yj α)wj .

(33) Let us insert Eq. (33) into the boundary conditions (26) and (27). Each of the obtained equation is multiplied by w(θm β), except of the second one, which is multiplied by w (θm β), and then integrated in the limits from 0 to s. As a result, the system of equations is obtained ⎧ 4 4 ⎪   ⎪ (m) (m) ⎪ ⎪ R1j wj = 0, R5j exp(zj ) wj = 0, ⎪ ⎪ ⎨ j=1 j=1 (34) m = 1, +∞ . 4 4 ⎪   ⎪ ⎪ (m) (m) ⎪ R2j wj = 0, R6j exp(zj ) wj = 0. ⎪ ⎪ ⎩ j=1

(m) R1j (m)

R5j

j=1

  B12  B12  B66 2 (m) 2 2 = + (β − ηm ), R2j = yj yj + β + η , B11 m B11 m B11 m B12 + B66  B66  (m) 2 = yj2 − (β − ηm ), R6j = yj βm , zj = θm yj l , j = 1, 4 . B11 m B11 yj2

(35)

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G. R. Ghulghazaryan and L. G. Ghulghazaryan

By equating the determinant e of the system (34) to zero and performing elementary operations with columns of the determinant the following dispersion equation is obtained  4 e = exp(−z1 − z2 ) (y2 − y1 )2 Det lij i,j = 0 . (36) l11 = R(m) 11 , l12 = y1 + y2 , l13 = l11 exp(z1 ) , l14 = l12 exp(z2 ) + l11 [z1 z2 ];

(m) 2  2 l21 = R21 , l22 = y1 y2 + B11 B22 βm − B12 βm − B12 B66 βm /(B11 B66 ) − ηm , l23 = −l21 exp(z1 ), l24 = −l22 exp(z2 ) − l21 [z1 z2 ]; B66  2 n31 = R31 = yj2 − (β − ηm ), n32 = y1 + y2 , B11 m l31 = n31 exp(z1 ), l32 = n32 exp(z2 ) + n31 [z1 z2 ]; l33 = n31 , l34 = n32 , B12 + B66 B12 + B66  n41 = y1 βm , n42 = βm ; B11 B11 l41 = n41 exp(z1 ), l42 = n42 exp(z2 ) + n41 [z1 z2 ]; l43 = −n41 , l44 = −n42 , zj = θm yj l , [z1 z2 ] = θm l (exp(z2 ) − exp(z1 ))/(z2 − z1 ).

(37)

The Eq. (36) is equivalent to the equation  4 2 2 Det lij i,j=1 =((B12 + B66 )/B11 )2 K2m (ηm )Q(ηm )(1 + βm exp(2(z1 + z2 ))) − 4( l21 l11 n32 n42 + l12 l22 n31 n41 ) exp(z1 + z2 ) + (l11 l22 + l21 l12 )(n32 n41 + n31 n42 )(exp(2z1 ) + exp(2z2 )) + 2(l11 l21 (n31 n42 + n32 n41 ) + n31 n41 (l11 l22 + l21 l12 ))(exp(z2 ) − exp(z1 ))[z1 z2 ] + 4 l11 l21 n41 n31 [z1 z2 ]2 = 0 .

(2) K2m ηm

(38)



  − B2 β  B B β 11 22 m 2 2 2 12 m = (βm − ηm ) − ηm y1 y2 , − ηm B11 B66

2 ) = y1 y2 + Q(ηm

B66  2 (β − ηm ). B11 m

If y1 and y2 are the roots of Eq. (31) with negative real parts, then, at θm l → ∞, the roots of Eq. (38) are approximated by the roots of the equation  

2 β B11 B22 βm − B12 m (2)  2 2 2 K2m ηm = (βm − ηm ) − ηm − ηm y1 y2 = 0 . (39) B11 B66 The Eq. (39) is an analogue of the Rayleigh equation for a long enough orthotropic rectangular plate with a free side (compare with [8–11]). Thus, the eigenfrequencies of the problem (25–28) can be found from (38). To find the corresponding eigenmodes, the coefficients wj , j = 1, 4 have to be determined from the system of Eq. (34) and inserted into (33). We can take w1 =

(m) (m) (m) (m) (m) R(m) 1 exp(z1 + 2z2 ) + R52 R2 exp(z1 ) − 2R12 R22 R51 exp(z2 ) (m) (m) (m) (m) (m) (m) R(m) 51 R1 − R51 R2 exp(2z1 ) + 2R11 R21 R52 exp(z1 + z2 )

,

Free Vibrations of Thin Elastic Orthotropic Cantilever (m) (m)

w2 =

R51 R2

(m) (m)

(m) (m) (m)

− R51 R1 exp(2z1 + z2 ) − 2R11 R21 R52 exp(z1 )

(m) (m) (m) (m) (m) (m) R(m) 51 R1 − R51 R2 exp(2z1 ) + 2R11 R21 R52 exp(z1 + z2 ) (m) (m)

w3 = −

(m) (m)

,

(m) (m) (m)

R52 R1 exp(z1 ) + R52 R2 exp(z1 + 2z2 ) − 2R12 R22 R51 exp(2z1 + z2 )

w4 = exp(z2 ),

451

(m) (m) (m) (m) (m) (m) (m) R51 R1 − R51 R2 exp(2z1 ) + 2R11 R21 R52 exp(z1 + z2 ) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) R1 = R11 R22 − R12 R21 , R2 = R11 R22 + R12 R21 .

,

(40)

as solutions to the system of Eq. (34) at a given dimensionless eigenfrequency characteristic ηm .

5 Bending Vibrations of an Orthotropic Cantilever Rectangular Plate Consider an orthotropic rectangular plate with thickness h, width s, and length l (Fig. 2). Consider now the problem of the existence of free bending vibrations of a cantilever rectangular plate. Let us start with the equation of low-amplitude bending vibrations of the classical theory of orthotropic plates [17]   ∂ 4 u3 ∂ 4 u3 ∂ 4 u3 4 (41) μ B11 4 + 2(B12 + 2B66 ) 2 2 + B22 4 = λu3 , ∂α ∂α ∂β ∂β where α (0 ≤ α ≤ l) and β (0 ≤ β ≤ s) are the orthogonal rectilinear coordinates of a point of the median plane of the plate; u3 is the normal component of the displacement vector of a point of the median plane; Bik , i, k = 1, 2, 6 are the elasticity coefficients; μ4 = h2 /12; λ = ω2 ρ, where ω is the natural frequency; ρ is the density of the material. The boundary conditions are given as follows:   B12 ∂ 2 u3  ∂ 3 u3 B12 + 4B66 ∂ 3 u3  ∂ 2 u3 + = + = 0, (42) ∂α 2 B11 ∂β 2 α=0 ∂α 3 B11 ∂α∂β 2 α=0  ∂u3  u3 |α=l = = 0, (43) ∂α α=l   ∂ 2 u3  ∂ 3 u3 B12 + 4B66 ∂ 3 u3  B12 ∂ 2 u3 + = + = 0, (44) B22 ∂α 2 ∂β 2 β=0,s ∂β 3 B22 ∂β∂α 2 β=0,s Here the conditions (42) and (44) mean that the edges α = 0 and β = 0, s are free; while the conditions (43) indicate that the edge α = l is rigid-clamped. The problem (41–44) does not allow separation of variables. The differential operator corresponding to this problem is self-conjugate and nonnegative definite. Therefore, the generalized Kantorovich-Vlasov method of the reduction to ordinary differential equations can be used to find the vibration eigenfrequencies and eigenmodes [12–16]. The solution of the system (41) is searched in the form u3 = wm (θm β) exp(θm yα),

m = 1, +∞ ,

(45)

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G. R. Ghulghazaryan and L. G. Ghulghazaryan

where wm (θm β) is defined in (2). The conditions (44) are satisfied automatically. Substitute (45) into Eq. (41). After multiplying the resulting equation by wm (θm β) and integrating it in the limits from 0 to s the characteristic equation is obtained   B11 4 2(B12 + 2B66 )  2 B66 2 y − βm y + βm βm − η = 0 , m = 1, +∞ , Rmm = a2 B22 B22 B22 m (46) λ , a2 = θm2 h2 /12 , θm2 B66

2 ηm =

(47)

where θm and βm , βm are defined in Eqs. (2) and (4), respectively. Let y3 and y4 be various roots of Eq. (46) with non-positive real parts, y2+j = −yj , j = 3, 4. The solution of the problem (41–44) is searched in the form u3 =

6 j=3

wm (θm β) exp(θm yj α) wj .

(48)

By inserting Eq. (48) into the boundary conditions (42) and (43), and after multiplying the resulting equations by wm (θm β), and integrating them in the limits from 0 to s, the system of equations is obtained ⎧ 6 6 ⎪   ⎪ (m) (m) ⎪ ⎪ R3j Wj = 0, R4j Wj = 0, ⎪ ⎪ ⎨ j=3 j=3 (49) 6 6 ⎪  (m)  ⎪ ⎪ (m) ⎪ R7j Wj = 0, R8j Wj = 0 . ⎪ ⎪ ⎩ j=3

(m)

R3j = yj2 − (m)

R7j

j=3

B12  B12 + 4B66  (m) βm , R4j = yj3 − βm yj , B11 B11 (m)

= exp(zj ) , R8j = yj exp(zj ) ; zj = θm yj l , j = 3, 6 .

(50)

By equating the determinant of system (49) b to zero and performing elementary operations on the columns of the determinant, the dispersion equation is obtained  4 b = exp(−z3 − z4 ) (y4 − y3 )2 Det bij i,j = 0 , (51) b11 = R(m) 33 , b12 = y3 + y4 , b13 = b11 exp(z3 ) , b14 = b12 exp(z4 ) + b11 [z3 z4 ]; (m)

b21 = R43 , b22 = y3 y4 + βm B12 /B11 , b23 = −b21 exp(z3 ), b24 = −b22 exp(z4 ) − b21 [z3 z4 ] b31 = exp(z3 ), b32 = [z3 z4 ], b33 = 1, b34 = 0; b41 = y3 exp(z3 ), b42 = exp(z4 ) + y3 [z3 z4 ],

Free Vibrations of Thin Elastic Orthotropic Cantilever

453

b43 = −y3 , b44 = −1; [z3 z4 ] = θm l (exp(z4 ) − exp(z3 ))/(z4 − z3 ); zj = θm yj l , j = 3, 4 . (52) The Eq. (51) is equivalent to the equation  4 Det bij i,j=1 = − K1m (ηm ) (1 + exp( 2 (z3 + z4 ))) − 4y3 b12 b22 exp (z3 + z4 ) + (b11 b22 + b21 b12 )(exp(2z3 ) + exp(2z4 )) + 2[b11 b21 + y3 (b11 b22 + b21 b12 )(exp(z4 ) − exp(z3 ))[z3 z4 ] + 4y3 b11 b21 [z3 z4 ]2 = 0, m = 1, ∞ .

(53)

  B66  B12 2 2 K1m ηm β . = y32 y42 + 4 βm y3 y4 − B11 B11 m

(54)

If y3 and y4 are the roots of Eq. (46) with negative real parts, then, at θm l → ∞, the roots of Eq. (53) are approximated by the roots of the equation   B66  B12 2   2 2 = y32 y42 + 4 K1m ηm βm = 0 , m = 1, ∞ . βm y3 y4 − B11 B11

(55)

The Eq. (55) is an analogue of the Konenkov equation for a long enough orthotropic rectangular plate with a free side (compare with [8–11, 19, 20]). Thus, eigenfrequencies of the problem (41–44) can be found from (53). To find the corresponding eigenmodes, the coefficients wj , j = 3, 6 have to be determined from the system of Eq. (49) and inserted into Eq. (48). As solutions to the system of Eq. (49) at a given dimensionless eigenfrequency characteristic ηm , it can be taken W3 =

(m) (m) (m) R(m) 3 exp(z3 + z4 ) − R4 exp(z3 ) + 2R34 R44 exp(z4 ) (m)

R3

(m)

(m) (m)

− R4 exp(2z3 ) + 2R33 R43 exp(z3 + z4 )

(m)

W4 =

(m)

(m) (m)

R3 exp(2z3 + z4 ) − R4 exp(z4 ) + 2R33 R43 exp(z3 ) (m)

R3

(m)

exp(z3 )[R3 (m)

R3

(m) (m)

− R4 exp(2z3 ) + 2R33 R43 exp(z3 + z4 ) (m)

W5 =

,

(m)

(m)

(m) (m)

+ R4 exp(z4 ) − 2R34 R44 exp(z3 )] (m) (m)

− R4 exp(2z3 ) + 2R33 R43 exp(z3 + z4 ) (m)

W6 = − exp(z4 ); R3

,

(m) (m)

(m) (m)

(m)

= R33 R44 − R34 R43 , R4

, (m) (m)

(m) (m)

= R33 R44 + R34 R43 . (56)

6 Asymptotics of Dispersion Eq. (24) 6.1 Asymptotics of Dispersion Eq. (24) at εm → 0 Using the previous formulas, we assume that η1m = η2m = η3m = ηm . Then, as εm → 0 , Eq. (12) transform into

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G. R. Ghulghazaryan and L. G. Ghulghazaryan

cm = y4 − B2 y2 +

 Rmm = a2

  B11 + B66 2 2 B22  B66 2 ηm y + (βm − η2m ) βm − ηm = 0 , m = 1, +∞ , (57) B11 B11 B11

 B11 4 2(B12 + 2B66 )  2 B66 2 y − βm y + βm βm − η = 0 , m = 1, +∞ . B22 B22 B22 m (58)

Here the limiting process εm → 0 is understood in the sense that by fixing the radius R and b– the distance between the boundary generatrices of the cylindrical panel, a  = 1/(nθ R) = transition to a cylindrical panel of radius R = nR and to the limit εm m εm /n → 0 at n → ∞ is performed. The Eqs. (57) and (58) are characteristic equations for the equations of planar and bending vibrations of orthotropic cantilever plates, respectively. The roots of the Eqs. (57) and (58) with non-positivec real parts, as in Sects. 4 and 5, are denoted by y1 , y2 and y3 , y4 , respectively. In the same way as in [19], it is proved that for εm 0, namely  > 0 and −1 < η < 1, that characterise the microstructure. With these and following [10], a free-energy density U (ε, χ) is constructed such that ∂U , ∂ε ∂U , μ= ∂χ

σ=

2.2

⇒

σ = 2Gε + Λ(tr ε)1,

(5a)

⇒

  μ = 2G2 χT + ηχ .

(5b)

Equations of Motion

The equations of motion, in the absence of body forces, read div s = ρ¨ u, ¨ 2 axial τ + div μ = J ϕ,

(6a) (6b)

having indicated time differentiation with a superposed dot. Here, ρ and J ≥ 0 are the mass density and the rotational inertia per unit volume, respectively. Besides, (axial τ )i = 12 Eijk τjk denotes the axial vector attached to the skew-symmetric tensor τ . Equation (6b) may be solved for τ to yield ¨ . τ = − 12 E (div μ − J ϕ) 2.3

(7)

Antiplane Shear Deformations

Two half-spaces, named A and B, are considered, in perfect contact along a plane surface, see Fig. 1. A right-handed Cartesian coordinate system (O, x1 , x2 , x3 ) is introduced whose axes are directed along the relevant unit vectors (e1 , e2 , e3 ). The co-ordinate system is located in such a way that the plane x2 = 0 corresponds to the contact surface between A and B. Both half-spaces possess a microstructure, which is described within the theory of linear couple stress (CS) elasticity.

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Fig. 1. Two half-spaces, named A and B, in perfect contact along the joining surface x2 = 0

Antiplane shear deformations are considered, such that, in each half-space, the displacement field u reduces to the out-of-plane component only u3 (x1 , x2 , t), and there is no dependence on the x3 co-ordinate. As a result, the only nonvanishing components of the strain and of the curvature tensors are ε13 = 12 u3,1 , ϕ1 =

ε23 = 12 u3,2 ,

1 2 u3,2 ,

χ11 = −χ22 =

ϕ2 = 1 2 u3,12 ,

χ21 =

(8a)

− 21 u3,1 , − 21 u3,11 ,

(8b) χ12 =

1 2 u3,22 .

(8c)

Also, accounting for the constitutive relations (5) and in light of (8), it is σ13 = Gu3,1 , μ11 = −μ22 = G() (1 + η)u3,12 , 2

σ23 = Gu3,2 ,

(9a)

μ21 = G() (u3,22 − ηu3,11 ),

(9b)

2

μ12 = −G() (u3,11 − ηu3,22 ), 2

(9c)

regardless of Λ. The equilibrium equations (6) lend σ13,1 + σ23,2 + τ13,1 + τ23,2 = ρ¨ u3 ,

(10a)

μ11,1 + μ21,2 + 2τ23 = J ϕ¨1 , μ12,1 + μ22,2 − 2τ13 = J ϕ¨2 .

(10b) (10c)

Substituting Eqs. (8b) and (9) into (7), the skew-symmetric part of the stress tensor is obtained ˆ 3,1 + J u τ13 = − 12 G2 u 4 ¨3,1 ,

ˆ 3,2 + J u τ23 = − 12 G2 u 4 ¨3,2 ,

(11)

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ˆ indicates the two-dimensional Laplace operator in the dimensional wherein  co-ordinates x1 and x2 . For homogeneous media, Eqs. (9a, 10a) and (11) provide a single governing equation in terms of displacement G 2.4



1 2ˆ ˆ 2  u3

 ˆ 3 − J ¨ ˆ u3 + ρ¨ − u u3 = 0. 4

(12)

Reduced Force and Couple Stress Traction Vectors

¯ , acting The reduced force and couple stress traction vectors, respectively p and q across a surface with unit normal n, are given by p = sT n +

1 2

grad μnn × n,

¯ = μT n − μnn n, q

(13)

where μnn = n · μn = q · n and × denotes the cross product between vectors. For the boundary surface x2 = 0 separating the two half-spaces, it is nA = e2 = −nB , and consequently pA3 = sA23 + 12 μA22,1 ,

q¯1A = μA21 ,

with

q¯2A = 0,

(14a)

for A, and   pB3 = − sB23 + 12 μB22,1 ,

q¯1B = −μB21 ,

with

q¯2B = 0,

(14b)

for B. 2.5

Nondimensional Form of the Governing Equations

For the sake of definiteness, quantities are normalized against the half-space A. Accordingly, the dimensionless coordinate ξ=

x , ΘA

x = [x1 , x2 ],

is let together with the reference time T A = A /cAs , whereby the dimensionless time is introduced as t τ = A. T  /ρA is the shear wave speed of classical elasticity (CE) for material A cAs = GA B and cs = GB /ρB the corresponding wave speed for material B. Further, let β=

B , A

υ=

TA , TB

(15)

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whereby υβ = cs B /cs A is the bulk shear wave ratio in CE. Substituting these nondimensional variables in Eq. (12), provides the governing equations, holding in A and B, in nondimensional form  A 2 (0 ) A A u − u uA3 − 2Θ2 uA3 − 2Θ4 (16a) 3,τ τ 3,τ τ = 0, Θ2  (B0 )2 Θ2 1 B B uB3 − 2 2 uB3 − 2Θ4 u − u = 0, (16b) 3,τ τ β Θ2 υ 2 β 2 υ 2 β 4 3,τ τ where  indicates the two-dimensional Laplace operator in the dimensionless coordinates ξ1 and ξ2 and

A,B 1 J A,B A,B A,B d . 0 = A,B with d =  2 ρA,B

3

Analysis in the Frequency Domain

For time-harmonic straight-crested antiplane wave solutions moving in the sagittal plane (ξ1 , ξ2 ), it is uA,B (ξ1 , ξ2 , τ ) = W A,B (ξ1 , ξ2 ) exp(−ıΩτ ), 3

(17)

√

being ı = −1 is the imaginary unit and Ω = ωT A > 0 the nondimensional frequency. Substituting the solution form (17) into Eqs. (16), a pair of metabiharmonic partial differential equations (PDEs) is arrived at    (18a)  − 2 1 − (A0 )2 Ω 2 Θ2  − 2Ω 2 Θ4 W A = 0, 

 2 B 2 4 Θ Θ ( )  − 2 1 − 02 Ω 2  − 2Ω 2 2 4 W B = 0. (18b) 2 υ β υ β Equation (18a) is easily factored out [13]    + δ 2 ( − 1) W A = 0, having let

 Θ2 =

(19)

2

(1 − (A0 )2 Ω 2 ) + 2Ω 2 − 1 + (A0 )2 Ω 2 2Ω 2

.

(20)

By Vieta’s theorem for polynomials, it is δ = 2δcr Θ2 , it being 1 A0 cr = √ , 2

and

Ω δcr = A0 cr Ω = √ . 2

Similarly, Eq. (18b) factors as     + δ12  − δ22 W B = 0,

(21)

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469

where the dimensionless wavenumbers for bulk travelling and bulk evanescent waves in medium B have been let δ12 =

δψ , β 2 υ2

δ22 =

δ . β2ψ

Turning to the boundary conditions, Eqs. (14a) are rewritten in the new symbols  GA  2 A A δ − 1 W,2A + (η A + 2)W,112 , + W,222 3 2Θ  GA A0  A A q¯1A = W,22 − ηW,11 , 2 Θ

pA3 = −

and the same goes with Eqs. (14b)  

  B κ22 ψ 2 GB B 2 B B B p3 = − 3 β (η + 2) W,112 + W,222 + 2 − 1 W,2 , 2Θ υ υ2  GB B0 2  B B . β W,22 − η B W,11 q¯1B = 2 Θ 3.1

(22a) (22b)

(23a) (23b)

Waves Localized at the Half-Spaces’ Interface

Waves propagating at the interface ξ2 = 0 have the form W A,B (ξ1 , ξ2 ) = A,B wA,B (ξ2 ) exp (ıκξ1 ) , with K = kA denoting the dimensionless (spatial) wavenumber in the propagation direction ξ1 and κ = ΘK. The dimensional phase speed in the propagation direction easily follows Ω ω ΘcAs . (24) c= = k κ Providing for decay, waves become localized at the interface wA (ξ2 ) = a1 exp (A1 ξ2 ) + a2 exp (A2 ξ2 ) , wB (ξ2 ) = b1 exp (−B1 ξ2 ) + b2 exp (−B2 ξ2 ) , with A1 = B1 =

 

κ2 − δ 2 ,

A2 =

κ2 − δ12 ,

B2 =

 

κ2 + 1, κ2 + δ22 .

Branch cuts for the square roots are taken parallel to the imaginary axis in antisymmetric fashion and the square root is made definite by taking the branch which warrants √ √ s → x, as s → x ∈ R+ .

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Hereinafter, a superscript asterisk denotes complex conjugation, i.e. given s = (s) + ı (s), it is s∗ = (s) − ı (s). Let’s define the antiplane Rayleigh function [14] 2 (25) R0 (κ, λ1 , λ2 , η) = (ηκ2 − λ1 λ2 )2 − λ1 λ2 (λ1 + λ2 ) , that is valid for either half-space. Indeed, for the half-space A, it is λ1,2 = A1,2 , η = η A , and one may define R0A (κ) = R0 (κ, A1 , A2 , η A ), to be compared with the corresponding expression in [12,13]. In similar fashion, for the half-space B, it is R0B (κ) = R0 (κ, B1 , B2 , η B ). 3.2

Dispersion Relation for Stoneley Waves

It is now possible to deduce the dispersion relation for antiplane Stoneley waves in couple-stress elasticity. To this aim, perfect contact conditions are imposed and a linear system in the amplitudes a1 , a2 , b1 , b2 is arrived at wA (0) = wB (0), dwA dwB (0) = (0), dξ2 dξ2 q¯1A (0) = q¯1B (0), pA3 (0) = pB3 (0). This homogeneous system system admits non-trivial solutions when the determinant of the linear system vanishes Δ(κ) = 0.

(26)

Equation (26) is the secular (or frequency) equation for Stoneley waves and, letting Γ = GB /GA , it reads Δ(κ) = Γ β 2 (A1 − A2 ) (B1 − B2 ) D0 (κ), having let 1 RA (κ) − 2D1 (κ) + Γ β 2 R0B (κ). (27) Γ β2 0 Here, R0A (κ) and R0B (κ) are the Rayleigh functions derived from (25), while D1 (κ) is a coupling term D0 (κ) =

   D1 (κ) = η A κ2 − A1 A2 η B κ2 − B1 B2 + 12 (A1 A2 + B1 B2 ) (A1 + A2 ) (B1 + B2 ) .

(28)

Equation (27) is the “Rayleigh function” for the antiplane Stoneley problem in CS media, whence it is named the Stoneley function. Its counterpart, within CE elasticity, is given in [6]. When A = B, whence (Γ, β, υ, ψ, η k ) = (1, 1, 1, δ, η A = η B ), one simply gets D0,A=B (κ) = −4A1 A2 (A1 + A2 )2 , which only allows antiplane bulk waves.

Stoneley Waves

3.3

471

Cuton Frequency for Antiplane Stoneley Waves

The frequency equation (26) admits a cuton frequency, beyond which Stoneley waves may propagate. To see this, it is enough to show that a real root is possible only inasmuch as (29) D0 (δM ) ≥ 0, with δM = max(δ, δ1 ). Indeed, for large wavenumbers κ, one deduces the asymptotics

 ηA + 1  B 3 − η A + β 2 Γ (η B + 1) κ4 + · · · < 0, D0 (κ) = − 3 − η + 2 β Γ as

κ → ∞. (30)

Besides, it is observed that, for a given triple δ, δ1 and δ2 , D0 (κ) is monotonic decreasing (for a proof of this see [14] and the argument principle therein). It is concluded that a single real zero for D0 is possible if condition (29) holds. This condition has a double purpose: 1. on the one hand, it may be employed as a propagation condition, which provide the minimum frequency Ω beyond which propagation is possible, namely the cuton frequency Ωcuton ; 2. on the other hand, given an admissible propagation frequency Ω ≥ Ωcuton , it provides the range of stiffness ratios Γ for which propagation is possible. Figure 2 shows the cuton frequency as a function of Γ , in the assumption δM = δ > δ1 . Vertical or horizontal asymptotes appear, beyond which the opposite

Fig. 2. Cuton frequency as a function of the ratio Γ between the shear moduli of media B A A and B, with the parameter set υ = β = 1.1, B 0 = 0.5, η = 0.5 and 0 = 0.3 (solid, A A black), 0 = 0.4 (dotted, blue), 0 = 0.5 (dashed, red). Transition from an horizontal to a vertical asymptote occurs. The missing part of each curve, as well as the region beyond either asymptote, is due to the breakdown of the condition δ > δ1 , which is assumed in this plot, see [14] for further details.

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inequality holds, see Fig. 3. For a deeper investigation of the possible propagation scenarios, also concerning existence and uniqueness, see [14].

Fig. 3. Cuton frequency as a function of the ratio Γ for υ = β = 1.1, B 0 = 0.5, η B = 0.5 and A 0 = 0.3. The solid curve is obtained from (29), assuming δ > δ1 , while the converse gives the dashed curve

4

Dispersion Curves

Real zeros of Eq. (27) provide travelling wave solutions. Since these sit in the open interval κ > δM , Stoneley waves are slower than the slowest bulk wave. For instance, assuming δ > δ1 , then bulk waves in the half-space B are fastest (with wavenumber δ1 ) and, as described in [12], they originate the fastest Rayleigh waves (wavenumber κ1R ). Moving down speed-wise, Stoneley waves are met, that are faster than the slowest Rayleigh wave, whose wavenumber is located in close proximity of δ, that is the wavenumber of the slowest bulk waves, the latter taking place in the half-space A. This result has been pointed out in [7] and in [11], without providing a formal proof.

Stoneley Waves

473

Fig. 4. Frequency spectrum for antiplane Stoneley waves for υ = β = 1.1, A 0 = 0.3 < B B 0 = 0.5, η = 0.5, Γ = 0.1 (solid, black) and Γ = 0.3 (red, dashed). Curves almost overlap but start at widely different cuton frequencies, namely Ωcuton = 0.89 and Ωcuton = 2.1, respectively for Γ = 0.1 and Γ = 0.3

Fig. 5. Dispersion curves for antiplane Stoneley waves for υ = β = 1.1, A 0 = 0.3 < B B 0 = 0.5, η = 0.5, Γ = 0.1 (solid, black) and Γ = 0.3 (red, dashed)

Figure 4 plots the frequency spectrum of antiplane Stoneley waves having taken υ = β = 1.1, A0 = 0.3 < B0 = 0.5 and η B = 0.5, and Γ = 0.1 and Γ = 0.3. It appears that curves almost overlap, although they start from widely different cuton frequencies. Since curves are clearly non-linear, dispersion occurs. Indeed, the corresponding dispersion curves are plotted in Fig. 5.

5

Conclusions

In this paper, Stoneley waves are investigated within the context of couple stress theory with micro-inertia, in an attempt to incorporate the role of material microstructure into the wave pattern. This consideration brings substantial modification in the propagation pathways, for antiplane Stoneley waves appear to be sustained under broad conditions. This stands in sheer contrast with the state of the matter in classical elasticity, according to which antiplane Stoneley waves are not supported altogether, while in-plane Stoneley waves may propagate under rather restrictive conditions on the material constants of the media

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in contact, cf. [17]. Indeed, lack of propagation for antiplane waves, that is found in classical elasticity, is relaxed, by consideration of the material microstructure, into propagation beyond a cuton frequency. An explicit expression for the latter is given which may serve either as a propagation condition, that provides the cuton frequency for a given setup, or as an admissibility condition, which restricts the permitted range of material parameters. The dispersion relations is also discussed and reveals a complex wave pattern. The appearance of antiplane Stoneley waves propagating under general conditions possesses important downfalls in many areas. In seismology, it suggests that seismic energy may escape at the boundary between the Earth’s layers. In micro-device design, it provides new pathways for long-range interaction. Finally, in non-destructive testing of materials, it paves the way for novel approaches which rely on non-surface waves that, as such, are capable of capturing defects inside the material. 5.1

Funding

This research was supported under the grant POR FESR 2014-2020 ASSE 1 AZIONE 1.2.2 awarded to the project “IMPReSA” CUP E81F18000310009.

References 1. Abd-Alla, A.M., Ahmed, S.M.: Stoneley and Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity. Appl. Math. Comput. 135(1), 187–200 (2003) 2. Achenbach, J.: Wave Propagation in Elastic Solids. Applied Mathematics and Mechanics, vol. 16. North-Holland, Elsevier (1984) 3. Anh, V.T.N., Thang, L.T., Vinh, P.C., Tuan, T.T.: Stoneley waves with spring contact and evaluation of the quality of imperfect bonds. Z. Angew. Math. Phys. 71(1), 1–19 (2020). https://doi.org/10.1007/s00033-020-1257-1 4. Barnett, D.M., Lothe, J., Gavazza, S.D., Musgrave, M.J.P.: Considerations of the existence of interfacial (Stoneley) waves in bonded anisotropic elastic half-spaces. Proc. R. Soc. Lond. A Math. Phys. Sci. 402(1822), 153–166 (1985) 5. Biot, M.A.: The interaction of Rayleigh and Stoneley waves in the ocean bottom. Bull. Seismol. Soc. Am. 42(1), 81–93 (1952) 6. Cagniard, L.: Reflection and Refraction of Progressive Seismic Waves. McGraw-Hill (1962) 7. Hsieh, T.M., Lindgren, E.A., Rosen, M.: Effect of interfacial properties on Stoneley wave propagation. Ultrasonics 29(1), 38–44 (1991) 8. Ilyashenko, A.V.: Stoneley waves in a vicinity of the Wiechert condition. Int. J. Dynam. Control 9, 30–32 (2021). https://doi.org/10.1007/s40435-020-00625-y 9. Kiselev, A.P., Parker, D.F.: Omni-directional Rayleigh, Stoneley and Sch¨ olte waves with general time dependence. Proc. R. Soc. A Math. Phys. Eng. Sci. 466(2120), 2241–2258 (2010) 10. Koiter, W.T.: Couple-stress in the theory of elasticity. In: Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, vol. 67, pp. 17–44. North Holland Publishing Co. (1964) 11. Lim, T.C., Musgrave, M.J.P.: Stoneley waves in anisotropic media. Nature 225(5230), 372–372 (1970)

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12. Nobili, A., Radi, E., Signorini, C.: A new Rayleigh-like wave in guided propagation of antiplane waves in couple stress materials. Proc. R. Soc. A 476(2235), 20190822 (2020) 13. Nobili, A., Radi, E., Vellender, A.: Diffraction of antiplane shear waves and stress concentration in a cracked couple stress elastic material with micro inertia. J. Mech. Phys. Solids 124, 663–680 (2019) 14. Nobili, A., Volpini, V., Signorini, C.: Antiplane Stoneley waves propagating at the interface between two couple stress elastic materials. Acta Mech. 232(3), 1207– 1225 (2021). https://doi.org/10.1007/s00707-020-02909-y 15. Ottosen, N.S., Ristinmaa, M., Ljung, C.: Rayleigh waves obtained by the indeterminate couple-stress theory. Eur. J. Mech. A. Solids 19(6), 929–947 (2000) 16. Owen, T.E.: Surface wave phenomena in ultrasonics. Prog. Appl. Mater. Res. 6, 71–87 (1964) 17. Scholte, J.G.: The range of existence of Rayleigh and Stoneley waves. Geophys. Suppl. Mon. Not. R. Astron. Soc. 5(5), 120–126 (1947) 18. Stoneley, R.: Elastic waves at the surface of separation of two solids. Proc. R. Soc. Lond. Ser. A Containing Pap. Math. Phys. Charact. 106(738), 416–428 (1924) 19. Stoneley, R.: Rayleigh waves in a medium with two surface layers (first paper). Geophys. Suppl. Mon. Not. R. Astron. Soc. 6(9), 610–615 (1954) 20. Tang, X.M., Cheng, C.H., Toks¨ oz, M.N.: Dynamic permeability and borehole Stoneley waves: a simplified Biot-Rosenbaum model. J. Acoust. Soc. Am. 90(3), 1632–1646 (1991)

Assessment of the Behaviour of Low-Modulus Polyurethane Foams Subjected to Severe Shear Deformation Conditions Cesare Signorini(B) Centro di Ricerca Interdipartimentale “En&Tech”, University of Modena and Reggio Emilia, via Amendola 2, Reggio Emilia, Italy [email protected]

Abstract. Polymeric materials are broadly employed as buildings materials because of a number of interesting properties for specific applications. Among them, polyurethane (PU) takes advantage of outstanding mechanical properties, such as high deformability and dissipation, as well as remarkable thermal and chemical stability. As a foam, PU is arguably the most popular insulating materials, used as a supplementary layer in precast concrete panels, infill walls and roofs. Experimental assessment of the mechanical behaviour of PU foams is therefore a long-standing issue, which is demanded to validate analytical models and provide reliable parameters in FEM modelling. In particular, reliable experimental assessment to large deformations is still difficult to attain. In the present study, we carry out a preliminary mechanical characterisation of a single low-modulus PU foam by means of a testing machine prototype, which performs simple shear and shear-per-traction deformations of a square-shaped sample, according to the restraining system adopted. Simple curve-fitting of the response leads to different mechanical parameters for the same material. Shear test results are related to compressive tests and microstructural investigation of the PU foam, through Scanning Electron (SEM) microscopy. The proposed polynomial laws for the tangential and normal net forces are applicable for calibrating FEM models aimed to predict the behaviour of soft materials subjected to high deformations. Keywords: Polyurethane foam

1

· Simple shear · Cyclic tests

Introduction

Polyurethane (PU) is a thermosetting foam that derives from the reaction between alcohol and isocyanate. PU can be regarded as a versatile polymeric material that find transversal application in Engineering. Indeed, the choice of the raw materials and the design of the foaming reactions can easily tailor the characteristics of the finished product and govern the physical properties c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 476–486, 2022. https://doi.org/10.1007/978-3-030-92144-6_36

Polyurethane Shear Behaviour

477

of the PU, such as its flexibility and density [4]. For instance, using aliphatic isocyanates in designing polyurethane coatings prevents from discolouration [7]. Owing to the multiple possible forms of PU, it can be employed in a variety of applications. Indeed, PU takes advantage of low heat conduction capability and water absorption, good resistance to heat and fire as well as a good mechanical response, associated to its extremely low density, especially in the case foams are concerned [5,8]. The moderate load-bearing capacity and the outstanding insulation properties are often exploited in sandwiches elements, where PU acts as core material between, for instance, steel plates. According to recent statistical studies, gathered in the exhaustive review by Akindoyo et al. [2], a significant fraction of the production of PU is earmarked for civil engineering applications (25% as rigid foams for insulation in construction and 9% as sealants, adhesives and coatings). Furthermore, a remarkable additional 31% is produced in the form of flexible foams, devoted to the manufacturing of mattresses, cushions and paddings, related to the wellness of people living in buildings. Automotive is the other main application field for PU, eminently in the form of moulded foams. The scientific literature has developed a lot since the invention of PU dated back to 1937. However, the mechanical properties of modified rigid and flexible PU foams is still a concern, as well as the properties of materials under large deformation regimes. In the paper by Menges and Knipschild [6], an equation correlating the mechanical behaviour of PU foams with foam density is proposed both for compressive and shear performance. A more recent study [3] presents an experimental campaign to determine the compressive and tensile behaviour of flexible PU foams (density of 62 kg m−3 ), also in combination with fatigue, for nautical purposes. Besides, the shear strength was investigated through a shear-per-traction set-up. Saint-Michelle et al. [9] have investigated the effect of the density in the modification of the compression properties of high-density PU foams possessing a closed-cell morphology (density ranging between 0.30 and 0.85), by proposing and calibrating a phenomenological predictive model. In more recent years, the scientific community has been concentrating on the possibility to modify the foamy reactions and/or add functional fillers to provide the PU with specific properties [10–15]. Apart from PUs, the issue of accurately measuring mechanical properties of soft materials is quite important, since it encompasses a huge range of materials, involving the fine characterisation of biological ones, especially as far as severe deformations are concerned. Indeed, soft materials stresses are extremely low in the presence of extreme deformations and the importance of providing reliable testing protocols is advocated. Such set-ups should be able to warrant the homogeneity of deformation, as pointed out by Horgan and Murphy [17]. Several scholars have been investigating the shear behaviour of soft biological matter, like for instance Destrade et al. [1]. In this case, shear tests are carried out on two different samples of brain matter, compared with a silicone gel used as per reference. However, the set-up herein proposed lack of lateral constraints for the prismatic samples and hence it is not properly compliant with the specifications for simple shear traditionally codified by Ogden [16]. In general, the standard practice for characterising soft matter

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consists in measuring the shear properties by means of shear-per-traction tests, combining the results with compression and tensile tests as well [19]. However, it is proven that deformation homogeneity is hardly preserved, especially in the proximity of the clamps [18], unless the thickness of the sample is significantly lower than its planar dimensions, as in the case of the study by Nunes [20]. In this work, we present simple shear and shear-per-traction tests on a low density PU foam, by means of an on-purpose testing prototype, till severe amount of shear levels. The differences between the two testing approaches is addressed and it is found that the common shear-per-traction test significantly underestimates the shear response of the sample, with respect to simple shear test set-up, owing to the more compliant constraining system. Remarkably, shear and normal stresses can be fitted with a simple second-order expression. Cyclic simple shear tests are also carried out to investigate the viscoelastic behaviour of the material. A Digital Image Correlation (DIC) set-up is also utilised to prove the fulfilment of the homogeneity of the simple shear test set-up during testing.

2 2.1

Materials and Methods Polyurethane Foam

A low-density polyurethane foam is tested, whose properties are reported in Table 1, as provided by the manufacturer (Olmo Giuseppe Spa, Italy). Such kind of foam is commonly adopted in furniture and paddings. Table 1. Mechanical properties of the polyurethane foam considered, as declared by the manufacturer Property

Unit

Density

kg m−3 24

Compression stress

2.2

Value Test reference EN ISO 845

kPa

4.5

EN ISO 3386-1

Tensile strength

kPa

105

EN ISO 1798

Dynamic fatigue

%

35

UNI 6356 - part 2

(at ε = 40%)

The Shear Testing Machine

A specifically designed testing machine is employed to characterise soft materials in high-deformation regimes. It consists of a sample casing, connected with an electro-mechanical actuator, which deforms the casing at a fixed displacement rate, imposed through a controller. A linear variable differential transformer (LVDT) is connected with the movable bar to track the displacement and hence the amount of shear deployed by the sample. The resultant shear force is measured by a sensor in direct contact with the actuator. The normal resultant force in the vertical direction is also measured by two sensors placed aside the

Polyurethane Shear Behaviour

479

casing. The system is governed by an on-purpose LabView script. The shear testing machinery is displayed in Fig. 1. Samples are treated on the surface with a stochastic (speckled) pattern to be analysed through a digital image correlation (DIC) device during testing.

Fig. 1. Shear machine and zoomed detail of the sample casing (right)

2.3

Specimen Preparation and Test Set-Up

Fig. 2. PU foam specimens for simple shear and shear-per-traction tests

Monotonic Tests. Prismatic cut-to-size PU specimens are adopted for testing, whose planar dimensions are 100 × 100 mm, as shown in Fig. 2. The thickness of the samples is equal to 50 mm. For simple shear (SS) tests, specimens are allocated within the casing of the shear machine and fixed at the whole boundary with glue, uniformly distributed on the interphase zone (Fig. 3(a)), whereas for shear-per-traction (SPT) tests, specimens are kept free at the lateral sides (Fig. 3(b)), as customary prescribed. The upper surface of the samples is sprayed with a matte black paint to obtain a stochastic pattern. Tests are conducted at a fixed displacement rate of the actuator equal to 2.0 mm s−1 , which corresponds to an actual displacement rate at the upper side of the casing equal to 0.58 mm s−1 . The non-linearity of the variation of the amount of shear (γ) as a function of time (γ˙ = dγ dt ) is put in emphasis by Eq. (1).

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γ˙ = arctan

δ˙ H

(1)

where δ˙ = dδ dt is the displacement rate of the actuator and H is the height of the sample casing (100 mm). The reference system is fixed according to the scheme reported in Fig. 4. A preliminary test is carried out on the casing with the sample dismounted, in order to discard the undesired contribution due to friction, albeit such aliquot is limited.

Fig. 3. Schematics for simple shear (SS) and shear-per-traction (SPT) tests

Fig. 4. Displacement (δ) and amount of shear (γ) rates for SS and SPT tests.

As shown in Fig. 4, at the initial stage of the tests the linearity of the amount of shear rate is obviously warranted. Then, at large deformations, the rate of shear is no longer a straight line. The actual rate of γ is therefore considered in the charts of Sect. 3. A stereoscopic 3 Mpixel Dantec Dynamics Q-400 Digital Image Correlation (DIC) set-up is placed over the sample, in order to monitor the displacement fields of the surface of the foam. Cyclic SS Tests. Alongside monotonic SS tests, a series of cyclic tests are carried out to investigate the viscoelastic properties of the PU foam. Specifically, a batch of PU prisms undertakes a cycle consisting in 4 loading and unloading stages. The loading-unloading switches are established at fixed values of the amount of shear, i.e. at 0.10, 0.20, 0.30, and 0.40 rad. The last loading cycle eventually attains the maximum measurable value of amount of shear, equal to 0.6 rad, as shown in Fig. 5. Stress is merely measured during the loading stages.

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Fig. 5. Loading programme for SS cyclic tests.

2.4

Microstructure Analysis

The investigation of the microstructure of the PU before and after the mechanical tests is performed through an environmental scanning electron microscopy (E-SEM) in high-vacuum mode (Quanta-200, Fei Company, The Netherlands). Indeed, PU is a structured medium and it may be modelled through polar theories [21–23].

3 3.1

Results and Discussion Monotonic Shear Tests

(a) Along x1

(b) Along x3

Fig. 6. DIC displacement fields at γ = 0.20.

Figure 6(a) provides evidence of the preservation of the deformation’s homogeneity, at moderate amount of shear values (γ = 0.2 rad). Indeed, the DIC colour map shows that the displacement along the x1 direction is uniform, since horizontal colour bands consistently appear on the specimen’s surface, being compliant with the imposed rigid motion of the aluminium constraining lateral

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bars. As shown in Fig. 6(b), slight slumps (x3 displacements lower than 2 mm) are measured at two opposite corners of the specimens. Figure 7 reports the average test results obtained for SS (a) and SPT (b) setups. Both shear stress and normal stress along x2 -axis are plotted. Remarkably, the reproducibility of the tests is evidenced by the surprisingly narrow standard deviation bands achieved for each test group.

(a) Simple shear

(b) Shear-per-traction

Fig. 7. Mean curves for shear tests alongside their relevant standard deviation bands (fine-dotted lines) for shear stress (τ12 , red solid lines) and normal stress (σ2 , garnet dash-dotted lines).

The mechanical behaviour of the foam is strongly affected by the test set-up. Indeed, the constraining system of the SS set-up is stiffer than the SPT one and hence at the same amount of shear, both shear and normal stresses detected for SS are significantly higher. This aspect should be thoroughly taken into account when the determination of the properties of soft materials largely deformed is looked for. In fact, the commonly adopted SPT set-up, which takes advantage of a simpler preparation, is proven to underestimate the internal stresses of the sample, as far as severe deformation configurations are considered. Interestingly, the mean strength curves, which as expected exhibit a nonlinear behaviour, can be fitted quite well with second-order polynomial expressions, as reported in detail in Table 2. In each context, the coefficient of determination is extremely close to the unity. The parabolic fitting law that is valid for the generic stress function (σij ) is given by: σij (γ) = αγ 2 + βγ

(2)

In Eq. (2), σij is the generic stress function, where i, j = 1, 2. When i = j σii refers to a normal stress, when instead i = j, σij refers to a shear stress (and it is identified by the customary label τij ). The fitting parameters α and β are retrieved through interpolation. Such parameters are notably different for the two shear testing set-ups, especially for the evaluation of the shear stress τ12 .

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Table 2. Second-order polynomial fitting for SS and SPT strength curves. Test Stress function α [kPa rad−2 ] β [kPa rad−1 ] R2 τ12 σ22

70.2 147.0

60.0 7.7

0.997 0.999

SPT τ12 σ22

142.0 130.8

5.5 11.2

0.986 0.996

SS

3.2

Cyclic Shear Tests

(a) τ12

(b) σ2

Fig. 8. Typical curves for cyclic simple shear tests for shear stress τ12 (a) and normal stress σ22 (b). Red dashed lines are representative of the average curve for monotonic tests and red square markers indicate the maximum stress level for each cycle.

Figure 8 shows the strength curves for cyclic SS tests of a typical PU sample, both in terms of shear τ12 (a) and normal σ22 (b) stresses. The curves obtained for each loading cycle, together with their peaks are compared with the monotonic curves of Fig. 7, now plotted with a red dashed line. The curves put in evidence the shape-memory behaviour of the foam. Indeed, for each loading cycle, till samples surpass the maximum amount of shear reached hereinbefore, exhibit a less stiff behaviour than the one observed in monotonic tests. Interestingly, the stress peaks attained at the end of each cycle fall exactly on the virgin curve. In correspondence with the last cycle, the last part of the curve coincides with the virgin one, between γ = 0.4 ÷ 0.6 rad.

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Fig. 9. Residual amount of shear (γres ) at each cycle after unloading.

At the beginning of each loading cycle, a residual permanent deformation is observed, as clearly seen in Fig. 9, whose trend is increasing in an almost linear way. The fact can be ascribed to the irreversible deformation induced by the breakage of the cellular structure, which can only be observed by means of SEM analysis.

Fig. 10. Scanning electron microscopy at 100× of the PU foam.

Figure 10 shows the magnification of a portion of PU foam before (a) and after (b) the shear tests. In the former case, the microstructure of the foam appears undamaged, apart from some isolated cells broken as a consequence of manipulation. Conversely, the tested specimen presents diffuse areas comprising broken cells, responsible for the residual deformation.

4

Conclusions and Further Research Insights

This paper reports a mechanical characterisation on a low density polyurethane foam subjected to severe shear deformations (up to an impressed amount of shear of 0.6 rad). Both simple shear and shear-per-traction tests are conducted over

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prismatic specimens taking advantage of an on-purpose machinery designed to perform and compare the two testing set-ups. Both monotonic and cyclic tests for simple shear are conducted. The following set of conclusions can be drawn from the experimentation: – The developed machinery accurately depicts the shear behaviour of soft materials, as highlighted by the extremely narrow data dispersion. Indeed, coefficients of variance (CoV) never exceeds 10% for each batch of specimens in simple shear; – The testing set-up strongly affects the mechanical parameters describing the shear behaviour of foams. As defined by Ogden [16], the proper isochoric simple shear test requires a stiffer constraining system with respect to the most commonly used shear-per-traction one, in which the lateral sides are free to deform. Therefore, if shear-per-traction set-up is considered in place of the simple shear one, at fixed values of amount of shear, the stress is considerably underestimated. – The monotonic tests for both the set-ups reveal a behaviour that is easily interpreted with second-order polynomial fitting curves, both in terms of shear stress and normal stresses. – Cyclic tests emphasise the viscoelastic behaviour of the polyurethane foam. Indeed, at each loading stage, once the previous maximum stress is attained, the curves tend to lie on the virgin one. – Severe testing induces irreversible modifications on the microstructure of the foam, even though this is not plainly visible. Electron microscopy points out the breakage of the internal cells of the material, which are responsible for plastic deformation, as evidenced by cyclic tests (i.e. residual deformation at complete unloading). The successful validation of the testing device herein discussed paves the way for further simple shear tests on different sorts of soft materials, even biological matter and functionalised materials for biomedical applications. Among the latter category, a valuable example can regard accurate mechanical characterisation for polymeric substrates adopted for cellular growth.

References 1. Destrade, M., Gilchrist, M.D., Murphy, J.G., Rashid, B., Saccomandi, G.: Extreme softness of brain matter in simple shear. Int. J. Non-Linear Mech. 75, 54–58 (2015). Elsevier 2. Akindoyo, J.O., Beg, M., Ghazali, S., Islam, M., Jeyaratnam, N., Yuvaraj, A.R.: Polyurethane types, synthesis and applications – a review. RSC Adv. 6(115), 114453–114482 (2016). Royal Society of Chemistry 3. Witkiewicz, W., Zieli´ nski, A.: Properties of the polyurethane (PU) light foams. Adv. Mater. Sci. 6(2), 35–51 (2006) 4. Randall, D., Lee, S.: The Polyurethanes Book. Wiley (2002) 5. Ashida, K.: Polyurethane and Related Foams: Chemistry and Technology. CRC Press (2006)

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6. Menges, G., Knipschild, F.: Estimation of mechanical properties for rigid polyurethane foams. Polym. Eng. Sci. 15(8), 623–627 (1975). Wiley Online Library 7. Valentine, C., Craig, T., Hager, S.: Inhibition of the discoloration of polyurethane foam caused by ultraviolet light. J. Cell. Plast. 29(6), 569–588 (1993). Sage Publications Sage CA: Thousand Oaks, CA 8. Wang, S.X., et al.: Inherently flame-retardant rigid polyurethane foams with excellent thermal insulation and mechanical properties. Polymer 153, 616–625 (2018). Elsevier 9. Saint-Michel, F., Chazeau, L., Cavaill´e, J.Y., Chabert, E.: Mechanical properties of high density polyurethane foams: I. Effect of the density. Compos. Sci. Technol. 66(15), 2700–2708 (2006). Elsevier 10. Wang, T., Zhang, L., Li, D., Yin, J., Wu, S., Mao, Z.: Mechanical properties of polyurethane foams prepared from liquefied corn stover with PAPI. Biores. Technol. 99(7), 2265–2268 (2008). Elsevier 11. Rojek, P., Prociak, A.: Effect of different rapeseed-oil-based polyols on mechanical properties of flexible polyurethane foams. J. Appl. Polym. Sci. 125(4), 2936–2945 (2012). Wiley Online Library 12. Hamilton, A.R., Thomsen, O.T., Madaleno, L.A., Jensen, L.R., Rauhe, J.C.M., Pyrz, R.: Evaluation of the anisotropic mechanical properties of reinforced polyurethane foams. Compos. Sci. Technol. 87, 210–217 (2013). Elsevier 13. Ciecierska, E., et al.: Flammability, mechanical properties and structure of rigid polyurethane foams with different types of carbon reinforcing materials. Compos. Struct. 140, 67–76 (2016). Elsevier 14. Yang, R., Hu, W., Xu, L., Song, Y., Li, J.: Synthesis, mechanical properties and fire behaviors of rigid polyurethane foam with a reactive flame retardant containing phosphazene and phosphate. Polym. Degrad. Stab. 122, 102–109 (2015). Elsevier 15. Pawlik, H., Prociak, A.: Influence of palm oil-based polyol on the properties of flexible polyurethane foams. J. Polym. Environ. 20(2), 438–445 (2012). Springer 16. Ogden, R.W.: Nonlinear elastic deformations. Dover Publication Inc., New York (1985) 17. Horgan, C.O., Murphy, J.G.: Simple shearing of soft biological tissues. Proc. Roy. Soc. A Math. Phys. Eng. Sci. 467(2127), 760–777 (2011). The Royal Society Publishing 18. Guo, D.L., Chen, B.S., Liou, N.S.: Investigating full-field deformation of planar soft tissue under simple-shear tests. J. Biomech. 40(5), 1165–1170 (2007). Elsevier 19. Dokos, S., LeGrice, I.J., Smaill, B.H., Kar, J., Young, A.A.: A triaxial-measurement shear-test device for soft biological tissues. J. Biomech. Eng. 122(5), 471–478 (2000) 20. Nunes, L.: Mechanical characterization of hyperelastic polydimethylsiloxane by simple shear test. Mater. Sci. Eng., A 528(3), 1799–1804 (2011). Elsevier 21. Lakes, R.S.: Experimental microelasticity of two porous solids. Int. J. Solids. Struct. 22(1), 55–63 (1986). Elsevier 22. Nobili, A., Radi, E., Vellender, A.: Diffraction of antiplane shear waves and stress concentration in a cracked couple stress elastic material with micro inertia. J. Mech. Phys. Solids 124, 663–680 (2019). Elsevier 23. Nobili, A., Radi, E., Signorini, C.: A new Rayleigh-like wave in guided propagation of antiplane waves in couple stress materials. Proc. Roy. Soc. A 476(2235), 20190822 (2020). The Royal Society Publishing

Minisymposium “Earthquakes and Seismic Protection”

Numerical Methods of Structures Seismic Analysis I. K. Shanshin1(B) , A. V. Lukin2 , and I. G. Svyatogorov1 1 Corning Scientific Center, St. Petersburg, Russia

[email protected] 2 Polytechnic University, St. Petersburg, Russia

Abstract. Asia (China, Japan, South Korea) is a seismically active region. Accordingly, prior to building of expensive structures in these areas, it is very important to check how structures will perform during earthquake events. To do so, we need to choose a seismic analysis method, which can estimate stress-strain state of a structure in a reasonable time and with reasonable accuracy. Equivalent Static Method (which is still frequently used) can provide underestimated results: it is possible to miss resonance excitations. Consequently, it is necessary to use dynamic methods. In this study main dynamic seismic analysis methods (Response Spectrum, Modal Superposition, Transient) were considered. For comparison purposes simple beam problem was solved using these methods both numerically (FEA) and analytically. Then the problem of calculating in-structure response spectrums is considered. A new method proposed by V. Sorin is investigated. Results dependencies from structural damping were observed. Keywords: Response spectrum method · Random vibration method · In-structure response spectrum

1 Introduction A problem of seismic analysis for heavy structures is highly relevant nowadays. There are about one third of the Earth’s population live in seismic active regions. Moreover, a huge amount of manufacturing plants is located in such regions as, for example, China and Japan. Earthquakes as well as other natural events, which are caused by them (landfall, tsunami) can lead to huge economic losses and human casualties. Consequently, it is necessary to develop and verify seismic analysis methods. Equivalent Static, Response Spectrum and Transient Dynamic methods are the most common for the seismic analysis purposes now. They are included in the standards of the seismic analysis in many regions, for example, in ASCE-2016 [1]. Equivalent Static method is the simplest and fastest one and it uses only maximum acceleration from an earthquake recording which applies to the structure as a static load. A huge drawback is that we can use this method only if a structure is stiff enough. In most cases we can consider structure as stiff and applicable for equivalent static analysis when the first © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 489–500, 2022. https://doi.org/10.1007/978-3-030-92144-6_37

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natural frequency of the structure is above 33 Hz approximately, because in most cases major frequency of an earthquake accelerogram is below this value. The most accurate seismic analysis method is dynamic transient. There are two main types: Modal Superposition and Full Transient. Both of these methods use time-history records of an earthquake (accelerograms) and can provide time-dependent response. The main benefit of the Full Transient method is it applicability for any models, i.e. we can use material and geometrical nonlinearities (creep, plasticity, large deformations). But this method requires significant computational resources to be involved. Modal superposition transient method is much more computationally efficient, but it can be applied only for linear problems. The response spectrum method uses so-called response spectrum as input. Response spectrum – function of frequency or period, showing the peak response of a simple harmonic oscillator that is subjected to a transient event. In our case a transient event is an earthquake. So, in order to obtain response spectrum, we need to apply an earthquake accelerogram to simple oscillator and to calculate maximum acceleration over time. If we repeat this procedure for several oscillators with various natural frequencies, we obtain acceleration response spectrum. The main idea of Response Spectrum method is to calculate responses from each natural mode of the structure and to calculate overall response using certain rule. One of the important task nowadays is a seismic analysis of structure components. In many cases equipment installed inside building is subjected to the significant seismic loadings, which can cause large economical losses. Significant seismic loading arises because typically a structure acts as amplifier of motion and shifter of excitation frequencies. That is, in-structure seismic motion usually has higher amplitude values. Correct seismic analysis of in-structure components requires the input motion directly at the mounting point in the structure. More often initial seismic information is known only for a structure’s base. Several approaches can be used to calculate seismic information at the equipment mounting point. The first one is the time-history method. This method implies that structure itself and equipment are treated as one system. Then one should perform transient dynamic analysis of this system and extract the time-history record from equipment mounting point. This procedure can give the most accurate input seismic information, but it is computationally inefficient due to complexity of the transient dynamic analysis for large models. Moreover, the time-history method of calculating in-structure seismic input requires earthquake records which are rarely available. Most of the time the response spectra specific to current soil conditions is the only available information. That is, it is necessary to use a so-called “direct spectra-to-spectra” procedure, which can give spectra at the equipment mounting point. There are a plenty of these procedures developed nowadays. For example, Peters et al. 1977 [2] developed a method based on a on the modal analysis of a support structure with interaction-free, onedegree-of-freedom system attached. Calvi and Sullivan 2014 [3] proposed a procedure based on an empirical expression for the dynamic amplification factor. In this paper we considered a new method proposed by Sorin 2017 [4]. This method uses Random Vibration method in its foundation.

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The objective of this paper is to investigate V. Sorin method performance. The results of investigation are described in Sect. 3. In Sect. 2, the general verification of Response Spectrum method is presented.

2 Response Spectrum Method Verification 2.1 Problem Statement Our first task was to compare the response spectrum method with other transient seismic analysis methods on a simple beam task. In addition to numerical methods we used analytical solution based on the Bernoulli-Euler beam formulation with the viscous damping forces included for calculations. Problem statement was set according to [5]. Base excitation was used as external loading. The corresponding beam equation is as follows (IV )

EIvt

+ m¨vt + c˙vt = p(x, t)

(1)

where vt – total displacement of the beam, E – Young modulus, I – cross-section moment of inertia, m – mass per unit length, c – viscous damping coefficient, p(x, t) – external load. We assume that beam is uniform. Let’s consider that beam is loaded by specified support motions and rotations (Fig. 1) vt (0, t) = δ1 (t) vt (L, t) = δ2 (t) vt (x, t)|x=0 = 0 vt (x, t)|x=L = 0

(2)

Fig. 1. Problem statement

As the beam supports excitation, we used the accelerogram sample (Fig. 2). In this paper we considered a case when both δ1 (t) and δ2 (t) are equal.

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Fig. 2. Accelerogram sample

To solve beam Eq. (1), it is convenient to express the beam total displacement vt (x, t) as the sum of the quasi-static displacement vs (x, t), which correspond to static application of the support motion δ1 (t) and δ2 (t) plus displacement v(x, t) due to dynamic effects vt (x, t) = vs (x, t) + v(x, t)

(3)

Substituting (3) to (1) we obtain EIv(IV ) + m¨v + c˙v = peff (x, t)

(4)

peff (x, t) = −(m¨vs + c˙vs )

(5)

where

Beam quasi-static displacement can be expressed as vs (x, t) =

4 i=1

ψi (x)δi (t),

(6)

where ψi (x) – static influence functions which in case of uniform beam are the cubic Hermit polynomials. Dynamic part of beam total displacement can be found using standard modal superposition procedure. 2.2 Results of Calculations Beam relative midpoint displacement is shown below (Fig. 3)

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Fig. 3. Beam midpoint displacement over time (full time range)

To clearly see the difference between results let’s consider time range from 18 to 21 s where maximum displacement occurs (Fig. 4).

Fig. 4. Beam midpoint displacement over time (from 18 s to 21 s)

It is clear that analytical solution and numerical solution using the full transient method almost coincide with each other. Solution using the modal superposition transient method are slightly different. We continue comparing the results of these methods with response spectrum method by calculating maximum value of the beam midpoint displacement over time. Values are provided in Table 1. It is clear to see from Table 1 that the maximum displacement values over time for all methods are close to each other (relative error is above 3%). The difference in results between Full Transient and Modal Superposition methods is due to limited number of the natural modes, which were used in calculations in Mode Superposition method.

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Response spectrum method gives very close results to all other methods. Therefore, this method can be used in seismic analysis to predict response of the structures.

3 Direct Spectra-To-Spectra Method 3.1 Sorin Spectra-To-Spectra Method Description This method is based on the random vibration and response spectrum methods. The following presents a synthesis of the main parts of these methods. Response Spectrum and Random Vibration Methods Short Description. Response spectrum method based on mode superposition technique. Thus, it is necessary to perform modal analysis at first. After that respective eigenvectors φi and modal participation factors γi can be calculated. Next, using a given acceleration response spectra mode coefficient can be obtained Ai =

Sai γi ωi2

(7)

where Sai – acceleration response spectra value at frequency ωi . After that we can calculate modal response by using formula Ri = Ai φi . Finally, total response Ra can be obtain by using one of the mode combination methods, for example, square root of the n 1 2 2 sum of the squares (SRSS) Ra = . i=1 (Ri ) Random vibration method uses power spectral density (PSD) function instead of response spectrum as input. It also based on modal superposition technique. The main difference between these two methods is in the output: in response spectrum method we obtain deterministic maximum response of the structure, while in the random vibration method we obtain standard deviation response. Square of standard deviation for linear dissipative oscillator is written as [6]  ∞ Sp (ω) ∗ |K(jω)|2 d ω (8) σa2 = 0

where Sp (ω) – PSD function, |K(jω)|2 =

1 2 ωi2 −ω2 +4ξ 2 ωi2 ω2



– square of absolute value of

oscillator transfer function, ωi – oscillator natural frequency. PSD function for seismic excitation usually has the form of smoothly variating function. This type of PSD function corresponds to broadband smoothed spectra. On the other hand, oscillator transfer function generally has the sharp peak near natural frequency. Thus, this function can be

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approximately considered as constant on the interval (0, ωi ) and as Dirac delta-function near natural frequency. Using these assumptions, we can obtain approximate formula for σa2 :    ωi  ωi  ∞ π |K(jω)|2 d ω − ωi Spi + −1 + Sp d ω = Spi ωi Sp d ω σa2 ∼ = ωi4 Spi 4ξ 0 0 0 (9) where Spi = Sp (ωi ), ξ – damping ratio. Corresponding modal coefficient can be written as     ωi 1 2 π γi γi −1 + Spi d ω (10) Ai = 2 σa = 2 Spi ωi 4ξ ωi ωi 0 Using Eq. (10) we can calculate structure response in the same way as in response spectrum method. Procedure Description. Using random vibration theory allows us to calculate instructure PSD function Spout (ω) knowing ground PSD Spin (ω) and corresponding transfer function from ground to the point of interest inside structure |K(jω)| by using formula Spout (ω) = Spin (ω) ∗ |K(jω)|2

(11)

Numerically in-structure PSD can be obtained by using, for example, ANSYS MAPDL RPSD function. But our goal is to find in-structure response spectra by using ground response spectra. Consequently, we need a procedure for direct and reverse PSD to response spectra conversion. It can be done by equating expressions for modal coefficients (7) and (10). After that we obtain equation for PSD to response spectra conversion:

1  f  2 π − 1 + ∫ Sp (v)dv , (12) Sa (f ) = Sp (f ) ∗ f ∗ 4ξ 0 where f – is frequency in Hz, v – integration variable. However, this equation is obtained using approximate formula (7) for acceleration standard deviation. As it was mentioned before, this formula is valid only for broadband smoothed spectra and for low damping ratios. In-structure PSD is usually having sharp peaks at the structure natural frequencies. Thus, it is necessary to use exact equation for standard deviation (8). After corresponding calculations, we obtain exact equation for response spectra to PSD conversion   ∞ Sp (ω) ∗ f 4 d ω, (13) Sa (f ) =  2 0 ω2 − f 2 + 4ξ 2 ω2 f 2 Equation for reverse response spectra to PSD conversion can be obtained by solving integral Eq. (12) regarding to variable Sp . The solution is as follows:  f d  2  aa−1 4ξ −aa S (v) ∗ v ∗ dv, (14) Sp (f ) = f π − 4ξ 0 dv a

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π where aa = π −4ξ . Summarizing all above-mentioned information, we write V. Sorin in-structure response spectrum calculation algorithm:

1. Calculate PSD from input spectra using Eq. (14) 2. Calculate PSD in the point of interest using some method 3. Calculate corresponding response spectra using Eq. (12) or (13) 3.2 Test Problem Description Method performance was investigated on a beam problem similar to the second paragraph of this article (Fig. 5).

Fig. 5. Test problem description

Right base of the beam is subjected to the same seismic excitation as it was in second paragraph (see Fig. 2). We calculated in-structure response spectrum at the beam midpoint. For comparison purposes we used solution using time-history method of instructure response spectrum calculation, which was described in the introduction section. Corresponding response spectra for damping ratio ξ = 0.05 is as follows (Fig. 6).

Fig. 6. Input Response Spectrum, ξ = 0.05

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As we can see, main peak of the input accelerogram corresponds to 4.3 Hz frequency. Beam first natural frequency is chosen equal to 4.5 Hz to consider method performance in near resonance conditions. PSD function calculated using formula (14) is shown below (Fig. 7).

Fig. 7. Input PSD, ξ = 0.05

As one can see, this PSD is not smooth, it has sharp peaks. Thus, using approximate formula (12) is doubtful. Next, we will consider performance of V. Sorin method for both using exact and approximate formulae. 3.3 Results of Calculations We considered a few values of damping: ξ = 0.025, ξ = 0.05 and ξ = 0.1. Results for the first damping ratio value is as follows (Fig. 8). One can see sharp peak at the beam first natural frequency 4.5 Hz. As anticipated, approximate formula gives wrong results, we can see serious overestimation of peak acceleration comparing to exact formula and time-history method. Exact formula gives good approximation of the peak value and overall performance of the method is acceptable. Results for the next damping ratio ξ = 0.05 are showed below (Fig. 9). Here we see that approximate formula shows better result in peak approximation, but the error is still high. Results for damping ratio ξ = 0.1 are showed on the Fig. 10.

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Fig. 8. In-structure response spectrums comparison, ξ = 0.025

Fig. 9. In-structure response spectrums comparison, ξ = 0.05

One can see that solution using approximate formula has improved a little bit more. Solution using exact formula is still in a reasonable correlation with time-history method. Comparison of the in-structure response spectrum peak accelerations values is provided in Table 2. As one can see, the error of peak acceleration estimation using exact formula is below 8%. It shows that V. Sorin method for in-structure response spectrum calculation can be applied for the seismic analysis of structures components.

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Fig. 10. In-structure response spectrums comparison, ξ = 0.1 Table 2. Comparison of the in-structure response spectrum peak accelerations values

4 Conclusion In this paper we compared basic seismic analysis methods (Full Transient, Mode Superposition, Response Spectrum) on a simple beam task. In addition, analytical solution for beam subject to support motions was obtained. Results showed a good agreement between all methods (error < 3%). Subsequently, a problem of in-structure response spectrum calculation was considered. A new method, developed by V. Sorin, was described and investigated. Two separate approaches were used: first uses approximate formula for acceleration standard deviation calculation, second uses exact formula. Results dependencies from various damping coefficient values showed that using approximate formula gives serious overestimation of in-structure response spectrum peak acceleration. Overall performance of this method is reasonable and it can be applied for structure equipment seismic analysis.

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References 1. American Society of Civil Engineers: Seismic Analysis of Safety-Related Nuclear Structures/American Society of Civil Engineers. American Society of Civil Engineers, Reston, Virginia (2017) 2. Peters, K.A., Schmitz, D., Wagner, U.: Determination of floor response spectra on the basis of the response spectrum method. Nucl. Eng. Des. 44(2), 255–262 (1977). https://doi.org/10. 1016/0029-5493(77)90032-2 3. Calvi, P.M., Sullivan, T.J.: Estimating floor spectra in multiple degree of freedom systems. Earthq. Struct. 7(1), 17–38 (2014). https://doi.org/10.12989/eas.2014.7.1.017 4. Sorinm V.: Some Aspects of Seismic Power Spectral Density (PSD) and Response Spectrum Analyses (Support studies to seismic analysis of the Tokamak). Energopul, St. Petersburg (2017) 5. Clough, R., Penzien, J.: Dynamics of Structures, 3rd edn. Computers and Structures Inc., University Ave. Berkeley, CA 94704 USA (1995) 6. Vanmarcke, E.H.: Structural Response to Earthquakes. In: Seismic Risk and Engineering Decisions, pp. 287–337. Elsevier (1976). https://doi.org/10.1016/B978-0-444-41494-6.500 11-4

Minisymposium “New Approaches for Oil and Gas Reservoirs Simulation”

Modeling of Liquid Displacement from the Porous Medium Taking into Account the Presence of Hydraulic Fracture E. I. Kolenkina Skryleva1,2(B) , N. N. Smirnov1,2 , V. F. Nikitin1,2 , R. R. Fakhretdinova1,2 , and M. N. Makeeva1 1 Moscow M.V. Lomonosov State University, Leninskie Gory, 1, Moscow 119992, Russia 2 Scientific Research Institute for System Analysis of the Russian Academy of Sciences, 36-1

Nakhimovskiy pr., Moscow 117218, Russia

Abstract. The paper investigates the influence of the presence and location of hydraulic fractures on the dynamics and quality of oil displacement. Particular attention is paid to ascertaining whether the presence of a hydraulic fracture increases the oil recovery coefficient or only intensifies oil production. The paper considers numerical modeling of two-dimensional displacement of a fluid from a porous medium by a less viscous fluid. Moreover, the phase boundary is unstable: due to random perturbations superimposed on the distribution of permeability and porosity in the computational domain, the initially flat boundary loses its shape, the growth of “viscous fingers” begins, which significantly affects the quality of the displacement. The influence of the presence and size of hydraulic fractures on the dynamics and quality of oil displacement is considered. This paper also discusses the process of cleaning a fracture from hydraulic fracturing fluid by oil. Using numerical modeling on the basis of the constructed mathematical model, a relationship is established between the quality of hydraulic fracture cleaning and the geometrical parameters of the fracture and the region filled with hydraulic fracturing fluid. Keywords: Hydraulic fracture · Displacement instability · Seepage · Porous medium · Numerical simulation · Multiphase flows

1 Introduction One of the technologies for extracting liquid minerals involves displacing them from porous formations under the influence of a pressure gradient (Fig. 1). The process is complex, as multiphase seepage occurs. In addition, for the correct mathematical modeling of the seepage process in the oil reservoir, it is necessary to take into account many physical and chemical interactions. The first step in this direction took place in the 19th century, when the French hydraulic engineer Henri Darcy substantiated the relationship between the velocity of fluid flow through thin channels with a pressure gradient [1]. In 1942, S. Buckley and M. Leverett first proposed the modeling of one-dimensional seepage for incompressible, immiscible liquids, under conditions where capillary pressure © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 503–518, 2022. https://doi.org/10.1007/978-3-030-92144-6_38

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and the influence of gravity can be neglected [2]. To build the model, the generalized Darcy law was used, the concepts of saturation, relative phase permeability were introduced.

Fig. 1. Oil displacement by water with the development of instability at the displacement front.

One of the difficulties with multiphase seepage is the displacement instability. When a more viscous liquid is displaced with a less viscous one, displacing liquid tends to break through the displaced layer, forming channels called “viscous fingers” in it. The resulting instability leads to disruption of the initially flat shape of the interface and the breakthrough of individual tongues of the displacing fluid. This can lead to the entrapped of oil inside the porous reservoir and reduce the quality of oil displacement. The phenomenon of instability of the interface between two fluids during the displacement of a less viscous fluid by a more viscous fluid was discovered in 1958. This instability was subsequently named after the scientists who first discovered it (Saffman-Taylor instability [3]). So modeling of the joint seepage of two liquids requires taking into account the influence of phases on each other. Multiphase seepage in an oil-bearing porous formation is a very complex process, which is described by complex equations that cannot be solved analytically. Real experiments of such processes are expensive and difficult to visualize. One of the most effective methods for studying seepage is a numerical experiment: it is much cheaper and more affordable compared real experiments. Features of numerical modeling of unstable displacement of a viscous fluid from a porous medium are considered in [4–7]. The preparation and conduct of a numerical experiment requires less time, it is easily reproduced and gives more complete information. On the other hand, numerical modeling is often the only possibility, since the equations describing seepage are in most cases not solved by analytical and even approximate methods. As a result, there is a need to use computers. In this paper, we consider a numerical simulation of the process of unstable seepage in the area containing hydraulic fracturing. Hydraulic fracturing is one of the methods to increase oil recovery. Fracturing technology involves injecting fracturing fluid into a well using powerful pumping stations. Typically, fracturing fluid contains proppant (a granular material that serves to prevent fracture closure due to rock pressure). Many theoretical problems associated with the

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appearance and advancement of a hydraulic fracture in a porous reservoir were considered in [8–13]. An important stage of the procedure is the process of cleaning a hydraulic fracture from a hydraulic fracturing fluid: a hydraulic fracturing fluid is gradually replaced by the produced fluid.This paper considers two-dimensional modeling of a two-phase seepage process in a porous medium containing a hydraulic fracture. The fracture is modeled as an area with increased porosity and permeability. There may be situations in which hydraulic fracturing leads to the expected result (increase of oil production), but at the same time not only oil but also displacing water flows into the well, which leads to an in the percentage of water in the entire area of the well and can negate the positive method effect. One of the goals of this work is to investigate the effect of hydraulic fracture size and fracturing fluid viscosity on the process of cleaning hydraulic fracture. Another goal of this work is to study the influence of the presence and size of hydraulic fractures on the quality and dynamics of oil production during oil displacement by water. Details of numerical algorithms for modeling such processes are described in [14].

2 Mathematical Model We will solve the problem with the following assumptions: • • • • • • •

All liquids are incompressible; The viscosity of the displaced fluid is greater than the viscosity of the displacing fluid; Capillary effects at the interface are not taken into account; Permeability and porosity are location dependent; Thermal effects and gravity are not taken into account; The outer boundary is impermeable; Seepage is modeled by Darcy’s law, taking into account the relative permeabilities of the phases • Hydraulic fracture - area of increased permeability and porosity • Permeability has a random “ripple” that contributes to the onset of instability of displacement. The following system of equations is solved. For each fluid phase, the law of mass conservation is written:  ∂  ∂ϕsk k + k uk,j = 0 ∂t ∂xj

(1)

Here ϕ is porosity, s is saturation,  is density (intrinsic), uk,j is j-th component of the seepage velocity of the k-th phase. The volumetric averaged seepage velocity of the fluid is defined as:  uj = ukj k

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The summation of Eq. (1), taking into account the fact that liquids are incompressible and porosity is constant in time, brings to: ∂uj =0 ∂xj

(2)

Darcy’s law for each phase is as follows: uk,j = −

KKkR ∂pk μk ∂xj

(3)

Here μk is dynamic viscosity of the phase, K is absolute permeability of the medium, KkR is the relative permeability of k-th phase, p is the pressure in the pores. Summation of Eq. (3) leads to uj = −K

 k

mk

∂p ∂xj

Substituting expression (3) into Eq. (2) gives an equation for the pressure:    ∂p ∂ mk K =0 ∂xj ∂xj k

Equation (1), taking into account the introduced definitions, leads to equation for saturation dynamics:  ∂  ∂sk fk uj = 0 + ϕ −1 ∂t ∂xj kr

k – proportion of relative phase Here mk = μkk – relative mobility, fk = m1m+m 2 mobility. The relative permeabilities KkR are calculated using the Brooks – Corey model:  0 0 nk res sk − skres R Kk = kk Sk , sk ≥ sk , Sk = 1 − s1res − s2res 0, sk < sres

k

Here kk0 > 0 and n0k > 0 are the model parameters, and the effective saturation Sk is determined by the residual saturations of 0 ≤ skres ≤ 1 (s1res + s2res < 1). Hydraulic fracturing is modeled as an area of increased permeability and porosity: Absolute permeability:  Kf , (x, y) ∈ F K(x, y) = /F K0 , (x, y) ∈ Porosity:  ϕ(x, y) =

ϕf , (x, y) ∈ F /F ϕ0 , (x, y) ∈

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F – points lying inside the fracture. It is also worth noting that a weak random “ripple” is superimposed on the permeability, so that finally the absolute permeability is defined as: Kˆ = Kexp(δξ ) Here ξ is a random variable uniformly distributed in the interval [−1; +1], δ is a rather small quantity. This random “ripple” contributes to the onset of instability of displacement.

3 Results 3.1 Hydraulic Fracture Cleaning The first model problem considered in this work is the process of cleaning a hydraulic fracture (Fig. 2)

Fig. 2. Calculation area for modeling the fracture cleaning process

The reservoir is initially saturated with pore fluid (oil). There are 4 wells in the corners with constant pressure in it. In the center of the area there is a production well and a hydraulic fracture filled with proppant and hydraulic fracturing fluid. It is assumed that part of the fracturing fluid has leaked into the surrounding rock. It is assumed here that the region impregnated with the hydraulic fracturing fluid is elliptical (region A). Due to the pressure drop, pore fluid is filtered in an isotropic porous medium to the producing well and to the outside. The viscosity of the fracturing fluid is greater than the viscosity of the oil. So the process is unstable. The outer walls are impermeable.

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The boundary conditions: in :

s = 0, P = Pin

out :

P = Pout < Pin

w :

un = 0 →

∂p 0 ∂n =

The initial conditions: 

s = 1, (x, y) ∈ A s = 0, (x, y) ∈ /A

Effect of Fracture Length on Displacement Dynamics The purpose of the first numerical experiment is to investigate the effect of fracture length (Lfracture ) on dynamics. Two calculations were carried out with different fracture lengths. The calculation parameters are given in the table:

Figure 3 shows the dependence of mean saturation of hydraulic fracturing fluid in elliptic zone on time.

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Fig. 3. The dependence of mean saturation of hydraulic fracturing fluid in elliptic zone on time.

The dependence in Fig. 3 initially behaves linearly, after that it continues to decrease, but never reaches a constant: the asymptote in this case is s = 0. Therefore, any sufficiently small or satisfactory saturation value can be taken as a criterion when it is considered that the fracture is already cleaned. If we assume that the fracture is already cleaned when the saturation of the hydraulic fracturing fluid reached 0.2, then it took about 5 days 8 h to clean the fracture 20 m long, and it took 9 days 13 h to clean the fracture 30 m long. It turns out that the ratio is 1.79. Figures 4, 5, and 6 show the distribution patterns of the oil seepage velocity for various moments of time. The instability of displacement is observed in all figures.

Fig. 4. Oil seepage velocity approximately 5 h after the start of cleaning.

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In Fig. 4 the largest flows are observed in the fracture, as well as near the production well. It can be seen that for a larger fracture, the fluid flows faster in the area near the well.

Fig. 5. Oil seepage velocity approximately 6 h after the start of cleaning.

Fig. 6. Oil seepage velocity approximately 15 h after the start of cleaning.

In Fig. 5 for the case when the length is 20 m, the flow occurs along the entire fracture. In the case of a longer fracture, a breakthrough occurred in the area of the production well. Thus, at this time moment, not the entire hydraulic fracture is involved in the seepage process. In Fig. 6 we can see that in almost the entire area, seepage occurs. Oil has accumulated strongly in the rock near the well, continues to flow into it. However, there are areas that have not changed much: in them, the accumulation of oil and the displacement of hydraulic fracturing occurs very slowly. Effect of Fracture Fluid Viscosity on Displacement Dynamics The purpose of the first numerical experiment is to investigate Effect of fracture fluid viscosity (µ2) on displacement dynamics. Two calculations were carried out with different viscosities. The calculation parameters are given in the table:

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In Fig. 7 we can see how the saturation of the displaced fluid in the elliptical region changes with time. It is noticeable that on each graph three parts can be distinguished. The first part – similar to linear, it ends when there is a breakthrough of oil to the production well. The second part is section, and the third – curved down section.

Fig. 7. The dependence of mean saturation of hydraulic fracturing fluid in elliptic zone on time.

The section curved up is the time interval when the displacement velocity increases. This phenomenon can be explained by the fact that during this time interval, liquid is displaced from those areas that were not previously involved. These areas are shown in Fig. 8. Then, when these hard-to-reach areas are partially cleared, displacement again slows down (third area). For the considered cases, the difference between two graphs (Fig. 7) is not significant. Throughout the entire cleaning period, the difference was no more than 3–4%

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Fig. 8. The distribution of saturation at different moments of time.

In Fig. 9 we can observe the presence of a breakthrough in the case of the high viscosity and the absence of a breakthrough in the case of the low viscosity. It turns out that a breakthrough for higher viscosity occurs earlier than for lower viscosity. When the fracturing fluid has a high viscosity, a breakthrough through areas near the production well is possible. That is, the hydraulic fracture does not work, the fracture is not connected to the seepage process.

Fig. 9. Oil seepage velocity approximately 5 h 38 min after the start of cleaning.

3.2 Unstable Displacement of Oil by Water, Taking into Account the Presence of a Hydraulic Fracture There are two wells in the computational area: injection and production(Fig. 10). One or two hydraulic fractures are possible near the production well. Fractures are considered completely cleaned. External walls are impermeable. Initially, the entire area is filled with oil. Low viscosity fluid (such as water) is pumped into the injection well. Oil is displaced due to the pressure drop across the injection and production wells. Oil viscosity is higher than water viscosity. Thus, the process is unstable.

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Fig. 10. Calculation area for modeling the unstable displacement of oil by water

All characteristics below are calculated per meter of reservoir power. Effect of Fracture Length on Displacement Dynamics The purpose of these calculations is to find out the effect of fracture length (Lfracture) on displacement dynamics. Two calculations were carried out with different fracture lengths. The calculation parameters are given in the table:

In the Fig. 11 we can see how the pattern of displacement has changed over time for each case. The instability of displacement can be observed. On the left side we see how the displacing fluid broke through the fracture to the production well.

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Fig. 11. Distribution of saturation of the displacing fluid in the computational area

Initially, only oil is extracted, but gradually the tongues of water break through to the production well and a mixture of liquids is extracted. In the Fig. 12 the dependence of the oil phase content in production on time is presented. According to this graph, we can conclude that the quality of oil in production in the case when the fractures are 200 m long decreases faster than in the second (when the fractures have a length of 100 m), but this difference is insignificant.

Fig. 12. Oil saturation in production versus time(years)

In the Fig. 13 we can see the dependence of total oil production on time. At the initial stage, by 6 years of operation of the well, in the case when the length of the fractures

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is 200 m, the total oil production is 41% more than in the case when the length of the fractures is 100 m. In the same time oil production per unit time is much higher for the case of a long fracture (Fig. 14).

Fig. 13. Total oil production (M3 ) versus time(years)

Fig. 14. Oil production per unit time (M3 /day) versus time (years)

Effect of the Presence of Fractures on Displacement Dynamics The purpose of these calculations is to find out how the presence or absence of a hydraulic fracture affects the quality and dynamics of oil recovery. The case when there is no fracture is compared with the case when the fracture is one and when there are two of them. The calculation parameters are given in the table:

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In Fig. 15 we can see that at the initial stage of well exploitation, the oil quality does not significantly differ in the case of one or two hydraulic fractures, but falls faster than in the case of absence of a fracture.

Fig. 15. Oil saturation in production versus time(years)

It can be seen that after 5 years of field operation, the total oil production is 3 times more in the presence of one fracture and 4 times more in the presence of two fractures, compared with the absence of hydraulic fracturing (Fig. 16). As expected, the highest oil production is observed when two hydraulic fractures are present (Fig. 17).

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Fig. 16. Total oil production(M3 ) versus time(years)

Fig. 17. Oil production per unit time (M3 / day) versus time (years)

4 Conclusions ✓ Instability takes place when a liquid with higher viscosity is displaced by a liquid of lower viscosity. ✓ In a small fracture, the cleaning process is faster, and the seepage process is more uniform. ✓ When the fracturing fluid has a higher viscosity than the displacing agent, it is possible that the breakthrough will happen near the well, and not through the fracture. That is, the hydraulic fracture does not work, the fracture is not connected to the seepage process. ✓ The breakthrough of the fluid to the production well is faster, the greater is the viscosity of the fracturing fluid. ✓ For the investigated hydraulic fracturing fluids with viscosities µ = 100 Mpa*s and µ = 80 Mpa*s, the difference between saturations hydraulic fracturing fluid at different moments of time is about 3–4%. ✓ When the fractures have a length of 200 m, the total oil production at the initial stage is more by 41% than the fractures have a length of 100 m. The quality of oil in production is not significantly different. ✓ In the first 5 years of field exploitation, it is more profitable to produce oil in the presence of one or two hydraulic fractures. At the same time, the quality of oil does not

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significantly differ in the case of one or two hydraulic fractures, but it decreases faster than in the absence of a fracture.

Acknowledgments. The authors wish to acknowledge the support by Russian Foundation for Basic Research (Grant initiative №20–07-00378).

References 1. Darcy, H.: The Public Fountains of the City of Dijon. Experience and Application, Paris (1856) 2. Buckley, S.E., Leverett, M.S.: mechanism of fluid displacement in sands. J. Pet. Technol. (1941) 3. Saffman, P.G., Taylor, G.J.: The penetration of a fluid into a porous medium of Hele-Shaw cell containing a more viscous fluid. Proc. R. Foc. Zond. A 245, 312 (1958) 4. Smirnov, N.N., Nikitin, V.F., Skryleva, E.I.: Microgravity investigation of seepage flows in porous media. Microgravity Sci. Technol. 31(5), 629–639 (2019) 5. Dushin, V.R., Skryleva, E.I., Nikitin, V.F.: Microgravity investigation of capillary forces in imbibition of fluid into porous media. In: Proceedings of 69th International Astronautical Congress, Bremen, Germany (2018) 6. Logvinov, O.A. Skryleva, E.I.: Displacement of a viscous fluid from a Hele-Shaw cell with a sink. Moscow Univ. Mech. Bull. 71(4), 77–81(2016) 7. Skryleva, E.I., Kozlov, I.V.: Mathematical modeling and processing of an experiment on the displacement of oil by water from Neocomian sandstones, Cybern. Bull. (2), 138–145 (2016) (in Russian: Ckpyleva E.I., Kozlov I.B. Matematiqeckoe modelipovanie i obpabotka kcpepimenta po vytecneni nefti vodo iz neokomckix pecqanikov. Bectnik kibepnetiki, № 2, c. 138–145) 8. Perkins, T.K., Kern, L.R.: Widths of hydraulic fractures. J. Petrol. Technol. paper SPE 89. 222, 937–949 (1961) 9. Nordgren, R.: Propagation of vertical hydraulic fractures. J. Petrol. Technol. 253, 306–314 (1972) 10. Khristianovich, S.A., Zeltov, Y.P.: Formation of vertical fractures by means of highly viscous liquid. Proc. Fourth World Petrol. Congress, Rome. 2, 579–586 (1955) 11. Lister, J.R.: Buoyancy-driven fluid fracture: the effects of material toughness and of low viscosity precursors. J. Fluid Mech. 210, 263–280 (1990) 12. Adachi, J.I., Detournay, E.: Self-similar solution of a plane-strain fracture driven by a power– law fluid. Int. J. Numer. Anal. Methods Geomech. 26, 579–604 (2002) 13. Hu, J., Garagash, D.I.: Plane-strain fluid-driven fracture propagation in a permeable rock of finite toughness. In: Proceedings of ASCE Engineering Mechanics Conference, Delaware, pp. 1–8 (2004) 14. Dushin, V.R., Nikitin, V.F., Skryleva, E.I.: Computational modeling of fluid displacement from a porous medium. Cybern. Bull. 4(28) (2017) (in Russian: Dyxin B.P., Hikitin B.F., Ckpyleva E.I.: Byqiclitelnoe modelipovanie vytecneni flida iz popicto cpedy. Bectnik kibepnetiki, tom 4, № 28)

Application of a* Algorithm for Tortuosity and Effective Porosity Estimation of 2D Rock Images Filippo Panini(B)

, Eloisa Salina Borello , Costanzo Peter , and Dario Viberti Politecnico Di Torino, 10129 Turin, Italy [email protected]

Abstract. Characterization and understanding of fluid flow phenomena in underground porous media at the micro and macro scales is fundamental in reservoir engineering for the definition of the optimal reservoir exploitation strategy. Laboratory analyses on rock cores provide fundamental macroscale parameters such as porosity, absolute and relative permeability and capillary pressure curves. In turn, macroscale parameters as well as flow behavior, are strongly affected by the micro geometrical features of the rock, such as pore structure, tortuosity and pore size distribution. Therefore, a thorough comprehension of single and multiphase flow phenomena requires analyses, observations and characterization at the micro scale. In this paper we focus on the analysis of a 2D binary image of a real rock thin section to characterize the pore network geometry and to estimate tortuosity, effective porosity and pore size distribution. To this end, a geometrical analysis of the pore structure, based on the identification and characterization of the set of the shortest geometrical pathways between inlets and outlets pairs, is implemented. The geometrical analysis is based on the A* path-finding algorithm derived from graph theory. The results provided by the geometrical analysis are validated against hydrodynamic numerical simulation via the Lattice Boltzmann Method (LBM), which is well suited for simulating fluid flow at the pore-scale in complex geometries. The selected rock for this analysis is Berea sandstone, which is recognized as a standard rock for various applications such as core analysis and flooding experiment. Results show that the path-finding approach provides reasonable and reliable estimates of tortuosity and can be successfully applied for analyzing the distribution of effective pore radius, as well as for estimating the effective porosity. Keywords: Tortuosity · Effective porosity · Path-finding algorithm · Lattice Boltzmann Method

1 Introduction An optimal reservoir exploitation strategy has to be defined taking into account of different aspects related to economical, technical and environmental issues [31] and requires a deep understanding of the fluid flow phenomena dominating the reservoir behavior. From © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 519–530, 2022. https://doi.org/10.1007/978-3-030-92144-6_39

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the viewpoint of the description and understanding of the reservoir dynamic behavior, all the information provided by technical disciplines (such as geology, geophysics, log interpretation, laboratory measurements of fluids and rock properties, well testing, reservoir engineering and geomechanics) have to be taken into account, compared, combined and properly integrated [5, 12, 28–30, 32, 37, 38, 41]. Furthermore, it is well known that any complex and nonlinear problem is affected by uncertainties. Therefore, the uncertainty associated to the interpretation provided by each discipline must be estimated [18, 39, 40] and, if possible mitigated by acquiring new and necessary information at all stages of reservoir life, even during the production stage of a mature field. In this view, characterization and understanding of fluid flow phenomena in underground porous media at the micro and macro scales is a fundamental piece of information that can minimize the uncertainties and contribute to maximize reservoir characterization and understanding. In many cases, the flow mechanism can be understood from porescale phenomena, allowing predictions at the macro-scale, which can then be compared with experimental results [4]. At the macro-scale, the fluid flow is modeled by averaging the microscopic continuity and momentum equations over a representative elementary volume (REV) [4] and the porous medium is parameterized mainly by porosity and permeability [13]; the fundamental equation of fluid motion in porous media under the assumption of small Reynolds numbers is Darcy’s equation [4]: k u = − ∇P μ

(1)

where u is the Darcy’s velocity, k is the permeability tensor, μ the viscosity and ∇P the pressure gradient. Porous media are complex materials characterized by a chaotic structure and tortuous fluid flow, with pore and grains dimension varying over a wide range [13]. To address the crooked fluid paths of pore structure, the concept of tortuosity (τ ) was introduced [9]. Two main types of tortuosity are defined in the literature: geometrical tortuosity (τg ) and hydraulic tortuosity (τh ). The geometrical tortuosity is defined as the shortest length between inflow and outflow points that avoids the solid obstacles divided by the distance between inlet and outlet [1, 11]. The hydraulic tortuosity is defined as the effective path length taken by the fluid divided by the length of the porous material measured along the flow direction [9]. Since the fluid flow path is always greater than the shortest geometrical path, hydraulic tortuosity is always greater than the geometrical tortuosity [11, 13]. For a complete review on the definitions of tortuosity, the reader can refer to [11] and to [13]. To account for pore space interconnections affected by the flow, the effective porosity (φe ) was introduced; this property is defined as the percentage of conductive pore space with respect to the bulk volume [20]: φe =

Vflow Vb

(2)

where Vb is the bulk volume and Vflow is the portion of volume contributing to the fluid flow. The impact of pore network geometry on the flow behavior is well recognized. In the literature, several analytical expressions have been provided to link permeability to

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the pore structure as a function of porosity (or effective porosity), tortuosity and grain or pore dimension. One of the most used expressions is the Kozeny-Carman equation [9, 44]: k=

2 φrH cτ 2

(3)

where c is the Kozeny’s constant and rH is the hydraulic radius, which gives a measure of the average pore dimension and is defined as [4]: rH = φ

Vb Aw

(4)

where Aw is the wetted surface. Several authors discussed the geometrical analysis of the pore structure from 2D and 3D images. Lindquist et al. [22] discussed the medial axes method to analyze structure properties such as pore throat and pore body size distributions and geometric tortuosity of a 3D digitalized image. Later, Sun et al. [35] used a shortest past approach based on Dijkstra’s algorithm to calculate the geometric tortuosity and connected porosity; then, they applied a multiscale method approach to upscale the permeability. Al-Raoush and Madhoun [2] presented an algorithm for calculating geometric tortuosity from 3D X-ray tomography images of real rocks based on a guided search for connected paths utilizing the medial surface of the void space of a 3D segmented image. In this paper we estimate tortuosity, effective porosity and permeability from a 2D binary images of rocks. To this end, we applied an approach based on geometrical analysis, developed based on a A* path-finding procedure taken from the graph theory, of the porous medium. This approach was validated by numerical simulation via the Lattice Boltzmann Method (LBM). We report the results referred to a 2D image of Berea sandstone. Even if not reported here, a preliminary validation of the methodologies was carried out on a set of simplified synthetic cases for which the true value of the parameters of interest was analytically computed [42].

2 Methodology Description Starting point for the presented methodology is a 2D binary image of a rock section. Such data can be obtained by image processing of the Scanning Electron Microscopy (SEM) image of a thin section or the slice of an X-ray micro-tomography image. In this paper we analyzed a binary image of Berea sandstone taken from [7]. On the 2D binary image we analyzed the pore structure in terms of pathways accessible by fluid flow in single-phase conditions and calculated the associated parameters: tortuosity, effective porosity and permeability. Two different approaches were applied to assess pathways: geometrical and hydrodynamic. The pore structure analysis was completed by calculation of pore radius variations along the path compared with the hydraulic radius. 2.1 Geometrical Path Calculation The pore structure of a binary 2D image is analyzed through a path-finding method based on graph representation. To this end, a set of inlets nin and outlets nout are placed along

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the inner and outer boundaries of the image of the porous domain orthogonal to the main flow direction (X or Y); the centroids of the pixels are used as reference grid node locations. These locations represent the points where the fluid enters and potentially leaves the porous system respectively. The shortest pathway between each couple of inlet-outlet points is obtained through the A* algorithm [17, 26]. A* is based on a cost function f (n) to determine the optimal path between two nodes: f (n) = g(n) + h(n)

(5)

where n indicates the considered node, g(n) the incremental distance from the considered node to the initial node and h(n) is a heuristic function used to obtain a prior estimation to reach the target from the considered node. The process of obtaining the optimal shortest path is achieved by steps. Starting at the initial node, the cost function f (n) is calculated at each adjacent nodes in order to identify the one having the minimum cost which will be used as reference for the next calculation. This process is progressively repeated until the target is reached. The output is represented as a graph G = (N, E) with N being a set of nodes with X − Y coordinates while E the edges connecting the nodes (Fig. 1).

Fig. 1. Schematic of pore size and effective porosity estimation from path-finding approach.

2.2 Hydraulic Path Calculation Hydrodynamic paths were obtained as a result of numerical simulation of single-phase fluid flow at the pore-scale. To this end, we implemented a discrete mesoscopic computational method based on the Lattice Boltzmann Method (LBM). LBM has been shown to be a powerful technique for the computational modeling of a wide variety of complex fluid flow problems including single and multiphase flows in complex geometries

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and porous media [7, 15]. The use of the LBM to evaluate the hydraulic tortuosity in synthetic porous media was presented by [25]. The LBM belongs to the family of discrete mesoscopic computational methods; unlike the conventional CFD methods, which numerically solve the Navier-Stokes (N-S) set of partial differential equations, the LBM solves the discrete lattice Boltzmann equation (LBE): fi (x + vi , t + t) − fi (x, t) = i (fi (x, t))

(6)

where fi is the density distribution function, vi the lattice velocity and i the collision operator. The fluid is modeled as consisting of fictive particles which perform consecutive streaming and collision processes over a discrete reticular grid called lattice mesh [16]. The propagation and interaction of the particles is simulated in terms of the time evolution of the density distribution function, representing an ensemble average of the particle distribution. The flow properties such as velocity, pressure or fluid density can be derived from the moments of the density distribution function. The rules governing the collisions are designed such that the time-average motion of the particles is consistent with the macroscopic hydrodynamics. Collision rules constitutes a simplified mesoscopic kinetic model based on a Boltzmann-type equation that incorporate only the essential physics of microscopic or mesoscopic processes, avoiding to follow each particle as in molecular dynamics simulations [43]. Due to its particulate nature and local dynamics, the LBM has several advantages over other conventional CFD methods, especially in dealing with complex boundaries, incorporating of microscopic interactions, and parallelization of the algorithm [34]. The implementation adopted in this paper for single-phase flow simulation in porous media is based on a single-relaxation time (SRT) approximation of the collision operator, called Bhatnagar-Gross-Krook model (BGK) [6]:   eq (7) i = −ωt fi (x, t) − fi (x, t) eq

where ω is the inverse of the relaxation time and fi the density distribution function at the equilibrium. The nine-velocity square lattice model D2Q9 was adopted to discretize the domain. At the fluid-solid interface, no-slip condition was imposed via halfway bounce-back [20]. Fixed pressure gradient between inlet and outlet was assumed as the boundary condition, which was implemented via the non-equilibrium bounce-back approach [45]. 2.3 Characterization of Pore Structure A quantitative characterization of the porous medium is carried out through the estimation of a series of parameters: pore size, tortuosity, effective porosity and permeability. In the following we will not parametrize the pore structure in terms of pore throat and pore body; we will refer to the local aperture between the pore walls as pore size or to the semi-aperture as pore radius. Along the geometrical pathways identified by the A* algorithm, the pore size is calculated at each node location by measuring the extension of the pore section length orthogonal to the local path direction (Fig. 1). The local pore size estimation allows monitoring the pore radius (rp ) evolution along each detected path. This information can

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be used for reconstructing 3D porous geometries [27] or for comparative analysis with hydrodynamic data [8]. The output is also analyzed by a statistical representation of the pore radius distribution of the sample, which is compared with the hydraulic radius (rH ). The hydraulic radius for a 2D section is defined as [42]: rH = φ

Ab Pw

(8)

where Ab is the bulk area of the sample and Pw is the wetted perimeter, i.e. the interface between grains and void, which can be easily calculated by image processing routine; we used the Image Processing Toolbox of MATLAB [24]. The geometrical tortuosity τg within a porous medium, in a given flow direction (dir), is calculated through the ratio between the average of the shortest pathway lengths in that direction (Lsh,dir ) by the length of the system domain along the fluid direction (Ldir ) [13]: τg,dir =

Lsh,dir Ldir

(9)

where Lsh,dir is calculated as the average of the shortest pathway lengths calculated. Hydraulic tortuosity was defined by Carman [9] as the ratio of the average length of the fluid paths divided by the length of the sample: τh =

Lh L

(10)

Koponen et al. [19] suggested to estimate tortuosity in a fixed direction from the velocity field simulated with a CFD numerical simulator as: τh,dir =

|v| vdir

(11)

where |v| is the absolute value of the local flow velocity, vdir is the directional component of that velocity and  denotes the spatial average over the pore space. We made use of a numerical simulator based on the Lattice Boltzmann Method (LBM) to simulate the fluid flow in the porous media at the pore-scale and obtain the velocity field and calculate the hydraulic tortuosity (τh ) with Eq. 11. We also verified the invariance of calculated tortuosity with respect to the variation of the applied pressure gradient until laminar flow conditions are guaranteed (Re ≤ 2) [3], where Re is the Reynolds number. In [42], we proposed a purely geometrical calculation of the effective porosity based on path-finding: φeg =

Npp Npx

(12)

where Npp is the number of image pixels belonging to the portion of pore channels crossed by a pathway (in X or Y direction) and Npx the total number of image pixels.

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The permeability can be calculated both applying the Darcy’s law (kh ) and the Kozeny-Carman equation (kKC,g ). The subscript h indicate that in the first case the permeability is calculated by using the outputs of the numerical simulation, while in the second case the subscript g indicate that values obtained by geometric approach were used as inputs in the Kozeny-Carman equation. In the first case, the Darcy’s velocity is computed from the hydrodynamical simulation velocity (v) as [14]:  1 v(x, y)dA (13) u= A A Then the permeability is estimated as: udir μ where dir = x, y (14) kh,dir = − ∇dir P In the second case, the Kozeny-Carman equation (Eq. 3) was modified substituting the porosity with the effective porosity calculated as in Eq. 12 and the tortuosity with the geometrical tortuosity calculated as in Eq. 9, obtaining an estimate of permeability only based on a geometrical analysis of the porous medium: kKC,g,dir =

2 φeg rH 2 cτg,dir

(15)

where in our case c = 5 [4].

3 Case Study: 2D Image of Berea Sandstone The geometrical and hydrodynamic characterization is carried out on an image obtained from thin sections of 3D samples of Berea sandstone (D50 = 23 μm [11]). Berea sandstone is characterized by high difference between pore throat and pore body dimension [10] and the high porosity and permeability values make this rock a potential source of oil and natural gas. The image used in this study is reported in Fig. 2. The image dimensions are 769 × 624 px2 (resolution of 4120 ppcm) and the 2D porosity calculated from the image is 0.3.

Fig. 2. 2D image of Berea sandstone

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3.1 Results and Discussion In order to observe hydrodynamic behavior, the smallest pore size should be 4–5 lattice units [34]. The parameters adopted for the numerical simulations are: nx = 1706 l.u., ny = 1367 l.u., x = 1.04 μm /l.u. The fluid used for the numerical simulation has the following properties: viscosity 0.5 cP, density 1050 kg/m3. A pressure gradient of 100 Pa/m was applied between inlet and outlet. Laminar flow occurs (Re = 3.18E-04). The case study was analyzed separately in the X and Y directions to assess eventual discrepancy in tortuosity and permeability which indicates anisotropy in the medium. Results are summarized in Table 1. Figure 3 compares the shortest paths obtained with the A* algorithm (red lines) with the velocity map resulting from the numerical simulation. The pore zones contributing to the flow and used for the calculation of the effective porosity are represented in red in Fig. 4. In Fig. 5 the pore size distribution (Fig. 5a) and the pore size variation along a path (Fig. 5b) are compared with the hydraulic radius. Table 1. Comparison of results obtained with the geometrical and hydrodynamic approaches Geometric

Hydrodynamic

τ x (−)

1.250

1.416

τ y (−)

1.437

1.542

φe (−)

0.277

0.256

Tortuosity values calculated with the two methods (Table 1) are comparable (the discrepancy is lower than 10%) and the geometrical tortuosity is slightly smaller than the hydraulic tortuosity, as expected [13]. Comparing our results with literature, the tortuosity results are coherent with the gamma-shaped distribution with a minimum value of 1.07 and a most probable value close to 2, reported in [22] and just below the range of 1.6–2.8 reported in [36]. The pore radius distribution obtained by path finding algorithm appears to be reliable. The distribution in Fig. 5a is in good agreement with the literature: 31% of pore throats diameter of about 10 μm [21]; 5 μm as the most frequent pore throat radius [27]; 37% of relative pore volume characterized by 7 − 10 μm of pore radius [33]. Reasonable agreement (order of magnitude) is observed between the hydraulic radius rp = 7.29 μm) obtained by (rH = 14.6 μm) and the mode of pore radius distribution ( the path-finding approach (Fig. 5), thus suggesting that hydraulic radius formula (Eq. 8) is reasonably representative of the effective pore radius of the sample. The values of permeability calculated with hydrodynamic approach (kh ) through Darcy’s law (Eq. 14) are of the order of magnitude of 1 D and comparable with those reported in [7]. As we expected [23], the permeability evaluated with the geometrical approach (kKC,g ) through Kozeny-Carman equation (Eq. 15) was significantly higher than kh , being of the order of magnitude of 6–7 D. It was observed that, if in KozenyCarman equation the pore radius mode ( rp ) is adopted instead of the hydraulic radius (rH ) the resulting permeability reduces to about 1.5 D, becoming comparable with the hydrodynamic one. Further investigation will be carried out in future works.

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Fig. 3. Simulated velocity map (|v|) vs. detected geometrical paths considering flow in the X (a) and Y (b) directions.

Fig. 4. Pore volume interested by flow (red) and dead zones (white) identified by applying the path-finding algorithm (on the left) vs. a cutoff on the simulated velocity (on the right)

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Fig. 5. Pore radius estimation from the path-finding approach: (a) pore radius distribution compared with hydraulic radius (red line); (b) pore radius variation along a single vertical path, compared with the mode of the pore radius along the selected path (green line) and the hydraulic radius (red line).

4 Conclusions In this paper we adopt a geometrical analysis based on a path-finding algorithm for the characterization of the pore network geometry and connectivity of 2D binary images of rock samples representative of real geological formations. In order to validate the results, we used a hydrodynamic numerical simulator based on the LBM and compared the results. Results showed that even if hydrodynamic simulation was more accurate in reproducing the flow behavior, the path-finding approach could give reasonable estimates of tortuosity and could also be successfully applied for analyzing the distribution of effective pore radius, as well as for estimating the effective porosity and for giving a reasonable order of magnitude of permeability.

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31. Rocca, V., Viberti, D.: Environmental sustainability of oil industry. Am. J. Environ. Sci. 9(3), 210–217 (2013). https://doi.org/10.3844/ajessp.2013.210.217 32. Salina Borello, E., et al.: Harmonic pulse testing for well monitoring: application to a fractured geothermal reservoir. Water Resour. Res. 55(6), 4727-4744. AGU Publications, Wiley (2019). https://doi.org/10.1029/2018WR024029 33. Shi, J.Q., Xue, Z., Durucan, S.: Supercritical CO2 core flooding and imbibition in Berea sandstone—CT imaging and numerical simulation. Energy Procedia 4, 5001–5008 (2011) 34. Succi, S.: The Lattice Boltzmann Method for Fluid Dynamics and Beyond. Oxford University Press (2001) 35. Sun, W.C., Andrade, J.E., Rudnicki, J.W.: Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability. Int. J. Numer. Meth. Eng. 88(12), 1260–1279 (2011) 36. Takahashi, M., Kato, M., Urushimatsu, Y.: Geometry of pore structure in pressurized Berea sandstone. eurock2009 rock engineering in difficult ground conditions. In: International Society for Rock Mechanics and Rock Engineering (2009) 37. Verga, F., Viberti, D., Salina Borello, E.: A new insight for reliable interpretation and design of injection tests. J. Pet. Sci. Eng. 78(1), 166–177 (2011). https://doi.org/10.1016/j.petrol. 2011.05.002 38. Verga, F., Salina Borello, E.: Unconventional well testing: a brief overview. In geoingegneria ambientale e mineraria (GEAM). Pàtron Editore S.r.l. 149(3), pp.45–54 (2016) 39. Viberti, D.: A rigorous mathematical approach for petrophysical properties estimation. Am. J. Appl. Sci. 7(11), 1509–1516 (2010). https://doi.org/10.3844/ajassp.2010.1509.1516 40. Viberti, D., Verga, F.: An approach for the reliable evaluation of the uncertainties associated to petrophysical properties. Math. Geosci. 44(3), 327–341 (2012). https://doi.org/10.1007/ s11004-011-9358-1 41. Viberti, D., Cossa, A., Galli, M.T., Pirrone, M., Salina Borello, E., Serazio, C.: A novel approach to a quantitative estimate of permeability from resistivity log measurements. In geoingegneria ambientale e mineraria (GEAM). Pàtron Editore S.r.l. 155(3), 17–24 (2018) 42. Viberti, D., Peter, C., Borello, E.S., Panini, F.: Pore structure characterization through pathfinding and Lattice Boltzmann simulation. Adv. Water Resour. 141, 103609 (2020). https:// doi.org/10.1016/j.advwatres.2020.103609 43. Jiyuan, T., Yeoh, G.-H., Liu, C.: Some advanced topics in CFD. In: Computational Fluid Dynamics, pp. 369–417. Elsevier (2018). https://doi.org/10.1016/B978-0-08-101127-0.000 09-X 44. Wyllie, M.R.J., Spangler, M.: B: Application of electrical resistivity measurements to problem of fluid flow in porous media. AAPG Bull. 36(2), 359–403 (1952) 45. Zou, Q., He, X.: On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids 9(6), 1591–1598 (1997)

Minisymposium “Geometry, Topology, Fractal and Multifractal Modeling in Geosciences”

Representative Elementary Volume via Averaged Scalar Minkowski Functionals M. V. Andreeva1 , A. V. Kalyuzhnyuk2(B) , V. V. Krutko3 , N. E. Russkikh1 , and I. A. Taimanov4 1

4

A.P. Ershov Institute of Informatics Systems of SB RAS, 630090 Novosibirsk, Russia 2 Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia kalyuzhnyuk [email protected] 3 Gazpromneft Science and Technology Center, 190000 St. Petersburg, Russia [email protected] Sobolev Institute of Mathematics of SB RAS, and Novosibirsk State University, 630090 Novosibirsk, Russia [email protected]

Abstract. Representative Elementary Volume (REV) at which the material properties do not vary with change in volume is an important quantity for making measurements or simulations which represent the whole. We discuss the geometrical method to evaluation of REV based on the quantities coming in the Steiner formula from convex geometry. For bodies in three-dimensional space this formula gives us four scalar functionals known as scalar Minkowski functionals. We demonstrate on certain samples that the values of such averaged functionals almost stabilize for cells for which the length of edges are greater than certain threshold value R. Therefore, from this point of view, it is reasonable to consider cubes of volume R3 as representative elementary volumes for certain physical parameters of porous medium.

Keywords: Representative elementary volume Convex geometry · Minkowski functionals

1

· Porous media ·

Introduction

There are few notions of representative elementary volumes (REV) [1–3] which have in common the condition that this is the minimal elementary volume which serves a value representative of a certain property of the whole media. Therewith it is important to mention in which respects it gives such a representation. We address the problem of elementary volumes which represent the permeability properties of porous medium. Such a problem was considered from different points of view, for instance, in [4–7]. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2022  D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 533–539, 2022. https://doi.org/10.1007/978-3-030-92144-6_40

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If we know the representative elementary volume then we can perform all hydrodynamic simulations on such volumes to evaluate effective characteristics of the medium. Unlike modeling on a full-scale model, this allows us to minimize the computations of such characteristics as the Darcy coefficient. For this reason, the challenge is to find geometric characteristics related to permeability to use them for evaluating representative elementary volumes. Recently different possible applications to estimating permeability of two- and three-dimensional porous media by using the scalar Minkowski functionals were discussed in [8–10]. Therefore we propose to use the averaged scalar Minkowski functionals as such geometric characteristics. In this article we show that such functionals detect certain REVs, including the REV of porosity. A porous medium is represented by a digital core consisting of voxels that correspond to elementary cells. Each cell corresponds to the value of the radiodensity. Picking up the excursion coefficient λ, 0 < λ < 1, we assume that the voxel is black if the proportion is greater than or equal to λ and it is white otherwise. As a result, we obtain a two-colored (binarized) digital core which, we assume, consists of N × N × N black or white voxels (see Fig. 1). Each voxel corresponds to a cubic cell of size L × L × L.

Fig. 1. Binarized digital core. 1400 × 1400 × 1400 voxels.

We take randomly a sample voxel and correspond to it the set of nested cubes centered at it. For each such a cube X we compute the averaged Minkowski functionals Wi (Y )/Vol(X), where Y is a subset of X formed by black voxels. Every such a cube consists of k × k × k voxels and we consider the values of these values on nested cubes as functions of k. If there exists a constant C such that for generic inital voxels the graphs of the averaged functional Wi go to the asymptotes as k → C we may consider a cube of volume R3 with R = CL as a candidate for a representative elementary volume for the property related to Wi . In the example considered in Sect. 3 we take a sample of medium with N = 1400 and the elementary length L = 1.5 µm. For sample voxels we consider

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the graphs of the averaged functionals on nested cubes. It appears that for all sample initial voxels and for the averaged functionals W0 and W1 the graphs go to the asymptote as k → C ≈ 200, for the averaged W2 functional it goes to the asymptote as k → C ≈ 100, and for the averaged W3 functional C ≈ 150.

2

The Scalar Minkowski Functionals

Let X be a convex body with a regular boundary in three-dimensional space R3 and B be the ball formed by all vectors of length not greater than one. We denote by X + εB all points of the form x + εb where x is a point from X, b is a vector from B and ε is a positive constant: x = (x1 , x2 , x3 ), b = (b1 , b2 , b3 ), x + εb = (x1 + εb1 , x2 + εb2 , x3 + εb3 ). Since the zero vector belongs to B the original set X is a subset of X + εB and moreover X + ε1 B lies in X + ε2 B if ε1 ≤ ε2 . It is also easy to check that the bodies of the form X + εB are convex. We denote by Vol(X) the volume of a body X in three-dimensional space. The famous Steiner formula reads that 3    3 Vol(X + εB) = Wk (X)εk . k k=0

The quantities Wk , k = 0, 1, 2, 3, are called the k-th quermassintegrals, or the scalar Minkowski functionals. The latter name demonstrates that there are their generalizations to higher ranks (for instance, vector and rank two functionals). We shall not discuss them here because the investigation of their possible applications to our main problem is in progress. We recall that the analog of the Steiner formula is valid for bodies in the space Rn of arbitrary dimension n. In such a case Vol(X + εB), the n-th dimensional volume of X +εB, is a polynomial in ε of degree n. Since for ε = 0 the polynomial gives us the volume of X, we have W0 (X) = Vol(X). Other quantities Wk , k > 0, give us nontrivial scalar functionals. To guess their meanings we write down the Steiner formula for a ball X = DR of radius R. In this case X + εB is a ball DR+ε of radius R + ε and we have Vol(DR + εB) =

4πR3 4π(R + ε)3 4 = + 4πR2 ε + 4πRε2 + πε3 . 3 3 3

We see that these quantities have the following interpretations which, in fact, are valid for all convex bodies with regular boundaries: W1 (X) =

1 Area(∂X), 3

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where Area(∂X) is the area of the boundary ∂X of X,  1 W2 (X) = H dA, 3 ∂X i.e. one third of the integral of the mean curvature H (for the sphere of radius R, which is ∂DR , it is equal to R1 ), over ∂X,  1 K dA, W3 (X) = 3 ∂X i.e. one third of the integral of the Gaussian curvature K, which is K = R12 for ∂DR , over ∂X. We do not dwell here on the standard notions of the curvatures referring to textbooks on geometry (for instance, [11]). By continuity, the definitions of these quantities are uniquely extended to valuations of all convex bodies. This is important for us because from a digital core of a porous medium we construct a union of cubes which may intersect only by their faces or vertices. In this case every cube corresponds to a voxel from a digital core. Such unions are not necessarily convex however the notions of these functionals are uniquely extended to non-convex bodies by using the additivity property: W (X ∪ Y ) = W (X) + W (Y ) − W (X ∩ Y ). Therefore, the quermassintegrals (of the scalar Minkowski functionals) are defined for all unions of cubes which are naturally constructed from digital cores of porous media. For instance, for bodies with piece-wise linear triangulated boundaries the explicit formulas for computations in terms of combinatorial data are given in [12]. W0 (Y ) measure the volume of the space compleThe functional P (Y ) = 1 − Vol(X) mented to the media Y ⊂ X and it is equal to the porosity of Y . Certain topological characteristics of the media, i.e. the Betti numbers of porous media (in particular, oil and gas reservoirs) weighted by volumes were considered in [13]. The functional W3 is expressed in terms not of the media but of its boundary as follows: W3 (Y ) = 2π 3 χ(∂Y ), where the Euler characteristic of ∂Y is the alternated sum of the Betti numbers bi of ∂Y : χ(∂X) = b0 − b1 + b2 .

3

Evaluation of REV

The method of evaluating REV was sketched in Introduction and it is as follows: – take a digital core sample which consists of N × N × N voxels such that every voxel corresponds to the elementary cubic cell of size L × L × L; – take randomly a sample voxel and correspond to it the set of nested cubes centered at it; – for each such a cube X we compute the averaged Minkowski functionals Wi (Y )/Vol(X), where Y is a subset of X formed by black voxels;

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– every such a cube consists of k × k × k voxels and we consider the values of functionals on nested cubes as functions of k; – given sufficiently many initial sample voxels, if the graphs of these functions go to the asymptotes as k → C we consider a cube of volume R3 with R = CL as a representative elementary volume. As an example we consider a sample consisting of 1400 × 1400 × 1400 voxels (N = 1400) such that every voxel corresponds to a cube of size 1.5 µm × 1.5 µm × 1.5 µm. We present below the graphs of the averaged Minkowski functionals. For every family of nested cubes the values of the averaged functionals are given by a certain (colored) line (see Fig. 2, 3, 4 and 5).

Fig. 2. W0 .

Fig. 3. W1

Fig. 4. W2

Fig. 5. W3

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We see that for the functionals which are the averages of W0 and W1 and for all initial voxels the graphs go to the asymptotes as k → C ≈ 200. For W2 the graphs go to the asymptote as k → C ≈ 100, and for W3 we have C ≈ 150. This allows us to consider cubes of volume R3 with R = CL and L = 1.5 µ m as the representative elementary volumes of this medium for the physical parameters related to W0 , W1 , W2 , and W3 , where C ≈ 200 for W0 and W1 , C ≈ 100 for W2 , and C ≈ 150 for W3 . Final Remarks We conclude that the averaged Minkowski functionals have a tendency to go to asymptotes at some volumes. Therefore the proposed method leads to reasonable candidates for the representative elementary volumes for certain properties of the media using geometric characteristics of binarized digital core model. Applying this procedure to W0 we obtain the REV for porosity. We need to compare the results with the others obtained by conventional methods of REV evaluation, such as computational fluid dynamics simulations, to understand their meaning. Since the averaged functionals stabilize at different scales, the corresponding physical parameters have to be different. To our opinion it is worth to extend the study to Minkowski tensors which in addition reflect the anisotropy of media. These ideas may be helpful for finding REV for permeability. For instance, it was demonstrated in [3,4] by using numerical simulation that for certain materials the spatial scales of REVs for permeability are approximately 1.5–2 times larger than those for porosity. We are also left to understand how the results obtained by the proposed method vary with the excursion coefficient λ.

References 1. Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11(5), 357–372 (1963). https://doi.org/10.1016/0022-5096(63)90036x 2. Drugan, W.J., Willis, J.R.: A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Solids 44(4), 497–524 (1996). https://doi.org/10.1016/0022-5096(96)000075 3. Zhang, D., Zhang, R., Chen, S., Soll, W.E.: Pore scale study of flow in porous media: scale dependency, REV, and statistical REV. Geophys. Res. Lett. 27(8), 1195–1198 (2000). https://doi.org/10.1029/1999GL011101 4. Mostaghimi, P., Blunt, M.J., Bijeljic, B.: Computations of absolute permeability on micro-CT images. Math. Geosci. 45(1), 103–125 (2013). https://doi.org/10.1007/ s11004-012-9431-4 5. Du, X., Ostoja-Starzewski, M.: On the size of representative volume element for Darcy law in random media. Proc. R. Soc. Lond. A 462, 2949–2963 (2006). https:// doi.org/10.1098/rspa.2006.1704

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6. Sun, W.C., Andrade, J.E., Rudnicki, J.W.: Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability. Int. J. Numer. Meth. Eng. 88, 1260–1279 (2011). https://doi.org/10.1002/nme. 3220 7. Bear, J.: Dynamics of fluids in porous media. Courier Corporation Chelmsford 120, 162–163 (2013). https://doi.org/10.1097/00010694-197508000-00022 8. Scholz, C., et al.: Direct relations between morphology and transport in Boolean models. Phys. Rev. E 92, 043023 (2015). https://doi.org/10.1103/PhysRevE.92. 043023 9. Armstrong, R.T., et al.: Porous media characterization using Minkowski functionals: theories, applications and future directions. Transp. Porous Media 130, 305– 335 (2018). https://doi.org/10.1007/s11242-01-1201-4 10. Slotte, P.A., Berg, C.F., Khanamiri, H.H.: Predicting resistivity and permeability of porous media using Minkowski functionals. Transp. Porous Media 131(2), 705– 722 (2019). https://doi.org/10.1007/s11242-019-01363-2 11. Novikov, S.P., Taimanov, I.A.: Modern Geometrical Structures and Fields. American Mathematical Society, Providence (2006). https://doi.org/10.1090/gsm/071 12. Schr¨ oder-Turk, G.E., et al.: Minkowski tensors of anisotropic spatial curvature. New J. Phys. 15, 083028 (2013). https://doi.org/10.1088/1367-2630/15/8/083028 13. Gilmanov, R.R., Kalyuzhnyuk, A.V., Taimanov, I.A., Yakovlev, A.A.: Topological characteristics of digital oil reservoir models at different scales. In: Proceedings of the 20th Annual Conference of the International Association for Mathematical Geosciences, IAMG 2019, State College, Pennsylvania, USA, 10–16 August 2019, pp. 94–99 (2019)

The Fractal Model of Fractured Reservoirs Based on Pareto Distribution and Integrated Investigations A. V. Petukhov1,2,3,4(B) 1 NGT-Engineering, LLC, Ufa, Russia 2 Ukhta State Technical University, Ukhta, Russia 3 Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia 4 St. Petersburg State University, Saint Petersburg, Russia

Abstract. Horizontal wells and hydraulic fracturing made a significant contribution to the US unconventional oil and gas production. However, due to complex fracture networks forecasting oil and gas production is a challenging task. This paper derives characteristic distribution of block patterns and fractures that are located between the blocks in traditional and unconventional reservoirs. By analyzing historical well production data on tens fields in different basins using distribution of initial and cumulative production and decline curve analysis of thousands wells from 1960 to 2019 we show that the Hyperbolic model, Pareto and Arps Laws can apply on the well-by-well production data on traditional and unconventional fields including shale and tight reservoirs. Physical and mathematical modeling and studies conducted in many oil and gas fields show that multi-scale rock fractures divide productive rocks into blocks of various sizes, which are complex self-similar fractal structures, and their behavior is described by a power law distribution and a general universal Pareto law. The generator of power law distribution and Pareto’s Law is an algorithm for cascading splitting of the continuum. It was found that the linear dimensions of different-scale blocks in fractured reservoirs asymptotically approach the value of the Golden Sect. (1.618). Based on this law has been developed fractal model of fractured reservoir. It includes several hierarchical levels of matrix blocks and fractures, sometimes ten and more. In the proposed model, not only the sizes of the blocks are in the ratio of 1.618, and permeability of the fractures changes in the ratio of 1.618, which allows to reproduce the daily and cumulative oil and gas well production according to power law distribution and Pareto’s law. The proposed model not only contributes to a more accurate mathematical description of the development of fractured oil and gas reservoirs in hydrodynamic modeling, but also allows us to explain in a new way the appearance of ring structures and vortex spiral objects in the earth’s crust, the formation of which is still debatable. Using of the special developed AVP program allows you to significantly improve the accuracy of hydrodynamic calculations of reservoir development and obtain all three types of Decline Curve wells in accordance with the Arps law.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 540–549, 2022. https://doi.org/10.1007/978-3-030-92144-6_41

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1 Introduction The experience of studying of oil and gas fields in various regions of the world shows that all reservoir rocks, both traditional reservoirs (sandstone, limestone, dolomites, etc.) and unconventional (shale, basement rock, coal seam, etc.) are fractured and within their limits there are always local areas that occupy no more than 20% of the area, which provide more than half of the hydrocarbon production. Despite such a wide spread, both in the lithological composition and age of reservoir rocks containing both traditional and unconventional oil and gas resources, they are united by the fact that they are all fractured to a certain extent and have a complex block discrete structure [1, 2, 7, 10]. Fracturing of rocks is a widespread phenomenon, so well-known Russian geologists such V. V. Belousov, Yu. A. Kosygin, V. E. Hain, etc. in their scientific works, they have repeatedly noted that all rocks, both in geosynclinal and platform areas, have a common planetary fracture. Considering a variety of oil and gas deposits, we can say that the fracturing of rocks causes the possibility of the existence of various natural reservoirs, including those containing unconventional hydrocarbons in shale formations and in tight reservoirs. It follows from the above that without a developed system of fluid-conducting fractures in rocks, it would be impossible for hydrocarbon fluids to get into tight reservoirs and filter in them. This raises the question of the possibility of modeling the structure of such complex fractured reservoirs using modern software systems used in oil and gas production to calculate the development and creation of hydrodynamic models of deposits.

2 Distribution Daily and Cumulative Production in Wells at the Different Fields The distribution of coal methane wells in the San Juan basin (USA) shows that approximately 22.2% (600) of producing wells produced 75% (12.6 billion m3 ) of coal methane, and 77.8% (2100) of producing wells produced 25% (4.2 billion m3 ) of coal methane. At the White Tiger field in Vietnam, approximately 30% of production wells are characterized by increased productivity and yield more than 60% of the produced oil. At the Yarega heavy oil field in Timan-Pechora basin, which is developed by the thermalmine method, back in 1941, the geologists have found that about 5% of high productive wells that intersect sub-vertical fractures in the productive Devonian sandstone reservoir produce approximately 35% of all oil production, while the remaining 95% of wells produce only 65% of oil [1]. The statistical distribution of about 50,000 oil wells operating in the largest oil-gas producing region of Russia – Western Siberia shows that 19.86% (9524) of drilled wells yield 62.51% (9,202 million metric tons) of oil produced in Western Siberia per month, and 80.13% (38433) of drilled wells yield only 37.49% (5.519 million metric tons) of oil produced in Western Siberia per month [5, 6]. 2.1 Physical and Mathematical Modeling Physical and mathematical modeling of the distribution over hierarchical levels using the Cayley tree shows that the evolution of branched self-similar systems, such as a

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productive fractured layer, divided into numerous blocks by multi-scale fractures, is reduced to an anomalous diffusion in the ultrametric space of the hierarchical system. The main feature of such random branched hierarchical networks is that when moving to a lower (deep) level, the system is divided into smaller subsystems, which consist of even smaller sub-assemblies of the next level, and so on. From a statistical point of view, the set of hierarchical ensembles and subsystems is determined by the complexity of system C, which, by analogy with entropy, characterizes the disorder of hierarchical communication. By the nature of the distribution, three types of complex networks can be distinguished: exponential, Poisson, and power networks. However, the first two types belong to scaled graphs, and the last one does not have any scale in the random variable spread, so the main problem of statistical description of complex hierarchical systems that are determined by Pareto’s law for large energy values of the i-th microstate εi is to determine exponential and power law distribution [9]. Research shows that the transition from exponential to power law distribution is provided by the deformation of logarithmic and exponential functions. This deformation leads to the Tsallis distribution and corresponds to a self-similar statistical system, in which there is no scale of change of the random variable εi . In this case, the value of εi corresponds to the fracturing of rocks. Indeed, the self-similar (more precisely, self-affine) probability distribution p (εi ) is defined by the equality. p(εi /ε) = ετi · p(εi )

(1)

according to which the variation of the scale ε, which characterizes the change in the variable εi , leads to the factorization of the probability density by the same scale, deformed by the similarity index τ. Using the scaled variable pi ≡ εi /ε and the function π(εi ) ≡ ετi ·p(εi ) reduces the self-similarity condition to a homogeneous probability distribution. p(εi ) = εi−τ · π(εi ), εi ≡ εi /ε

(2)

meaning that reducing the argument of the function π (ei) by a factor of ε leads to factorization of the probability of the random variable εi , raised to the power –τ. The absence of scale corresponds to the limits of ε → ∞, pi → 0, at which the multiplier π (pi) tends to a constant value and the distribution (2) takes the power form of Pareto’s law. pi = C · εi−τ

(3)

with positive constants C and τ. 2.2 The Generator of Power Law Distribution and Pareto Law The behavior of complex systems described by a power law Pareto called simplistically “20 at 80”, characterizes the distribution of such values as wealth of different population groups in different countries, the intensity of earthquakes, the number of differentscale fractures formed during the deformation of rocks, the catastrophes at financial exchanges, etc. The cumulative and daily production of wells drilled within various fields are distributed in accordance with the Pareto law too (Fig. 1.).

8

Numder of wells (n)

Number of wells (n)

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y = 453.94x-0.449 R² = 0.9468

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y = 1929.9x-0.644 R² = 0.9896 0

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Logarithm of the number of wells- ln(n)

y = -0.449x + 6.118 R² = 0.9468

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Cumulave producon (Q), metric tons

Cumulave producon (Q), metric… 2.5 2 1.5 1 0.5 0

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2.5 2 1.5 1 0.5 0

y = -0.6443x + 7.5652 R² = 0.9896

8 10 12 Logarithm of the cumulave…

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Fig. 1. Distribution of wells by cumulative production in normal and logarithmic scales within the North-Yangtinskoe oil field (left) and the Kuyumbinskoye oil and gas condensate field (right).

The calculations performed using Pareto’s law show that establishing a hierarchical relationship in a complex system of fractured reservoir rocks quickly increases the complexity of the statistical ensemble. With increasing branchability of multi-scale fractures in oil-gas reservoirs, the maximum complexity of the system monotonically increases from zero at branchability a = 1 to infinite at branchability a → ∞.With the change of the dispersion  of the statistical ensemble behavior inherent in simple systems observed in the indicators of branchability of fractures in excess of the value of a+ = 1,618, and the decay of complexity with the increase in dispersion, characteristic of complex selfsimilar hierarchical systems manifests itself by branchability of fractures limited to the interval 1 < a < 1,618. Studies conducted in many oil fields in Texas, the South China sea, Western and Eastern Siberia, the Dnieper-Donets, Volga-Ural and Timan-Pechora basins show that multi-scale rock fractures divide productive rocks into blocks of various sizes, which are complex self-similar fractal structures, and their behavior is described by a power law distribution and a general universal Pareto law [2–8]. The generator of power law distribution is an algorithm for cascading splitting of the continuum. A continuum is any space or set - it can be a physical space, number of people living in different cities of a country, amount of money available in the economy, number of words in the text, a lot of different-scale fractures that occur in a tight rock when it is deformed, etc. Everything that can be repeatedly divided into parts and split this is a continuum. The basis for constructing a cascading splitting fractal of a continuum is a continuum of any size. In our example, this will be the square of the unit area, which, when cascading splitting, gives an ideal fractal, called the fractal of Turcotte by seismologists. (Fig. 2).

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Fig. 2. Stages of building an ideal fractal of cascading splitting.

The ideal fractal of cascading splitting considered above or the fractal of Turcotte has a stochastic analogue, when the continuum is not split into pieces of the same area, but, randomly, as in rock (Fig. 3). These stochastic version of the cascading splitting fractal are of particular interest to us, since they generate smooth power law distributions.

Fig. 3. Stochastic version of the fractal of cascading splitting, when the continuum is split randomly, as in rock.

Thus, the mathematical expression of self-similarity is power laws. If in a homogeneous power function f(x) = cx a , where c and a are constants, x is subjected to a similarity transformation by multiplying by some constant, then the function will still be proportional, although with a different proportionality coefficient. The power laws with integer or fractional exponents are self-similarity generators. Self-similar power laws are obeyed, for example, by growing cities and rocks that break up into separate fragments upon impact. At the same time, it is necessary to note the importance of the only prerequisite for the implementation of a self-similar law: the absence of internal scale objects for this type of object. Indeed, there are no real cities with fewer than 1 or more than 109 inhabitants. Similarly, the size of a rock fragment cannot be smaller than a molecule, or larger than a continent. Thus, if self-similarity is unlimited, it is only in limited areas. The fact that homogeneous power laws do not have natural internal scales

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causes another phenomenon – scale invariance. In other words, when the scale changes, power law distribution reproduce themselves (Fig. 4).

Fig. 4. Conceptual model of scale orders of fracturing on the example of the Cambrian and Ordovician sedimentary strata of South-Eastern Jordan. For simplification, only two orthogonal fracture systems are shown. 1-2-3-scale of fractures (according to Strijker et. al., 2012).

3 The Fractal Model of Fractured Reservoirs Based on the established regularities, a mathematical model was proposed that reflects the values identified in the ratio of linear sizes of blocks and fractures separating them, which asymptotically approach the value of the Golden section of 1.618 (0.618). The resulting model includes up to 10 or more hierarchical levels. In this model, not only the linear dimensions of rock blocks are in the specified ratio, but also the permeability of fractures separating these blocks changes in this ratio [2, 3, 5–8, 10]. In this case, for example, if the permeability of the largest fracture of the first hierarchical level in the model is 10 μm2 , then the permeability of the fracture of the last tenth hierarchical level, which limits the most extreme blocks, will be only 0.132 μm2 , which roughly corresponds to the matrix permeability of conventional productive reservoirs in the block itself, i.e. a complex system with this ratio of the permeability of fractures and the matrix at the last level will degenerate (Fig. 5 on the left). The area and layer-by-layer permeability of the matrix of productive reservoirs in the blocks themselves, when using this model, can be changed using traditional tensor averaging techniques. The proposed model not only contributes to a more accurate mathematical description of the development of fractured oil and gas reservoirs in hydrodynamic modeling, but also allows us to explain in a new

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way the appearance of ring structures and vortex spiral objects in the earth’s crust, the formation of which is still debatable (Fig. 5).

Fig. 5. The fractal hierarchical model of productive fractured layers proposed for simulating the development of oil and gas deposits in fractured reservoirs (left), an annular anomaly in the section of the earth’s crust according to seismic data (center), and a view from space of the Rishat ring structure (eye of the Sahara) in Mauritania (right).

Figure 5 on the left shows with a red oval line from above that when tangential deformation of such a multi-scale block system occurs due to a slight displacement of blocks around its axis, an annular spiral structure can be formed, which is characteristic of many natural objects known as ring anomalies. The use of irregular grids (Fig. 6) and developed software (AVP program) for correct modeling of the hierarchy of fractures in tight sandstones, carbonate rocks and shales allows for the correct forecast of field development indicators in hydrodynamic calculations, and also avoids repeated recalculations and constant updating of basic projects and other design and technical documentation [10].

Fig. 6. Uniform cellular model, traditionally used for hydrodynamic calculations (left), and irregular fractal cellular model, taking into account different-scale fractures and fractured corridors (right), used in the AVP program.

The fact that the sizes of blocks of different scales asymptotically approach the value of the golden section is evidenced not only by the mathematical calculations performed [2, 9], but also by examples of the distribution of rock blocks in the developed oil and gas

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fields established by different methods. As an example, Fig. 7 shows the linear dimensions (Li) of blocks of a productive reservoir of rocks located in the Pashninskoye field. One or more mathematical parameters that characterize relationships of neighboring blocks’ characteristic dimensions can be calculated, such as the ratios of large to small (X1 ) and smaller to larger (X2 ), as well as their sum (X1 + X2 ). Figure 7 shows that, in at least some cases, the distribution of these dimensions can have an order and the values obtained can be arranged in a specific order. Thus, a complex natural system of reservoir rocks, which can have a random distribution of multi-scale tectonic fractures dividing it into blocks of different dimensions, which also can have a random distribution, can be organized or characterized as a mathematical sequence. In other words, order can be created from chaos.

Fig. 7. Linear dimensions of drained blocks of productive sandstones of the Pashninskoye oil field in the Timan-Pechora basin, determined through cumulative oil production of every well with using the volumetric method for calculating oil reserves.

The right-most column of Fig. 7 shows the Fibonacci series for the subject example, which is but one of many. This series of numbers can be characterized by a certain selforganization, such as wherein each successive number of the set is obtained by adding the two previous numbers, and the ratio between these numbers can be characterized by the so-called “golden section” of 1.618. In the exemplary study of the Pashninskoye oilfield, eight linear dimensions of blocks were identified, i.e., eight hierarchical levels of the natural system of the productive reservoir. However, this need not be the case, and any number of linear dimensions or hierarchical levels can be identified in a reservoir. When comparing the set of linear block sizes with the Fibonacci series, it can be noted that the self-organization of the natural system of reservoir rocks in the Pashninskoye field (as developed by the producing wells located there) tend to the Fibonacci sequence, i.e.,

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in terms of the Fibonacci series synergy certain acts of selfattractor natural objects can result in each successive level of cells being similar to a previous one. Thus, the first two dimension (L1 and L2 ) correspond to values slightly higher than the Fibonacci sequence of numbers and the following six linear dimensions, on the contrary, correspond to values somewhat less than the corresponding properties of the Fibonacci series. In at least one embodiment, this may indicate that the first two dimensions have already exceeded the Fibonacci series capacity, but the following six dimensions can be characterized by a certain growth opportunity for achieving a value ratio of 1.618, which can be typical of the Fibonacci numbers. In fact, during the present author’s analyses of wells in the Pashninskoye field, he determined that production from blocks with the dimensions of the first two hierarchical levels had been exhausted and production operations had ceased, but that production from subsequent hierarchical levels continued, approaching closer to the modeled relation between adjacent levels of 1.618. Of course, the examples discussed above with regard to the Pashninskoye field are but some of many and are discussed herein for illustrative purposes. Other applications and outcomes are possible. For instance, the present author has conducted similar analyses and identified similar self-organizations and strict ordering of block sizes with regard to the total cumulative oil production from wells in the Kharyaga oilfield in the Timan-Pechora basin and the Krapivinskoe oilfield in the West Siberian basin. These studies show that natural oil and gas reservoirs can be considered complex self-similar fractal structures, and their behaviour can be modelled using one or more of the systems and methods of the present disclosure. In at least one embodiment, a method can include constructing one or more mesh models in which the linear dimensions of the blocks (aka cells or cellules) are in the ratio of 1.618 as between blocks in consecutive hierarchical levels. For more information, you can see [10]. Using the AVP program allows you to significantly improve the accuracy of hydrodynamic calculations of reservoir development and obtain all three types of curves of falling oil flow rates of producing wells, in accordance with the Arps law, depending on their location relative to the maximum fracture zones (fractured corridors) as shown in Fig. 6 on the right.

4 Conclusion 1. Based on the structural features of fractured reservoirs and local distribution of «sweet spots» in different reservoirs the daily and cumulative production of wells within the deposits have the power law distribution. This distribution is representative of the behavior of many complex nonlinear systems and can be described by Pareto’s law. 2. The power law distribution generator is fractures of various sizes that break rocks into different-scale blocks. It is the uneven blocks that can explain the smooth power law distributions of wells for daily and cumulative oil and gas production within the studied fields. 2. The discovered laws of many levels and quantity parameter distributions of different-scale fractures at the studied oil and gas fields can to describe by the fractal model of fracture reservoirs. This model can be used for simulation of oil and gas deposits development and illustrate the «sweet spots» term in relation to oil and gas reservoirs.

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3. The fractal model include a ratio of dimensions among hierarchical levels and different permeability of fractures for improved reservoir modeling. The proposed model makes it possible to substantiate the optimal dimensions and number of cells (blocks) and fractures for purposes of using various reservoir simulators (Petrel, Eclipse, Tempest MORE, etc.). This not only contributes to a more accurate mathematical description of fractured reservoirs but also makes it possible to come up with a new explanation of the origin of circular structures within the Earth’s crust that is split up into blocks, or torsion fields and other spiral objects whose origin is still open to debate.

References 1. Petukhov, A.V.: Theory and Methodology of Structural and Spatial Zoning Study in Fractured Oil and Gas Reservoirs. Ukhta State Technical University, Ukhta (2002) 2. Petukhov, A.V., Shelepov, I.V., Petukhov, A.A., Kuklin, A.I.: Development of a mathematical model of complex reservoirs containing unconventional oil and gas resources. J. Gas Industry 676, 64–70 (2012) 3. Petukhov, A.V., Egorov, S.I., Oppermann, R., Karyakov, A.N.: New Technolgies of development of Austin chalk and Buda low-porous carbonate reservoirs in south Texas. In: Technical report, International Conference on Hard-to-Recover Resources and Unconventional Sources of Hydrocarbons. Problems, Possibilities, Estimates. VNIGRI, St. Petersburg, Russia (2015) 4. Petukhov, A.V., Kuklin, A.I., Petukhov, A.A., Vasquez Cardenas, L.C., Roschin, P.V.: Origins and integrated exploration of sweet spots in carbonate and shale oil-gas bearing reservoirs of the timan-pechora basin. In: SPE/EAGE European Unconventional Resources Conference and Exhibition. Society of Petroleum Engineers (2014). https://doi.org/10.2118/167712-MS 5. Petukhov, A.V.: The role of Hypogenic karst in formation of carbonate reservoirs and development of oil deposits. In: AAPG Middle East Region GTW on Regional Variations in Charge Systems and their Impact on Petroleum Fluid Properties in Exploration, AAPG Online Journal for E&P Geoscientists (2019). https://doi.org/10.1306/42361Petukhov2019 6. Petukhov, A.V., Egorov, S.I., Karyakov, A.N., Oppermann, R., Petukhov, A.A.: Using phenomenology, holistic models and new technologies to Optimise the development of oil and gas reservoirs. In: 7th EAGE Saint Petersburg International Conference and Exhibition (2016). https://doi.org/10.3997/2214-4609.201600181 7. Petukhov, A.V., Dolgiy, I.E., Kozlov, A.V., Petukhov, A.A.: Features of hydrodynamic modeling of deposits in fractured carbonate reservoirs of the Yurubcheno-Tokhomsky oil and gas accumulation zone. J. Notes of the Mining Institute 200, 242–248 (2013) 8. Petukhov, A.V., Shelepov, I.V., Petukhov, A.A., Kuklin, A.I.: Power law and self-similarity principle in the study of fractured oil and gas reservoirs and hydrodynamic modeling of the development process. J. Neftegazovaya Geologiya. Theory Practice. 7(2). http://www.ngtp. ru/rub/3/33_2012.pdf (2012) 9. Petukhov, A.V., Nikitin, M.N., Petukhov, A.A., Kuklin, A.I.: Structural features of nonNewtonian oils and reservoir rocks containing them. In: International Seminar NonNewtonian Systems in the Oil and Gas Industry, pp. 31–40. Ukhta State Technical University, Ukhta (2012) 10. Petukhov, A.V.: Discrete irregular cellular models for simulating the development of fractured reservoirs. Patent WO 2016/011064, G06G7/48, pp. 45 (2016)

Multifractal Interpretation of Images of Coal Specimen Surfaces to Assess the Degree of Coal Tectonic Disturbance Malinnikova Olga1(B) , Malinnikov Vasiliy2 , Uchaev Denis2 and Uchaev Dmitry2

,

1 Academician Melnikov Institute of Integrated Mineral Development, Russian Academy of Sciences (IPKON RAS), 4 Kryukovsky Dumb, 111020 Moscow, Russia 2 Moscow State University of Geodesy and Cartography (MIIGAiK), 4 Gorokhovsky Pereulok, 105064 Moscow, Russia

Abstract. The possibility of using multifractal interpretation of microimages of coal specimen surfaces to quantify the degree of coal tectonic disturbance was studied. For this, for a series of scanning electron microscopy images (SEMI) of coal specimens taken from outburst-nonhazardous and outburst zones of coal seams, Chebyshev multifractal signatures were restored using generalized multifractal analysis (GMA) based on the use of Chebyshev polynomial kernels. The obtained signatures are considered as alternatives to traditional multifractal spectra for characterization of the surface structure of coals. In the course of the research, it was established that multifractal signatures for disturbed coals from outburst zones are usually greater than signatures for undisturbed coals from outburst-nonhazardous zones, which agrees well with our results obtained for model configurations of disturbed elements generated for different values of the disturbance threshold. The parameters of multifractal signatures that can be used to separate coals with different degrees of disturbance were revealed. The values of these parameters are found which are characteristics of highly disturbed coals from outburst zones and slightly disturbed coals from outburst-nonhazardous zones of coal seams. Based on the results of the study, it can conclude that a multifractal interpretation of SEMI of coal specimen surfaces can be used to determine the degree of coal disturbance. Keywords: Multifractal · Tectonically disturbed coal · Scanning electron microscopy

1 Introduction Coal is a naturally complex porous and fractured organic rocks. The pore-fracture system in coal provides channels for coal seam methane transportation. Physical structure of coal including pore diameter, volume, specific surface area, shape, connectivity, etc., represents a key factor that dominates its important transport properties such as porosity and permeability [1]. In the literature, significant efforts have been devoted to accurately probe and describe the microscopic pore structure of tectonically disturbed coal (TDC). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 550–562, 2022. https://doi.org/10.1007/978-3-030-92144-6_42

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TDC is a kind of coal in which, under mono- or multiphase tectonic stress fields, its primary texture and structure is significantly destroyed [2]. With respect to the terminology of TDC, different scholars have different descriptions, such as deformed coal, tectonically deformed coal, tectonically destroyed coal [18–20]. The major classification criterion of TDC is damage intensity (degree of disturbance) of coal body. Since the 50s of the XX century, when TDC classification based on the degree of disturbance was proposed [3], many quantitative estimates characterizing a coal body have been proposed. However, the answer to the question which of the coal body characteristics can be a measure of the intensity of coal body damage remains open. In last years, fractal analysis theory became a scientific tool that can be used to characterize coal physical-geometric properties, such as the structure of its micrometerscale pores and fractures, and permeability [4–6]. It have been widely accepted that the TDC solid-pore-fracture structure shows the self-similar properties and fractal characteristics [7–9]. Thus, the fractal theory can be effectively applied to quantitatively describe the solid-pore-fracture structure of coal. In recent years, with the development of cross-disciplines and the application of mathematical methods in researches of TDCs, multifractal methods open new paths for the study of TDCs [10–13]. The multifractal approach to assessing the degree of coal disturbance is a natural extension of the fractal approach used to describe the surface of the TDC solid-pore-fracture structure. This approach is based on the following statements: • the structural organization of coal surface elements can be represented by a multifractal with a corresponding spectrum of fractal dimensions; • the process of destruction of a coal seam, which is a process of multiple nucleation and development of micro-fractures, can also be considered as multifractal, since when moving from one scale defect level to another, the fractal dimension changes due to changes in microstresses in different structural areas. The multifractal theory has already been applied to characterize variations and heterogeneity of micro-fractures in TDCs. Li et al. [11] applied the multifractal method to analyze the heterogeneity of pores in TDCs. Yu et al. [14] further explored the significance of the multifractal method in charactering the heterogeneity of pore structures in TDCs. Exploring high-rank brittle TDCs, Li et al. [15] showed that micro-fractures exhibit multifractal behavior and, therefore, the obtained multifractal spectra (generalized dimension spectra D(q) and singularity spectra f (α)) and their characteristic parameters, including the capacity dimension D0 , information dimension D1 , correlation dimension D2 , D1 /D0 , width of multifractal spectrum α, and difference of fractal dimensions f (α), can be applied to accurately describe differences and variations of micro-fracture distributions in TDCs. Using 13 test scanning electron microscopy images (SEMIs) of surfaces of coal specimens obtained from outburst-nonhazardous zones and outburst zones, we showed that a relationship can exist between the disturbance of TDCs and the asymmetry of fractal spectra for coal SEMIs [16]. Further analysis [12] of more than 140 test SEMIs of coal specimens obtained from different mines confirmed the existence of this relationship. This article attempts to generalize the capabilities of multifractal analysis to quantitatively describe the degree of coal disturbance by SEMIs of coal specimens. For these

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purposes, it is proposed to use the multifractal interpretation (MI) methodology, originally proposed for aerospace images [17, 18]. Hereinafter, we assume that MI of SEMIs (SEMIMI) is a specific type of SEMI interpretation, which is based on the identification and description of natural systems by their SEMIs. The article show that results of MI of SEMIs of coal specimen surfaces can be used to separate coals with different degrees of disturbance. The paper is organized as follows. Section 2 presents possibilities of SEMIMI. The main attention in this section is given to recently proposed multifractal signatures [19], which have great potential in describing the multifractal properties of natural systems. Section 3 describes the source data. In Sect. 4, using the simplest multifractal model of a coal massif, it is shown that multifractal signatures can be used as differentiating characteristics of the studied coal specimens of different degrees of disturbance.

2 Semimi The MI methodology consists of the following components when applied to SEMIs: improving the quality of the SEMIs, providing an increase in the reliability of their multifractal characteristics (MCs); multifractal analysis (MA) of SEMIs; interpretation of SEMI (SEMII) by their multifractal characteristics (MCs). A basic component of the SEMIMI is MA of SEMIs. The main idea behind the any multifractal analysis of images is to make a description of the measure formed for the analyzed image and reflecting a spatial distribution of a certain characteristic for image objects. MA allows to describe the measure both locally and globally using a set of scaling exponents (MCs). Based on the paper [20], we can conclude that if the measure μ formed for an image I is multifractal, then the following fundamental scaling relations must be satisfied for any (x, y) ∈ I and sufficiently small ε and δ: με (x, y) εα(x,y) , ηε,δ [α(x, y)] εfG [α(x,y)] , where με (x, y) is the measure in a box of size ε centered on (x, y); α(x, y) is a Hölder exponent (local dimension) in (x, y); ηε,δ [α(x, y)] is the number of boxes of size ε, which form an image partition and have a coarse Hölder exponent αε (x, y) = ln με (x, y)/ ln ε close to α(x, y) up to a precision δ; fG [α(x, y)] is a 2D large deviation spectrum. We assume that SEMIMI is based on GMA of SEMI [17]. GMA is based on analysis of scaling properties of multiscale functions Zp,q,r (ε) and Wp,q,r (ε, δ) for correlations r [α(x, y)]) and kernels  (x, y) (hereinafter, p = 0, 1, . . . , p between μrε (x, y) (ηε,δ p,q max , q = 0, 1, . . . , qmax , r ∈ R, x = 0, 1, . . . , M − 1, y = 0, 1, . . . , N − 1 and ε, δ ∈ R) constructed using discrete orthonormal polynomials (OPs). For construction of OP kernels, various discrete orthogonal polynomials can be used including Chebyshev, Kravchuk, Hahn, Meixner and Charlier polynomials [21]. As can be seen from Fig. 1, if p and q are equal to zero, then the multiscale correlation functions Zp,q,r (ε) and Wp,q,r (ε, δ) are reduced to multiscale moments Zq (ε) and Wq (ε, δ) for με (x, y) and ηε,δ [α(x, y)], respectively (in Eqs. (D.1) and (D.2), q is used instead of r to denote moment orders). By analyzing scaling properties of the moments Zq (ε) and Wq (ε, δ) or partial derivatives of functions ln Zq (ε) and ln Wq (ε, δ) with respect to q, one-dimensional multifractal spectra of global scaling exponentials (1D MSGSE) can be obtained. In addition to the

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1D MSGSE, two-dimensional multifractal spectra of global scaling exponentials (2D MSGSE) are introduced by OMFA, which can be obtained by analyzing scaling properties of the aforementioned multiscale correlation functions defined by expressions (D.3) and (D.4) or their partial derivatives with respect to r. As can be seen from Figs. 1 and 2D MSGSE are related with 1D MSGSE using Hölder exponents (HEs), and with onedimensional multifractal spectra of local scaling exponentials (1D MSLSEs) through 2D large deviation spectra and 2D Legendre spectra, which together with HEs form a group of two-dimensional multifractal spectra of local scaling exponents (2D MSLSE). The practical implementation of OMA of SEMI of coal specimen surfaces can include the following steps: formation of image covering by boxes of fixed sizes; formation of a box measure that reflects a spatial distribution of a certain characteristic of image objects; testing the hypothesis on a multifractality of the box measure (local analysis) or the hypothesis on a scaling behavior of multiscale correlation functions or moments (global analysis); calculation of global MCs.Thus, using the methodology of SEMIMI, which based on the OMA, it is possible to calculate most of the currently widely used global multifractal descriptors of SEMI of coal surfaces both directly and through 2D MSGSE (Table 1). This allows not only to reduce the influence of errors of the selected method for calculating a particular MC, but, more importantly, to obtain a set of analytically related descriptors of multifractal properties of surfaces of the studied coal specimens. In this article, to describe the structural organization of surfaces of coals with different degree of disturbance, it is proposed to use so-called Chebyshev multifractal signatures, first described in [19]. Chebyshev multifractal signatures represent an alternative to traditional multifractal spectra: the Legendre spectrum fL (α), which is only the upper envelope of the theoretical (Hausdorff) multifractal spectrum fH (α) [22], and the large deviation spectrum fG (α), that is, in general, “closer” to fH (α) than fL (α), but it fails to obtain a good approximation of fH (α), if we have not strictly multiplicative cascades [23]. As shown in [19], for an image of size N × N , Chebyshev multifractal signatures can be defined as follows: A(s/smax ) =

1 smax

 p+q=s

|K(p, q)|, F(s/smax ) =

1 smax



|L(p, q)|,

(1)

p+q=s

where s = 1, 2, . . . , smax , smax = 2N − 2, K(p, q) and L(p, q) are two-dimensional multifractal spectra that are analogs of frequency spectra from Fourier analysis (they are frequency representations of widely used spectra of local scaling exponents for images, namely, the spectrum of Holder exponents α(x, y) and two-dimensional realization of a large-deviation spectrum fG (α)). Chebyshev multifractal signatures A(s/smax ) and F(s/smax ) have a number of properties suitable for describing objects identified by images: invariance to image scaling, sensitivity to image rotation, insensitivity of low-order signatures to high-frequency additive noise of α(x, y)- and fG [α(x, y)]-spectra.

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Fig. 1. Multifractal spectra generated by GMA. Similar 1D and 2D spectra of global scaling exponents are highlighted by borders of the same color. The blue arrows show that fG [α(x, y)] and fL [α(x, y)] can be obtained from fG (α) and fL (α) only using α(x, y) [17].

3 Source Data In this study, imaging of coal specimen surfaces was performed using scanning electron microscopes JEOL JSM 5910LV and JEOL 6610LV. The spatial resolution of the microscopes is more than 10 and 100 nm for secondary and reflected electrons, respectively. Low–energy secondary electrons are used in imaging surface topography. Natural–shape coal specimens were placed in a work camera via a gate. In the mode of registration of secondary electrons, the work camera was vacuumized (with >10–6 mm hg vacuum). Secondary electrons were recorded by a standard detector, which a type of a sweeping–field photomultiplier tube connected to scintillator. The source data for this study was the IPKON collection of coal specimens from Vorkuta and Kuzbass mines obtained from outburst–nonhazardous zones and outburst zones. For our research, we analyzed microstructure of coal surface in the images magnified 1000 times which showed coal grains with a characteristic size from 0.5 to a few microns. Methane can desorb from such grains, diffuse and flow in fractures as free gas [27]. Figure 2 a–c and 3 a–c show regions of some images used in this work.

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Table 1. The most informative multifractal descriptors of SEMI of coal surfaces calculated using SEMIMI. Multifractal descriptors

Formula

Description

References

D0

–

Fractal dimension

[15, 24]

D1

–

Information dimension [15]

D2

–

Correlation dimension [15]

q D

q D = D1 − Dq , q  1

Order parameter

α

α = αmax − αmin , where Width of multifractal αmin = spectrum fL (α) min

1≤x≤W , 1≤y≤H

[25, 26] [15, 24–27]

α(x, y),

αmax = max

1≤x≤W , 1≤y≤H

α(x, y), W

and H — width and height of SEMI fL (α)

fL (α) = fL (αmax ) − fL (αmin ), where fL (αmin ) =

Difference of fractal dimensions

[15]

inf {αmin q − τ (q)},

q∈R

fL (αmax ) = inf {αmax q − τ (q)}

q∈R

r α , Rα

L α rα =  , Rα L α−R α Rα =  L α+R α , where L α = α0 − αmin , R α = αmax − α0

r S , RS

R, rS = SSL , RS = SSL −S R L +SR  α0 where SL = αmin f (α)d α,  SR = αα0max f (α)d α

Asymmetry coefficient [13, 28] for fL (α)

Asymmetry coefficient [12] for fL (α)

4 Research Results At the first stage of our research, Chebyshev multifractal signatures were calculated for each of the original SEMIs. For these purposes, first each source image was binarized with an empirically selected threshold equal to 32 (Fig. 2, d–f and 3, d–f). When have been choosing the value of this threshold, two criteria were taken into account: (i) the achievement of a visual estimated balance when reflecting main structural elements of the coal surface (pores, fractures, coal particles) in binarized images and (ii) minimization of the effect of uneven illumination of coal specimen surfaces for some images. Further,

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Fig. 2. Regions of images of the surfaces of coal specimens obtained from the outburst–nonhazardous zones: region of SEMI 1 (a), region of SEMI 2 (b), region of SEMI 3 (c); corresponding binarized images (d–f) (the binarization threshold was chosen equal to 32).

Fig. 3. Regions of images of the surfaces of coal specimens obtained from the outburst zones: region of SEMI 1 (a), region of SEMI 2 (b), region of SEMI 3 (c); corresponding binarized images (d – f) (the binarization threshold was chosen equal to 32).

each of the binarized images was inverted so that non-zero pixels correspond to voids in the coal surface structure. Multifractal signatures were calculated using formula (1),

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while two-dimensional multifractal spectra K(p, q) were calculated using Hölder exponents of binarized images (D13) and L(p, q) through the calculation of one-dimensional (D13) and two-dimensional (D14) large deviation spectra (Fig. 1). A similar calculation scheme, as practice has shown, gives the most stable calculation results. At the next stage of research, to study the possibility of using multifractal signatures A(s/smax ) and F(s/smax ) to quantify the degree of coal disturbance, a series of experiments was carried out. The purpose of these experiments was to study of signatures obtained for different configurations of disturbed elements. For these aims, more than 40 model configurations of disturbed elements were formed (Figs. 4 and 5). We used an algorithm for generating random configurations of disturbed elements of a coal massif, which includes the following steps [29]. At the first step of formation of two-dimensional configurations of disturbed elements, the square block with size 1 × 1 was divided into M = b2 (where b is a priori scale factor) equal-sized blocks with size 1/b × 1/b. Then, each of the resulting blocks was marked with the weight (probability) w(i) (here i is 2 the  block number, 1 ≤ i ≤ b ) corresponding to the degree of disturbance of blocks ( w(i) = 1). It was assumed that the disturbance of each block is directly proportional to the number of disturbed elements inside this block. After that, the obtained blocks were assigned to a subset of disturbed (or undisturbed) blocks, based on the following: if block B(i) is such that w(i) > p (where p is a priori threshold value of block disturbance) then the block is disturbed, otherwise it is assumed that the block B(i) is undisturbed. At the second and subsequent steps of the algorithm, the sequence of actions described at the first step was repeated: each of the disturbed blocks was divided into b2 equalsized blocks; the resulted blocks were marked with weights w(i) (1 ≤ i ≤ b2 ); blocks with w(i) > p were combined with disturbed blocks, and blocks with w(i) ≤ p were combined with undisturbed blocks. At the nth step (where n is a priori stage of constructing random configurations) the process of forming a two-dimensional configuration of disturbed elements was completed. Further, comparing signatures A(s/smax ) and F(s/smax ) for model configurations of disturbed elements formed in numerical experiments (Figs. 6 and 7), we came to the following conclusions: • signatures A(s/smax ) and F(s/smax ) obtained for configurations generated for fixed values of p and b, but different values of n, are similar in shape and quite often overlap; • signatures A(s/smax ) and F(s/smax ) obtained for configurations generated for fixed values of b and n but different values of the disturbance threshold p differ from each other. It was established that for larger value of p higher signatures are obtained. The established relationships between disturbance threshold values for model configurations of disturbed elements and their Chebyshev multifractal signatures suggested that these signatures can be used to characterize the degree of disturbance for model configuration of disturbed elements. Based on this assumption, we analyzed the possibility of using SEMIMI methods to assess the disturbance of real coal specimens by their SEMI. For this purpose, we compared the calculated signatures A(s/smax ) and F(s/smax ) for coal specimens shown in Figs. 2 and 3 with signatures A(s/smax ) and F(s/smax ) calculated for model configurations with various values of the disturbance threshold p (Fig. 8). The results allowed us to draw the following conclusions:

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Fig. 4. Two-dimensional configurations of disturbed elements formed using fixed values of b and n (b = 3, n = 6) and different values of p: a) p = 3/9, b) p = 4/9, c) p = 5/9, d) p = 6/9, e) p = 7/9, f) p = 8/9.

Fig. 5. Two-dimensional configurations of disturbed elements formed using b = 3, n = 6 i p = 6/9: a) model 1, b) model 2, c) model 3 i d) model 4. White and black colors in the figure show disturbed and undisturbed blocks, respectively.

1. SEMI signatures of coal specimens obtained from outburst zones and outburstnonhazardous zones are statistically significantly different from each other. Moreover, it turns out that signatures for SEMIs of coal specimens obtained from

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outburst zones lie above signatures for SEMIs of coal specimens obtained from outburst-nonhazardous zones. 2. From the analysis of A(s/smax ), it follows that signatures for images of coal specimens obtained from outburst zones are close to the signature for the model configuration of disturbed elements generated using the disturbance threshold p equal to 6/9 (p1), and signatures for images of coal specimens from outburst-nonhazardous zones are located in the vicinity of the signature for the model configuration generated using p = 5/9 (p2). 3. From the analysis of F(s/smax ), one can draw a conclusion similar to item 2, with the only difference being p1 ≈ 7/9 and p2 ≈ 5.5/9. 4. Multifractal signatures calculated for all test images show a close shape with a distinct maximum at s/smax ∈ [0.45; 0.5].

Fig. 6. Chebyshev multifractal signatures A(s/smax ) (a) and F(s/smax ) (b), calculated for twodimensional configurations of disturbed elements shown in Fig. 4.

Fig. 7. Chebyshev multifractal signatures A(s/smax ) (a) and F(s/smax ) (b), calculated for twodimensional configurations of disturbed elements shown in Fig. 5.

Summarizing all above-mentioned, we can conclude that SEMIMI of coal specimen surfaces can be considered as an auxiliary tool for quantifying the degree of coal disturbance. In this case, to separate TDCs according to the degree of disturbance, it may be sufficient to use only one pair of multifractal signature values, which consists of the maximum signature value and its argument. Given the statistical nature of the calculated signatures, we can use the maximum value of a signature approximation (Aamax or Famax ) and its argument (sAa = argmax Aa (s/smax ) or sFa = argmax Fa (s/smax )). Based on all s

s

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Fig. 8. Parts of multifractal signatures A(s/smax ) (a) and F(s/smax ) (b) calculated for binarized images of coal specimens shown in Figs. 2 and 3 (d–f) as well as multifractal models for p = 5/9, 6/9 and 7/9. ONZ – outburst-nonhazardous zone, OZ – outburst zone.

results of the studies, we can draw a conclusion that values of signature approximation parameters, typical for highly disturbed coals from outburst zones and slightly disturbed coals from outburst-nonhazardous zones of coal seams, are approximately in the ranges shown in Table. 2. At the same time, it should be noted that the problem of choosing between multifractal signatures A(s/smax ) and F(s/smax ) for quantifying the degree of disturbance of coals by their SEMIs still requires further study. Table 2. Values of informative parameters of Chebyshev multifractal signatures A(s/smax ) and F(s/smax ) for coals with different disturbance degrees. Disturbance degree

sAa

Aamax

sFa

Famax

Highly disturbed (coals from outburst zones)

0.44 ± 0.01

0.11 ± 0.01

0.43 ± 0.01

0.07 ± 0.01

Slightly disturbed (coals from outburst-nonhazardous zones)

0.47 ± 0.05

0.08 ± 0.02

0.48 ± 0.04

0.04 ± 0.01

Acknowledgments. The study devoted to multifractal signatures which can be used to distinguish structures with different disturbances was supported by the Russian Foundation for Basic Research (Project No. 19-05-00330 A). The study on a multifractal interpretation of SEMIs of coal specimen surfaces used to determine the degree of coal disturbance was supported by the Russian Foundation for Basic Research (Project No. 19-05-00824 A).

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Author Index

A Andreeva, M. V., 533 Atroshenko, S. A., 261 B Babenkov, M. B., 381 Banichuk, Nikolay, 197 Belousova, E. A., 381 Belyaev, Alexander, 296 Bessonov, Nikolay, 202 Borovskoy, Alexey, 278 Bulygin, Anatolii N., 409 Bykov, Nikolay, 278 D Denis, Uchaev, 550 Dmitry, Uchaev, 550 Dragunov, Sergey, 288 Dyakova, O. A., 15 F Fakhretdinova, R. R., 503 Fedotov, Aleksandr V., 234 Fekry, Montaser, 175 Filippenko, George V., 245 Frolov, O. Yu., 15 G Galyautdinova, Aliya, 296 Ghulghazaryan, G. R., 441 Ghulghazaryan, L. G., 441 Goryachev, V. D., 98 Goryacheva, Irina, 350

I Igumnova, Vasilisa, 159 Islamov, B. A., 381 Iulmukhametova, Regina, 23, 117 Ivanova, Svetlana, 197 K Kalyuzhnyuk, A. V., 533 Knyazeva, A. G., 126 Kobelev, Anton, 278 Kolenkina Skryleva, E. I., 503 Kovaleva, Liana, 23, 117 Krutko, V. V., 533 L Lebedev, S. F., 381 Leontiev, Victor L., 151 Lomonosov, Alexey M., 426 Lukin, A. V., 489 Lukin, Alexey, 108, 159 Lychev, Sergei A., 175 Lychev, Sergey, 331 Lycheva, Tatiana, 331 Lyubicheva, Anastasia, 305 M Makeeva, M. N., 503 Mayer, Andreas P., 426 Meshcheryakova, Almira, 350 Mikhalchenko, E. V., 98 Mullayanov, Almir, 117 Müller, Wolfgang H., 363 Musin, Airat, 23, 117

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2020, LNME, pp. 563–564, 2022. https://doi.org/10.1007/978-3-030-92144-6

564 N Nikiforov, D. A., 397 Nikitin, V. F., 98, 503 Nobili, Andrea, 463 O Obraztsov, Nikita, 278 Olga, Malinnikova, 550 P Panini, Filippo, 519 Parfenova, E. S., 126 Pavlov, Yurii V., 409 Peter, Costanzo, 519 Petukhov, A. V., 540 Polyanskiy, Vladimir, 202 Popov, Ivan, 108, 159 Pupyrev, Pavel D., 426 R Rickert, Wilhelm, 363 Russkikh, N. E., 533 Rybakovskaya, A. A., 397 S Salina Borello, Eloisa, 519 Sedova, Yulia, 202 Senkin, I. S., 397 Shanshin, I. K., 489 Shtukin, Lev, 159 Signorini, Cesare, 476 Smirnov, N. N., 503

Author Index Smirnov, Nikolay N., 3 Spannan, Lars, 223 Stepanov, Fedor I., 320 Surov, Alexander, 278 Svyatogorov, I. G., 489 T Taimanov, I. A., 533 Timoshenko, V. A., 381 Torskaya, Elena V., 320 Tretyakov, Dmitriy, 296 Tsukanov, Ivan Y., 314 Tsykunov, O. I., 397 U Udalov, Pavel, 108 V Valiullina, Vilena, 117 Vasiliy, Malinnikov, 550 Vasilyeva, A. U., 381 Vedeneev, Vasily, 35 Vershinin, Vladislav V., 82 Viberti, Dario, 519 Volpini, Valentina, 69 W Woschke, Elmar, 223 Z Zamula, Yuriy, 117 Zinovieva, Tatiana V., 245