Advanced Problem in Mechanics III: Proceedings of the XLIX International Summer School-Conference “Advanced Problems in Mechanics”, 2021, St. Petersburg, Russia [1 ed.] 303137245X, 9783031372452

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Lecture Notes in Mechanical Engineering

D. A. Indeitsev A. M. Krivtsov   Editors

Advanced Problem in Mechanics III Proceedings of the XLIX International Summer School-Conference “Advanced Problems in Mechanics”, 2021, St. Petersburg, Russia

Lecture Notes in Mechanical Engineering Series Editors Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Francesco Gherardini , Dipartimento di Ingegneria “Enzo Ferrari”, Università di Modena e Reggio Emilia, Modena, Italy Vitalii Ivanov, Department of Manufacturing Engineering, Machines and Tools, Sumy State University, Sumy, Ukraine Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Editorial Board Francisco Cavas-Martínez , Departamento de Estructuras, Construcción y Expresión Gráfica Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany 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 Jinyang Xu, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China

Lecture Notes in Mechanical Engineering (LNME) publishes the latest developments in Mechanical Engineering—quickly, informally and with high quality. Original research reported in proceedings and post-proceedings represents the core of LNME. Volumes published in LNME embrace all aspects, subfields and new challenges of mechanical engineering. To submit a proposal or request further information, please contact the Springer Editor of your location: Europe, USA, Africa: Leontina Di Cecco at [email protected] China: Ella Zhang at [email protected] India: Priya Vyas at [email protected] Rest of Asia, Australia, New Zealand: Swati Meherishi at [email protected] Topics in the series include: • • • • • • • • • • • • • • • • •

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D. A. Indeitsev · A. M. Krivtsov Editors

Advanced Problem in Mechanics III Proceedings of the XLIX International Summer School-Conference “Advanced Problems in Mechanics”, 2021, St. Petersburg, Russia

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 978-3-031-37245-2 ISBN 978-3-031-37246-9 (eBook) https://doi.org/10.1007/978-3-031-37246-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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

Organisation

Scientific Committee D. A. Indeitsev A. M. Krivtsov P. A. Dyatlova 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 M. E. Frolov S. N. Gavrilov I. G. Goryacheva

IPME RAS, Peter the Great St. Petersburg Polytechnic University, Russia—Co-chairman Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia—Co-chairman Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia—Scientific Secretary 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 FarEastern Branch of RAS, Komsomolsk-na-Amure, Russia University of Utah, Salt Lake City, USA Università di Roma La Sapienza and MEMOC S Center, 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 Peter the Great St. Petersburg Polytechnic University, Russia IPME RAS, St. Petersburg, Russia Institute for Problems in Mechanics RAS, Moscow, Russia

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vi

E. F. Grekova

Organisation

IPME RAS, St. Petersburg, Russia; University of Seville, Spain N. Gupta Indian Institute of Technology Delhi, India M. A. Guzev Institute of Applied Mathematics of Far-Eastern Branch of RAS, Russia H. E. Huppert University of Cambridge, UK H. Irschik Johannes Kepler University of Linz, Austria M. L. Kachanov Tufts University, Medford, USA B. L. Karihaloo Cardiff University, UK D. Koroteev Center for Hydrocarbon Recovery, Skoltech (Digital Petroleum), Russia S. V. Kuznetsov IPMech RAS, Russia V. A. Kuzkin Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia V. A. Levin Lomonosov Moscow State University, Russia A. M. Linkov IPME RAS, Russia; Rzeszow University of Technology, Poland I. I. Lipatov Moscow Institute of Physics and Technology, Russia O. S. Loboda Peter the Great St. Petersburg Polytechnic University, IPME RAS, Russia E. V. Lomakin Lomonosov Moscow State University, Russia G. Mishuris Aberystwyth University, UK N. F. Morozov St. Petersburg State University, IPME RAS, Russia W. H. Müller Technical University of Berlin, Germany U. Nackenhorst Leibniz University of Hannover, Germany E. Pavlovskaia University of Aberdeen, UK S. V. Petinov IPME RAS, Peter the Great St. Petersburg Polytechnic University, Russia Y. V. Petrov St. Petersburg State University, IPME RAS, Russia A. V. Porubov IPME RAS, St. Petersburg, Russia D. Prikazchikov Keele University, UK J.-N. Roux Universi´te Paris-Est, Laboratoire Navier, France M. B. Rubin Israel Institute of Technology, Haifa, Israel A. I. Rudskoi Peter the Great St. Petersburg Polytechnic University, Russia S. Rudykh University of Wisconsin-Madison, USA S. H. Sargsyan Gyumri State Pedagogical Institute, Armenia V. V. Sergeev Peter the Great St. Petersburg Polytechnic University, Russia I. Sevostianov New Mexico State University, USA M. Simonov Gazpromneft Science and Technology Center, Russia A. Sokolov TU Berlin, Germany M. Wiercigroch Aberdeen University, Scotland H. A. Wu University of Science and Technology of China, Chinese Academy of Sciences P. Venkitanarayanan Indian Institute of Technology, India

Organisation

E. N. Vilchevskaya

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

Local Organizing Committee Polina Dyatlova

Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Mikhail Speranskii Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Varya Shubina Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Maria Loboda Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Ekaterina Saprykina Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Anastasia Vilchevskaya Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Yekaterina Moisseyeva Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia Kristina Plokhova 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

Scientific Program Presentations devoted to fundamental aspects, or widening the field of applications of mechanics, are invited. We are particularly keen to receive contributions that show new effects and phenomena or develop new mathematical models. The topics of the conference cover all fields of mechanics, including, but not restricted, to • complex media: micropolar theory, chemomechanics, biomechanics, acoustic metamaterials, etc. • fluid mechanics • nano-, micro- and mesomechanics • solids and structures • nonlinear and multibody dynamics, chaos and vibration • mechanical and civil engineering applications • heat transfer • vibration in science and technology.

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Organisation

Minisymposia MS1 “Minisymposium on biomechanics” Organizers Olga S. Loboda Peter the Great St. Petersburg Polytechnic University, Russia Svetlana M. Bauer St. Petersburg State University, Peter the Great St. Petersburg Polytechnic University, Russia MS2 “Extreme loading on structures” Organizers Danila Prikazchikov Keele University, UK Nikita F. Morozov St. Petersburg State University, IPME RAS, Russia Vladimir A. Bratov IPME RAS, Russia MS3 “Nonlinear waves in continuous media” Organizers Vladimir I. Erofeev Mechanical Engineering Research Institute of the Russian Academy of Sciences or MERI RAS, Nizhniy Novgorod, Russia Alexey V. Porubov IPME RAS, St. Petersburg, Russia MS4 “Earthquakes and Seismic Protection” Organizers Tzu-Kang Lin National Taiwan University, Taiwan Sergey V. Kuznetsov IPMech RAS, Russia Vladimir Bratov IPME RAS, Russia MS5 “New approaches for multiphase gas condensate flow simulation” Organizers Yekaterina Moisseyeva Peter the Great St. Petersburg Polytechnic University, Russia Falk Ahnert Wintershall Holding GmbH, Germany MS6 “Mathematical modeling in petroleum engineering” Organizers Liliana Rybarska-Rusinek Rzeszow University of Technology, Poland

Organisation

Alexander M. Linkov Vitaly A. Kuzkin

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

MS7 “Anomalous energy transfer on a nano-scale” Organizers Wolfgang H. Mu¨ller Technical University of Berlin, Germany Alexey V. Porubov IPME RAS, Russia A. Sokolov TU Berlin, Germany

Preface

General Information The international conference “Advanced Problems in Mechanics 2021” is the fortyninth 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. The conference is organized by Peter the Great St. Petersburg Polytechnic University (SPbPU), Russian National Committee on Theoretical and Applied Mechanics (RNCTAM) and Institute for Problems in Mechanical Engineering of Russian Academy of Sciences (IPME RAS) under the patronage of Russian Academy of Sciences (RAS). General sponsor of the conference “Advanced Problems in Mechanics—2021”—company Gazpromneft Science and Technology Centre. In connection with the COVID-19 pandemic, the annual international conference “Advanced Problems in Mechanics” was held in an unusual online format last summer and in a mixed format last fall. It was an unforgettable experience that demonstrated that the conference can be successful even under quarantine conditions. 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 such domains 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 transfer of scientific experience from well-known scientists to their young colleagues. The special scientific session “Vibration in Science and Technology” dedicated to the memory of outstanding Russian researcher in the area of mechanics, applied mathematics and engineering, Prof. Iliya Izrailevich Blekhman (29.11.1928–03.02.2021), will be held within the framework of the conference.

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Preface

Session Organizers: Scientific adviser of IPME RAS, Corresponding member of RAS Prof. Dmitry Indeitsev, Professor of Structural Dynamics at the Karlsruhe Institute of Technology, Alexander Fidlin IPME RAS Head of Lab., IEEE Life Fellow, Prof. Alexander Fradkov.

History of the School 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. The School specialized in this way because at that time in Russia (USSR) there were held regular National Meetings on Theoretical and Applied Mechanics, and there existed many conferences on mechanics with a more particular specialization. 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 the 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 associated into interdisciplinary problems.

Contents

Complex Media: Micropolar Theory, Chemomechanics, Acoustic Metamaterials etc. Moment-Membrane Dynamic Theory of Elastic Thin Shells and Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. H. Sargsyan

3

Fluid Mechanics Effect of Thermal Wall Boundary Conditions on a Viscous Fluid Flow Through an Abrupt Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kira Ryltseva and Gennady Shrager

19

Study on Application of Tangential Jet Blowing on Adaptive High-Lift Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladislav Alesin, Olga Pavlenko, Albert Petrov, and Evgeny Pigusov

30

Numerical Study of the Deformable Particle Dynamics in Microchannel with Hydrodynamic Traps . . . . . . . . . . . . . . . . . . . . . . . . . . N. B. Fatkullina, O. A. Solnyshkina, A. Z. Bulatova, and V. A. Andryushchenko The Algorithm of the Path Length Optimization on the Polyhedron Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alia Gumirova, Ilia Marchevsky, and Yurii Safronov Barnes–Hut/Multipole Fast Algorithm in Lagrangian Vortex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Julia Chernyh, Ilia Marchevsky, Evgeniya Ryatina, and Alexandra Kolganova

43

55

69

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Contents

Heat Transfer and Wave Motion Simulation of a Detonation Engine on an Acetylene-Oxygen Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. V. Mikhalchenko, V. F. Nikitin, and V. D. Goryachev

85

Feedforward Optimal Control with Constraints for a Cylindrical Thermoelectric System Actuated by a Peltier Element . . . . . . . . . . . . . . . . Alexander Gavrikov and Georgy Kostin

95

Nano-, Micro- and Mesomechanics Modeling of Domain Structure Evolution in Ferroelectroelastic Crystals Under Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 S. M. Lobanov, A. Mamchits, and A. S. Semenov Solids and Structures Accounting for the Skin Effect in Hydrogen-Charged Samples in the HEDE Model of Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Yu. S. Sedova, V. A. Polyanskiy, and N. M. Bessonov Equal-strength Design of Axially Symmetric Thin-Walled Shell Moving in Deformed Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Nikolay Banichuk and Svetlana Ivanova Numerical Aspects of the J-Integral Estimation for Thermomechanical Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 A. V. Savikovskii, A. N. Gordeev, A. A. Michailov, O. V. Antonova, and A. S. Semenov Wear Resistance of Fe66 Cr10 Nb5 B19 Coatings Obtained by Detonation Spraying at Different Explosive Charges . . . . . . . . . . . . . . . 176 I. D. Kuchumova, V. S. Shikalov, Igor S. Batraev, Tatiana A. Borisenko, Guilherme Yuuki Koga, and Alberto Jorge Moreira Jr. On the Lifetime of Spherical Vessels Subjected to Mechanochemical Corrosion and Temperature Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Olga Sedova and Yulia Pronina Exact and Approximate Solutions of the Modified System of Interrelated Kinetic Equations for Damage Parameter and Creep Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 A. R. Arutyunyan and R. R. Saitova Using of Modified Maxwell Equation to Describe Experimental Creep Curves of Carbon Fiber Reinforced Plastic Composites After Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 A. R. Arutyunyan

Contents

xv

Residual Stresses in a Thermo-viscoelastic Additively Manufactured Cylinder Subjected to Induction Heating . . . . . . . . . . . . . . 208 Sergei A. Lychev and Montaser Fekry Mechanical and Civil Engineering Applications Influence of Aero-thermoacoustic Treatment on the Characteristics of Casting Aluminum Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 S. A. Atroshenko and E. M. Nikolaeva Insectomorphic Robot Rescue from an Emergency on the Back Under the Interference of Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Yury F. Golubev, Victor V. Koryanov, and Elena V. Melkumova Stability, Instability Study and Control of Autonomous Dynamical Systems Based on Divergence Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Igor B. Furtat and Pavel A. Gushchin Vibration in Science and Technology Energy-Optimal Control by Boundary Forces for Longitudinal Vibrations of an Elastic Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Georgy Kostin and Alexander Gavrikov Self-synchronization of Inertial Vibration Exciters in a System with an Elastic Limiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Grigory Panovko and Alexander Shokhin Modelling the Dynamics of a Large-Scale Industrial Manipulator for Precision Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 M. P. Cartmell, I. Gordon, D. Johnston, S. Liang, L. McIntosh, and B. Wynne The Effect of Hydrogen Embrittlement on the Torsional Oscillations of the Pipe as a Multilayer Shell . . . . . . . . . . . . . . . . . . . . . . . . . 319 George V. Filippenko and Tatiana V. Zinovieva Minisymposium on Biomechanics Wireless ECG Monitoring Device with the Ability to Collect and Analyze the Received Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 V. N. Malysheva, A. O. Stosh, D. M. Zaynullina, and G. A. Cherepennikov Algorithmic Modelling of the Heart Based on CT Scans . . . . . . . . . . . . . . . 341 I. A. Kulczycki, O. S. Loboda, and V. V. Suvorov

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Contents

Numerical Simulation of the Stress Nature of the Stresses Arising in the Bone, When Installing Different Structures for Temporary Epiphysiodesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 V. M. Ivanov, A. V. Kiyuts, O. S. Loboda, R. Yu. Shipov, A. V. Smirnov, V. M. Kenis, and D. A. Petrova Minisymposium “Extreme Loading on Structures” Modeling of Functional Behavior of Cu-Al-Be Shape Memory Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Fedor S. Belyaev, Aleksandr E. Volkov, Margarita E. Evard, and Maria S. Starodubova Minisymposium “Nonlinear Waves in Continuous Media” Analytical Description of the Die Movement on Semi-space Surface on the Basis of Nonlinear Model of Crystal Medium Deformation . . . . . . 375 Anatolii N. Bulygin and Yurii V. Pavlov Minisymposium “Mathematical Modeling in Petroleum Engineering” Modeling the Displacement of Oil From a Porous Medium Taking into Account Thermochemical Interactions Between Phases . . . . . . . . . . . 391 V. R. Dushin, N. N. Smirnov, V. F. Nikitin, E. I. Skryleva, and M. N. Makeeva Coupled Model of Deep-bed-filtration and Cake Filtration in Porous Media for Gravel Pack Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 410 Vsevolod D. Pashkin and Konstantin E. Lezhnev Minisymposium “Anomalous Energy Transfer on a Nano-Scale” Simulation of Inelastic Response of Polycrystalline Nickel Based on Micromechanical Model Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . 427 I. R. Murtazin, B. E. Melnikov, and A. S. Semenov Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Complex Media: Micropolar Theory, Chemomechanics, Acoustic Metamaterials etc.

Moment-Membrane Dynamic Theory of Elastic Thin Shells and Variational Principles S. H. Sargsyan(B) Shirak State University, Gyumri, Armenia [email protected]

Abstract. The paper considers a system of governing equations and boundary conditions of the three-dimensional dynamic moment theory of elasticity with independent fields of displacements and rotations. Hypotheses are applied, which are of both kinematic and static character and which correspond to the qualitative results of the asymptotic method of integration of the mentioned boundary problem for the region of the three-dimensional thin shell. A two-dimensional moment-membrane dynamic theory of thin shells is constructed. Further, variational principles of Hu-Washizu type, the virtual work principle, principles when the causes of the shell’s vibrations change harmonically in time and the Hamilton principle are established for the constructed theory. Keywords: Moment theory of elasticity · Thin shells · Hypothesis · Moment-membrane dynamic theory · Variation principles

1 Introduction Papers [1, 2] demonstrate that crystalline nanostructures, in particular, two-dimensional nanomaterials (rapheme, nanotube, fullerene) are described continuously by the equations of the macroscopic moment theory of elasticity with independent fields of displacements and rotations. In paper [3], on the basis of theoretical and experimental studies, it is shown that at the meso-level, as well as at the nano-level, the deformation in the crystalline fits to the concept of shear-plus-rotation. These considerations lead to the conclusion that for the continuous deformations of two-dimensional nanomaterials it is relevant to construct a model of thin shells and plates based on the moment theory of elasticity with independent fields of displacements and rotations, where the deformations fit to the concept of shear-plus-rotation. In paper [4], such a model for statics of elastic thin shells is constructed on the basis of the moment theory of elasticity with independent fields of displacements and rotations [5]. Considering that the distribution of stresses and moment stresses in this model are uniform along the shell thickness, this model of the elastic thin shell is also referred to as the moment-membrane theory of a thin shell. This paper generalizes the idea and the approach of paper [4] and demonstrates the construction of the moment-membrane dynamic theory of elastic thin shells; a general © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. A. Indeitsev and A. M. Krivtsov (Eds.): APM 2021, LNME, pp. 3–15, 2024. https://doi.org/10.1007/978-3-031-37246-9_1

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S. H. Sargsyan

variational principle of Hu-Washizu type and, in particular, the virtual work principle are established for the mentioned theory; the main energy equality is derived—the law of conservation of energy. Further, a case is considered when the causes of deformation and movement of the shell are changed harmoniously in time: 1) the amplitudes of displacements and free rotations are subject to variation, 2) the amplitudes of internal forces and moments are subject to variation. Further, the extreme properties of the corresponding functionals are established. A variational principle of Hamilton’s type is established for the moment-membrane dynamic theory of elastic thin shells.

2 Basic Equations and Relations of the Three-Dimensional Moment Dynamic Theory of Elasticity with Independent Fields of Displacements and Rotations As a basis we shall consider three-dimensional equations and relations of the linear moment dynamic theory of elasticity with independent fields of displacements and rotations [5], in the region of a shell with a thickness 2h. We shall proceed from the general Hu-Washizu-type variational principle for the linear moment dynamic theory of elasticity with independent fields of displacements and rotations (the corresponding variational principle for the static problem is used, with the involvement of D’Alembert principle): ¨ h δI = δ

  { −σmn γmn − (∇m vn − εkmn ωk ) − μmn (χmn − ∇m vn )}H1 H2 d α1 d α2 dz

(s) −h

¨ +

[σ2n (vn − v n ) + μ2n (ωn − ωn )]d 1 − 1

− (s) −h

¨

− (s+ ) ¨   1

[σ1n (vn − vn ) + μ1n (ωn − ωn )]d 2 2

¨ h 

+



¨



∂ 2 vk ∂ 2 ωk Fk − ρ 2 δvk + Ck − I δωk H1 H2 d α1 d α2 dz ∂t ∂t 2

 + + qn δvn + m+ n δωn ds + 

¨

 − − qn δvn + m− n δωn ds

(s− ) ¨  σ 2n δvn + μ2n δωn d 1 − σ 1n δvn + μ1n δωn d 2 = 0.

  2

(1)

Here m, n, k accept the values 1, 2, 3; σmn , μmn are components of stress and moment stress tensors; γmn , χmn − components of tensors of deformations and bending-torsions; Vn , ωn − components of vectors of displacements and free rotations; εkmn − Levi-Civita tensor components;  − volume density of potential energy of deformation: (s) (s)

(a) (a)

(s) (s)

(a) (a)

 = μγmn γmn + αγmn γmn + γ χmn χmn + εχmn χmn +

λ β γ γnn + χkk χnn . 2 kk 2

(2)

μ, λ, α, γ , ε, β are elastic constants; the subscript (s)− marks the symmetric components, and the subscript (a) marks the antisymmetric components of the corresponding tensors.

Moment-Membrane Dynamic Theory of Elastic Thin Shells

5

It must be also noted that α1 , α2 are the lines of the main curvatures of the shell mid surface (z = 0), and the rectilinear axis z is directed along the normal to this surface. Lamé coefficients of the triorthogonal coordinate system are:

 z (i = 1, 2), H3 = 1, Hi = Ai 1 + Ri

(3)

where Ai , Ri —coefficients of the first quadratic form and the main radii of curvature of the shell mid surface. From the general Hu-Washizu variational principle (1) follow the basic equations (equations of motion, elasticity and geometric relations) of the moment theory of elasticity with independent fields of displacements and rotations, as well as natural boundary conditions on the front surfaces of the shell (S + and S − ) and on the shell lateral surface ( = 1 + 2 , where i = i + i , i = 1, 2). In the moment dynamic theory of elasticity with independent fields of displacements and rotations, the variational principle of Hamilton type is also established: t1 δ t0

( − K)dt = 0,

(4)

 = U − A,

(5)

¨ h U =

 H1 H2 d α1 d α2 dz,

(6)

(S) −h

potential energy of deformation of the whole body: 1 K= 2

¨ h (S) −h

⎡

⎤ 2 2

3  ∂v ∂ω k k ⎣ ⎦H1 H2 d α1 d α2 dz, ρ +I ∂t ∂t

(7)

k=1

kinetic energy of body movement: ¨ h A=

¨ (Fk vk + ck ωk )H1 H2 d α1 d α2 dz +

(S) −h

¨

− (S − )

 − − qn vn + m− n ωn dS +

¨ 2

 + + qn vn + m+ n ωn dS

(S + ) 

σ 1n vn + μ1n ωn d 2 −

¨



σ 2n vn + μ2n ωn d 1 .

(8)

1

Hamilton’s principle contains the differential equations of motion of the moment theory of elasticity with independent fields of displacements and rotations, as well as force and moment boundary conditions on S + , S − , 1 , 2 .

3 Basic Hypotheses Displacements and Rotation Deformations and Bending-Torsions, Stresses and Moment Stresses The content of hypotheses accepted below can be considered as kinematic and static. 1. The essence of the main kinematic hypothesis is the assumption of the constancy of all components of the vector of displacement and the vector of independent rotation along the coordinate z:

6

S. H. Sargsyan

Vi = ui (α1 , α2 ), V3 = w(α1 , α2 ) ωk = k (α1 , α2 ) (i = 1, 2), (k = 1, 2, 3).

(9)

2. Along with the kinematic hypothesis (9), there are static assumptions: in the corresponding physical relations about the smallness of the normal stress σ33 in comparison with the stress σii ; moment stress μ33 in comparison with the stress μii ; stresses σ3i (i = 1, 2) in comparison with σi3 ; moment stresses μ3i (i = 1, 2) in comparison with μi3 (i = 1, 2). 3. We shall assume that the shell is thin ( Rh