Proceedings of the 5th International Conference on Numerical Modelling in Engineering: Volume 2: Numerical Modelling in Mechanical and Materials Engineering, NME 2022, 23–24 August, Ghent University, Belgium 9819903726, 9789819903726

Citation preview

Lecture Notes in Mechanical Engineering

Magd Abdel Wahab   Editor

Proceedings of the 5th International Conference on Numerical Modelling in Engineering Volume 2: Numerical Modelling in Mechanical and Materials Engineering, NME 2022, 23–24 August, Ghent University, Belgium

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 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 Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia 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: • • • • • • • • • • • • • •

Engineering Design Machinery and Machine Elements Automotive Engineering Engine Technology Aerospace Technology and Astronautics Nanotechnology and Microengineering MEMS Theoretical and Applied Mechanics Dynamical Systems, Control Fluid Mechanics Engineering Thermodynamics, Heat and Mass Transfer Manufacturing Precision Engineering, Instrumentation, Measurement Tribology and Surface Technology

Indexed by SCOPUS and EI Compendex. All books published in the series are submitted for consideration in Web of Science. To submit a proposal for a monograph, please check our Springer Tracts in Mechanical Engineering at https://link.springer.com/bookseries/11693.

Magd Abdel Wahab Editor

Proceedings of the 5th International Conference on Numerical Modelling in Engineering Volume 2: Numerical Modelling in Mechanical and Materials Engineering, NME 2022, 23–24 August, Ghent University, Belgium

Editor Magd Abdel Wahab Faculty of Engineering and Architecture Ghent University Ghent, Belgium

ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-981-99-0372-6 ISBN 978-981-99-0373-3 (eBook) https://doi.org/10.1007/978-981-99-0373-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Organising Committee

Chairman Prof. Dr. Ir. Magd Abdel Wahab, Laboratory Soete, Ghent University, Belgium

International Scientific Committee Prof. D. Ribeiro, School of Engineering, Polytechnic of Porto (ISEP-IPP), Portugal Prof. J. Santos, University of Madeira, Portugal Prof. J. Toribio, University of Salamanca, Spain Prof. B. B. Zhang, Glasgow Caledonian University, UK Prof. V. Silberschmidt, Loughborough University, UK Prof. T. Rabczuk, Bauhaus University Weimar, Germany Prof. L. Vanegas Useche, Universidad Tecnológica de Pereira, Colombia Prof. N. S. Mahjoub, Institut Préparatoire aux Etudes d’Ingénieurs de Monastir, Tunisia Prof. A. Cheknane, Amar Telidji University of Laghouat, Algeria Prof. E. N. Farsangi, Kerman Graduate University of Advanced Technology (KGUT), Iran Prof. N. A. Noda, Kyushu Institute of Technology, Japan Prof. K. Oda, Oita University, Japan Prof. S. Abdullah, Universiti Kebangsaan Malaysia, Malaysia Prof. C. Zhou, Nanjing University of Aeronautics and Astronautics, China Prof. B. Bhusan Das, National Institute of Technology Karnataka, India Prof. R. V. Prakash, Indian Institute of Technology, India Prof. H. N. Xuan, Hutech University, Vietnam Prof. Giuseppe Carbone, University of Calabria, Italy Prof. Fadi Hage Chehade, Lebanese University, Lebanon Prof. Sohail Nadeem, Quaid-i-Azam University, Pakistan

v

vi

Organising Committee

Dr. A. San-Blas, Miguel Hernández University of Elche, Spain Dr. G. Minafo, University of Palermo, Italy Dr. A. Caggiano, Technische Universität Darmstadt, Germany Dr. S. Khatir, Ghent University, Belgium Dr. T. Yue, Ghent University, Belgium Dr. A. Rudawska, Lublin University of Technology, Poland Dr. L. V. Tran, Sejong University, South Korea Dr. X. Zhuang, Leibniz Unversität Hannover, Germany Dr. I. Hilmy, International Islamic University Malaysia, Malaysia Dr. C. Wang, Liaocheng University, China Dr. M. Mirrashid, Semnan University, Iran Prof. A. G. Correia, University of Minho, Portugal Dr. M. Wang, Los Alamos National Laboratory, USA Dr. Filippo Genco, Adolfo Ibáez University, USA Dr. Denis Benasciutti, University of Ferrara, Italy Dr. Y. L. Zhou to Xi’an Jiaotong University, China

Preface

This volume contains the Proceedings of the 5th International Conference on Numerical Modelling in Engineering—Volume 2: Numerical Modelling in Mechanical and Materials Engineering. Numerical Modelling in Engineering NME 2022 is the 5th NME conference and was held online via MS Teams, during the period 23–24 August 2022. Previous NME conferences were celebrated in Ghent, Belgium (2018), Beijing, China (2019), and Ghent, Belgium (2020–2021). The overall objective of the conference is to bring together international scientists and engineers in academia and industry in fields related to advanced numerical techniques, such as FEM, BEM, and IGA, and their applications to a wide range of engineering disciplines. The conference covers industrial engineering applications of numerical simulations to Civil Engineering, Aerospace Engineering, Materials Engineering, Mechanical Engineering, Biomedical Engineering, etc. The presentations of NME 2022 are divided into two main sessions, namely (1) Civil Engineering and (2) Mechanical and Materials Engineering. This volume is concerned with the applications of Mechanical and Materials Engineering. The organising committee is grateful to keynote speaker, Prof. Timon Rabczuk, Bauhaus Universität Weimar, Chair of Computational Mechanics, Germany, for his very interesting keynote speech entitled ‘Machine Learning-based solutions of Partial Differential Equations’. Special thanks go to members of the Scientific Committee of NME 2022 for reviewing the articles published in this volume and for judging their scientific merits. Based on the comments of reviewers and the scientific merits of the submitted manuscripts, the articles were accepted for publication in the conference proceedings and for presentation at the conference venue. The accepted papers are of very high scientific quality and contribute to the advancement of knowledge in all research topics relevant to the NME conference.

vii

viii

Preface

Finally, the organising committee would like to thank all the authors, who have contributed to this volume, and those who have presented their research work at the conference in MS Teams. Ghent, Belgium

Prof. Magd Abdel Wahab Chairman of NME 2022

Contents

1

2

3

4

5

6

7

8

Optimization of Ship Energy Efficiency Considering Navigational Environment and Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . Min Hyok Jon and Chung Il Yu

1

Study on the Consistency of a Phase Field Modeling Method and the Determination of Crack Width . . . . . . . . . . . . . . . . . . . . . . . . . . Feiyang Wang, Youliang Chen, Tongjun Yang, and Hongwei Huang

17

Numerical Simulation of Shear Wave Propagation Through Jointed Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resmi Sebastian and Kallol Saha

27

A Quasi-static Computational Model for Fracture in Multidomain Structures with Inclusions . . . . . . . . . . . . . . . . . . . . . . Roman Vodiˇcka

41

Updated Lagrangian Curvilinear Beam Element for 2D Large Displacement Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Iandiorio and Pietro Salvini

61

Modelling and Simulation of Micro-electro-Mechanical Systems for Energy Harvesting of Random Mechanical Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kailing Song and Michele Bonnin Revisiting a Model that Describes the Process of the Vocal Oscillation During Phonation, a Numerical Approach . . . . . . . . . . . . M. Filomena Teodoro

81

93

Shear Stress and Temperature Analysis of Inconel 718 During the Backward Flow Forming Process Using the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Acar Can Kocabıçak and Magd Abdel Wahab

ix

x

Contents

9

Mechanical Design and Optimization of Large-Scale Parabolic Trough Solar Collectors for Industrial Applications . . . . . 113 Ossama Mokhiamar, Mohammed Siddeq, and Osama Elsamni

10 Finite Element Modeling of Ultrasonic Nanocrystalline Surface Modification Process of Alloy 718 . . . . . . . . . . . . . . . . . . . . . . . 125 Chao Li, Ruslan Karimbaev, Auezhan Amanov, and Magd Abdel Wahab

Chapter 1

Optimization of Ship Energy Efficiency Considering Navigational Environment and Safety Min Hyok Jon

and Chung Il Yu

Abstract The attention to ship energy efficiency and CO2 emission is significantly increasing. Both are related to fuel consumption and can be assessed by ship energy efficiency operational indicator (EEOI). The aim of this research is to develop a formula for estimation of operational carbon intensity indicator (CII) and an optimal model of ship’s route and operational speed to minimize the EEOI considering navigational environment and ship’s safety. The formula for CII is given assuming to be a function of a ship’s main particular, i.e. block coefficient, and ratio of operating speed to design speed of the ship. For navigational environment, wave and wind, which influence the ship’s performance especially including resistance and seakeeping, are considered. For ship’s safety, motion sickness incidence (MSI) which is one of seakeeping indices is considered. Particle Swarm Optimization (PSO) algorithm is adopted to solve the model. The proposed method is illustrated with a numerical example, comparing with full-scale data. The comparing results show the proposed method can effectively reduce the CO2 emission and improve the ship energy efficiency. Keywords Energy efficiency operational indicator (EEOI) · Carbon intensity indicator (CII) · Navigational environment · Marine safety · Seakeeping

1.1 Introduction The excessive greenhouse gas (GHG) emission from shipping, which results in the speed-up in global warming, caused the pressing concern around the world. Anderson and Bows [1] showed that shipping should reduce its CO2 emissions by more than 80% by 2050 compared to 2010 levels for achieving the 2 °C climate goal. International Maritime Organization (IMO) established the Initial IMO Strategy on M. H. Jon (B) · Chung Il Yu Faculty of Naval Architecture and Ocean Engineering, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_1

1

2

M. H. Jon and C. I. Yu

Reduction of GHG Emissions from Ships in 2018. In accordance with the strategy, CO2 emissions per transport work as an average across international shipping are to be reduced by at least 40% by 2030, pursuing towards 70% by 2050, compared to 2008 [2]. Under the circumstances, useful indices such as Energy Efficiency Design Index (EEDI), Energy Efficiency Existing Ship Index (EEXI) and Energy Efficiency Operational Indicator (EEOI) were introduced to indicate CO2 emissions of ships. All of them reflects the effective energy usage on ships, especially EEOI being an operational carbon intensity indicator related to CO2 emission from ship operation activities. That is, energy efficiency management and improvement are fundamental issues to account for fuel costs savings and CO2 emissions reduction. In order to reduce the GHG emission from ships and improve the energy efficiency of ships, many researches were carried out proposing some technical and operational measures during a few decades [3–5]. Regarding technical measures, some researches focused on improvement of ship’s performance, which is closely related to many design parameters of ships’ hull and propeller. From the point of view of hull form optimization, Hou [6] proposed a uncertainty ship hull design method to achieve a minimum EEOI with a focus on the influence of speed perturbation. Hou et al. [7] introduced a mixed aleatory/epistemic uncertainty analysis and optimization method based on probability and evidence theory and demonstrated excellent adaptability and reliability in minimum EEDI ship hull lines’ designs. Niese et al. [8] proposed a novel ship design evaluation framework rooted in Markov decision analysis and derived metrics, examining scenarios subject to carbon emission regulations and uncertainty surrounding enforcement of the EEDI. Other technologies including energy-saving, use of renewable energy (e.g. wind engine, solar panels) and use of alternative fuels (e.g. LNG, hydrogen, ammonia) are also developed and costs and GHG emission reduction potential of the technologies were discussed [9–13]. For instance, Bøckmann et al. [14] performed route simulations on a general cargo ship equipped with retractable bow-mounted foils for resistance reduction and motion damping in waves, showing the result of the average fuel saving, while Zhao et al. [15] investigated propulsion system optimization design. Some researchers focused on seakeeping to optimize the ship’s performance considering ship’s safety and human comfort [16, 17]. Scamardella and Piscopo [18] presented a method to achieve the best seakeeping performances for passenger vessels by only varying several hull form parameters. Regarding operational measures, there are researches on navigation speed and route optimization for in-service ships in order to improve energy efficiency [19–25]. Wang et al. [26] proposed a dynamic optimization method adopting the model predictive control (MPC) strategy to optimize ship energy efficiency accounting for timevarying environmental factors such as weather. Hou et al. [27] introduced uncertainty analysis and optimization theory when considering ice loads and other important stochastic factors, taking the main engine speed of ships as the design variable to solve the minimum EEOI optimization model. Tran [28] addressed a numerical method to decrease the fuel consumption of diesel engine and restrict the exhaust gases emission

1 Optimization of Ship Energy Efficiency Considering Navigational …

3

from the ship operational activities focusing on EEOI. Aiming at optimal path and speed profile for a ship voyage on the basis of weather forecast maps, ship motions and human comfort was also taken into account to minimize fuel consumption [29]. Wang et al. [30] established ship energy efficiency real-time optimization model to determine the best engine speed under different working conditions which were predicted by the method of wavelet neural network in short distance ahead of the ship. Different studies were conducted regarding the vessel’s power management. Kanellos [31] pointed out a three stages method based on the operational cost minimization to find out an optimal power management solution under the limitation of the greenhouse gas emissions without changing the technical and operational constraints. Baldi et al. [32] proposed a method to model the power plant of an isolated system with mechanical, electric and thermal power demands and to optimize the load allocation of the different components. Anconaa et al. [33] proposed an optimization framework based on genetic algorithms in order to maximize the energy efficiency and minimize both the fuel consumption and the thermal energy dissipation, by optimizing the load allocation of the ship energy systems. There are various papers about shipping emissions where some tried to develop emission factors and estimate current and future emissions and some focused on single emission type and its impacts on human health and environment [34–36]. Shipping emission estimations are realized by using two basic methods: Top-down and bottom-up approaches. Bilgili and Celebi [37] developed some equations in order to estimate the potential airborne emissions of a bulk carrier based on two main characteristics (DWT and CB) during pre-design, by applying a regression analysis to the three-year operation data of nine bulk carriers. In the calculation of the EEDI for ships, the ship speed/power curves is of great significance. Tu et al. [38] suggested a more accurate estimation method of speed/power curves for container ships based on investigation of the influence of three important factors (i.e. the main engine power, the deadweight and the reference speed) on the EEDI value. In light of the above, there is lack of researches on calculation of the operational carbon intensity indictor for individual ships and on optimization of ship energy efficiency. This paper aims at developing a formula in order to estimate the operational carbon intensity ahead of navigation and a method to optimize the ship energy efficiency during navigation considering navigational environment and safety. The rest of paper is organized as follows. In Sect. 1.2, a simple formula to estimate CII is developed and a method is proposed to optimize the ship energy efficiency during navigation considering navigational environment and safety. Section 1.3 shows the case study results to demonstrate the feasibility and efficiency of the proposed formula and method. Finally, Sect. 1.4 contains conclusions and potential contributions of the paper.

4

M. H. Jon and C. I. Yu

1.2 Methods 1.2.1 Formula to Estimate the Operational Carbon Intensity In this section, a formula to estimate the CII over a year is suggested. In the Initial IMO Strategy on Reduction of GHG Emissions from Ships, the goal level on carbon intensity of international shipping is quantified by the CO2 emissions per transport work, as an average across international shipping. The indices indicating the average CO2 emissions per transport work of a ship are generally referred to as operational carbon intensity indicator (CII) [39]. A specific CII calculated based on the actual or estimated mass or volume of the shipment carried on board a ship is generally referred to as demand-based CII. The typical one for demand-based CII is Energy Efficiency Operational Indicator (EEOI), which is given as follows. m C O2 m pl × D pl ∑ = FC j × E F j

EEOI = m C O2

(1.1) (1.2)

j

where m C O2 is the sum of CO2 emissions (in grams) from all the fuel oil consumed on the voyage, m pl is the mass of transported cargo, D pl is the voyage distance with the cargo, j is the fuel oil type, FC j is the total consumption of the fuel j, E F j is the emission factor of the fuel j. A specific CII, in which calculation the capacity of a ship is taken as proxy of the actual mass or volume of the shipment carried on board, is generally referred to as supply-based CII. The supply-based CII which uses DWT as the capacity is referred to as AER, and the supply-based CII which uses GT as the capacity is referred to as cgDIST [39]. AER is expressed as follows. AE R =

m C O2 DWT × Dt

(1.3)

where Dt is the total voyage distance. To estimate the operational carbon intensity, AER is chosen as an index to be formulated by statistical analysis method. To start with, let a vessel with main engine’s power of P0 kW navigate the total distance of Z n.mile during the total time of Ta hours over a year. Assuming that the amount of CO2 emission will be proportional to P0 and Ta , Eq. (1.4) is given as follows. M = k · E F · Ta · P0 · S FC

(1.4)

1 Optimization of Ship Energy Efficiency Considering Navigational …

5

where M is the total amount of CO2 emission over the one-year period and k is coefficient, S FC and E F being the specific fuel consumption and emission factor of the fuel respectively. Letting the average operating speed over the year be υoper , Ta = Z /υoper

(1.5)

By substituting Eq. (1.5) into Eq. (1.4), P0 M = k · EF · · S FC DW T · Z DW T · υoper

(1.6)

where DWT is a term to identify the movable load, such as cargo, passengers, crew, fuel and stores. Basically, DWT is the difference in tonnes between the displacement of a ship at the maximum summer load draught and the lightweight of the ship. Then it can be seen that the left side of Eq. (1.6) is the same as AER. As a result, AE R = k · E F ·

P0 · S FC DW T · υoper

(1.7)

In Eq. (1.7), the coefficient k, which varies depending on the hull form and navigational condition, is assumed to be a function of a ship’s main particular, i.e. block coefficient, C B and ratio of operating speed to design speed of the ship, υoper /υdesign and to be different depending on the type of ships. C B was chosen as a term which describes the fineness of a ship and gives important information about the ship hull and type. υoper /υdesign was chosen as a term which describes how fast the vessel goes during the navigation. Even though there might be many other factors related to the coefficient, but we focused on only two parameters in this paper. To determine k, statistical analysis method is used [37, 40]. It needs enough data on voyages for regression analysis. Although the uncertainties, this research collected the necessary data on fuel consumption, mass of cargo, voyage distance and so on from ship’s navigation reports. We investigated the relation between k and C B in a given range of υoper /υdesign and the relation between k and υoper /υdesign in a given range of C B from the collected data (Figs. 1.1 and 1.2). From the Figs. 1.1 and 1.2, it can be seen that k nearly has the quadratic form with respect to C B and υoper /υdesign . As a result, k is assumed to be expressed as follows. k = β0 + β1 C B +

β2 C B2

( ) υoper 2 υoper + β3 + β4 υdesign υdesign

(1.8)

where βi (i = 0–4) are constant coefficients, which vary from ship’s kind (e.g. general cargo ship or bulk carrier). Those coefficients can be estimated by regression analysis

6

M. H. Jon and C. I. Yu

Fig. 1.1 Relation between k and C B

Fig. 1.2 Relation between k and υoper /υdesign

from the collected voyage data for a certain kind of ship. The uncertainty analysis associated with this formula for AER is discussed in Sect. 1.3 in detail.

1.2.2 Optimization of Ship Energy Efficiency Considering Navigational Environment and Safety In this section, a method to optimize the ship energy efficiency considering navigational environment and safety is presented. Target function for optimization is just evaluation of the EEOI during navigation with constraints involving consideration of navigational environment and safety. For

1 Optimization of Ship Energy Efficiency Considering Navigational …

7

navigational environment, which what is more affects the evaluation of EEOI, wave and wind are considered, while ship’s seakeeping for safety. To evaluate the EEOI, the most important parameter is just fuel consumption, which is mainly used to overcome ship’s resistance. After calculating the ship’s resistance under different environmental conditions, the fuel consumption of the main engine can be obtained by analyzing the energy transmission between ship hull, propeller and main engine [26]. The ship’s resistance can be given as the sum of calm water resistance, wave resistance and wind resistance. Those resistances can be calculated by using methods in Holtrop and Mennen [41], Kwon [42] and Townsin et al. [43], respectively. Then the ship’s total resistance can be given as a function of vessel’s speed, wind’s speed and direction and wave’s height, speed and direction (Eq. (1.9)). RT (Vs , Vwind , Vwave , H ) = Rcalm + Rwave + Rwind

(1.9)

where RT stands for ship’s total resistance, Rcalm for calm water resistance, Rwave for wave resistance, Rwind for wind resistance, Vs for vessel’s speed, Vwind for wind speed, Vwave for wave propagation speed and H for significant wave height. The propeller performance is simulated by using the open water diagrams of thrust and torque coefficients, K T and K Q (Eqs. (1.10) and (1.11)). KT =

T ρn 2 D 4

(1.10)

KQ =

Q ρn 2 D 5

(1.11)

where ρ is the density of water, T and Q are thrust and torque of open water propeller, n is the propeller revolution speed and D is the propeller diameter, respectively. The K T and K Q can be obtained as functions of propeller advance coefficient J by the interpolation polynomials. That is, J=

Vs Dn

K T = f T (J ), K Q = f Q ( J )

(1.12) (1.13)

Open water efficiency of the propeller is ηO =

KT · J K Q · 2π

(1.14)

1−t 1−w

(1.15)

while hull efficiency being ηH =

8

M. H. Jon and C. I. Yu

Letting k be the number of propellers, η S be the shaft transmission efficiency and ηG be the gearbox efficiency, the necessary output power of the main engine can be given as follows. PM E =

RT · Vs RT · Vs · K Q · 2π · (1 − w) = kη S ηG η O η H kη S ηG · K T · J · (1 − t)

(1.16)

The specific fuel consumption (SFC) of the main engine is assumed to vary as a function of engine load as shown by the following empirical equation [2]. qload = qb ∗ (0.455 ∗ load 2 − 0.71 ∗ load + 1.28)

(1.17)

where qload is the specific fuel consumption at a given engine load, qb is the lowest specific fuel consumption for a given engine and load is just engine load. The qload curves for marine engines are u-shaped: it gets lower until it reaches a minimum and begins to increase again at higher loads. Equation (1.17) shows how SFC changes as a function of engine load with SFCs being generally optimized (i.e., lowest) at 80% load. qbase varies based on engine age and engine type (e.g., SSD, MSD, HSD), which reflects the SFC at the engine’s most efficient load. From Eqs. (1.9) to (1.17), fuel consumption per unit time on a voyage segment Q is given as follows. Q = k · PM E · qload

(1.18)

Dividing the whole voyage route into several subvoyage segments in which vessel’s speed, engine load, etc. are invariable, the total fuel consumption on the voyage is given as sum of fuel consumptions on the subvoyage segments as follows. M=

∑ i

Mi =

∑ i

Qi ·

Di Vwavei − Vsi · cos μi

(1.19)

where M is fuel consumption and subscript i denotes ith subvoyage segment (e.g. Mi is one on the ith subvoyage segment). As a result, ship’s energy efficiency EEOI is given as follows. ∑

Mi × E F m C O2 i = EEOI = m pl × D pl m pl × D pl

(1.20)

Equation (1.20) involves not only time-varying engine load but also the navigational environment. To optimize the ship energy efficiency is just to minimize the value of EEOI in Eq. (1.20) on the voyage. Given the number of segments, vessel’s speed and navigation distance in each segment, Vsi and Di are chosen for optimization parameters.

1 Optimization of Ship Energy Efficiency Considering Navigational …

9

Constraints is to reflect the safety i.e. seakeeping performance of the ship. There are several indices in seakeeping including “Subjective Magnitude”(SM) and “Motion Sickness Incidence” (MSI). Motion sickness generally indicates discomfort on a moving environment, having the peak of different associated symptoms in vomiting [44]. MSI is the percentage of subjects who vomit in the specified time that subjects are exposed to the motions. There are some different ways in which MSI is calculated, while the method described in Lloyd [45] is used to evaluate MSI in this research. That is, { MSI (%) = 100 × ϕ

lg( |¨sg3 | )−μMSI

}

0.4

μMSI = −0.819 + 2.32(lg ωe )2

(1.21) (1.22)

where ϕ(x) is the cumulative normal distribution function up to x for a normal distribution with zero mean and unity standard deviation and ωe is the encounter frequency in rad/s. By O’Hanlon and McCauley [46], μMSI = 0.654 + 3.697(lg f e ) + 2.32(lg f e )2

(1.23)

with f e in Hz. √ In / which, the acceleration |¨s3 | is given by |¨s3 | = 0.798 m 4 and ωe = 2π f e ,

m4 . ωe = m 2 Given wave energy spectrum Sζ (ω), the nth spectral moment is calculated using the equation below.

{∞ mn =

ωn Sζ (ω)dω (n = 0, 1, 2, . . . .)

(1.24)

0

Finally, the navigation optimization system, which is just a nonlinear optimization model, is comprised of the optimization target and constraints as expressed in the following equations. ∑ min EEOI =

Mi × E F

i

m pl × D pl

(1.25)

10

M. H. Jon and C. I. Yu

⎧ N ∑ ⎪ Di ⎪ ⎪ ≤ Tlim ⎪ ⎪ ⎪ V − Vsi · cos μi wave i ⎪ i=1 ⎪ ⎪ ⎪ ⎨∑ N Di = Dt ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ ⎪ n min < n engi < n max , i =1∼N ⎪ ⎪ ⎩ MSI < MSImax

(1.26)

The optimal solution will be found by following recursive several steps: Step 1: to divide the whole voyage route in N segments. Step 2: to give speeds in each segment. Step3: to calculate the amounts of fuel consumptions in each segment. Step4: to calculate the EEOI. Setp5: to check out if the constraints are satisfied. Setp6: to remember the distances and speeds in each segment in case that the value of EEOI is smaller than the previous one. Setp7: to finish the loop in case that stopping criteria are satisfied, otherwise to go to Step 1. To realize the optimal solution, particle swarm optimization (PSO) algorithm is used. Particle swarm algorithm is a population-based algorithm. In this respect, it is similar to the genetic algorithm. Compared with the genetic algorithm, PSO has the advantage of faster convergence and fewer control parameters, and it has been widely used for global optimization in cases of complicated and non-linear objective functions or constraints [47, 48]. Thus, we apply PSO algorithm to determine the optimal distances and speeds in each segment.

1.3 Results and Discussion In this section, the proposed method is illustrated through a case study. First of all, the estimates of AER are calculated. We referred to report on actual ship navigation data and energy efficiency by a port authority. The reported data were appropriate to the proposed method. In the formula for AER, i.e. Eq. (1.7), the coefficient k as a function of C B and υoper /υdesign varies from kind of ships. That is, the coefficients β i (i = 0–4) in Eq. (1.8) for k depend on the vessel’s type. Data on voyages of more than 60 general cargo ships over a period of 5 years were collected and analyzed. Block coefficients, ratios of operating speed to design speed and fuel consumptions of some ships are presented in Table 1.1. Figure 1.3 shows the distribution of coefficient k with C B and υoper /υdesign . derived from data on fuel consumptions.

1 Optimization of Ship Energy Efficiency Considering Navigational …

11

Table 1.1 Block coefficients, ratios of operating speed to design speed and fuel consumptions of 3 ships over a 5-year time Speed ratio, υoper /υdesign

Block coefficient, CB

2017

2018

2019

2020

2021

2017

Fuel consumption, M(t) 2018

2019

2020

2021

Ship 1

0.86

0.82

0.83

0.81

0.89

0.79

78

120

125

110

102

Ship 2

0.79

0.85

0.93

0.91

0.89

0.92

88

110

115

97

56

Ship 3

0.88

0.92

0.81

0.78

0.80

0.82

95

105

135

107

98

Fig. 1.3 Distribution of coefficient k with C B and υoper /υdesign

Table 1.2 Coefficients β i (i = 0–4) for general cargo ship β0

β1

β2

β3

β4

0.396

3.022

−1.919

−4.023

3.196

Coefficients in Eq. (1.8) can be determined from regression analysis based on collected data. Table 1.2 shows the coefficients β i (i = 0–4) in the formula for AER in case of general cargo ship. In order to evaluate the effectiveness of the formula, standard deviation of differences between actual ship data and formula’s results for AER is investigated. Figure 1.4 shows the differences of AER values between actual ship data and results computed by using the formula, while Table 1.3 shows the standard deviation and variation. As can be seen in Table 1.3, AER values can be estimated by using the proposed formula with the correctness of more than 91%. Next, optimal navigation scheme is demonstrated through a case study. The navigation data is given as follows (Table 1.4). Navigational environment varies over time along the ship’s route, which can be obtained by using INMARSAT or NOAA data. For simplicity, the whole voyage is divided into 3 subvoyage by time interval where navigation environment data is almost invariable, as shown in Table 1.5. For convenience, wind direction and wave direction are set with respect to ship’s forward route.

12

M. H. Jon and C. I. Yu

Fig. 1.4 Differences of AER values between actual ship and formula

Table 1.3 Standard deviation of differences of AER values between actual ship and formula

Table 1.4 Navigation data for case study

Number of ships

Mean of AER

Standard deviation of AER

Uncertainty (%)

214

20.17

1.79

8.87

Length (m)

78

Deadweight (t)

5600

Breadth (m)

14.5

Main engine power (kW)

1520

Draft (m)

6.3

Cargo (t)

3529

Block coefficient

0.8

Voyage distance (n.mile)

720

Table 1.5 Environment data on the voyage

Subvoyage 1 Subvoyage 2 Subvoyage 3 Wind speed (m/s)

5

10

Wind direction (°) 135–145

8.2

170–180

180–225

Wave height (m)

3.5

4.1

6.4

Wave speed (m/s)

5.7

5.2

6.5

Wave direction (°) 120

160

180

For optimization, the number of segments is set between 2 and 10, which is normally given by the captain or ship operator. That is, the whole voyage is divided into several subvoyages in which distances and speeds are optimized according to ways described in Sect. 1.2.2. The optimization results are shown in Table 1.6. As a result, fuel consumption was smaller than the usual navigation. This technology intends to save fuel consumption during navigation by speed reduction. It should be noted that owing to a dispersion in actual speed of ships, even if the type and size of the ships are the same, there is also a little variation in the CO2 reduction

1 Optimization of Ship Energy Efficiency Considering Navigational …

13

Table 1.6 Optimization results Subvoyage 1

Speed (kn)

Time (h)

Distance (n.mile)

Fuel (t)

15.25

20

305.08

6.7

Subvoyage 2

14.90

24

357.66

8.03

Subvoyage 3

14.68

3.9

57.36

1.24

Optimal EEOI (g/t-n.mile)

20.15

effect. We estimated CO2 reduction effect obtained by the proposed method and the results showed, on the average, almost 3–5% of speed reduction compared to usual navigation.

1.4 Conclusions In this paper, a formula for AER was developed to estimate the energy efficiency ahead of the navigation, assuming to be functions of block coefficient and ratio of operating speed to design speed of the ship, which is novel. A method to optimize the ship energy efficiency on the voyage in consideration of navigational environment and safety was also presented. For this, it’s novel to combine the minimization of EEOI with constraints of seakeeping (i.e. MSI). Besides in evaluation of fuel consumptions, the engine load according to the necessary output power was taken into account. The contribution of this research is to help an effective operation as well as fuel saving on the voyage. Because of the uncertainty of each individual ship data and navigational environment data, the result of this research might be also correlated to some degree. In addition, the proposed method would offer more accurate results as more navigational data accumulate. Detailed records of navigation including fuel consumptions are updated regularly over time (e.g. annually). Further studies would be focused on determining the range of sizes of ships to be investigated for the proposed method, and on what to do in case that navigational environment data is not available. The authors will continue to research for elimination of uncertainty as well. Acknowledgements This work was supported by the Korea Classification Society. The authors would like to thank the expert reviewers for their helpful comments on our work. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

14

M. H. Jon and C. I. Yu

References 1. Anderson K, Bows A (2012) Executing a Scharnow turn: reconciling shipping emissions with international commitments on climate change. Carbon Manag 3:15–28 2. IMO.: Reduction of GHG Emissions from Ships-Fourth IMO GHG Study 2020. Marine Environment Protection Committee 75th session (2020) 3. Wang H, Zhou P, Wang ZC (2017) Reviews on current carbon emission reduction technologies and projects and their feasibilities on ships. J Marine Sci Appl 16:129–136. https://doi.org/10. 1007/s11804-017-1413-y 4. Nuchturee C, Li T (2018) A review of energy efficient methods for all-electric ships. In: IOP conference series: earth and environmental science, vol 188, p 012050. https://doi.org/10.1088/ 1755-1315/188/1/012050 5. Perera LP, Mo B (2016) Emission control based energy efficiency measures in ship operations. Appl Ocean Res 60:29–46 6. Hou YH (2017) Hull form uncertainty optimization design for minimum EEOI with influence of different speed perturbation types. Ocean Eng 140:66–72 7. Hou YH, Li YJ, Liang X (2019) Mixed aleatory/epistemic uncertainty analysis and optimization for minimum EEDI hull form design. Ocean Eng 172:308–315 8. Niese ND, Kana AA, Singer DJ (2015) Ship design evaluation subject to carbon emission policymaking using a Markov decision process framework. Ocean Eng 106:371–385 9. IMarEST.: Marginal Abatement Costs and Cost Effectiveness of Energy-Efficiency Measures. MEPC 62/INF.7 (2011) 10. Eide et al (2011) Future cost scenarios for reduction of ship CO2 emissions. Marit Policy Manag 38(1):11–37 11. Schultz MP, Bendick JA, Holm ER, Hetel WM (2011) Economic impact of biofouling on a naval surface ship. Biofouling 27(1):87–98 12. Elgohary MM, Seddiek IS, Salem AM (2015) Overview of alternative fuels with emphasis on the potential of liquefied natural gas as future marine. Proc Inst Mech Eng Part M: J Eng Maritime Environ 229(4):365–375. https://doi.org/10.1177/1475090214522778 13. Levander O (2011) Dual fuel engines latest developments. In: Proceeding of ship efficiency 14. Bøckmann E, Yrke A, Steen S (2018) Fuel savings for a general cargo ship employing retractable bow foils. Appl Ocean Res 76:1–10 15. Zhao F, Yang W, Tan WW, Yu W, Yang J, Chou SK (2016) Power management of vessel propulsion system for thrust efficiency and emissions mitigation. Appl Energy 161(7):124–132 16. Tezdogan T, Incecik A, Turan O (2014) Operability assessment of high speed passenger ships based on human comfort criteria. Ocean Eng 89:32–52 17. Bhattacharyya A, Steen S (2014) Influence of ducted propeller on seakeeping in waves. Ocean Eng 91:243–251 18. Scamardella A, Piscopo V (2014) Passenger ship seakeeping optimization by the overall motion sickness incidence. Ocean Eng 76:86–97 19. Norstad I, Fagerholt K, Laporte G (2011) Tramp ship routing and scheduling with speed optimization. Transport Res Part C: Emerg Technol 19(5):853–865 20. Magirou EF, Psaraftis HN, Bouritas T (2015) The economic speed of an oceangoing vessel in a dynamic setting. Transport Res Part B: Methodol 76:48–67 21. Corbett JJ, Wang H, Winebrake JJ (2009) The effectiveness and costs of speed reductions on international shipping. Transp Res Part D: Transp Environ 14:593–598 22. Chang CC, Wang CM (2014) Evaluating the effects of speed reduce for shipping costs and CO2 emission. Transp Res Part D: Transp Environ 31:110–115 23. Chang CC, Chang CH (2013) Energy conservation for international dry bulk carriers via vessel speed reduction. Energy Policy 59(1):710–715 24. Psaraftis HN, Kontovas CA (2013) Speed models for energy-efficient maritime transportation: a taxonomy and survey. Transport Res Part C: Emerg Technol 26:331–351 25. Psaraftis HN, Kontovas CA (2014) Ship speed optimization: concepts, models and combined speed-routing scenarios. Transp Res Part C: Emerg Technol 44(4):52–69

1 Optimization of Ship Energy Efficiency Considering Navigational …

15

26. Wang K, Yan XP, Yuan YP, Jiang XL, Lind X, Negenborn RR (2018) Dynamic optimization of ship energy efficiency considering time-varying environmental factors Transport. Res Part D 62:685–698 27. Hou YH, Kang K, Liang X (2019) Vessel speed optimization for minimum EEOI in ice zone considering uncertainty. Ocean Eng 188:106240 28. Tran TA (2020) The novelty numerical simulation method for reducing the fuel oil consumption and the greenhouse gas emission in shipping transportation industry. Cogent Environ Sci 6(1):1809072 29. Zaccone R, Ottaviani E, Figari M, Altosole M (2018) Ship voyage optimization for safe and energy-efficient navigation: a dynamic programming approach. Ocean Eng 153:215–224 30. Wang K, Yan XP, Yuan YP, Li F (2016) Real-time optimization of ship energy efficiency based on the prediction technology of working condition. Transport Res Part D 46:81–93 31. Kanellos FD (2014) Optimal power management with GHG emissions limitation in all electric ship power systems comprising energy storage systems. IEEE Trans Power Syst 29(1):330–339 32. Baldi F, Ahlgren F, Melino F et al (2016) Optimal load allocation of complex ship power plants. Energy Convers Manag 124:344–356 33. Anconaa MA, Baldib F, Bianchic M et al (2018) Efficiency improvement on a cruise ship: load allocation optimization. Energy Convers Manag 164:42–58 34. Wang C, Corbett JJ (2007) The costs and benefits of reducing SO2 emissions from ships in the US West Coastal waters. Transp Res Part D 12:577–588 35. Wang C, Corbett JJ, Firestone J (2008) Improving spatial representation of global ship emissions inventories. Environ Sci Technol 42(1):193–199 36. Jalkanen JP, Johansson L, Kukkonen J, Brink A, Kalli J, Stipa T (2012) Extension of an assessment model ship traffic exhaust emissions for particulate matter and carbon monoxide. Atmos Chem Phys 12:2641–2659 37. Bilgili L, Celebi UB (2018) Developing a new green ship approach for flue gas emission estimation of bulk carriers. Measurement 120:121–127 38. Tu H, Yang YF, Zhang L, Xie D, Lyu XJ, Song L, Guan YM, Sun JL (2018) A modified admiralty coefficient for estimating power curves in EEDI calculations. Ocean Eng 150:309–317 39. MEPC (2021) Reduction of GHG Emissions from Ships: Report of the eighth meeting of the Intersessional Working Group on Reduction of GHG Emissions from Ships (ISWG-GHG 8). Marine Environment Protection Committee 76th session Agenda item 7, MEPC 76/WP.4 40. Ye X, Kang Y, Zuo B, Zhong K (2017) Study of factors affecting warm air spreading distance in impinging jet ventilation rooms using multiple regression analysis. Build Environ 120:1–12 41. Holtrop J, Mennen GGJ (1982) An approximate power prediction method. Int Shipbuild Prog 29(7):166–170 42. Kwon YJ (2008) Speed loss due to added resistance in wind and waves. Nav Archit 3:14–16 43. Townsin RL, Moss B, Wynne JB (1975) Monitoring the speed performance of ships. University of Newcastle, England 44. Reason JT, Brand JJ (1975) Motion sickness. Academic Press, London 45. Lloyd ARJM (1998) Seakeeping ship behaviour in rough water. Ellis Horwood series in Marine technology 46. O’Hanlon JF, McCauley ME (1974) Motion sickness as a function of the frequency and acceleration of vertical sinusoidal motion. Aerosp Med 45:366–369 47. Kornelakis A (2010) Multiobjective particle swarm optimization for the optimal design of photovoltaic grid-connected systems. Sol Energy 84(12):2022–2033 48. Wang K, Yan X, Yuan Y, Tang D (2018) Optimizing ship energy efficiency. In: Application of particle swarm optimization algorithm. Proc Inst Mech Eng Part M J Eng Maritime Environ 232(4):379–391. https://doi.org/10.1177/1475090216638879

Chapter 2

Study on the Consistency of a Phase Field Modeling Method and the Determination of Crack Width Feiyang Wang , Youliang Chen , Tongjun Yang , and Hongwei Huang

Abstract The phase field modeling method has been proven to be a powerful and robust tool for the numerical simulation of discontinuous problems by regularizing the topology of sharp cracks into diffusive cracks with a continuous phase-field function. A new framework of phase-field modeling method for mixed-mode fracture of brittle materials is proposed based on the spectral decomposition with a thermodynamic consistent projection operator, which subtly unifies the critical energy release rate and crack driving strain energy in the phase-field evolution equation. The crack-driving strain energy is determined by the spectral decomposition, and the crack expands in the direction perpendicular to the maximum principal stress. The proposed phase-field modeling method is realized by a staggered algorithm, which is implemented by the subroutines UEL and UMAT of the commercial finite element software ABAQUS. The thermodynamic consistency of the proposed phase-field modeling method is verified by simulating the uniaxial tensile test and mixed-mode fracture test under the loading path and loading–unloading path. Then, an evaluation formula based on a level-set method is put forward to determine the crack width. A case study indicates that the evaluation formula is reasonable and accurate for the recommended length-scale parameter within the range of 2.0–3.0 times the characteristic size of the mesh element. Keywords Phase-field model · Mixed-mode crack · Thermodynamic consistency · Level-set method · Crack width

F. Wang (B) · Y. Chen Department of Civil Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China e-mail: [email protected] T. Yang · H. Huang Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China T. Yang Southwest Transportation Construction Group Co. Ltd, Kunming 650032, Yunan, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_2

17

18

F. Wang et al.

2.1 Introduction Phase-field theory is widely used to solve discontinuous crack problems without the need to track sharp crack interfaces. Even though several frameworks of phase-field modeling method for mixed-mode fracture of brittle materials have been proposed [1–4], there are still some issues worth discussing: (1) the consistency of the governing equation, especially the phase-field evolution equation, (2) thermodynamic consistency, (3) the phase-field crack width. A phase-field modeling method for mixed-mode fracture of brittle materials proposed by Wang et al. [4] is first introduced and the consistency of the governing equation is illustrated. Then, the thermodynamic consistency of the proposed phasefield modeling method is verified by two study cases. Finally, a calculation method is established to calculate the phase-field crack width.

2.2 A New Phase-Field Modeling Method 2.2.1 Governing Equation According to the maximum normal stress criterion, cracks in brittle materials always expand in the direction perpendicular to the maximum normal stress [5]. As revealed by the unified tensile fracture criterion of Eq. √ (2.1), when the ratio of critical shear stress to critical normal stress should not be < 2/2, the brittle material follows the maximum normal stress√criterion [6]. The schematic of crack direction in brittle materials with τ c /σ c ≥ 2/2 is shown in Fig. 2.1. σ2 τ2 + ≥1 σc2 τc2 where σ c is the critical normal stress and τ c is the critical shear stress. Fig. 2.1 Schematic of crack direction in brittle √ materials with τ c /σ c ≥ 2/2

(2.1)

2 Study on the Consistency of a Phase Field Modeling Method …

19

Considering the above fracture behavior of brittle materials, a new framework of phase-field modeling method is established based on the spectral decomposition. Wang et al. [4, 7] derived the governing equation in detail using the Hamilton’s principle. The differential equation of stress equilibrium and phase-field evolution equation are expressed as follows. ∇ · σ+b = 0

(2.2)

Gc ∂g(φ) H + φ − G cl△φ = 0 ∂φ l

(2.3)

where σ and b are the stress and body force, φ is the phase field, g(φ) represents a degradation function, H is the crack-driving strain energy, Gc and l are the critical energy release rate and length-scale parameter. In the governing equation, Gc and l are the model parameters. The degradation function depends on the material properties, and usually takes some polynomials [8]. In the works of Wang et al. [4], a general form of the degradation function is proposed. ( ) 1 − e−αφ +k g(φ) = (1 − k)(1 − φ) 1 − 1 − e−α

(2.4)

where k is a small-value parameter to ensure numerical convergence, and α is a dimensionless material parameter.

2.2.2 The Consistency of the Governing Equation As for the differential equation of stress equilibrium, the stiffness of cracking material is determined by the energy equivalence principal and the spectral decomposition with a thermodynamically consistent projection operator. The stiffness of cracking material is derived as C=

(√

) ) (√ g(φ)R+ + R− : C0 : g(φ)R+ + R−

(2.5)

where C0 is the stiffness of the undamaged material, and R+ and R− are thermodynamically consistent projection operators. {

R+ = R+ 0 +N − R = I − R+

(2.6)

where tensor, R+ 0 = H (σ n )Mnn ⊗ Mnn , N = ∑ s,t I the fourth-order identity 1 2 a σ n −σ a Mna ⊗ Mna , Mab = 2 (na ⊗ nb + nb ⊗ na ), na and nb (a = n, s, t and b

20

F. Wang et al.

= n, s, t) are the eigenvectors of the stress σ, and n, s and t denote the corresponding axes. To ensure the consistency of the differential equation of stress equilibrium and the phase-field evolution equation, the crack-driving strain energy takes the following form. ⎧ ⎨ ψ (φ, ε) = 1 R+ : σ : R+ : ε ψ (φ, ε) > H + i + 2 H= (2.7) ⎩ Hi ψ+ (φ, ε) ≤ Hi where H i is the maximum crack-driving strain energy in history, and ψ + (φ, ε) represents the current crack-driving strain energy. The phase-field evolution equation in Eq. (2.3) contains the crack criterion through the relationship between the crack-driving strain energy and the critical energy release rate. As recommended by the International Union of Laboratories and Experts in Construction Materials (RILEM), the critical energy release rate is usually determined through various indoor tests, such as single edge-notched plate, and threepoint bending beam. The existing calculation models for critical energy release rate of tensile crack (mode I) and shear crack (mode II or mode III) are directly related to the corresponding component of stress [9]. The existing phase-field modeling methods are established on the volumetric-deviatoric strain [3, 10], which makes the form of crack-driving strain energy different from the critical energy release rate. However, it is easy to find that the crack-driving strain energy in Eq. (2.7) derived from the spectral decomposition is consequentially consistent with the critical energy release rate obtained by the test. Therefore, the proposed phase-field modeling method not only ensures the consistency between the differential equation of stress equilibrium and the phase-field evolution equation, but also unifies the crack-driving strain energy and the critical energy release rate in the phase-field evolution equation.

2.3 Thermodynamic Consistency The single-edge notched specimen and four-point bending notched beam are simulated by the proposed phase-field modeling method. The thermodynamic consistency of the proposed phase-field modeling method is verified by comparing the mechanical behaviors of the loading path and the loading–unloading path.

2.3.1 A Uniaxial Tensile Fracture Test A single-edge notched tensile test [1, 4, 11] is taken as a benchmark to verify the thermodynamic consistency of the proposed phase field model. The specimen is

2 Study on the Consistency of a Phase Field Modeling Method …

21

Fig. 2.2 A uniaxial tensile test, (a) single-edge notched specimen, (b) loading path

fixed on the bottom, and the vertical displacement load is applied on the top. The single-edge notched specimen is shown in Fig. 2.2(a). In the phase-field model, the elastic modulus is set to E = 210 kN/mm2 , Poisson’s ratio to μ = 0.3, the critical energy release rate to Gc = 0.0027 kN/mm, and the length scale parameter to lc = 0.0075 mm. To confirm the thermodynamic consistency of the proposed phase-field model, the displacement–time curves shown in Fig. 2.2(b) are taken as the load of the specimen. The simulation results of the single-edge notched specimen under the specified loading path are shown in Fig. 2.3. The vertical axis refers to the applied load, and the horizontal axis refers to the average phase field along the crack trajectory. The notched specimen undergoes two stages, elastic stage and cracking stage, with loading. In the elastic stage, the unloading has little effect on both the load-average phase field curves and the load-time curves, which indicates the thermodynamic consistency of the proposed phase-field model. However, in the cracking stage, there is a negligible difference between the loading path and the loading–unloading path. The difference is mainly caused by the crack tip propagation during loading and unloading.

2.3.2 A Mixed-Mode Fracture Test A mixed-mode fracture test of four-point bending notched beam conducted by Gálvez et al. [12] is simulated by the proposed phase-field modeling method. The four-point bending notched beam with a cross-section of H × T (75 mm × 50 mm) is shown in Fig. 2.4. The width and depth of the notch are 2 mm and 0.5 H, respectively. The elastic modulus and Possion’s ratio are 23 GPa and 0.25. Comprehensively considering the interaction between the length-scale parameter and the critical energy release rate [4], the phase-field model parameters are l = 2 mm, Gc = 0.05 kN/m and α = 10.

22

F. Wang et al.

Fig. 2.3 Mechanical behavior, (a) load-average phase field curves (The black line in the contour is the crack trajectory, and the horizontal axis refers to the average phase field along the crack trajectory), (b) load-time curves

Fig. 2.4 A mixed-mode fracture test, (a) four-point bending notched beam, (b) loading path

The numerical simulation results and test results of the mixed-mode fracture test under different loading paths are shown in Fig. 2.5. The horizontal axis in Fig. 2.5(a) is the displacement at the loading point. It can be seen that the load–displacement curves and load-time curves of the phase-field model are in good agreement with the test curves. Moreover, in the elastic stage and the cracking stage, the load–displacement curve and load-time curve under loading–unloading path can be restored to the original curves under loading path. Therefore, there is no spurious energy dissipation except the strain energy released by cracking, which implies thermodynamic consistency for the mixed-mode fracture.

2.4 Phase-Field Crack Width 2.4.1 A Level-Set Based Calculation Method Crack width is a very important index of discontinuous deformation. In the phase-field modeling method, the phase-field crack is continuous and diffusive, which makes it

2 Study on the Consistency of a Phase Field Modeling Method …

23

Fig. 2.5 Mechanical behavior, (a) load–displacement curves, (b) load-time curves

difficult to determine the crack width. Therefore, a formula based on the level-set method is presented to calculate the width of the phase-field crack. For the diffusive phase-field crack, the level-set method is used to identify the interface between the crack domain and the uncrack domain [13]. ⎧ ⎪ ⎨ f (φ) < 0, x ∈ ΩU f (φ) = 0, x ∈ ┌ F (2.8) ⎪ ⎩ f (φ) > 0, x ∈ Ω F where ΩU := {x ∈ Ω | φ(x, t) < φ F }, ΩF := {x ∈ Ω | φ(x, t) > φ F }, ┌ F := {x ∈ Ω | φ(x, t) = φ F }. φ(x, t) denotes the phase field at the position x and time t. Herein, φ F ∈ [0, 1] is the specified value of the phase field to define the fracture boundary. To be simplified, the phase-field based level-set function f (φ) is constructed as the following form. f (φ) = φ(x, t) − φ F

(2.9)

The differential of the crack domain in the current configuration is written as follows. nT du ≃ nT (F · n0 )dx

(2.10)

where F is the deformation gradient, and n0 and n are phase-field gradients in the reference configuration and the current configuration, respectively. The crack width is derived from the displacement integral along the characteristic width of the crack domain. To be simplified, the characteristic width of the crack domain takes l f = 2 l in the reference configuration. The characteristic width lf of the crack domain in the current configuration is expressed as

24

F. Wang et al.

{l df =

{l n du ≃ T

−l

( T ) n · F · n0 dx =

−l

{l λ f dx

(2.11)

−l

where l is the length-scale parameter, and λf denotes the projection of the stretch vector λ = F·n0 onto the current crack direction, λf = nT ·F·n0 . ( ) ∇φ T · ∇φ | λ2f = nT · F · n0 · n0T · FT · n = | T |∇φ · C−1 · ∇φ |

(2.12)

where C = FT F denotes the right Cauchy–Green tensor. The crack width wf refers to the difference between the width of the crack domain in the reference configuration and the current configuration. wf =

⎧∑( ) ⎨ λm − 1 h m m

⎩0

f

iff φm >φlim

(2.13)

otherwise

where m is the element number perpendicular to the crack direction in the crack domain, hm is the characteristic size of elements in the reference configuration, λmf is the average value of integral points in the element, and φ lim is the critical phase field.

2.4.2 Case Study The case in Sect. 3.1 is used to illustrate the validity and accuracy of Eq. (2.13) for the crack width. The characteristic size h of elements in the phase-field model is 0.002 mm. The crack width of each element in the phase-field model is shown in Fig. 2.6(a). It can be seen from the nephogram that the crack domain is diffusive, and the crack width varies in the crack domain. The jump displacement d refers to the displacement difference between A and B on both sides of the crack profile. Actually, the jump displacement d is supposed to be the true value, and the crack width w calculated from Eq. (2.13) is the estimated value. The curves of crack width are plotted in Fig. 2.6(b). With the increase of the length-scale parameter, both the jump displacement and the crack width increase nonlinearly. The relative error between the estimated value and the true value increases over time, but it is still accurate and acceptable. As the length-scale parameter increases, the diffusive crack widens [4], which may result in a relatively large calculation error. Therefore, it is recommended to set the lengthscale parameter within the range of 2.0–3.0 h, so as to obtain accurate and reasonable crack width by using Eq. (2.13).

2 Study on the Consistency of a Phase Field Modeling Method …

25

Fig. 2.6 Crack width, (a) nephogram, (b) crack width during evolution under different length-scale parameters

2.5 Conclusion A new framework of phase-field modeling method for mixed-mode fracture of brittle materials is introduced in brief. By spectral decomposition, the stress equilibrium differential equation is consistent with the phase-field evolution equation, and the crack-driving strain energy in the phase-field evolution equation is unified with the critical energy release rate. The new phase-field modeling method is thermodynamic consistency, which has been proven by the uniaxial tensile fracture test and mixedmode fracture test. A level-set based calculation method is presented to estimate the phase-field crack width, providing a basis for the multi-field coupling analysis. Acknowledgements This research is sponsored by Science and Technology Commission of Shanghai Municipality, China (Grant No. 22YF1429900). The authors are grateful for the substantial support of this program.

26

F. Wang et al.

References 1. Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45–48):2765–2778 2. Steinke C, Kaliske M (2019) A phase-field crack model based on directional stress decomposition. Comput Mech 63(5):1019–1046 3. Zhang X, Sloan SW, Vignes C, Sheng D (2017) A modification of the phase-field model for mixed mode crack propagation in rock-like materials. Comput Methods Appl Mech Eng 322:123–136 4. Wang F, Shao J, Huang H (2021) A phase-field modeling method for the mixed-mode fracture of brittle materials based on spectral decomposition. Eng Fract Mech 242:107473 5. Courtney TH (2012) Mechanical behaviour of materials. Springer, Netherlands 6. Zhang Z, Eckert J (2005) Unified tensile fracture criterion. Phys Rev Lett 94(9):094301 7. Wang F, Huang H, Zhang D, Zhou M (2022) Cracking feature and mechanical behavior of shield tunnel lining simulated by a phase-field modeling method based on spectral decomposition. Tunn Undergr Space Tech 119:104246 8. Kuhn C, Schlüter A, Müller R (2015) On degradation functions in phase field fracture models. Comput Mater Sci 108:374–384 9. Li W, Huang P, Chen Z, Zheng X (2021) Testing method of critical energy release rate for interfacial mode II crack. Eng Fract Mech 248(1):107708 10. Ambati M, Gerasimov T, Lorenzis LD (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55:383–405 11. Molnár G, Gravouil A (2017) 2D and 3D Abaqus implementation of a robust staggered phasefield solution for modeling brittle fracture. Finite Elem Anal Des 130:27–38 12. Gálvez J, Elices M, Guinea G, Planas J (1998) Mixed mode fracture of concrete under proportional and nonproportional loading. Int J Fract 94(3):267–284 13. Osher S (2003) Level set methods and dynamic implicit surfaces. Springer, Netherlands

Chapter 3

Numerical Simulation of Shear Wave Propagation Through Jointed Rocks Resmi Sebastian and Kallol Saha

Abstract Shear wave propagation through jointed rock mass is a complex phenomenon as the waves get transmitted and reflected at the joints. The seismic wave propagation results in the development of strains of varying levels, depending on the distance of the source of the vibration from the point of interest and the properties of the material through which it propagates. This paper describes the numerical simulation of a test facility that generates shear waves in rock plates. The numerical simulations have been developed with the help of a distinct element code program, three-dimensional distinct element code (3DEC). The test facility has an incident and transmitted plates along with a friction bar that produces shear waves in the incident plate, which are transmitted across the joint to reach the transmitted plate. The numerical simulation is validated by comparing the maximum particle velocities and maximum particle displacements developed in the incident and transmitted plates due to the propagation of waves, in the laboratory and in the numerical model. The wave energy that is transmitted and reflected at the rock joints has been calculated from the particle velocities obtained at the monitoring points. The ratio of the shear stresses due to the wave propagation at the monitoring points and the wave velocity reduction across the joint also have been calculated. A parametric study on the shear wave propagation has been conducted by varying the normal stress, properties of the joint and the magnitude of the load applied and the results are presented in this paper. Keywords Shear wave · Particle velocity · Jointed rocks · Wave propagation · Wave amplitude

R. Sebastian (B) · K. Saha Department of Civil Engineering, Indian Institute of Technology Ropar, Rupnagar, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_3

27

28

R. Sebastian and K. Saha

3.1 Introduction Analysis of dynamic behavior of jointed rock mass due to seismic wave propagation is very important for determining the seismic response of a structure situated in them. The presence of faults, fissures, and bedding planes makes structures founded in rock mass vulnerable to seismic loading. Keeping this in mind, several researchers have used experimental, analytical and numerical techniques to study shear wave propagation through jointed rock mass. Several experiments have been done on jointed rocks to study wave velocity and wave attenuation due to presence of rock joints. Ultrasonic pulse velocity (UPV) test, bender element (BE) test and Resonant column (RC) test have been used by different researchers [2, 8, 13, 15, 19] to study the effect of frictional joints during wave propagation. Split shear plate (SSP) apparatus was developed by Liu et al. [10] to study the wave propagation across rock joints having filled materials. It is very difficult to simulate the real world wave propagation in laboratory experiments. Therefore, analytical and numerical techniques have usually been adopted by researchers [1, 4, 6, 9, 11, 14, 20, 21] for conducting wave propagation. Tiwari et al. [18] conducted dynamic tests on the weathered rocks of a tunnel subjected to blast loading and validated with finite element analysis. Although popular continuum numerical methods like Finite Element Method (FEM) and Boundary Element Method (BEM) are available for the analysis of jointed rock mass, discontinuum numerical methods are preferred for this purpose due to their capability of representing discontinuous nature of rock mass accurately. Discrete Element Method (DEM) is a numerical technique for solving applied mechanics problems involving discontinuous materials. The presence of interfaces or contacts between the discrete bodies separates a discontinuous medium from a continuous medium. DEM considers the domain as an assemblage of discrete particles. It considers particle to particle interaction through their contact points only. DEM allows finite displacements and rotations of discrete bodies. Newton’s second law is used to perform the calculations for particles and force- displacement law for contacts. DEM can be used for modeling tunnels, caverns, hydroelectric projects, analyzing stability of rock slope and mines, analyzing seismic and blast loading in rocks and many more. 3-Dimensional Distinct Element Code (3DEC) and Universal Distinct Element Code (UDEC) software have an advantage over others as they are specifically designed for rock engineering project simulation. For modeling wave propagation in rocks, Cai and Zhao [3] employed 3DEC and UDEC software. Sebastian and Sitharam [15–17] used laboratory testing and numerical models generated using 3DEC to investigate the wave propagation mechanism in rocks at low strain levels. The present study describes the numerical simulation of the experiments conducted using the test facility of SSP apparatus, validation and the parametric study conducted on the numerical model developed with 3DEC software. The influence of different joint parameters i.e., joint stiffness, joint cohesion, joint friction, normal stress and load magnitude on the transmission of propagating shear waves across rock joints are discussed in this paper.

3 Numerical Simulation of Shear Wave Propagation Through Jointed Rocks

29

3.2 Laboratory Experiments Using Split Shear Plates Experiments were conducted on the split shear plates set up that has incident and transmitted plates separated by a frictional planar joint having a joint roughness coefficient (JRC) of 0–2. A dynamic triggering system was used to apply a compression force of 5 kg/cm2 on the friction bar, which in turn generates shear wave in the incident plate due to sliding movement of the friction bar with respect to the incident plate. Friction bar is a cuboidal bar of dimensions 600 mm * 100 mm * 100 mm (length * width * height). Both the incident and transmitted plates have the width of 300 mm and thickness of 30 mm. Length of the incident and the transmitted plates are 670 mm and 630 mm respectively. The schematic diagram of the whole facility is shown in Fig. 3.1. Both these plates were supported by four cuboid shaped stainless steel (SS) bars beneath them. These SS bars were further supported by cylindrical mild steel (MS) bars, using which total length of rock plates were adjusted. On the inner side of friction bar, grooving was done to generate significant friction at the intersection of incident plate and friction bar. Shear wave in the incident plate caused particles to vibrate in the perpendicular direction of the generated shear wave. Piezoelectric accelerometers were installed at specific locations on the rock plates to monitor propagation of shear waves across the frictional joint. Four accelerometers were installed at locations A–D (shown in Fig. 3.1.) for conducting this study. Location of A was on the friction bar. Locations of B, C and D were located on rock plates at a distance of 30 mm, 640 mm and 670 mm from the start of incident plate. The experimental set up is shown in Fig. 3.2. The acceleration-time (a-t) history was obtained using a data acquisition (DAQ) system having a maximum sampling frequency of 125 kHz. The a-t history obtained for the applied load was integrated to obtain the velocity-time (v-t) history and the displacement-time (d-t) history at the above- mentioned monitoring points. Figure 3.3 shows the a-t history on the friction bar due to generation of compression wave using dynamic triggering system. Hardened gypsum samples were used as the incident and transmitted plates to conduct laboratory experiments, as they are easily available and can be reproduced easily. The engineering properties of the prepared samples are given in Table 3.1. Samples having 90° joint orientation have been made by placing a smooth sided partitioner (for attaining a JRC of 0–2) of 300 mm length within the mould during the preparation of samples. The effect of 90° frictional joint on the peak particle velocity, peak particle displacement and energy of wave propagation at three locations, i.e., B–D of the rock mass have been obtained and are explained in the following sections. For this study, the effect of 90° joint orientation on shear wave propagation across frictional joints was obtained by comparing the transmission amplitude coefficient (with respect to displacement and velocity) of the wave at two locations, i.e., C and D (shown in Fig. 3.1). These monitoring points were located at 640 and 700 mm from the start of incident plate respectively. The a-t history obtained from the accelerometers were integrated to get the particle v-t history in a particular direction and particle d-t history.

30

R. Sebastian and K. Saha

Fig. 3.1 Schematic diagram of the split shear plates [10]

Fig. 3.2 Entire test setup with incident plate, transmitted plates, friction bar and dynamic triggering system

3 Numerical Simulation of Shear Wave Propagation Through Jointed Rocks

31

Fig. 3.3 Acceleration-time history of dynamic load measured in friction bar

Table 3.1 Engineering properties of gypsum samples used for the study

Properties

Values

Density

1150 kg/m3

Uniaxial compressive strength

8 MPa

P wave velocity

2500 m/s

S wave velocity

1200 m/s

Poisson’s ratio

0.35

Shear modulus

1.58 GPa

Young’s modulus

4.26 GPa

From the particle velocities obtained at the rock plates (using both experimental and numerical simulation), energy flux associated with the propagating waves were calculated, using the equation from Miller [12]. T p +t

EI/R/T = ρVShear

∑

2 VParticle

(3.1)

t

where, EI/R/T denotes energy flux per unit area per cycle of oscillation for incident, reflected and transmitted wave respectively. VShear denotes velocity of propagating shear wave. VParticle denotes particle velocity (in perpendicular to the direction of wave propagation). T denotes time period of the incident, reflected and transmitted wave. From the energy fluxes of transmitted and reflected waves, the Transmission energy coefficients can be obtained as Miller [12]. ( T =

ET EI

) 21 (

γ1 γ2

) 21 (3.2)

32

R. Sebastian and K. Saha

where, γ1 and γ2 indicate seismic impedance of medium 1 (incident plate) and medium 2 (transmitted plate) respectively as shown in Eqs. (3.3) and (3.4). 1

γ1 = (G 1 ρ1 ) 2

1

γ2 = (G 2 ρ2 ) 2

(3.3) (3.4)

G and ρ denote shear modulus and density of the rock plates respectively. Suffix 1 and 2 denote the incident plate and transmitted plate respectively. As same material is being used in both the rock plates, γ1 = γ2 for this study. In this paper, energy coefficient for reflection (R) has been defined by ( R=

Energy remaining at location C for 90◦ jointed sample Incident Energy at location B for 90◦ jointed sample )1 ( )1 Energy remaining at location C for intact sample 2 γ1 2 − Incident Energy at location B for 90◦ intact sample γ2

(3.5)

Shear modulus (G) in both plates has been obtained using the shear wave velocities. (VShear ) in the respective plates and this velocity has been determined using first arrival of waves in a-t history obtained from the accelerometers located on both ends of each plate, as given in Eqs. (3.6) and (3.7). VShear =

Distance between two accelerometers located on incident or transmitted plate Time required for the shear wave to travwel between two accelerators

(3.6) 2 G = ρVShear

(3.7)

The transmission amplitude coefficients were determined from the displacements and velocities obtained in the incident and transmitted plates across the joint, as given in Eqs. (3.8) and (3.9). Transmission amplitude coefficient with respect to displacement Particle displacement amplitude after the joint (in transmitted plate) (3.8) = Particle displacement amplitude before the joint (in incident plate) Similarly,

3 Numerical Simulation of Shear Wave Propagation Through Jointed Rocks

Transmission amplitude coefficient with respect to velocity Particle velocity amplitude after the joint (in transmitted plate) = Particle velocity amplitude before the joint (in incident plate)

33

(3.9)

3.3 Numerical Simulation Using 3-Dimensional Distinct Element Code 3DEC (3-Dimensional Distinct Element Code) made by Itasca Consulting Group Inc. has been used for performing numerical simulation of laboratory tests. The numerical model of SSP apparatus used in the laboratory is shown in Fig. 3.4. The testing sample (gypsum plaster samples), friction bar and supporting block were made deformable by finite difference meshing. As per suggestion of Deng et al. [5], Kuhlemeyer and Lysmer [7], for the proper simulation of wave transmission through the model, the mesh size or average edge length of tetrahedral element was selected to be 20 mm. Elastic isotropic model has been chosen for all the materials used in the numerical simulation. Particle velocities in the direction perpendicular to the direction of shear wave propagation were allowed. An in-situ stress of 1 MPa was applied to the numerical model for simulating the stress applied by the supporting block to the rock plates in the laboratory testing. The velocity–time history obtained by integrating the acceleration- time history at the friction bar was provided on the face of the friction bar, such that a shear wave gets generated at the start of the simulated incident plate, due to the movement of the friction bar. The particle displacements and velocities at the top surface of the samples were monitored at three different locations of B–D. The monitoring points of the sample for evaluating the velocity and displacements are shown in Fig. 3.4. Properties of the blocks and the joints used to describe the SSP setup in the numerical simulation are shown in Tables 3.2 and 3.3. The validation of the numerical simulation, analysis of SSP test on intact sample and sample with joint orientation of 90° are described in the following sections.

Fig. 3.4 Numerical simulation of SSP set up

34

R. Sebastian and K. Saha

Table 3.2 Properties of blocks used to describe the SSP setup in numerical model Sl no.

SSP apparatus components

Material

Density (kg/m3 )

Shear modulus (GPa)

Bulk modulus (GPa)

1

Friction bar

Mild steel

7850

80

140

2

Incident and transmitted plates

Mixture of gypsum powder and water

1150

1.58

4.73

3

Supporting block

Mild steel

7850

80

140

Table 3.3 Properties of joints used to describe the SSP setup in numerical model Joint

Location

Joint normal stiffness (GPa/m)

Joint shear stiffness (GPa/m)

Joint friction angle (°)

Joint 1

Between friction bar and incident plate

200

100

45

Joint 2

Between transmitted plate and supporting block

200

100

45

Joint 3

Between incident plate and transmitted plate

2

0.36

25

Joint cohesion (MPa/m)

2.50

3.4 Results and Discussions 3.4.1 Validation of the Numerical Model The validation was done by comparing the maximum particle velocities and maximum particle displacements obtained at the three monitoring locations (B–D) in the numerical model and in the plates in the laboratory testing during the propagation of shear wave. The comparison of the maximum particle velocities and the maximum particle displacements are shown in Table 3.4. The properties of the joint (joint number 3 given in Table 3.3) used for the validation were obtained by trial and error, performed by comparing the results obtained from numerical simulation and the laboratory experiment. The comparison of results obtained from laboratory experiment and numerical simulation is shown in Table 3.4.

3 Numerical Simulation of Shear Wave Propagation Through Jointed Rocks

35

Table 3.4 Comparison of results obtained from numerical simulation and laboratory experiment Parameters

Location

Results Laboratory experiment

Peak particle velocity Incident plate (mm/s) Transmitted plate Peak particle displacement (mm)

Numerical simulation

178

165

61

150

Incident plate

5

14.4

Transmitted plate

4.5

9.6

Table 3.5 Results of energy coefficients obtained from numerical simulation

Energy coefficients for

Results obtained from numerical simulation

Transmission

0.70

Reflection

0.27

3.4.2 Calculation of Energy Coefficients Using the Numerical Simulation Equation (3.1) has been used to determine the energy flux at all specific locations. Energy flux of the incident and transmitted waves were obtained by using velocity– time data at B and D locations respectively. Separating the reflected wave is very difficult in the incident plate at the location before the joint. Therefore, the reflected energy flux was calculated by comparing the energy flux calculated at the same location of an intact plate and jointed plate. Energy flux associated with reflected wave, i.e., ER was determined by subtracting the percentage of energy flux at location C of intact samples from percentage of energy flux at location C of the jointed samples (the percentage of energy flux remaining at the point C is higher for 90° jointed plates than that for intact plates, due to the reflection at the joint interface). Table 3.5 shows results of energy coefficients obtained from the numerical simulation.

3.4.3 Parametric Study Using Numerical Simulations Parametric studies were conducted to assess the significance of joint stiffness, joint friction, joint cohesion, normal stress and velocity coefficient in obtaining the particle velocities and particle displacements from numerical simulations. Particle velocities and particle displacements were used for determining the transmission amplitude coefficient.

36

3.4.3.1

R. Sebastian and K. Saha

Joint Shear Stiffness

To validate the results of laboratory experiments, the shear stiffness of joint in the numerical model was considered as 360 MPa/m. The parametric study for shear stiffness of joint was conducted by varying the joint shear stiffness, ±10% of the joint shear stiffness adopted for the validation. Therefore, the joint shear stiffness was varied from 240 to 480 MPa/m and transmission amplitude coefficients were determined (in Fig. 3.5a) using Eqs. (3.8) and (3.9). Transmission amplitude coefficients were observed to be increasing with increase in joint shear stiffness. The particle velocities obtained in the incident and transmitted plates for the joint shear stiffness of 240 and 480 MPa/m are shown in Fig. 3.5b. The figure shows that multiple reflections of the wave are present for the joint shear stiffness of 480 MPa/m, due to which high transmission amplitude coefficients have been obtained.

3.4.3.2

Joint Normal Stiffness

To validate the results of laboratory experiments, the normal stiffness of joint in the numerical model was considered as 2 GPa/m. The parametric study on joint normal stiffness was conducted by varying the joint normal stiffness, ±10% of the joint normal stiffness used for validation. Joint normal stiffness was varied from 1.40 to 2.60 GPa/m and transmission amplitude coefficients were determined (in Fig. 3.5c) using Eqs. (3.8) and (3.9). No significant changes in transmission amplitude coefficients were observed with change in joint normal stiffness.

3.4.3.3

Joint Cohesion

For validating the results of laboratory experiments, the joint cohesion in the numerical model was considered as 2.50 MPa/m. The parametric study on joint cohesion was conducted by varying the joint cohesion ±10% of the joint cohesion adopted for the validation. Joint cohesion was varied from 1.75 to 3.25 MPa/m and transmission amplitude coefficients were determined (in Fig. 3.5d). Transmission amplitude coefficients were observed to be unchanged with change in joint cohesion.

3.4.3.4

Joint Friction

The joint friction of 25° was used in the numerical model for validating the results of laboratory experiments. The parametric study on joint friction was conducted by varying the joint friction ±10% of the joint friction used for validation. Joint friction was varied from 17.5° to 32.5° and transmission amplitude coefficient were determined (in Fig. 3.5e). Transmission amplitude coefficients were observed to be unchanged with change in joint friction.

3 Numerical Simulation of Shear Wave Propagation Through Jointed Rocks

37

Transmission amplitude coefficient

1.0 0.8 0.6 0.4

Particle displacement Particle velocity

0.2 0.0 210

260

310 360 410 460 Joint shear stiffness (MPa/m)

510

(a)

Transmission amplitude coefficient

(b)

1.0 0.8 0.6 0.4

Particle displacement Particle velocity

0.2 0.0 1250

1500

1750 2000 2250 Joint normal stiffness (MPa/m) (c)

2500

2750

38

R. Sebastian and K. Saha

Transmission amplitude coefficient

Transmission amplitude coefficient

◀Fig. 3.5 a Transmission amplitude coefficients for different joint shear stiffness. b Particle velocities monitored in the incident and transmitted plates (i) Joint shear stiffness of 240 MPa/m (ii) joint shear stiffness of 480 MPa/m. c Transmission amplitude coefficients for different joint normal stiffness. d Transmission amplitude coefficients for different joint cohesion. e Transmission amplitude coefficients for different joint friction angles. f Transmission amplitude coefficients for different normal stress. g Transmission amplitude coefficients for different velocity coefficient

1.0 0.8 0.6 0.4

Particle displacement Particle velocity

0.2 0.0 1.5

2

2.5 Joint cohesion (MPa)

3

3.5

1.0 0.8 0.6 0.4

Particle displacement Particle velocity

0.2 0.0 15

20

(d)

(e) Transmission amplitude coefficient

Transmission amplitude coefficient

35

1.0

1.0 0.8 0.6 0.4

Particle displacement Particle velocity

0.2 0.0 0.60

25 30 Joint friction angle (degrees)

0.70

0.80

0.90 1.00 1.10 Normal stress (MPa)

1.20

(f)

1.30

0.8 0.6 0.4

Particle displacement Particle velocity

0.2 1.40

0.0 0.0030

0.0040

0.0050

0.0060

0.0070

Velocity coefficient

(g)

Fig. 3.5 (continued)

3.4.3.5

Normal Stress

The normal stress of 1 MPa was used in the numerical model for validating the results of laboratory experiments. The parametric study on normal stress was conducted by varying the normal stress ±10%of the normal stress used for validation. Normal stress was varied from 0.70 to 1.35 MPa and transmission amplitude coefficient were determined (in Fig. 3.5f). Transmission amplitude coefficients were observed to be unchanged with change in normal stress.

3.4.3.6

Velocity Coefficient

To validate the results of laboratory experiments, the velocity coefficient, which was multiplied with the applied velocity data at the face of friction bar, in the numerical model was considered as 0.005. The parametric study on velocity coefficient was conducted by varying the velocity coefficient, ±10% of the velocity coefficient, used

3 Numerical Simulation of Shear Wave Propagation Through Jointed Rocks

39

for validation. Velocity coefficient was varied from 0.0035 to 0.0065 and transmission amplitude coefficients were determined (in Fig. 3.5g). Transmission amplitude coefficients with respect to particle displacements were observed to be in no certain pattern with change in velocity coefficient. Transmission amplitude coefficients with respect to particle velocities were observed to be almost constant with a change in velocity coefficient.

3.5 Conclusions The study of shear wave propagation in jointed rock mass is an important topic in rock mechanics as the failure of the structures constructed in rock during earthquakes is a common phenomenon. The laboratory experiments on propagation of shear waves across jointed rock plates using SSP facility have been numerically simulated using 3DEC and the results have been obtained. The major conclusions that are drawn from this study are: • DEM can be successfully used for simulating the wave propagation in jointed rock mass. • Joint shear stiffness is an important parameter for determining the transmission of shear waves across rock joints. • As the joint shear stiffness increases, waves are reflected multiple times, leading to more transmission amplitude coefficients. • Joint normal stiffness, joint friction, joint cohesion, velocity coefficient and normal stress do not have significant influence on the shear wave transmission, for the values used for the simulation. Acknowledgements This research project is funded by Science and Engineering Research Board (SERB), Department of Science and Technology (DST), India, through the research project, No: ECR/2018/001966.

References 1. Aki K, Richards PG (1980) Quantitative seismology: theory and methods, vol 1. W.H. Freeman, San Francisco. https://doi.org/10.1002/gj.3350160110 2. Arroyo M, Pineda JA, Romero E (2010) Shear wave measurements using bender elements in argillaceous rocks. Geotech Test J, ASTM 33(6):1–11 3. Cai JG, Zhao J (2000) Effects of multiple parallel fractures on apparent wave attenuation in rock masses. Int J Rock Mech Min Sci 37(4):661–682. https://doi.org/10.1016/S1365-1609(00)000 13-7 4. Chen SG, Cai JG, Zhao J, Zhou YX (2000) Discrete element modelling of an underground explosion in a jointed rock mass explosion in a jointed rock mass. Geotech Geol Eng 18:59–78. https://doi.org/10.1023/A:1008953221657

40

R. Sebastian and K. Saha

5. Deng XF, Zhu JB, Chen SG, Zhao J (2012) Some fundamental issues and verification of 3DEC in modelling wave propagation in jointed rock masses. Rock Mech Rock Eng 45(5):943–951 6. Frazer LN (1990) Dynamic elasticity of microbedded and fractured rocks. Journal of GeophysicalResearch 95(B4):4821–4831 7. Kuhlemeyer RL, Lysmer J (1973) Finite element method accuracy for wave propagation problems. J Soil Mech Found Div, ASCE 99(SM5):421–427 8. Kurtulu¸s C, Üςkarde¸s M, Sari U, Güner SO ¸ (2012) Experimental studies in wave propagation across a jointed rock mass. Bull Eng Geol Environ 71(2):231–234 9. Li JC, Ma GW (2009) Experimental study of stress wave propagation across a filled rock joint. Int J Rock Mech Min Sci 46(3):471–478. https://doi.org/10.1016/j.ijrmms.2008.11.006 10. Liu T, Li J, Li H, Li X, Zheng Y, Liu H (2017) Experimental study of S-wave propagation through a filled rock joint. Rock Mech Rock Eng 50(10):2645–2657. https://doi.org/10.1007/ s00603-017-1250-y 11. Miller RK (1977) An approximate method of analysis of the transmission of elastic waves through a frictional boundary. J Appl Mech 44:652–656. https://doi.org/10.1016/0148-906 2(79)90465-0 12. Miller RK (1978) The effects of boundary friction on the propagation of elastic waves. Bull Seismol Soc Am 68(4):987–998. https://doi.org/10.1016/0148-9062(79)90465-0 13. Pineda JA, Romero EE, Alonso EE (2011) Tracking degradation of argillaceous rocks using bender elements. In: Proceedings of international symposium on deformation characteristics of geomaterials, Seoul, 1–3 September, pp 240–245 14. Pyrak Nolte LJ, Cook NGW (1987) Elastic interface waves along a fracture. Geophys Res Lett 11(14):1107–1110. https://doi.org/10.1029/GL014i011p01107 15. Sebastian R, Sitharam TG (2014) Transmission of elastic waves through a frictional boundary. Int J Rock Mech Min Sci, Elsevier 66:84–90 16. Sebastian R, Sitharam TG (2016a) Transformations of obliquely striking waves at rock joint: numerical simulations. Int J Geomech, ASCE 16(3):04015079. https://doi.org/10.1061/(ASC E)GM.1943-5622.0000575 17. Sebastian, Sitharam (2016b) Long wavelength propagation of waves in jointed rocks—study using resonant column experiments and model material. Geomech Geoeng 11(4):281–296. https://doi.org/10.1080/17486025.2016.1139753 18. Tiwari R, Chakraborty T, Matsagar V (2016) Dynamic analysis of tunnel in weathered rock subjected to internal blast loading. Rock Mech Rock Eng 49:4441–4458 19. Zhao XB, Zhao J, Hefny AM, Cai JG (2006) Normal transmission of S-wave across parallel fractures with coulomb slip behaviour. J Eng Mech 132(6):641–650 20. Zhao XB, Zhao J, Cai JG, Hefny AM (2008) UDEC modeling on wave propagation across fractured rock masses. Comput Geotech 35(1):97–104. https://doi.org/10.1016/j.compgeo.2007. 01.001 21. Zhu JB, Zhao J (2013) Obliquely incident wave propagation across rock joints with virtual wave source method. J Appl Geophys 88:23–30. https://doi.org/10.1016/j.jappgeo.2012.10.002

Chapter 4

A Quasi-static Computational Model for Fracture in Multidomain Structures with Inclusions Roman Vodiˇcka

Abstract A computational model for dealing with fracture in materials with inclusions is considered. The proposed model allows to predict crack initiation and growth in quasi-brittle materials. The inclusions cause that cracks may appear inside the matrix materials or along matrix-inclusion interfaces. The present model can treat them both using two internal variables in a form considered in damage mechanics so that crack formation process is a consequence of a material degradation. The first of the damage variables is defined at matrix-inclusion interfaces which are represented by a thin degradable adhesive layer so that an adequate stress-strain relation is rendered as in common cohesive zone models. The second variable is defined in the structural domains, matrix plus inclusions, as a phase-field fracture variable which causes domain elastic properties degradation in a narrow material strip that results in a diffuse form of a crack. Both these damaging schemes are expressed by a quasistatic energy evolution process. The numerical solution approach is thus rendered from a variational form obtained by a staggered time-stepping procedure related to a separation of deformation variables from the damage ones and using sequential quadratic programming algorithms implemented together within a MATLAB finite element code. The numerical simulations with the model include simplified structural and material elements containing one or more inclusions. Keywords Phase-field fracture · Interface damage · Inclusions · Quadratic programming · Quasi-static evolution · Staggered approach

R. Vodiˇcka (B) Faculty of Civil Engineering, Technical University of Košice, Vysokoškolská 4, 042 00 Košice, Slovakia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_4

41

42

R. Vodiˇcka

4.1 Introduction Modern materials in structural engineering are composite. They contain fibres or other inclusions randomly distributed in a matrix material. The high loads on structures may lead to substantial changes in material structure and cause loss of structural element functionality. In such material or between such material components, macroscopically evident cracks appear. The mentioned issues may be described in mechanical models by damage of materials leading subsequently to fracture, see [3, 13]. Successful computational approaches analysing the element material or material interface degradation are of high importance. The present one naturally and in an analogous way captures both in-material and inter-material cracks. Beginning with Griffith [11], the modern analysis of brittle or quasi-brittle material fracture was developed. There was postulated that sufficient release of energy per crack area increment is needed for an existing crack to grow, this critical value is called fracture energy. Nevertheless, to treat the issues of crack nucleation, a fracture formulation was introduced by [9] which describes the brittle fracture by an energy minimisation algorithm. Therefore, the focus on energy formulations is natural and will be followed below. In a variation formulation, the determination of crack path was resolved by incorporating some internal parameters into the model in fashion of damage mechanics, as documented by [4], which provoked development of various phase-field fracture models (PFM), see [15, 16, 22, 32, 33]. The computational model of material damage based on cumulation of microscopic faults is described by internal variables as introduced in [10] and also described and used elsewhere, e.g. [12, 14]. It can also be applied to interfaces considered as zero thickness layers of an adhesive. These internal variables express the current state of material or of an adhesive along materials interfaces. A limit value of such variables reflects total degradation of the material or of the joint, which is explained as fracture. The case of interface can be understood within the theory of interface damage or adhesive contact as developed in [6, 18] and followed by [20] to account for the cracks in a manner of cohesive zone models (CZM) as subsequently analysed in [25, 29]. Anyhow, an internal parameter of damage along the interface is introduced in such models: its varying from an intact state to full separation of surfaces is related to required energy release. Using the same philosophy, an in-material crack can be considered through a change of another internal variable and a regularisation process of [9] variational formulation trying to extend the classical energy-based Griffith criterion. The new damage parameter have to be of a non-local character, these introduces PFM models as mentioned above. To this end, the model requires a predefined crack width parameter, which may be used to identify crack formation process as presented in [22, 24] and related also to problems in an interface as in [17]. Those cracks are also called smeared as the theory relates continuous distribution of the internal variable to the presence of a crack in the material. Advantageously, they are able to identify direction in which the crack propagates. In computations, there are several possibilities to follow in order to obtain a physically reasonable data. Here, the solution is approximated by a semi-implicit time

4 A Quasi-static Computational Model for Fracture in Multidomain . . .

43

stepping algorithm which guarantees a variational structure to the solved problem by two recursively solved minimisations with respect to separated deformation and damage variables which was devised in [19, 21] and also described and used in the previous author’s works [25, 28, 29, 31]. Another numerical tools implemented in the model include discretisation by the Finite Element Method (FEM) used in an author own code, which exploits simple implementation in Matlab environment based on [1], it was also used in authors previous papers [26, 27]. Naturally, it is used to determine actual state of the solved structure at every selected instant. The evolution of those states then utilises Quadratic Programming (QP) algorithms (possibly in a sequential modification) to calculate the solutions for energy minimisations in a variationally based approach [7, 8]. The paper is structured as follows. Section 4.2 introduces relations for energies which are used to describe the model and relations which govern evolution of the model. Section 4.3 explains some aspects of the numerical solution and its particular implementation. Finally, Sect. 4.4 presents numerical results which demonstrate the properties of the model and reveal its characteristic response to selected material elements containing some inclusions.

4.2 Description of the Model The present modelling of fracture is based on the variational formulation introduced in [9]. It relies on energetic assumptions and conditions for crack propagation which were stated already by Griffith. Therefore, the focus on such an energy formulation is natural. Let the problem be solved in a bounded domain Ω which contains at least one interface dividing the domain into several subdomains as it is shown in Fig. 4.1 for a domain with two subdomains Ω A and Ω B . The respective boundaries of the subdomains are denoted Γ A and Γ B . The interface, the common part of the subdomain boundaries, or a contact zone, is denoted Γ i . Degradation of interface or of the domain material leading to a crack, appears under an increasing hard-device loading. Therefore, it considers quasi-static conditions under which the state of the structure evolves in time t. The bulk crack is denoted Γc , while a crack developed along Γi is marked as Γic . The boundary conditions can be introduced for displacement field u by a time dependent function u(t)|ΓD = g(t)

Fig. 4.1 Description of the domain, cracks, boundary conditions and constraints

g

B

nA sA

Γ DB ΩB

Γi

Γ ic ΩA B n s B 2ε

Γ NB Γc

x2 B 0 x1 s Γ DB nB

gB = 0

44

R. Vodiˇcka

characterising the hard-device on a part of the domain boundary Γ DA and Γ DB . The rest of the outer boundary is supposed to be traction free as only displacement loading is considered. These boundaries are denoted Γ NA and Γ NB . The state of the structure at particular time instant t now needs a description in terms of energies. The variables of state include first of all deformation quantities: the displacement field u in interior of the domains which takes also the gap of displacements [[u]] along interfaces Γi into account. The formulation of stored energy in a manner of [9] includes elastic stored energy and energy accumulated in formed cracks   1 e(uη ) : Cη : e(uη ) dΩ E(t; u, Γc , Γic ) = η 2 Ω \Γ c η=A,B   G c dΓ + G ic dΓ (4.1) + Γc

Γic

for an admissible state u η |Γ η = g η , D

      u = u A −u B , u n ≥ 0 on Γi , u = 0 on Γi \ Γic ,

(4.2)

where the subscript n refers to the outward normal vector n of pertinent domain (distinguished by a superscript, if necessary) and [[u]]n is meant in the sense: [[u]]n = [[u]] · n B . The energy accumulated in cracks has two resources, therefore we also distinguish between fracture energy for bulk cracks G c , and fracture energy for the interface cracks G ci . The problem in finding actual position of the crack at current step of load has two principal aspects: an unknown position and an unknown length of the crack. For interface cracks, there  remains only the second aspect. Thus for determining the interface crack integral Γic G ic dΓ , there is a need to determine extension of the crack along the interface Γi . It can be seen within the theory of adhesive contact shown in [18], developed also in [29, 31] where it was used to treat the interface stress-strain relation like for cohesive zone models. In such models, an interface  damage parameter ζ is defined so that the aforementioned integral is converted to Γ i G ic (1 − ζ) dΓ , where ζ ∈ [0; 1] is defined so that ζ = 1 pertains to the intact interface and ζ = 0 reflects the actual crack. Additionally, the interface is considered as an infinitesimally thin adhesive layer with its own stiffness κ, which is degraded introducing a degradation function φ having the properties φ(1) = 1, φ(0) = 0, φ (x) > 0 for all x ∈ [0; 1]. This requires toadd a term corresponding to the elastic energy in such a definition  of the adhesive Γi 21 φ(ζ)κ[[u]] ·[[u]] dΓ .  In a similar way, the term Γc G c dΓ can be replaced using an internal parameter α defined in the bulk domain, which however should be available at each point of the domain as the crack position is not known. Such a replacement can be obtained by a functional of Ambrosio and Tortorelli [2], which allows continuously connect displacement across   a crack, with a length  parameter ε controlling amount of regularisation: Ω 38 G c 1ε (1 − α) + ε (∇α)2 dΩ. This introduces so called smeared crack

4 A Quasi-static Computational Model for Fracture in Multidomain . . .

45

as due to the regularisation the crack is diffused to exhibit a finite width determined by ε. The reasoning behind the functional may naively be explained so that the term 1 (1 − αη ) guarantees in energy minimisation that for really small ε the variable α ε tends to be mostly equal to 1 for the integral not to be so large. It then provides the domains where α = 1 in a narrow strips of small width controlled by ε. The term ε (∇αη )2 guarantees continuity of α though possibly with high gradients due to small ε. Simultaneously, as presented by [4, 22], the parameter ε can be used to control a stress criterion in damage and crack propagation. Though with finite ε, it is a bit far from classical Griffith crack interpretation, it was shown that for ε approaching zero the functional reverts to the formulation in Eq. (4.1) in the sense of Γ -convergence, see [5]. That is also the reason for including factor 83 as explained e.g. in [24]. The energy functional (4.1) with described replacements sounds as follows: E (t; u, α, ζ) =

  η=A,B

Ωη





2 Φ(αη ) K pη sph+ e(uη ) + μη |dev e(uη )|2





2 3 1 + K pη sph− e(uη ) + G cη (1 − αη ) + ε (∇αη )2 dΩ 8 ε      1  − 2 1 κφ(ζ) u · u + κG u n + 2 Γi 2   2 i (4.3) + G c (1 − ζ) + εi (∇s ζ) dΓ , for an admissible state u η |Γ η = g η , D

  u = u A −u B , 0 ≤ αη ≤ 1, 0 ≤ ζ ≤ 1.

(4.4)

It can be considered that the non-admissible states have infinite energy E. The term containing new parameter κG was added here to replace contact conditions (including [[u]]n ≥ 0 in Eq. (4.2)) by a penalisation term enabling small interpenetration (say, explained by asperities on the surface), κG introduces normal compression stiffness as a large number. Further, εi is a small number accounting for a non-local character of ζ and allowing for high (surface) gradients ∇s in ζ distribution (as in gradient damage theory, see [31]), and v ± = max(0, ±v). The interface gradient term also leads to the same structure in the interface and bulk damage terms. The bulk elastic energy term was written here in a more specific form then in Eq. (4.1). Namely, the isotropic material was used with the parameters: the (plain strain) bulk modulus K p and the shear modulus μ. Additionally, the parameter α is a damage-like parameter, thus it causes degradation of the material via the function Φ (analogous to the interface function φ) which has the properties Φ(1) = 1, Φ(0) = δ, Φ  (0) = 0 (for computational purposes it is also assumed Φ  (x) > 0 for all x ∈ [0; 1]), δ being a small positive number to guarantee positiveness of bulk energy also in the case α → 0. The elastic energy formulation includes the additive orthogonal split of the (small) strain tensor e into spherical sph e and deviatoric dev e parts to have a possibility

46

R. Vodiˇcka

of different material degradation related to volumetric or shear strain (though not considered in the formula), and a split of the spherical part into tensile (+) and compressive (–) parts due to a common assumption that there is no degradation in pure compression. It is also important not to forget that damaging and crack propagation is a unidirectional process. It means that both introduced internal parameters may only decrease by satisfying the conditions ζ˙ ≤ 0 on Γi , and α˙ ≤ 0 in Ω, where ’dot’ means time derivative. This unidirectionality can be introduced into present energy formulation ˙ =0 ˙ ζ) in a form of dissipation. This can be expressed as a (pseudo)potential R(α, provided that ζ˙ ≤ 0 on Γc , and α˙ ≤ 0 in Ω. As above, other states are not admissible, which can be enforced by infinite value of R. Finally, if there were external forces, energy pertinent to them should be added to the system, too. Here, we do not consider such a case. The relations which govern the quasi-static evolution in a deformable structure with cracks can be generally written in a form of nonlinear variational inclusions

∂ u E(t; u, α, ζ) ˙ ˙ ζ)+ ∂ α E(t; u, α, ζ) ∂ α˙ R(α, ˙ ∂ ζ E(t; u, α, ζ) ∂ ζ˙ R(α, ˙ ζ)+

0, 0,

(4.5)

0,

where ∂ denotes a partial subdifferential as the functionals does not have to be smooth, e. g. R jumps from zero to infinity at zero arguments. For smooth functionals, the subdifferentials can be replaced by Gateaux differentials and the inclusions by equations. The first relation in fact determines the deformation state, the other two are flow rules for propagation of internal parameters of phase-field damage and interface damage. Along with the described relations, initial conditions for the state variables have to be taken into account: η

η

uη (0, ·) = u0 , αη (0, ·) = α0 = 1

in Ω η ,

ζ(0, ·) = ζ0 = 1

on Γc .

(4.6)

The initial values for damage parameters correspond to an intact state.

4.3 Numerical Solution and Computer Implementation The described evolution problem requires numerical procedures for time discretisation and spatial discretisation. Both they are applied in appropriate algorithms. Some aspects of the used algorithms are described in what follows.

4 A Quasi-static Computational Model for Fracture in Multidomain . . .

47

4.3.1 Time Discretisation The computational scheme for the time discretisation uses a staggered algorithm. Though in actual quasi-static solution the time step can be adapted, here, we describe the procedure for a fixed time step size τ . Then the solution is obtained at the instants t k = kτ for k = 1, . . . , Tτ . Therefore, the relations from Eq. (4.5) have to be written for separate time instants, using the approximation of the rates of the state variables k k−1 , where αk denotes the solution at the instant by the finite difference, e.g. α˙ ≈ α −α τ k ˙ is substituted by differentiation t . The differentiation with respect to the rates, say α, with respect to αk at the instant t k . From Eq. (4.5) we obtain

∂ uk E(t k ; uk , αk−1 , ζ k−1 ) 0,



αk −αk−1 ζ k −ζ k−1 , + ∂ αk E(t k ; uk , αk , ζ k ) τ τ k

α −αk−1 ζ k −ζ k−1 , + ∂ ζ k E(t k ; uk , αk , ζ k ) τ ∂ζk R τ τ

τ ∂ αk R

0,

(4.7)

0,

so that the initial conditions (4.6) read u0 = u0 , α0 = α0

in Ω η ,

ζ 0 = ζ0

on Γc .

(4.8)

In the proposed numerical scheme, the inclusions are solved separately in u and α, ζ. The separation of variable provides a variational structure to the solved problem. In each time step, two minimisations have to be resolved: first, the minimisation of the functional Hku (u) = E(t k ; u, αk−1 , ζ k−1 )

(4.9)

provides uk , and then the minimisation of the functional

Hkd (α, ζ)

α−αk−1 ζ−ζ k−1 , = E(t ; u , α, ζ) + τ R τ τ k

k

(4.10)

provides αk , ζ k . It should be noted that due to a simplified form of the functional R, which is zero provided that the state is admissible, satisfying unidirectional character of the damage process (α˙ ≤ 0, ζ˙ ≤ 0), the functional in Eq. (4.10) is reduced to the first right-hand side term only, E, equipped with constraints 0 ≤ α ≤ αk−1 , 0 ≤ ζ ≤ ζ k−1 induced by Eq. (4.4) and definition of the functional R.

48

R. Vodiˇcka

4.3.2 Remarks on FEM and the Minimisation Algorithms The problem is numerically solved by recursive minimisation of the functionals (4.9) and (4.10). In the former functional, we have quadratic functionals for whose minimisation quadratic programming algorithms are applied. The latter functional is generally a convex functional to which the QP algorithm is applied sequentially. However, it remains quadratic if the functions Φ(α) and φ(ζ) are quadratic. After the discretisation, all variables are approximated by the pertinent finite element mesh, with typical mesh size h, introduced by adequate implementation of FEM, see [23, 30]. The approximation formula at the time instant t k can be written in the schematic form  whk (x) = Nn (x)wnk , (4.11) n

where whk can be uk , αk , ζ k , wnk are the nodal unknowns associated to the node xn . In what follows, the nodal values wnk are grouped into a column vector wk . The functional (4.9) can be expressed after discretisation in a general matrix form as 1 k  k−1 k Hku,h (uh ) = u C h (αk−1 (4.12) h , ζh )uh − (bh ) uh + ch . 2 h The stiffness matrix C h depends on the actual state of damage as can be guessed from Eq. (4.3). The minimum obtained by a QP algorithm (implemented by a conjugate gradient based scheme with bound constraints [8]) is denoted by ukh . Similarly, the discrete form of the functional (4.10) can be written as Hkd,h (αh , ζh ) = Ah (ukh ; αh ) + K p h (ukh ; ζh ) − (Gkh ) αh − (Gihk ) ζh + ckh , 0 ≤ αh ≤ αk−1 h ,

0 ≤ ζh ≤ ζhk−1 ,

(4.13)

where the constraints guarantee unidirectionality of the damage process and also the bounds for the interface-damage and phase-field-damage parameters. The functions Ah , K p h reflect the dependence on the actual deformation state as can be read from Eq. (4.3) so that Ah (ukh ; ·) and K p h (ukh ; ·) can be quadratic as the first term in Eq. (4.12), if Φ and φ in Eq. (4.3) depend quadratically on α and ζ, respectively. Otherwise, they are considered convex. Anyhow, the constrained minimum provides nodal values αkh , ζhk . In the general case, the QP algorithm is implemented sequentially, see [25], by an approximation using the quadratic Taylor polynomial. Say, for Ah (ukh ; αh ) abbreviated to Akh (αh ) it sounds Akh (αh ) ≈ Akh (α0 h ) + (Ak )h (α0 h ) (αh − α0 h ) 1 + (αh − α0 h ) (Ak )h (α0 h ) (αh − α0 h ) , 2

(4.14)

4 A Quasi-static Computational Model for Fracture in Multidomain . . .

49

where α0 h is known, in the algorithm it is initiated by the values rendered from the previous iteration. It should be noted that the second derivative is positive due to required convexity of the Ah and K p h functions restricted to damage variables. Therefore, in each time step k we first find ukh by QP . Then we apply SQP k−1 and iteratively find the by starting with m = 1 and putting α0 h = αk−1 h , ζ0 h = ζh k,m minimum of Hd,h (αh , ζh ), the minimiser being denoted αm h , ζm h until a convergence criterion is met for αm¯ h , ζm¯ h . Then, we put αkh = αm¯ h , ζhk = ζm¯ h . This process is applied recursively up to given time T .

4.4 Examples The described fracture model is tested for materials with inclusions. As rather academic examples two structural elements have been considered. One of them contains one inclusion, the other takes three inclusion into account, unevenly placed within the matrix domain. The intention for the one inclusion case is to compare crack initiation process with changing relative values of fracture energies along interface and in the domain. The other example is focused on presenting kinking of interface cracks into matrix material and their subsequent propagation till total rupture of the element occurs.

4.4.1 One Inclusion The scheme for the problem with one inclusion is shown in Fig. 4.2. The parameters of the computational model include the material stiffness matrix C, which is calculated from the (plane) bulk modulus K p = 192 GPa, the shear modulus μ = 77 GPa for the inclusion, and K p = 22 GPa, μ = 12 GPa for the matrix. Next, for the inter  face κ = 01 01 · 6.88 TPam−1 , κG = 1 PPam−1 . The minimal element size of the FEM mesh is 0.25 mm. The fracture energy in the matrix domain is G c = 10 Jm−2 , considering the phase-field length parameter ε = 0.5 mm, and at interfaces it is G ci = {0.8, 20, 500} Jm−2 . With the domain fracture energy and a basic degradation function Φ(α)  = α2 + 10−6 , the critical stress in domain for uniaxial stress state would be σc =

3G c εΦ  (1)

·

Kpμ , K p +μ

which numerically gives σc = 15.26 MPa. Similarly

βζ provides the at the interface, the used interface degradation function φ(ζ) = 1+β−ζ  i 2G c κ critical stress σc = φ (1) which gives σc = {1, 5, 25} MPa, respectively to G ci . The structural element is loaded by displacement loading so that g(t) = v0 t, at the velocity v0 = 1 mm s−1 . This load is applied incrementally by time steps starting at 1 ms and refined down to 10 µs.

50

R. Vodiˇcka

g(t)

Fig. 4.2 Description of the domain with an inclusion

80

40

ω

S

x2 x1

Fig. 4.3 Working diagram of the one inclusion problem for various critical interface stresses

Let us first check the total response of the element. The graphs in Fig. 4.3 present relation between the total force applied to the top face of the matrix domain F and the prescribed displacement g. Three options for interface fracture energy have been chosen to compare the competitiveness of the interface and domain crack propagation. In two of them, the interface crack is first initiated and only afterwards the crack kinks into the matrix causing finally total rupture of the element. It is seen in the picture where the curves decrease their slopes as the structure softens after debonding of the matrix from the fibre. Details can be observed in the next pictures. For the case with σc = 1 MPa in the interface, reaching of this critical value can be seen in the first graph of Fig. 4.4. The interface damage parameter can be seen to start decreasing from one. The parameter ω measures the angle between the current point, the centre of the interface circle, and the rightmost point of that circle as seen in Fig. 4.2. The next used instant documents approaching of the damage parameter to zero which also corresponds to vanishing of the stress. The last used instant reflects the evolution of the interface damage at the time where also phase-field damage starts to evolve. These facts can be compared in Fig. 4.5. The first instant is the same as the last one in Fig. 4.4. The stress distribution exhibits a stress concentration at the interface crack tip. As the crack is opening, the stress trace is used as a stress measure. The second one pertains to the state

4 A Quasi-static Computational Model for Fracture in Multidomain . . .

51

Fig. 4.4 Distribution of normal stress (left) and interface damage parameter (right) along the interface for various time instants and σc = 1 MPa

Fig. 4.5 Stress (trace) distribution (top) and crack propagation as distribution of the phase-field variable (bottom) for σc = 1 MPa at instants 14, 14.6, 15.8 ms. Deformation at the bottom plots is magnified 100×

where phase-field damage variable has just reached its limit value and the final one documents a total separation of the element into two parts. Naturally the stress in this state vanishes. The next case with σc = 5 MPa in the interface is documented in Figs. 4.6 and 4.7. The first instant used in Fig. 4.6 shows the state where the critical value has been reached. The second one is just an intermediate state to present how both distributions in damage and stress evolve. The last two curves pertain to the states of initiating phase-field damage. Here, the matrix degradation starts before the end of interface

52

R. Vodiˇcka

Fig. 4.6 Distribution of normal stress (left) and interface damage parameter (right) along the interface for various time instants and σc = 5 MPa

Fig. 4.7 Stress (trace) distribution (top) and crack propagation as distribution of the phase-field variable (bottom) for σc = 5 MPa at instants 17.8, 18.3, 19.3 ms. Deformation at the bottom plots is magnified 100×

crack propagation, therefore these two instants are the same as used in Fig. 4.7. Anyhow, the matrix stress distribution exhibits a stress concentration near the interface crack tip. Also here, the second one pertains to the state where phase-field damage variable has reached its limit value. The final one shows how a crack is developed to break the element. The last case with σc = 25 MPa in the interface is shown in Figs. 4.8 and 4.9. The value of the critical stress at the interface is not reached as shows the first of the figures. Therefore, the value of the interface damage remains at ζ = 1 and it is

4 A Quasi-static Computational Model for Fracture in Multidomain . . .

53

Fig. 4.8 Distribution of normal stress along the interface for various time instants and σc = 25 MPa. Here, the interface damage is unaltered from the original intact state

Fig. 4.9 Stress (trace) distribution (top) and crack propagation as distribution of the phase-field variable (bottom) for σc = 25 MPa at instants 27, 27.3, 28 ms. Deformation at the bottom plots is magnified 100×

not shown. The matrix degradation starts above the top and below the bottom of the inclusion and it is propagated in a direction to break the element under tension. Three selected instants are shown in Fig. 4.9. At the interface, it is seen that the maximal stress is lower then the maximum allowed at the moment of the initiation of matrix degradation. Then also the stress at the interface is decreased as the material neighbour to the interface is degraded: the last used instant in Fig. 4.8 is the same as the second one in Fig. 4.9.

54

R. Vodiˇcka

g(t)

Fig. 4.10 Description of the domain with three inclusions

80

S2

S S3 20

x2

5

5 5 10

20

S1

15

x1

4.4.2 Three Inclusions The scheme for the three inclusion problem is shown in Fig. 4.10. The parameters of the computational model again include the material stiffness matrix C, which is calculated from the bulk modulus K p = 52 GPa, the shear modulus μ = 29 GPa for the inclusions, and K p = 3 GPa, μ = 1 GPa for the matrix. Next,  0  PPam−1 , κG = 100 PPam−1 . The minimal element for the interfaces κ = 01 0.5 size of the FEM mesh is 0.25 mm. The fracture energy in the matrix domain is G c = 1.25 kJm−2 , and at all interfaces it is G ci = 1 Jm−2 . With the domain fracture energy and the same basic degradation function Φ as above, the critical stress in the matrix domain for uniaxial stress state would be σc = 44.23 MPa. At the interexp −γ −1 (βζ+δ) −exp −γ −1 (δ) face, the used interface degradation function φ(ζ) = exp( −γ −1 (β+δ) )−exp (−γ −1 (δ) ) with ( ) ( )   γ(z) = exp(−z) 1 + z + 21 z 2 provides the maximum interface stress σc = 13.84 MPa, where the parameters were set as follows: β = 0.99, δ = 0.005, see [25]. It should be noted that with this degradation function the stress maximum occurs for ζ < 1. The structural element is loaded by displacement loading so that g2 (t) = v0 t, at the velocity v0 = 1 mm s−1 . This load is applied incrementally in time steps refined to 0.1 ms. The global response of the structural element is shown in Fig. 4.11. As in the previous example, the relation between the total force applied at the top face of the matrix domain F and prescribed displacement g is shown. The graph corresponds to abrupt crack propagation, though a small decrease of stiffness can be identified right prior to the jump decrease, which corresponds to crack initiation along interfaces. It can also be identified in comparing the working diagram with the time instants used in Fig. 4.12 and also the first two used in Figs. 4.13 and 4.14. The particularly chosen values of fracture energies at interface and in the bulk provide first initiation of an interface crack and afterwards the interface cracks are kinked into the matrix and finally lead to total rupture.

4 A Quasi-static Computational Model for Fracture in Multidomain . . .

55

Fig. 4.11 Working diagram for the three-inclusion case including a purely elastic response (dashed line)

Fig. 4.12 Distribution of normal stress (left) and interface damage parameter (right) along the interface around the centre S1 (top) or S3 (bottom), see Fig. 4.10, for various time instants

The graphs in Fig. 4.12 show also other details from the interface crack propagation interval. Results from two interfaces are depicted. The parameter ω measures the angle in the same sense as shown in Fig. 4.2. The instants, respectively to the values in the picture, pertain to the moments when the interface damage is initiated, the interface stress reaches its maximum value, the interface damage reaches the extreme value, and the phase-field damage variable reaches the extreme value. The final state of damage variable reveals the interface cracks at top and bottom parts of the interfaces. Near the crack tips, stress is concentrated so that the cracks propagate into the matrix.

56

R. Vodiˇcka

Fig. 4.13 Stress (trace) distribution at instants 260, 271, 273.4, 275.7, 276.8, 820 ms

Fig. 4.14 Crack propagation as distribution of the phase-field variable at the same instants as in Fig. 4.13. Deformation is magnified 10×

Pictures in Figs. 4.13 and 4.14 show how the material in these locations is degraded and provokes arising of the cracks in the matrix. Though, initial material degradation occurs at several places, the actual crack then propagates only from the critical ones

4 A Quasi-static Computational Model for Fracture in Multidomain . . .

57

and they finally join all interface cracks into one global crack. Also here, the instants presented in pictures pertain respectively to particular situations: interface crack initiating, bulk crack initiation, a crack between two inclusions, the crack reaches outer contour(s), final rupture of bridged material.

4.5 Conclusions A model for solving problems with cracks along interfaces combined with those in the material has been presented and tested. It is clear that such a model needs several parameters related to a crack formation process. The values of the parameters may change character of degrading processes in materials so that their adjustments have to be done in comparison with experimental measurements. It was intended to show the effect of such changes by modifying the values of fracture energy. From the computational sight, a semi-implicit time discretisation was implemented to obtain a variational structure of the numerical approximation in the solution of an evolutionary problem. The approximated model then utilises the method of sequential quadratic programming with a common FEM space discretisation implemented in an own MATLAB code. Anyhow, the results of the current study can be considered satisfactory, so that the proposed computational approach will be in the future successfully implemented in more complex practical engineering calculations. Acknowledgements The author acknowledges support from The Ministry of Education, Science, Research and Sport of the Slovak Republic by the grants VEGA 1/0374/19 and VEGA 1/0363/21.

References 1. Alberty J, Carstensen C, Funken S, Klose R (2002) Matlab implementation of the finite element method in elasticity. Computing 69:239–263. https://doi.org/10.1007/s00607-002-1459-8 2. Ambrosio L, Tortorelli VM (1990) Approximation of functional depending on jumps by elliptic functional via γ-convergence. Commun Pure Appl Math 43(8):999–1036. https://doi.org/10. 1002/cpa.3160430805 3. Besson J, Cailletaud G, Chaboche J, Forest S (2010) Non-linear mechanics of materials. Springer, Dordrecht 4. Bourdin B, Francfort GA, Marigo JJ (2008) The variational approach to fracture. J Elast 91:5– 148 5. Dal Maso G (2012) An introduction to -convergence, vol 8. Springer Science & Business Media 6. Del Piero G (2013) A variational approach to fracture and other inelastic phenomena. J Elast 112:3–77 7. Dostál Z (2006) An optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum. Computing 78(4):311–328 8. Dostál Z (2009) Optimal quadratic programming algorithms, springer optimization and its applications, vol 23. Springer, Berlin

58

R. Vodiˇcka

9. Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342. https://doi.org/10.1016/S0022-5096(98)00034-9 10. Frémond M (1995) Dissipation dans l’adhérence des solides. CR Acad Sci, Paris, Sér.II 300:709–714 11. Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A 221:163–198 12. Kružík M, Roubíˇcek T (2019) Mathematical methods in continuum mechanics of solids. Interaction of mechanics and mathematics. Springer, Switzerland 13. Lemaitre J, Desmorat R (2005) Engineering damage mechanics. Springer, Berlin 14. Maugin G (2015) The saga of internal variables of state in continuum thermo-mechanics. Mech Res Commun 69:79–86 15. Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Method Appl Mech Eng 199(45–48):2765–2778. https://doi.org/10.1016/j.cma.2010.04.011 16. Molnár G, Gravouil A (2017) 2D and 3D Abaqus implementation of a robust staggered phasefield solution for modeling brittle fracture. Finite Elem Anal Des 130:27–38. https://doi.org/ 10.1016/j.finel.2017.03.002 17. Paggi M, Reinoso J (2017) Revisiting the problem of a crack impinging on an interface: a modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model. Comput Method Appl Mech Eng 321:145–172. https:// doi.org/10.1016/j.cma.2017.04.004 18. Raous M, Cangemi L, Cocu M (1999) A consistent model coupling adhesion, friction and unilateral contact. Comput Meth Appl Mech Eng 177(6):383–399 19. Roubíˇcek T, Panagiotopoulos C, Mantiˇc V (2015) Local-solution approach to quasistatic rateindependent mixed-mode delamination. Math Models Methods Appl Sci 25(7):1337–1364 20. Roubíˇcek T (2013) Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity. SIAM J Math Anal 45(1):101–126. https://doi.org/10.1137/12088286X 21. Roubíˇcek T, Panagiotopoulos C, Mantiˇc V (2013) Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity. Zeitschrift angew Math Mech 93:823–840 22. Sargado JM, Keilegavlen E, Berre I, Nordbotten JM (2018) High-accuracy phase-field models for brittle fracture based on a new family of degradation functions. J Mech Phys Solids 111:458– 489. https://doi.org/10.1016/j.jmps.2017.10.015 23. Sutradhar A, Paulino G, Gray L (2008) The symmetric Galerkin boundary element method. Springer, Berlin 24. Tanné E, Li T, Bourdin B, Marigo JJ, Maurini C (2018) Crack nucleation in variational phasefield models of brittle fracture. J Mech Phys Solids 110:80–99. https://doi.org/10.1016/j.jmps. 2017.09.006 25. Vodiˇcka R (2016) A quasi-static interface damage model with cohesive cracks: SQP-SGBEM implementation. Eng Anal Bound Elem 62:123–140 26. Vodiˇcka R (2019) On coupling of interface and phase-field damage models for quasi-brittle fracture. Acta Mechanica Slovaca 23(3):42–48 27. Vodiˇcka R (2020) A computational model of interface and phase-field fracture. AIP Conf Proc 2309:020002 28. Vodiˇcka R, Kormaníková E, Kšiˇnan F (2018) Interfacial debonds of layered anisotropic materials using a quasi-static interface damage model with coulomb friction. Int J Frac 211(1– 2):163–182. https://doi.org/10.1007/s10704-018-0281-z 29. Vodiˇcka R, Mantiˇc V (2017) An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law. Discrete Cont Dyn Syst–Ser S 10(6):1539–1561 (2017) 30. Vodiˇcka R, Mantiˇc V, París F (2007) Symmetric variational formulation of BIE for domain decomposition problems in elasticity—an SGBEM approach for nonconforming discretizations of curved interfaces. CMES—Comp Model Eng 17(3):173–203 31. Vodiˇcka R, Mantiˇc V, Roubíˇcek T (2014) Energetic versus maximally-dissipative local solutions of a quasi-static rate-independent mixed-mode delamination model. Meccanica 49(12):2933– 296

4 A Quasi-static Computational Model for Fracture in Multidomain . . .

59

32. Wang Q, Feng Y, Zhou W, Cheng Y, Ma G (2020) A phase-field model for mixed-mode fracture based on a unified tensile fracture criterion. Comput Methods Appl Mech Eng 370:113270. https://doi.org/10.1016/j.cma.2020.113270 33. Wu JY (2017) A unified phase-field theory for the mechanics of damage and quasi-brittle failure. J Mech Phys Solids 103:72–99. https://doi.org/10.1016/j.jmps.2017.03.015

Chapter 5

Updated Lagrangian Curvilinear Beam Element for 2D Large Displacement Analysis Christian Iandiorio and Pietro Salvini

Abstract This paper presents a 2D curvilinear beam finite element model focusing the interest on its use for non-linear analysis caused by very large displacements, addressed with the Update Lagrangian strategy. The method allows using very long curvilinear beams even when high geometric nonlinearities occur. This is due inasmuch the proposed formulation does not require any pre-set shape function that would inevitably force to use a huge number of elements to achieve reliability. The lack of shape-functions is overcome using the integration of the compatibility equations, that provide the whole internal displacement field from the only knowledge of the element nodal degree of freedom. Section-slices subdivision allows to sum, not to assemble, the flexibility contribute of each slice and consequently to build up the end-to-end tangent stiffness matrix of a generic curvilinear beam element. Moreover, the flexibility feature of every slice can be deduced analytically once and for all. To validate the proposed element some comparisons are carried out with analytical and numerical solutions obtained with Runge–Kutta integration method or cubic isoparametric finite elements. Keywords Large displacement analysis · Finite element method · Curvilinear beam element

5.1 Introduction Large displacements, large deflections or geometric nonlinearities are the names whereby is defined the analysis of the nonlinear behaviour of high-flexible structures. The investigation of the structural response of slender elements is of considerable interest. The results are used in many classic and more recent engineering applications such as leaf springs [1], micro-electric switches [2], mechanical systems [3] and compliant mechanisms [4]. C. Iandiorio (B) · P. Salvini Department of Enterprise Engineering, University of Rome “Tor Vergata”, Via del Politecnico 1, 00133 Rome, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_5

61

62

C. Iandiorio and P. Salvini

The shape assumed by an elastic beam subjected to applied loads is known as elastic curve; this name turns into Elastica when large displacements and rotations occur, so that the equilibrium on the deformed configuration is no longer equivalent with that occurring on the undeformed structure. The Elastica problem has been analytically addressed by several authors. Its mathematical modeling combines a system of nonlinear ODE with a boundary value problem. For the case of straight or constant curvature beams (arcs) only subjected to concentrated loads, the solution can be found analytically involving Elliptic Integrals, Jacobi elliptical functions [5–7], or hypergeometric functions [8–11]. When generic distributed loads occur analytical solutions are not known. Perturbative approaches allow getting approximated solutions [5, 12, 13], which unfortunately require a huge number of terms to obtain reliable results; however, they do not guarantee the series convergence at the end-boundary [14]. To overcome these difficulties a new method called CAMM [15–17] has recently been proposed; this allows to lead back the solutions of the Elastica problem with distributed loads to a form that is similar to those adopted for concentrated loads. In some determined case where analytical solutions are known, they imply a sturdy benchmark and a deep understanding of the modelling. The solution is given in a one-shot loading, with concentrated loads, both constant direction or follower ones. Differently from numerical methods, they do not require a loading path and provide all possible configurations (both energetically stable and unstable). Furthermore, they allow plotting input–output in the dimensionless form to build up design charts [10]. Unfortunately, only elementary structures can be easily addressed with analytical solutions, especially when static indeterminate structures are considered. Namely, the large displacement analysis of complex structures is generally faced with computerbased automated procedures. There is a large literature concerning the numerical methods used to solve geometric nonlinear problems of beam structures, e.g.: R-K integration scheme with shooting method [18] and Newton method [19], Automatic Taylor expansion technique [20], Homotopy analysis method [21]. On the other hand, a huge number of papers regarding nonlinear geometric analyses of beam structures can be found, but the Finite Element Method is generally recognized as the most effective and adaptable numerical formulation. Three main formulations are used in FEA to address geometric non-linearities: Total Lagrangian (TL), Updated Lagrangian (UL) and the Co-Rotational (CR). They differ in the definition of adopted reference system. In the TL [22–25] the kinematic equations defining the element formulation refer to the coordinate system of the undeformed geometry. The advantage is that moderate changes of the element configuration are accepted at every step, but very highly nonlinear expressions may result. It is worth pointing out that the extent of the configuration changes is anyhow limited by the updating of load locations, which force to proceed by incremental steps. The UL formulation [26–29] considers the updated geometry to derive the kinematic expressions in the updated reference system, resulting by the last converged iteration. The kinematic expressions that emerge are much easier to manage than for

5 Updated Lagrangian Curvilinear Beam Element for 2D Large …

63

previous TL, but they assume that the element configuration change between two successive increments keeps small. The CR method [30–36] can be taken as a miscellany of above formulations, where the total change of configuration is shared into a rigid body aliquot and a purely deformation one. The rigid body contribution does not deform the element and is associated to a global coordinate system. The purely deformation aliquot is referred to a co-rotational coordinate system, where displacements and rotations are made small by appropriate refinement allowing a linear formulation for strains. On the other hand, the CR method transfers the non-linearity from the strain-kinematic relationships to the changes of the reference systems. Large Displacements involve a non-linear analysis, which is performed with the FEM (for any type of formulation) by repeating numerous times the solution of a linear system [37, 38]. For structures modelled with beam elements (but it is also true for other types of elements), the above aim is generally achieved with a mesh made by many non-curved elements. Therefore, a high number of elements is required to reliably follow the deformed shape of the structure. With the aim to minimize the d.o.f. required in non-linear large displacements analysis, in [39] it is developed a curved filiform beam FE using a cubic shape function not based on displacements but on the curvature radius; this allows to model a generic filiform structure with a few numbers of long curvilinear elements. The seeks of large-size elements is a field of interest also for linear (small displacements) problems that involve straight or curved structures with variable sections (or thicknesses for shell elements) [40–45] with a particular focus on the use of NURBS functions. This paper presents a new approach to modelling a structure with long 2D curvilinear elements, also valid for large displacement analyses. With the proposed element it is possible to solve a geometric non-linear analysis using a minimal number of d.o.f.; the element nodes are positioned as it would be done in the linear solution of a straight beam, i.e. in correspondence of concentrated loads, imposed displacements or geometric discontinuities. To do this, the element is based on a coupling between the force-flexibility approach and compatibility condition [46]. Through the integrations of the compatibility equations, it is possible to recover the whole internal displacements field of the element from the knowledge of nodal displacements, without any restriction due to shape functions. According to this force-flexibility approach, in a generic iteration of the incremental analysis each element is curvilinear. It is internally subdivided into sectionslices that allow to build up the node-to-node tangent stiffness matrix as the simple sum (not the assemble) of their contributions to flexibility. In the present paper, the local formulation of the proposed element is intended in the updated reference geometry, namely the Update Lagrangian strategy is followed. The fact remains that it would still be possible to formulate the proposed element with another strategy, as Total Lagrangian or Co-rotational.

64

C. Iandiorio and P. Salvini

5.2 Fundamentals Hereinafter the fundamental relations to formulate the large size beam element in its local reference are derived. The configuration taken into consideration should be intended as generic, namely as the ones resulting after a converged iteration of a non-linear incremental analysis. For sake of clarity, in what follows the general vector and tensor quantities/variable are written in bold font; the quantities directly related to the FE implementation are evidenced with curly brackets {·} for vectors, and square bracket [·] for matrices.

5.2.1 Equilibrium This paragraph derives the equilibrium relations which must be satisfied for the generic planar configuration of the beam. Figure 5.1 shows an element having two extremal nodes I and J and two reference systems: the element reference ex , ey , ez and the intrinsic mobile frame es , eξ , eη , following beam axis and cross-section. At node I are applied the force and moment vectors, written with the classical FE notation: F 0 = (F1 , F2 , 0)T = F1 ex + F2 e y , M 0 = (0, 0, F3 )T = F3 ez ; similarly, at the node J: F L = (F4 , F5 , 0)T = F4 ex + F5 e y , M 0 = (0, 0, F6 )T = F6 ez . Along the ( )T element length distributed forces and moments q(s) = qx (s), q y (s), 0 , m(s) = (0, 0, m z (s))T are applied. By virtue of the Euler’s principle of sectioning, any portion of the element is in equilibrium; therefore, at the generic point P along the curvilinear abscissa s the ( )T vectors force and moment F P (s) = Fx (s), Fy (s), 0 , M P (s) = (0, 0, Mz (s))T , referred to the element coordinate system, maintain equilibrium with both ends. Imposing the balance at the generic position s, assuming as momentum pole of equilibrium the node I, it results:

Fig. 5.1 Generic configuration of the curvilinear beam element

5 Updated Lagrangian Curvilinear Beam Element for 2D Large …

65

{s F 0 + F P (s) +

q(˜s )d s˜ = 0

(5.1)

0

{s M 0 + M P (s) + r c (s) × F P (s) +

{s m(˜s )d s˜ = 0

r c (˜s ) × q(˜s )d s˜ + 0

(5.2)

0

where r c (s) = (x(s), y(s), 0)T is the radius vector of the point P by respect to element reference, while s˜ is a dummy variable. The Eqs. (5.1), (5.2) can be expressed in the typical FE compact form notation, useful for element assembly, where vectors and matrix quantities are indicated with curly {·} and square [·] brackets, respectively: [ { ] } {F(s)} = − T F (s, 0) {F I } − F q (s)

(5.3)

where: {

}

{s

F (s) = q

[

] T F (s, s˜ ) {q(˜s )}d s˜

(5.4)

0

[

⎡

1 0 T F (s, s˜ ) = ⎣ 0 1 y(s) − y˜ (˜s ) x(˜ ˜ s ) − x(s) ]

⎤ 0 0⎦ 1

(5.5)

{ }T The vector {F(s)} = Fx (s), Fy (s), Mz (s) contains the forces acting on the section at the generic abscissa s, referred to element coordinate system; it is possible to compute it from the knowledge of the vector {F I } = {F1 , F2 , F3 }T that includes the forces applied at the node I , and the vector {F q (s)} relating to accrued distributed loads. [ ] The matrix T F (s, s˜ ) , downstream the form of Eq. (5.3), can be defined as the { }T force transfer matrix, {q(˜s )} = qx (s), q y (s), m z (s) is the distributed load vector. The force vector {F J } = {F4 , F5 , F6 }T related to the node J is simply found by computing Eq. (5.3) in s = L: [ { ] } {F J } = − T F (L , 0) {F I } − F q (L)

(5.6)

In which: {

}

{L

F (L) = q

0

[

] T F (L , s) {q(s)}ds

(5.7)

66

C. Iandiorio and P. Salvini

5.2.2 Section’s Stiffness Response The one-dimensional beam model can be interpreted as the behaviour of an infinite number of cross-sections connected each other with virtual internal joints. If the hypothesis of warping absence is assumed, the kinematic of all points along the section is known by only three pieces of information: two translations of the centroid and the section rotation. Therefore, the displacements field is assumed as: U(r) = u + ϑ × r cp

(5.8)

where u is the centroid displacement vector, ϑ is the section rotation vector, r cp is the distance of a generic point p on the section from the centroid. By means of the relations of solid mechanics, starting from the three kinematic d.o.f., three deformation mechanisms are found for each section: an axial strain ε namely collinear with the tangent vector es (Fig. 5.1), a shear (distortional) strain γ directed as eξ on the cross section, and a curvature strain κ around eη . All are incorporated in the strain vector {ε} = {ε, γ, κ}T . Forces and the moment are given in the same reference. The vector of the resulting internal forces {N(s)} = {N (s), T (s), M(s)}T related to the cross-section attitude can be derived from {F(s)} by: {N(s)} = [R(s)]{F(s)}

(5.9)

where [R(s)] is the common change of basis (direction cosines) matrix that transforms the components referred to the element reference to the mobile one; for curvilinear elements each cross-section has a variable attitude function of the curvilinear abscissa. The relationship between {N} and {ε} defines the cross-section compliance. As discussed in [47], the response of the cross section depends on the material property and on how the section area is distributed; although, for a curved beam, an important role is played by the actual curvature, which modifies the effective moment of inertia. As demonstrated in [47], this effect is dominated by the ratio between the point that have the maximum distance from the centre of rotation and the actual curvature radius. When this ratio is small (≤0.1) the beam can truly be considered slender (or thin); namely, it is lawful to assume the force–deformation relationship as if each cross-section behaves like a straight piece. This paper, focused on the behaviour of thin structures, assumes this condition thus the relationship between resultant force and strain turns out independently of the actual curvature: [ ] {N} = K s {ε}

(5.10)

For a beam made with an isotropic material and a reference located at the section centroid (also coincident with the shear centre) and oriented along the principal [ ] inertia axes, the section’s stiffness matrix is simply a diagonal matrix: K s = diag[E A, G A/χ , E I ], where E, G are the elastic properties of the material, i.e.

5 Updated Lagrangian Curvilinear Beam Element for 2D Large …

67

the Young’s and shear elastic modulus, A, I are the translational and rotational cross-section inertia properties and χ is the shear correction factor. In order to have a concise notation in the following the cross-section flexibility [ ] [ ]−1 matrix is used instead C s = K s = diag[1/E A, χ /G A, 1/E I ]. [ ] It is worth pointing out that for thick curvilinear beam, the terms in K s would change considering also the curvature, as deduced in [47], but the proposed FE formulation that follows would remain the same.

5.2.3 Compatibility Equations and Their Integration A solid body can be thought as connected but not overlapped sets of infinitesimal volumes. In the deformed configuration as well as the non-overlapping (interpenetrations), there must also be no voids. From the mathematical point of view, a compatibility strain field exist when six compatibility equations are respected. The necessity of these latter equations were first deduces by B. de Saint–Venant in 1861 [48, 49], who obtained them by carrying out a manipulation of sums and derivatives of equations obtained through derivation of the strain field (congruence); the sufficiency of these conditions was obtained by E. Beltrami in 1886 [50]. Some years later, in a number of papers, V. Volterra deduced these equations starting from an integral form, valid for a non-infinitesimal continuum [51–54], which was soon strongly simplified by Cesaro [55]. This last is often referenced in nowadays books [56]: {

{ dU = L

∇Ud r = 0

(5.11)

L

where r is the radius vector that starts from the origin of the reference system and points to the generic point of the circuit line. Cesaro states that the strain field due to displacements U respects the compatibility if Eq. (5.11) holds for any internal closed path L (path-independence) on the continuum; Eq. (5.11) emphasized the connection with the conservativeness concept of the strain field, providing a geometric view of the compatibility. Therefore, the key-point is to make explicit the Eq. (5.11), assuming a generic non-closed integration path, so that one can apply the compatibility equations to the beam along a line that moves from the first node to another internal one. For a generic continuum, being r 0 a reference point, Eq. (5.11) turns out: {r U(r) = U(r 0 ) +

∼

{r

{r ∼ Dd r + ω × d r

r0

r0

∇Ud r= U(r 0 ) + r0

∼

(5.12)

68

C. Iandiorio and P. Salvini

where the displacement gradient is additively decomposed in its symmetric part D = Sym(∇U), which represents a pure deformation (volume and shape change) associated to the point, and an asymmetric part W = Asym(∇U) representing rigid rotations. Furthermore, being W a skew-symmetric matrix, it is possible to use the equivalence W d r = ω × d r, where ω is the generator (axial) rotation vector of W . Manipulating the Eq. (5.12), after some steps, this can be expressed in a convenient form for FE manipulations, that separate the rigid motion into a translational displacement and a pure rotation: {r U(r) = U(r 0 ) + ω(r 0 ) × (r − r 0 ) +

{r ( ) ∼ ∼ Dd r + ∇ωd r × r− r ∼

r0

(5.13)

r0

{r ω(r) = ω(r 0 ) +

∼

∇ωd r

(5.14)

r0

These equations admit a simple physical meaning. Equation (5.13) states that the translational displacements of a generic point r is given by the sum of the translational displacements computed at r 0 , a term which takes into account the product of the arm and the rotation in r 0 , and other two contributes: the first accounts of the accumulated translational strains, while the second considers the effect of accumulated rotational strains along the path. Equation (5.14) defines the total rotation account of all rotations along the path. It is relevant to observe that no assumptions are made on the magnitude of the displacements gradient in Eq. (5.11) so that Eqs. (5.12), (5.13) still hold under large displacements, rotation and strains. Focussing on one-dimensional (engineering/structural) beam model, Eqs. (5.13), (5.14) reduces as follow on paths that follows the beam axis: {s U(s) = U(s0 ) + ω(s0 ) × (r(s) − r(s0 )) +

d+

dω × (r(s) − r(˜s ))d s˜ (5.15) ds

s0

{s ω(s) = ω(s0 ) +

dω d s˜ ds

(5.16)

s0

where d = D · es . Applying the displacement fields of Eq. (5.8) in Eqs. (5.15), (5.16), the components of the {ε} vector defined in Sect. 5.2.2 turn out. With the aim to obtain a compact FE form related for each axial point at the element reference system, Eqs. (5.15), (5.16) should be multiplied for the rotation matrix transpose [R(s)]T (to transform the components referred to the mobile reference to the element one); it is useful to observe that using Eqs. (5.9), (5.10):

5 Updated Lagrangian Curvilinear Beam Element for 2D Large …

69

[ ] [ ] [R(s)]T {ε} = [R(s)]T C s {N} = [R(s)]T C s [R(s)]{F(s)} Now, with the aim to form the FE element, it is mandatory to choose the node J as initial point for the compatibility equations so that s0 = L; Eqs. (5.15), (5.16) turns into: [ ] {U(s)} = T U (s, L) {U J } − {U ε (s)}

(5.17)

where the order of the extreme integration is permuted by respect to Eqs. (5.15), (5.16): {L

ε

{U (s)} =

[

[ ] ] T U (s, s˜ ) [R(s)]T C s [R(s)]{F(s)}d s˜

(5.18)

s

[

⎤ 1 0 y˜ (˜s ) − y(s) T U (s, s˜ ) = ⎣ 0 1 x(s) − x(˜ ˜ s) ⎦ 00 1 ]

⎡

(5.19)

{ }T where the vector {U(s)} = Ux (s), Vy (s), ψz (s) contains the displacements (translations and rotations) of the cross-section at the generic abscissa s; it is referred to the element coordinate system. {U J } = {U4 , U5 , U6 }T which includes the displacements of the node J , while the vector {U ε (s)} accounts of the accrued translational and [ rotational ] strain at s. The matrix T U (s, s˜ ) , similar to that introduced in Eq. (5.5), can be defined as the displacements transfer matrix.

5.3 Finite Element Formulation: The Section-Slices Idea The terms in Eq. (5.18) are nonlinear functions of s; furthermore, for a generic curvilinear configuration (i.e. for a converged iteration of the FE incremental analysis) the trend of these terms by respect to the curvilinear abscissa is not deducible analytically. In order to easily compute the Eq. (5.18), the beam is subdivided into section-slices formed by consecutive straight cylinders of variable attitude (Fig. 5.2). If the beams present a non-small (≥0.1) thickness-curvature radius ratio (Sect. 5.2.2) the section properties must be computed accounting the curvature [46]. The straight cylinder subdivision allows performing Eq. (5.18) as a sum of trivial terms inasmuch: (i) the rotation matrix is constant along each section-slice and straightforward to derive knowing only the end positions of every section-slice; (ii) the coordinates x(s), y(s) changes linearly within a section slice so that the integration of the transfer matrix [ U ] T (s, s˜ ) is trivial; (iii) x(˜ ˜ s ), y˜ (˜s ) can be easily parametrized in s˜ using the cosine directors of the section slices. Hence the Eq. (5.18) turns out:

70

C. Iandiorio and P. Salvini

Fig. 5.2 Section-Slices subdivision of the curvilinear element

k(L) { ∑ [ sk+1

ε

{U (s)} =

[ ] ] T U (s, s˜ ) k [R]kT C s k [R]k {F(s)}k d s˜

(5.20)

k(s) s k

where k(s) identifies the section-slice at the abscissa s and k(L) refers to the last section-slice. Note that the integrals in Eq. (5.20) involve only a linearly changing of the vari˜ s ), y˜ (˜s ) within each section-slices, so that the solution is easy to be deterables x(˜ mined analytically. In the following, for sake of brevity, the subscripts “k” inside integrals are omitted. Applying the Eq. (5.3) in (5.20), the Eq. (5.17) becomes: { [ ] } {U(s)} = T U (s, L) {U J } + [C I I (s)]{F I } + U qI (s)

(5.21)

where: k(L) { ∑ [ sk+1

[C I I (s)] =

[ ] [ ] ] T U (s, s˜ ) [R]T C s [R] T F (˜s , 0) d s˜

(5.22)

k(s) s k

{

} q U I (s)

k(L) { ∑ [ sk+1

=

[ ] { ] } T U (s, s˜ ) [R]T C s [R] F q (s) d s˜

(5.23)

k(s) s k

The matrix [C I I (s)] contains the flexibility referred only to the node I , if a { q terms, } constrain is clamped in s. The vector U I (s) contains the displacements at node I only due to the applied distributed load when a constrain is clamped in s.

5 Updated Lagrangian Curvilinear Beam Element for 2D Large …

71

Equation (5.21) is fundamental, inasmuch it allows to recover the internal displacements of the element by the nodal solutions. More precisely, the Eq. (5.21) requires the knowledge of the displacements at the node J , the force applied at the node I and the acting distributed loads. The same equation is also crucial to compute the actual stiffness matrix of every curvilinear element. Considering Eq. (5.21), when s = 0, i.e. on node I, the displacement vector is {U (0)} = {U I } = {U1 , U2 , U3 }T so that: [ { } ] {U I } = T U (0, L) {U J } + [C I I ]{F I } + U qI

(5.24)

where: k(L) { ∑ [ sk+1

[C I I ] =

[ ] [ ] ] T U (0, s) [R]T C s [R] T F (s, 0) ds

(5.25)

k(0) s k

{

q} UI

k(L) { ∑ [ sk+1

=

[ ] { ] } T U (0, s) [R]T C s [R] F q (s) ds

(5.26)

k(0) s k

It is straightforward to note that the reduced flexibility [C I I ] is a symmetric matrix. Assuming fixed the node J , i.e. {U J } = 0, the Eq. (5.24) turns out the classical displacement-force [C I I ] is the flexibility matrix associated to { qequation; } { q therefore } the node I , and U I = U I (0) is the displacements of the node I caused by the distributed loads. Equation (5.24), as well as the (5.21), contain the information regarding compatibility, congruence (strains-displacements relationship) and constitutive relations; the Eq. (5.3) encompasses the equilibrium information. Therefore, with Eqs. (5.3), (5.24) one has all the necessary ingredients to build-up the curvilinear beam Element. Putting in evidence {F I } from Eq. (5.24): { } [ ] {F I } = [K I I ]{U I } − [K I I ] T U (0, L) {U J } − Q I

(5.27)

[K I I ] = [C I I ]−1

(5.28)

} { q} Q I = [K I I ] U I

(5.29)

where:

{

Equations (5.3), (5.27) contains all the necessary information and can be expressed in a compact form as follow: [K ]{U} = {F} + { Q}

(5.30)

72

C. Iandiorio and P. Salvini

or [

[K I I ] [K I J ] [K J I ] [K J J ]

]{

{U I } {U J }

}

{ =

} {{ }} {F I } Q + { I} {F J } QJ

where: [ ] [K I J ] = −[K I I ] T U (0, L)

(5.31)

[ ] [K J I ] = [K I J ]T = − T F (L , 0) [K I I ]

(5.32)

[ [ ] ] [K J J ] = T F (L , 0) [K I I ] T U (0, L)

(5.33)

{

} { } [ ]{ } Q J = F q (L) − T F (L , 0) Q I

(5.34)

Equation (5.30) is the fundamental FE relation, where [K ] is the tangent stiffness matrix of the generic curvilinear element, { Q} is the vector of the condensed distributed loads. Using Eq. (5.27) in (5.21), one obtains an alternative equation to (5.21) which is more useful for the recovery of the incremental internal displacements in view of the incremental analysis: } { {U I } + {U q (s)} {U(s)} = [N(s)] {U J }

(5.35)

where: [N(s)] = [[N I (s)][N J (s)]] [N I (s)] = [C I I (s)][K I I ]

{

[N J (s)] = [T U (s, L)] + [C I I (s)][K J I ]

(5.36)

} { q } { q} U q (s) = U I (s) − [C I I (s)][K I I ] U I

(5.37)

The Eq. (5.35) can be interpreted in a certain sense as tangent shape functions, inasmuch it returns the internal displacement given the converged shape of the beam element axis in the previous configuration, the nodal displacement vector and the aliquot due to distributed loads. Equation (5.30) cannot be solved in one shot because the tangent stiffness matrix continuously modifies inasmuch the section-slices change their attitude while loads increase. Therefore, an incremental procedure based on Newton–Raphson algorithm [30, 37, 38] is followed; the loading is subdivided into sub-steps modulated by an incremental load multiplier △λ; convergence in the sub-step is reached through

5 Updated Lagrangian Curvilinear Beam Element for 2D Large …

73

iterations, here represented with subscripti. Equation (5.30), after the assemblage procedure of all elements forming the structure, can be partitioned in constrained “c” and free (non-constrained) “ f ” d.o.f., where for the latter the incremental Eq. (5.27) results: {

△U f

} i+1

]−1 [ = K f f i {R}i

(5.38)

Solving Eq. (5.38) the total nodal displacement vector results {U}i+1 = {U}i + {△U}i+1 , while the n(L) internal displacements are updated using Eq. (5.35). Known the entire displacements field, hence the new configuration of every beam, it is possible to compute the new tangent stiffness matrix [K ]i+1 from Eq. (5.25) using Eqs. (5.28)–(5.31) and[ taking into ] [ account ]the updated attitude of each section-slice in the matrices [R], T U (0, s) , T F (s, 0) . { el } Before a new iteration, the internal force vector F int, f i+1 , element by element, (5.27) and therefore the can be deduced using the Eqs. (5.6), { { } } overall residual { }(unbalance) force vector {R}i+1 = F ext, f i+1 − F int, f i+1 where F ext, f i+1 = ) ({ } { } [ ] △λ F f + Q f i+1 − K f c i+1 {U c } is computed. This iterative scheme continues until a convergence-criterion is satisfied and a new sub-step starts until the total application of the loads is reached.

5.4 Numerical Validations In this sub-paragraph three numerical applications are shown. To validate the proposed procedure exposed in Sect. 5.3 the results are compared with analytical solutions and numerical methods, the latter obtained by both Runge–Kutta–Fehlberg method and a F.E. software. Runge–Kutta–Fehlberg 45 method integrates the nonlinear system of ODE that governs large displacement problems [5, 7, 10]. It cannot be applied directly, inasmuch the boundary conditions form a boundary value problem which implies that the RKF algorithm needs to be supported by an attempt (shooting) method, here driven by bisection method [15, 16]. The admissible error tolerance on the end curvature is set to 10−6 N /m, the RKF absolute and relative tolerance are set 10−8 . The others numerical comparisons are carried out with a commercial F.E. software. In all comparisons the material has E = 210 GPa, the section is rectangular, 5 · 10−3 m wide and 3 · 10−3 m thick. The stop-criterion for the non-linear analysis is that the relative tolerance of the L2 norm of the residual vector is lower than to 10−3 . The test cases in Figs. 5.3 and 5.4 concerns an arch cantilever beam with constant curvature k0 = −π/2m −1 and a length of L = 2m. The first case in Fig. 5.3 regards moment loads M applied at the free end, and the proposed model are compared with the trivial analytical solution [9, 10]:

74

C. Iandiorio and P. Salvini

Fig. 5.3 Comparison between the analytical solution and the one obtained with the proposed element for a line-structure having constant curvature and subjected to end moment loads

Fig. 5.4 Comparison between the proposed element result and the isoparametric elements for a line-structure having constant curvature and subjected to end force loads

5 Updated Lagrangian Curvilinear Beam Element for 2D Large …

75

Table 5.1 Cumulative iterations required by the proposed element for the beam with initially constant curvature subjected to end moment loads 1 curvilinear element sampled in 100 section-slices M [Nm]

– 3

– 1

2

4

Cumulative Iterations

37

5

22

55

[( ) ] ) ] [( sin k0 + EMI s M ) ( s ds = cos k0 + EI k0 + EMI

(5.39)

[( ) ] ) ] [( 1 − cos k0 + EMI s M ( ) s ds = sin k0 + EI k0 + EMI

(5.40)

{s x(s) = 0

{s y(s) = 0

Table 5.1 indicates the cumulative iterations occurring to get convergence with the unique proposed element. In the second case (Fig. 5.4) force loads at the free end are applied, in the x and y directions; for these load conditions the comparisons are carried out using a F.E. model formed by 100 isoparametric beam elements based on cubic shape functions [37]. Even in these cases, the use of the proposed element allows to recur to only one element (only two end nodes are present) to get the nonlinear solution. In order to compare the required computational effort due to the number of iterations, the section-slice subdivision is set equal to the number of isoparametric elements, namely 100. The results exposed in Figs. 5.3 and 5.4 show a strong convergence of the solutions computed with the four methods. Table 5.2 shows the numbers of substeps and iterations required for each applied load by the use of the proposed or the isoparametric beams. From the comparison in Table 5.2 between the two FE methods, where the substeps are forced to be equal in number, it emerges that the proposed method is advantageous in terms of overall iterations. Furthermore, regarding the computational costs, it is important to note that the proposed element involves at each iteration the inversion of a stiffness matrix formed only by 3 d.o.f., instead of 300 d.o.f. Table 5.2 Substeps and cumulative iterations required by the proposed isoparametric elements for the curved cantilever subjected to end force loads 100 isoparametric beam elements 1 proposed element, 100 section-slices F[N]

Fx

Fy

Fx

Fy

5

30

5

30

5

30

5

30

Substeps

3

27

4

11

3

27

4

11

Cumulative iterations

37

214

39

110

15

79

16

48

76

C. Iandiorio and P. Salvini

The test case in Fig. 5.5 regards an initial straight structure of length L = 1m subjected to distributed loads along the y direction. The two models are formed by a unique proposed element or by 100 isoparametric elements, very close results are shown. The steps required for incremental analysis are summarized in Table 5.3, the reduction in the number of overall iterations is quite similar to the concentrated load cases. The stiffness matrix modifies during the non-linear evolution of the structure, but even the distributed load vector modifies during the iterations because its contribution in the mobile frame reference changes in accordance with the updating of the internal

Fig. 5.5 Comparison between the proposed and isoparametric elements ones for a initially straight beam subjected to distributed loads in y direction

Table 5.3 Substeps and cumulative iterations required by the proposed and isoparametric elements for the beam with initially constant curvature subjected to distributed loads along y direction 100 isoparametric beam elements

1 proposed element, 100 section-slices

q[N/m]

5

10

30

60

5

10

30

60

Substeps

2

3

5

15

2

3

5

15

Cumulative Iterations

8

18

55

107

5

10

26

47

5 Updated Lagrangian Curvilinear Beam Element for 2D Large …

77

geometry. This causes a continuous variation of the loads applied to the ending nodes which can be made explicit only if the internal shape is known.

5.5 Conclusion This paper introduces a new way to model 2D curvilinear beam elements. The core is that the proposed method allows to consider a single and long (end-to end) element, even if nonlinearities due to large displacements that continuously modify the shape of the structure occur. This implies the minimal use of d.o.f. and the possibility to mesh a generic curvilinear beam structure for a non-linear analysis as would be done for a linear analysis involving straight beams: namely the element nodes may be positioned only in correspondence to concentrated loads or imposed displacements. This is feasible because the proposed element formulation does not require any pre-set shape function that would otherwise limits the shape of the element forcing to adopt the necessity of mesh refinement. The shape functions are unnecessary because the internal displacement field is directly computed through the integration of the compatibility equations. These last provide a strong condition through which all the internal displacements field are only given by end-nodal displacements. The compatibility and equilibrium equations, together with the elastic stiffness characteristics of the cross-section, allow to derive the flexibility matrix of the element. In order to extend the flexibility matrix derivation to a new configuration at the end of a converged iteration, section-slices subdivision is performed. It allows to evaluate the flexibility matrix as a simple sum of slice flexibility contributions, that can be evaluated analytically once and for all. The d.o.f. of the curvilinear element keeps to 6 whatever is the varying curvature of the line. Consequently, the build-up of the end-to-end tangent stiffness matrix results immediate. The strategy engaged to address the large displacement analysis is the Update Lagrangian, but it would still be possible to employ the present method with other strategies as the Total Langrangian or Co-Rotational. Three numerical comparisons are showed that involve both analytical and numerical solutions obtained using RKF 45 method with a shooting technique and cubic isoparametric finite elements. The validations show a close overlap of the results. We want to point out that the proposed element requires a much lower number of iterations, probably in spite of the d.o.f. reduction of the modelling. Furthermore, at every iteration the computational cost is strongly reduced inasmuch the size of the stiffness matrix to invert is minimal (two nodes for the whole curvilinear beam).

References 1. Kadziela B, Manka M, Uhl T, Toso A (2016) Validation and optimization of the leaf spring multibody numerical model. Arch Appl Mech 85:1899–1914

78

C. Iandiorio and P. Salvini

2. Grade J., Hal Jerman T.W.: A Large-Deflection Electrostatic Actuator for Optical Switching Applications. 2000 Solid-State, Actuators, and Microsystems Workshop (2000). 3. Wagner T, Vella D (2013) Switch on, switch off: stiction in nano electro-mechanical switches. Nanotechnology 24:275501 4. Howell L (2001) Compliant mechanisms. Wiley 5. Frisch-Fay R (1962) Flexible bars. Butterworths, London 6. De Bona F, Zelenika S (1997) A generalized elastica-type approach to the analysis of large displacements of spring-strips. Proc Inst Mech Eng Part C- J Mech Eng Sci 211(7):509–517 7. Batista M (2014) Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions. Int J Solids Struct 51(13):2308–2326 8. Mingari Scarpello G, Ritelli D (2011) Exact solutions of nonlinear equation of rod deflections involving the Lauricella hypergeometric functions. Int J Math Math Sci, Hindawi, 1–22 9. Iandiorio C, Salvini P (2019) An analytical solution for large displacements of end-loaded beams. In: Proceedings of the 1st international conference on numerical modelling in engineering, vol 2, pp 320–338 10. Iandiorio C, Salvini P (2020) Large displacements of slender beams in plane: analytical solution by means of a new hypergeometric function. Int J Solids Struct 185–186:467–484 11. Barbieri E (2020) Analytical solution of the cantilevered elastica subjected to a normal uniformly distributed follower load. Int J Solids Struct 202:486–494 12. Rohde F (1953) Large deflections of a cantilever beam with uniformly distributed load. Quart Appl Math 11:337–338 13. Da Deppo DA, Schmidt R (1971) Analysis of nonlinear deflections of fibers. Text Res J 41(11):911–915 14. Wang CY (1986) A critical review of the heavy elastica. Int J Mech Sci 28(8):549–559 15. Iandiorio C, Salvini P (2020) Heavy Elastica soil-supported with lifting load and bending moment applied to an end: a new analytical approach for very large displacements and experimental validation. Int J Solids Struct 206:153–169 16. Iandiorio C, Salvini P (2022) Inflectional heavy elastica with unilateral contact constraint: analytical solution through the curvilinear abscissa mapping approximation. Int J Solids Struct 234–235:111258 17. Marotta E, Iandiorio C, Salvini P (2021) Experimental setup for the evaluation of large displacements in the inflected beams sustained to ground. In: IOP conference series: materials science and engineering, vol 1038 18. Banerjee A, Bhattacharya B, Mallik AK (2008) Large deflection of cantilever beams with geometric nom-linearity: analytical and numerical approaches. Int J Non Linear Mech 43(5):366–376 19. Sitar M, Kosel F, Brojan M (2014) A simple method for determining large deflection states of arbitrarily curved planar elastica. Arch Appl Mech 84:263–275 20. Tari H (2013) On the parametric large deflection study of euler-bernoulli cantilever beams subjected to combined tip point loading. Int J Non-Linear Mech 49:90–99 21. Masjedi PK, Weaver PM (2022) Analytical solution for arbitrary large deflection of geometrically exact beams using the homotopy analysis method. Appl Math Model 103:516–542 22. Simo JC, Vu-Quoc L (1986) On the dynamics of fexible beams under large overall motions-Te plane case: part I. J Appl Mech 53(4):849–854 23. Nanakorn P, Vu LN (2006) A 2D field-consistent beam element for large displacement analysis using the total Lagrangian formulation. Finite Elem Anal Des 42:1240–1247 24. Beheshti A (2016) Large deformation analysis of strain-gradient elastic beams. Comput Struct 177:162–175 25. Vo D, Nanakorn P (2020) A total Lagrangian Timoshenko beam formulation for geometrically nonlinear isogeometric analysis of planar curved beams. Acta Mech 231:2827–2847 26. Bathe KJ, Bolourchi S (1979) Large displacement analysis of three-dimensional beam structures. Int J Numer Methods Eng 14:961–986 27. Yang YB, Leu LJ (1991) Force recovery procedures in nonlinear analysis. Comput Struct 41(6):1255–1261

5 Updated Lagrangian Curvilinear Beam Element for 2D Large …

79

28. Yang YB, Kuo SR, Wu YS (2002) Incrementally small deformation theory for nonlinear analysis of structural frames. Eng Struct 24(6):783–798 29. Hosseini Kordkheili SA, Bahai H, Mirtaheri M (2011) An updated Lagrangian finite element formulation for large displacement dynamic analysis of three-dimensional flexible riser structures. Ocean Eng 38(5–6):793–803 30. Crisfield MA (1991) Non-linear finite element analysis of solids and structures: non-linear finite elements analysis, vol 1. Wiley 31. Felippa CA, Haugen B (2005) A unified formulation of small-strain corotational finite elements: I. Theory. Comput Methods Appl Mech Eng 194(21–24):2285–2335 32. Urthaler Y, Reddy JN (2005) A corotational finite element formulation for the analysis of planar beams. Int J Numer Methods Biomed Eng 21(10) 33. Battini J (2008) Large rotations and nodal moments in corotational elements. CMES-Comput Model Eng Sci 33(1):1–16 34. Nguyen DK (2013) A Timoshenko beam element for large displacement analysis of planar beams and frames. Int J Struct Stabil Dyn 12(06) 35. Elkaranshawy HA, Elerian AAH, Hussien WI (2018) A corotational formulation based on Hamilton’s principle for geometrically nonlinear thin and thick planar beams and frames. In: Mathematical problems in engineering, Hindawi, pp 1–22 36. Tang YQ, Du EF, Wang JQ, Qi JN (2020) A co-rotational curved beam element for geometrically nonlinear analysis of framed structures. Structures 27:1202–1208 37. Zienkiewicz OC, Taylor RL (2000) The finite element method, solid mechanics, vol 2. Wiley 38. Bathe KJ (1997) Finite element procedures. Prentice Hall, New Jearsey 39. Marotta E, Massimi L, Salvini P (2020) Modelling of structures made of filiform beams: development of a curved finite element for wires. Finite Elem Anal Des 170:103349 40. Ray D (2015) Computation of nonlinear structures: extremely large elements for frames. Wiley, Plates and Shells 41. Radenkovi´c G, Borkovi´c A (2018) Linear static isogeometric analysis of an arbitrarily curved spatial Bernoulli-Euler beam. Comput Methods Appl Mech Eng 341:360–396 42. Radenkovi´c G, Borkovi´c A (2020) On the analytical approach to the linear analysis of an arbitrarily curved spatial Bernoulli–Euler beam. Appl Math Model 77(Part 2):1603–1624 43. Molins C., Roca P., Barbat A.H.: Flexibility-based linear dynamic analysis of complex structures with curved-3D members. Earthquake Engineering and Structural Dynamics, 27(7) (1998). 44. Molari L, Ubertini F (2006) A flexibility-based finite element for linear analysis of arbitrarily curved arches. Numer Methods Eng 65(8) (2006). 45. Jafari M, Mahjoob MJ (2010) An exact three-dimensional beam element with nonuniform cross section. J Appl Mech 77(6):061009 46. Iandiorio C, Salvini P, Pre-integrated beam finite element based on state diagrams with elastic perfectly-plastic flow. (Awaiting Publication) 47. Iandiorio C, Salvini P (2022) An engineering theory of thick curved beams loaded in-plane and out-of-plane: 3D stress analysis. Eur J Mech A Solids 92:104484 48. de Saint-Venant B (1861) Note of conditions of compatibility. L’Institut 28:294–295 49. Todhunter I (1889) The elastical researches of Barré de Saint-Venant. Cambridge University Press, London 50. Beltrami E (1886) Sull’interpretazione meccanica delle formule di Maxwell. Rendiconti del Circolo Matematico di Palermo 3 51. Volterra V (1905) Sulle distorsioni generate de tagli uniformi. Rendiconti del Circolo Matematico di Palermo 5(14):329–342 52. Volterra V (1905) Sulle distorsioni dei solidi elastici più volte connessi. Rendiconti del Circolo Matematico di Palermo 5(14):351–356. Rendiconti del Circolo Matematico di Palermo 5(14):431–438 (1905) 53. Volterra V, Sulle distorsioni dei corpi elastici simmetrici 54. Volterra V (1907) Sur l’équilibre des corps élastiques multiplement connexes. Annales Scientifiques de l’Ecole Normale Supérieure, Paris 24(3):401–518

80

C. Iandiorio and P. Salvini

55. Cesàro E (1906) Sulle formole del Volterra fondamentali nella teoria delle distorsioni elastiche. Il Nuovo Cimento 12(1):143–154 56. Barber JR (2004) Elasticity, 2 edn. Kluwer Academic Publishers

Chapter 6

Modelling and Simulation of Micro-electro-Mechanical Systems for Energy Harvesting of Random Mechanical Vibrations Kailing Song and Michele Bonnin

Abstract We consider the problem of modelling and analyze nonlinear piezoelectric energy harvesters for ambient mechanical vibrations. The equations of motion are derived from the mechanical properties, the characterization of piezoelectric materials, and the circuit description of the electrical load. For random ambient vibrations, modelled as white Gaussian noise, the performances are analyzed through MonteCarlo simulations. Recently proposed solutions inspired by circuit theory, aimed at improving the power performances of energy harvesters are tested. Our results show that circuit theory inspired solutions, permit to design energy harvesters with significant improved performances, also in the case of random vibrations. Keywords Energy harvesting · Micro-electro-mechanical systems · Random vibrations · Nonlinear dynamical systems · Circuit theory · Noise · Stochastic processes · Stochastic differential equations

6.1 Introduction Over the past two decades, network technologies have become an integral part of our daily life. Networks of computers, routers, printers, mobile phones, and smart household appliances, that are wireless connected and exchange information through the Internet, have become common and are widespread. Wireless sensor and actuators networks (WSAN) are used both in work and home environments, to gather K. Song (B) IUSS, University School for Advanced Studies, Pavia, Italy e-mail: [email protected] M. Bonnin Department of Electronics and Telecommunication, Politecnico di Torino, Turin, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_6

81

82

K. Song and M. Bonnin

information about the surrounding ambient, and respond to the sensed events/data by performing appropriate actions. These networks may involve hundreds of nodes, made up by multi-domain systems, with both electrical and mechanical components. Among the many challenges posed by network technologies, the problem of how supply power to the nodes is of fundamental importance. WSAN often include nodes that are remotely located and difficult to access, making battery replacement inconvenient. Furthermore, the progressive miniaturization of electronic components discourage the use of batteries, that remain bulky and relatively heavy. Systems capable of self-powering, collecting energy from the surrounding environment whenever available, would represent the ideal solution. Energy harvesting refers to a set of technical solutions, to realize micro-scale systems that are able to scavenge energy from ambient sources, like mechanical vibrations, thermal gradients, dispersed electromagnetic radiations, etc., and convert it into usable electrical power [1–5]. Kinetic energy, in the form of ambient mechanical vibrations, is particularly interesting, because it is widespread and it has relatively high power density. Kinetic energy from ambient mechanical vibrations can be converted into electrical power using different physical principles. In particular, piezoelectric materials offer a relatively cheap, easy to implement and reliable mechanism [6–9]. If the energy of ambient vibrations is concentrated at a single frequency, or in a sufficiently narrow frequency band, they can be modelled by a simple sinusoidal force [10]. Conversely, if vibrational energy is spread over a relative wide frequency interval, ambient vibrations are best described as a stochastic process. If the spectrum is reasonably flat, white Gaussian noise is a convenient approximation, because the theory of stochastic differential equation driven by white Gaussian noise is well developed. Alternatively, if the finite, non null noise correlation time is taken into account, a colored noise process can be used [11, 12]. The most important factor limiting the performances of a piezoelectric energy harvester, is the impedance mismatch between the mechanical and the electrical parts. Impedance mismatch introduces a lag between current and voltage, that can be reduced introducing a proper reactive element in parallel with the load, a procedure known in circuit theory as power factor correction [10, 13]. In this work, we consider the problem of modelling a piezoelectric energy harvester subject to random mechanical vibrations, and its numerical analysis. The mathematical model is derived from the properties of the mechanical part, the constitutive equations of linear piezoelectric materials, and the circuit description of the electrical part. The model includes nonlinearities, which take into account nonlinear contributions in the mechanical stiffness of a cantilever beam. Ambient mechanical vibrations are modelled as a white Gaussian noise process. Consequently, the equations of motion are stochastic differential equations, that are solved using various numerical integration schemes. Inspired by recent work on the application of circuit theory to improve the efficiency of energy harvesting systems, we apply a power factor correction solution to the load [10, 14]. We evaluate the advantage offered by the modified load in terms of output average voltage, output average power and power efficiency. The numerical simulations shows that, even for the case of ran-

6 Modelling and Simulation of Micro-electro-Mechanical Systems …

83

dom mechanical vibrations, the application of power factor correction improves the performances by a significant amount.

6.2 Nonlinear Piezoelectric Energy Harvesting: Modelling A piezoelectric energy harvester is a multi-domain system, with both mechanical and electrical parts. The mechanical structure is responsible for capturing the kinetic energy of parasitic mechanical vibrations. It is composed by a cantilever beam, fixed at one hand to a vibrating support, with an inertial mass at the opposite end. The cantilever beam is covered by a layer of piezoelectric material, that converts mechanical stress and strain into electrical power. A schematic representation of a piezoelectric cantilever beam energy harvester is shown in Fig. 6.1. The equations of motion for the mechanical part are readily derived using classic mechanics (6.1) m x¨ + γ x˙ + U ' (x) + F pz = Fm (t) where m is the inertial mass, x is the displacement with respect to the equilibrium position, dots denote derivative with respect to time, γ is the damping coefficient, U (x) = 21 k1 x 2 + 41 k3 x 4 is a nonlinear elastic potential, and the ' denotes derivation with respect to the argument. For k3 = 0 the harvester is linear, otherwise it is nonlinear. Finally, F pz is the force exerted by the piezoelectric layer, and Fm (t) is the external force modelling ambient vibrations. The piezoelectric layer constitutes a mechanical-to-electrical power transducer. The governing equation for the transducer can be derived from the constitutive equations for a linear piezoelectric material [16]. Through spatial integration of the local variables, and neglecting the stiffness of the piezoelectric layer, the following relationships between mechanical and electrical variables are obtained

Fig. 6.1 Schematic representation of a piezoelectric cantilever beam energy harvester

V ibrating support Clamped end Load

P iezoelectric layer Inertial mass

m M otion

Cantilever beam

84

K. Song and M. Bonnin

Fig. 6.2 Electrical load connected to the piezoelectric transducer PZ. Mechanical quantities are applied to the left port of the transducer and electrical quantities to the right. a Resistive-inductive load. b Resistive load

F pz = α e q pz = −α x + C pz e

(6.2) (6.3)

where α is the electro-mechanical coupling (in N/V or As/m), C pz is the electrical capacitance of the piezoelectric layer, and e is the output voltage. The voltage is used to supply power to an electrical load, typically modelled as a resistor (see Fig. 6.2b). One of the most important factors limiting the harvested power, is the impedance mismatch between the electrical and the mechanical parts. In [10], it was proved, for the case a single frequency external excitation, that the harvested average power and the power efficiency can be significantly increased placing a reactive element in parallel with the resistive load, as shown in Fig. 6.2a. The role of the reactive element, is to reduce the lag between the current and voltage induced by the capacitive reactance of the piezoelectric transducer, a method known in circuit theory as power factor correction. For the resistive-inductive load of Fig. 6.2a, application of Kirchhoff current law to the node a yields q˙ pz + i + Ge = 0 (6.4) where G = 1/R is the load conductance. Introducing the linear momentum y = x/m, ˙ ˙ Eq. (6.1) and the and using the constitutive relationship for a linear inductor e = L i, time derivative of (6.3) give the state equations x˙ =

1 y m

y˙ = −U ' (x) −

(6.5a) γ y − α e + Fm (t) m

1 i˙ = e L ) 1 (α e˙ = y−i −Ge C pz m

(6.5b) (6.5c) (6.5d)

6 Modelling and Simulation of Micro-electro-Mechanical Systems …

+ vC (q) − ϕ

q˙ R1

+ − vin(t)

Ambient vibrations

L1

M echanical domain

iP + vP −

1:n

iS + vS −

P iezoelectric transducer

c

85

q˙pz i Cpz L

R

+ e −

Load

Fig. 6.3 Equivalent circuit for a piezoelectric nonlinear energy harvester with a resistive-inductive load

It is convenient to transform the electromechanical system into an equivalent electrical circuit. Equivalent means that the two systems are described by the same differential equations, although with different meaning of the state variables. Exploiting mechanical-to-electrical analogies, displacement x is replaced by charge q, momentum y by flux linkage ϕ, mass m by inductance L 1 , stiffness constants k1 , k3 by the inverse of capacitance 1/C1 , 1/C3 , friction γ by resistance R1 , and force Fm (t) by voltage vin (t). The equivalent circuit for the electromechanical system, obtained after transformation of variables is shown in Fig. 6.3. The nonlinear stiffness of the beam is accounted for by a nonlinear charge controlled capacitor, with characteristic relationship vC (q) = C11 q + C13 q 3 . The piezoelectric transducer is modelled by an ideal transformer with turns ratio 1:n, in parallel with a capacitor of capacitance C pz . It is straightforward to derive the equations for the piezoelectric transducer model. Using the constitutive relationships for an ideal transformer, vs = nv p and i s = − n1 i p , where v p , vs , i p , i s are the voltages and currents at the primary and secondary windings, respectively, and applying Kirchhoff current law to node c, we have e = n vp 1 q˙ pz = − q˙ + C pz e˙ n

(6.6) (6.7)

that correspond to (6.2) and (6.3), where mechanical quantities have been replaced by their electrical analogous. It is worth mentioning that the turns ratio n = α−1 is an a-dimensional parameter. Rigorously speaking, n should have dimension V/N. However in mechanical-to-electrical analogies voltage and force are equivalent entities, known as “effort”. To assess the advantage in harvested power offered by a resistive-inductive load in the case of random mechanical vibrations, we shall consider also a simple resistive ˙ in load, as the one shown in Fig. 6.2b. Introducing the linear momentum y = x/m

86

K. Song and M. Bonnin

Eq. (6.1), taking derivatives with respect to time in (6.3) and using the Ohm’s law q˙ pz = −G e yields the state equations x˙ =

1 y m

γ y˙ = −U ' (x) − y − αe + Fm (t) m ) 1 (α y−Ge e˙ = C pz m

(6.8a) (6.8b) (6.8c)

The equivalent circuit for the energy harvester with resistive load is analogous to the one shown in Fig. 6.3, but without the inductor L in parallel with the resistor R.

6.3 Stochastic Differential Equations Ambient mechanical vibrations are best described as stochastic processes. When the external force Fm (t) is modelled as a random noise, the differential equations (6.5) or (6.8) become stochastic differential equations (SDEs). Let (Ω, F , P) be a probability space, where Ω is the sample space, F = (Ft )t≥0 is a filtration, e.g. the σ-algebra of all the events, and P is a probability measure. Let Wt = W (t) be a one dimensional Wiener process, characterized by = 0 (symbol denotes mean value), cov(Wt , Ws ) = = min(t, s) and Wt ∼ N (0, t), where symbol ∼ means “distributed as”, and N (0, t) denotes a normal distribution, centered at zero. A d-dimensional system of SDEs driven by the one-dimensional Wiener process Wt is1 d X t = a(X t )dt + B(X t )dWt (6.9) X t : Ω |→ Rd is a vector valued stochastic process, e.g. a vector of random variables parametrized by t ∈ T . The parameter space T is usually the half-line [0, +∞[. Alternatively, the stochastic process can be thought of as the function X t : Ω × T |→ Rd . The vector valued functions a : Rd |→ Rd , B : Rd |→ Rd , called the drift and the diffusion, respectively, are measurable functions satisfying a global Lipschitz and a linear growth conditions, to ensure the existence and uniqueness solution theorem [15]. If the function B(X t ) is constant, then noise is said to be un-modulated or additive, otherwise it is said to be modulated or multiplicative. For numerical simulations, it is often convenient to work with a-dimensional variables and time scaled equations. In the special case of a nonlinear piezoelectric energy harvester, the d-dimensional system of SDEs can be recast in the form (see Sect. 6.4): 1

We adopt the standard notation used in probability: Capital letters denote random variables, lower case letters denote their possible values.

6 Modelling and Simulation of Micro-electro-Mechanical Systems …

d X t = ( A X t + n(X t )) dt + B dWt

87

(6.10)

where the constant matrix A ∈ Rd,d and the function n : Rd |→ Rd collect linear and nonlinear terms of the drift, respectively, and the diffusion matrix is assumed to be constant (un-modulated noise). A-dimensional equations are obtained as a special case of linear transformed variables y = P x, where the matrix P ∈ Rd,d is a constant matrix. In order for the transformation to be invertible, P must be regular. The new variables are stochastic processes with differential ( ) dY t = P A P −1 Y t + P n( P −1 Y t ) dt + P B dWt

(6.11)

Linear time scaling is obtained introducing in Eq. (6.11) the time transformation t → t ' = τ (t) = t/T , that implies dt = T dt ' . Using the change of time theorem for Itô integrals (see [15], p. 156) the time scaled Wiener process is Wτ (t) =

√ 1 τ ' (t) Wt = √ Wt T

with differential dWt =

√

T dWt '

(6.12)

(6.13)

Finally the SDEs for the transformed variables with time scaling is dY t ' = T

(

√ ) P A P −1 Y t ' + P n( P −1 Y t ' dt ' + T P B dWt '

(6.14)

6.4 Nonlinear Piezoelectric Energy Harvesting: Stochastic Model Analysis The assumption that ambient mechanical vibrations can be modelled as white Gaussian noise, leads to substitute the external voltage source vin (t) in the equivalent circuit, with a stochastic voltage source vin (t)dt = D dWt , where D > 0 is a constant that measure the noise intensity and variance. The system of differential equations (6.5) is rewritten as a system of SDEs in the form (6.10), with X t = [q, ϕ, i, e]T , n(X t ) = [0, −q 3 /C3 , 0, 0]T , B = [0, D, 0, 0]T and ⎡ ⎤ 1 0 0 0 L1 ⎢ ⎥ ⎢ 1 1 ⎥ ⎢ − C − LR1 ⎥ 0 − n ⎥ 1 1 ⎢ ⎥ (6.15) A=⎢ ⎢ 1 ⎥ ⎢ 0 ⎥ 0 0 ⎢ L ⎥ ⎣ ⎦ 0 nC pz1 L 1 − C1pz − CGpz

88 Table 6.1 Values of circuit components, based on [17]

K. Song and M. Bonnin Parameter

Value

R1 C1 L1 C pz R n D

6.9366 Ω 5.874 µF 1H 80.08 nF 1 MΩ 37.4254 10 mV

The SDEs for a-dimensional variables are obtained introducing the diagonal −1 −1 facmatrix P = diag[Q −1 , T Q −1 L −1 1 , T Q , C 1 Q ], where Q is a normalization √ tor that has dimension of a charge, and T is the time scaling factor T = L 1 C1 . Similarly, for the resistive load case the system of differential equations (6.8) can be rewritten in the form (6.10), with X t = [q, ϕ, e]T , n(X t ) = [0, −q 3 /C3 , 0]T , B = [0, D, 0]T and ⎡ ⎤ 1 0 0 L1 ⎢ ⎥ ⎢ ⎥ R A = ⎢ − C11 − L 11 − n1 ⎥ (6.16) ⎣ ⎦ 0 nC pz1 L 1 − CGpz SDEs for the a-dimensional variables are obtained using√the matrix −1 P = diag[Q −1 , T Q −1 L −1 L 1 C1 . 1 , C 1 Q ], and the time scaling factor T = To verify whether the power factor corrected load increases the harvested power also for the white Gaussian noise case, we have performed Monte Carlo simulations on the SDEs system (6.14), for both the resistive-inductive and the resistive loads. The SDEs have been solved numerically using various numerical integration schemes: Euler-Maruyama, strong order 1 stochastic Runge–Kutta, and weak order 2 stochastic Runge–Kutta [18]. Time simulation length was △t = 104 s, with a constant time integration step δt ≈ 30 µs. Given the relatively small time step, different numerical schemes do not show significant differences between each other. Expected values were calculated averaging over 100 different realizations of the Wiener process Wt . In our simulations, we used values of the circuit components taken from [17], and summarized in Table 6.1. For the nonlinearity, we assumed C3 = Q 2 · C1 , where Q = 1C. √ Figure 6.4 shows the√root mean square values for the current ir ms = and the output voltage er ms = , versus the value of the inductance L, for the resistiveinductive load case. The output voltage has a maximum at L = 73.3934 H. We stress that the high value of the optimum inductance is a consequence of the fact that inductance L 1 is normalized to 1H (see Table 6.1). The maximum value for the output voltage is emax R L ,r ms = 5.6423 V. By comparison, for the same values of the components, the resistive load offers a maximum output voltage emax R L ,r ms = 2.0263 V.

6 Modelling and Simulation of Micro-electro-Mechanical Systems …

89

Fig. 6.4 Root mean square values for the current i (left) and the output voltage e (right) versus the value of the inductance L, for the resistive-inductive load

The average power absorbed by the load is Pout = G er2ms

(6.17)

Figure 6.5 shows the output average power (on the left) and the power efficiency (on the right), versus the value of the inductance. Obviously, the maximum output average power and maximum output voltage are achieved for the same value of the inductance L. It is worth noting that, because the power Again, the resistiveinductive load offers a much higher output power with respect to the simple resistive load. To determine the power efficiency, the average power injected into the system by the noise is needed. Tellegen’s theorem implies that the only elements absorbing average power in the equivalent circuit of Fig. 6.3 are the resistors R1 and R. Using the constitutive relationship for the inductor L 1 implies q˙ = ϕ/L 1 , we have that the input average power is R1 (6.18) Pin = 2 ϕr2ms + G er2ms L1 √ where ϕr ms = . The power efficiency is then given by ε = Pout /Pin . Table 6.2 summarizes the main results for both a resistive load and a resistiveinductive load. The latter solution offers better performances in all aspect. In particular, the output voltage is increased by almost three times, while the output power and power efficiency are increased by almost eight times.

Table 6.2 Performances comparison for a simple resistive load, and a resistive-inductive load, with L = 73.3634H ermax Pout ε (%) Load ms Resistive Resistive-inductive

2.0263 V 5.6423 V

4.1059 µW 31.835 µW

8.217 63.67

90

K. Song and M. Bonnin

Fig. 6.5 Output average power (on the left) and power efficiency (on the right) versus the value of the inductance L, for the resistive-inductive load

Fig. 6.6 Marginal stationary probability distribution for the output voltage. Resistive load (left) and resistive-inductive load (right). Value of the inductance is L = 73.3934H

Finally, Fig. 6.6 shows the asymptotic marginal probability distribution for the output voltage pst (e) = lim p(e, t). The probability to find a voltage value in t→+∞

the interval e + de is evaluated as the fraction of samples falling in such an interval, normalized to the total number of samples. The stationary distribution for the resistive load (on the left) is compared to the one the resistive-inductive load (on the right). Positioning the inductor in parallel with the resistor, the variance of the output voltage is increased, as it can be clearly seen from the stationary distributions (note the different x-scales).

6.5 Conclusions Piezoelectric energy harvesters are micro-electro-mechanical systems, that are capable to convert ambient mechanical vibrations into usable electrical energy. They can be used as a power source for electronic circuits, sensors and actuators, and are par-

6 Modelling and Simulation of Micro-electro-Mechanical Systems …

91

ticularly well suited to supply power to wireless networks of miniaturized sensors and actuators. The most important limiting factor for vibrational energy harvesting is the impedance mismatch between the mechanical and the electrical parts. A possible solution is to use power factor correction method of circuit theory. Positioning a reactive element in parallel with the load, the lag between the voltage across and the current through the load can be reduced, thereby increasing the absorbed power. In this work we analyzed a nonlinear piezoelectric energy harvester for ambient mechanical vibrations. The equations of motion have been derived from the mechanical properties, the characteristic equations of linear piezoelectric materials and the electrical circuit description of the load. In the case of random ambient vibrations described as white Gaussian noise, the resulting stochastic differential equations have been solved numerically, and expected quantities have been calculated using Monte-Carlo simulation methods. Our analysis shows that the power factor corrected solution offer better performances in terms of output voltage, output average absorbed power and power efficiency. The output voltage is increased by almost three times, while absorbed power and power efficiency are increased by almost eight times.

References 1. Roundy S, Wright PK, Rabaey JM (2003) Energy scavenging for wireless sensor networks. Springer, Boston 2. Paradiso JA, Starner T (2005) Energy scavenging for mobile and wireless electronics. IEEE Pervas Comput 4:18–27 3. Beeby SP, Tudor MJ, White N (2006) Energy harvesting vibration sources for microsystems applications. Meas Sci Technol 17:R175 4. Mitcheson P, Yeatman E, Rao G, Holmes A, Green T (2008) Energy harvesting from human and machine motion for wireless electronic devices. Proc IEEE 96:1457–1486 5. Lu X, Wang P, Niyato D, Kim DI, Han Z (2015) Wireless networks with RF energy harvesting: a contemporary survey. IEEE Commun Surv Tutor 17:757–789 6. Khaligh A, Zeng P, Zheng C (2009) Kinetic energy harvesting using piezoelectric and electromagnetic technologies-state of the art. IEEE Trans Ind Electron 57:850–860 7. Vocca H, Neri I, Travasso F, Gammaitoni L (2012) Kinetic energy harvesting with bistable oscillators. Appl Energy 97:771–776 8. Wen X, Yang W, Jing Q, Wang ZL (2014) Harvesting broadband kinetic impact energy from mechanical triggering/vibration and water waves. ACS Nano 8:7405–7412 9. Fu Y, Ouyang H, Davis RB (2018) Nonlinear dynamics and triboelectric energy harvesting from a three-degree-of-freedom vibro-impact oscillator. Nonlinear Dyn 92:1985–2004 10. Bonnin M, Traversa FL, Bonani F (2021) Leveraging circuit theory and nonlinear dynamics for the efficiency improvement of energy harvesting. Nonlinear Dyn 104:367–382 11. Daqaq MF (2010) Response of uni-modal Duffing-type harvesters to random forced excitations. J Sound Vib 329:3621–3631 12. Bonnin M, Traversa FL, Bonani F (2020) Analysis of influence of nonlinearities and noise correlation time in a single-DOF energy-harvesting system via power balance description. Nonlinear Dyn 100:119–133 13. Huang D, Zhou S, Litak G (2019) Analytical analysis of the vibrational tristable energy harvester with a RL resonant circuit. Nonlinear Dyn 97:663–677

92

K. Song and M. Bonnin

14. Bonnin M, Traversa FL, Bonani F (2022) An impedance matching solution to increase the harvested power and efficiency of nonlinear piezoelectric energy harvesters. Energies 15:2764 15. Øksendal B (2003) Stochastic differential equations. Springer, Berlin 16. Priya S, Inman DJ (2009) Energy harvesting technologies, vol 21. Springer, Boston (2009) 17. Yang Y, Tang L (2009) Equivalent circuit modeling of piezoelectric energy harvesters. J Intel Mat Syst Str 20(18):2223–2235 18. Särkkä S, Solin A (2019) Applied stochastic differential equations, vol 10. Cambridge University Press

Chapter 7

Revisiting a Model that Describes the Process of the Vocal Oscillation During Phonation, a Numerical Approach M. Filomena Teodoro Abstract A numerical scheme is revisited, which approaches the solution of a particular functional differential equation and the mucosal wave model of the vocal oscillation during phonation. The mathematical equation describes a superficial wave propagating through the tissues. The numerical scheme is adapted from the work developed previously by the author and collaborators. Keywords Mixed-type functional differential equations · Non linear equations · Vibration of elastics tissues · Numerical approximation · Method of steps · Human phonation · Mechanical system

7.1 Introduction Numerous mathematical models have the form of differential equations with delayed and advanced arguments usually denominated mixed type functional differential equations (MTFDEs) [1]. We can find MTFDEs in a wide range of study areas, namely modeling traveling waves in a spatial lattice [2, 3], in optimal control [4, 5], in nerve conduction [6–9], quantum photonic physics [10], in economics and financial dynamics [11], in physiology [12–18]. In last decade, we have focused our interest in a MTFDE with the form (7.1) with symmetric shifts ±τ1 x ' (t) = F(t, x(t), x(t − τ1 ), . . . , x(t + τ1 )), τ1 > 0.

(7.1)

M. F. Teodoro (B) CINAV, Center of Naval Research, Naval Academy, Portuguese Navy, 2810-001 Almada, Portugal e-mail: [email protected]; [email protected] CEMAT, Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, Lisbon University, 1048-001 Lisboa, Portugal © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_7

93

94

M. F. Teodoro

Some numerical schemes to solve numerically some particular cases of (7.1) such as for the linear case (authonomous and non authonomous equations) [2, 19–25] and non linear cases (in particular, the FitzHugh-Nagumo equation) [9, 26]. In this manuscript we consider another non-linear delayed-advanced functional differential equation with symmetric delay and advance which arose in acoustics and can represent the voice source production analyzing the dynamical behavior of a vocal fold. It consists in the interaction of a flowing fluid (air, blood,...) with an elastic structure tissue, usually defined as the aero-elastic oscillatory phenomena (AOP). The equation under study is also applied in other physiological systems such as avian syrinx, snore, or a flow passing a constricted channel (artery, lips, soft palate, nostrils). We considered the model presented by the author of [12, 27]. This model simulates the flow induced oscillations of a superficial wave propagating through the tissues driven by the airflow from the lungs: the mucosal wave model. Under the aim of dynamics phonation, the mucosal wave model consists in single coupled spring mass driven by the airflow. This model was first introduced in 1968 by the author of [28]). The present manuscript is summarized in five sections. In Sect. 7.2 we remember the method of steps in first Subsection, in second subsection is made a short state of art. In Sect. 7.3 details the problem of interest and analyzes the equation to solve. The numerical approach is described in Sect. 7.3. Section 7.4 we present some details about the numerical approximation and present some results. In fifth Section we get some conclusions.

7.2 Some Prelimimaries 7.2.1 Method of Steps Under the aim to solve a non linear MTFDE from acoustics we revisit the method of steps (MS) theorem verified in [29], for a linear MTFDE. Theorem 7.1 Assume the non-degeneracy condition that γ (t) /= 0, for t ≥ 0, so that the non authonomous MTFDE can be rewritten in the form (7.2) x(t + 1) = a(t)x ' (t) + b(t)x(t − 1) + c(t)x(t), t ≥ 0 and a(t) =

1 , γ (t)

(7.2)

b(t) = − γβ(t) and c(t) = − γα(t) with (t) (t) x(t) = ϕ(t), t ∈ [−1, 1],

where the function ϕ is defined by { ϕ1 (t), i f t ∈ [−1, 0], ϕ(t) = ϕ2 (t), i f t ∈ (0, 1].

(7.3)

(7.4)

7 Revisiting a Model that Describes the Process of the Vocal Oscillation …

95

Let x be the solution of problem (7.2), (7.4), where α(t), β(t), γ (t) ∈ C 2L ([−1, 2L + 1]), γ (t) /= 0, t ∈ [−1, 2L + 1], ϕ1 (t) ∈ C 2L+1 ([−1, 0]), ϕ2 (t) ∈ C 2L+1 ([0, 1]) f or some L ∈ N.

(7.5)

Moreover, suppose that ϕ1(l) (0− ) = ϕ2(l) (0+ ), ϕ2 (1) = a(0)ϕ1' (0− ) + b(0)ϕ1 (−1) + c(0)ϕ1 (0); )| l ( ϕ2(l) (1− ) = dtd l a(t)ϕ1' (t) + b(t)ϕ1 (t − 1) + c(t)ϕ1 (t) |t=0− , l = 0, 1, 2, . . . , 2L + 1.

(7.6)

Then there exist functions δi,l , εi,l , δ¯ i,l , ε¯ i,l ∈ C([−1, 2L + 1]), l = 1, . . . , L, i = 0, 1, . . . , 2l, such that the following formulae are valid: ∑2l−1 i=0

δi,l (t)ϕ1(i) (t − 2l) +

∑2l−1

εi,l (t)ϕ2(i) (t − 2l + 1), t ∈ [2l − 1, 2l]; ∑2l−1 ∑2l ε¯ i,l (t)ϕ2(i) (t − 2l) + i=0 δ¯ i,l (t)ϕ1(i) (t − 2l − 1), x(t) = i=0 t ∈ [2l, 2l + 1] l = 1, 2, . . . .

x(t) =

i=0

(7.7)

Moreover, the solution x, constructed according to the formulae (7.7), belongs to the class ∩ ∩ ∩ C 2L+1 ([−1, 1)) C 2L ([−1, 2)) ··· C 1 ([−1, 2L + 1)). (7.8) The MS is an adequate tool often used to extend the solution of a delay differential equation in a larger interval when the solution is known in in an initial interval providing sufficient conditions for the existence of solution. Notice that the MS is applied for autonomous and non autonomous linear MTFDEs. Our equation of study is a non linear MTFDE implying the imposition of reducing the equation of interest to the linear case by the use of Newton method, similarly to the technique used in [9, 30], using a uniform mesh, reducing the problem to a boundary value problem on a bounded interval. The non-lineariety of the equation under study implies the use of an adapted method of steps.

7.2.2 Some History We can find numerous models for the vocal folds at phonation. These models make part of a large set that try to represent the dynamical behavior of some aeroelastic

96

M. F. Teodoro

oscillators in physiology. In [12] the author consider the vocal fold oscillation as a flow induced mechanical system that can show some instabilities depending on the flow conditions being able to occur some frictional energy transfer from glotal airstream to tissue. At 1968 a low dimensional model was presented in [28] consisting in a coupled single spring mass driven by the airflow from the lungs. The model try to characterize the oscillation as superficial wave propagating through the tissues in the direction of the airflow. A two mass model was proposed in [31] exploring the fluid mechanical considerations of vocal cord vibration. In 1977 the same author publishes a book about speech physiology and acoustic phonetics [32]. The same interest is revealed in [13, 33]. The one mass model and two mass model have suffered numerous improvements and/or variants since their initial proposal [34, 35]. The oscillation depends on the relation between inertial and elastics properties and the geometry issues. The author of [36] confirm such dependence studying the influence of geometrical and mechanical input parameters on theoretical models of phonation. In [34] Drioli simplifies the one mass model introducing a parametric nonlinear component maintaining the flow induced oscillation. Titze, in [27] consider one mass model with small amplitude oscillations. By opposite, in [37] is studied the effect of dehydration on properties of vocal folds for large amplitude oscillations case. The author of [38] analyzes a one mass models but with a non constant τ , in opposite of the work presented in [12]. In [39], a lumped mucosal wave model of vocal folds was revisitated and some extensions were proposed extensions and oscillation hysteresis analysis is performed. The effect of viscosity in the oscillation amplitude is studied in [40]. The authors of [17] study aerodynamically and acoustically driven modes of vibration. At the same time, in [41] is done a theoretical and experimental validation of some extended models. More recently, the authors of [42] use a three-dimensional vocal fold model to evaluate the vocal fold stiffness versus dehydration levels. Similar study is found in [43] where it is calculated the effect of four dehydration levels on the viscoelastic properties of the vocal fold mucosa. A combined model using a mechanical model with a machine learning effort to model physical systems with a Hopf bifurcation was introduced in [16].

7.3 The Problem We are interested in the numerical MTFDE that models the vibration of some elastics tissues, by the interaction of a flowing fluid (air, blood,...) with an elastic structure tissue, the aero-elastic oscillatory phenomena. This phenomena often happens in physiology, for example when the oscillation of a superficial wave propagates in the tissues on the direction of the flow. We consider a MTDE from acoustics denominated mucosal wave model of the vocal oscillation during phonation, a model proposed by Titze in [12, 27] for small amplitude oscillations. In Fig. 7.1 we can observe the one mass model of the vocal fold. x1 defines the displacement at the entrance of glottis relatively to vertical axis. x2 defines the displacement at the exit of glottis relatively to vertical axis. T is the thickness of fold, Pm the is driving pressure at the

7 Revisiting a Model that Describes the Process of the Vocal Oscillation …

97

Fig. 7.1 One mass model of the vocal fold. Adapted from [34]

entrance of glottis and its assumed that it equals the pressure of the lungs. Pg the is driving pressure at the exit of glottis and its assumed that it equals the atmospheric pressure. Also, like in [12, 38], it is supposed that airflow is incompressible and quasistationary, consequently, within the glottis we can consider the flow independent of the vertical position and also uni-dimensional. The mechanical system is focused at mid point of glottis and it described by the coupled mass springer equation (7.10), with x(t)s the displacement of tissue at the midpoint of the glottis. We can get the equation of motion, a nonlinear MTFDE with deviating arguments, given by M x '' (t) + Bx ' (t) + K x(t) = Fm ,

(7.9)

with M, B, K and kt are, respectively, the effective mass, damping factor and stiffness per area unit of vocal fold medial surface and the transglottal pressure coefficient. Fm the force that drives the fold. Taking Pav as the average glottal pressure w e get Fm = Pav . Equation (7.9) can be rewritten in form (7.10) M x '' (t) + Bx ' (t) + K x(t) =

PL x(t − τ ) − x(t + τ ) , kt x0 + x(t + τ )

(7.10)

where, by imposing the glottis is open, (x0 + x(t + τ ) > 0 and defining τ as the time that the wave travels to the edges of vocal fold at z = ±T /2 (upper and lower edges respectively), τ is given by τ = T /2c, with c is the velocity of the wave.

98

M. F. Teodoro

7.4 Numerical Scheme Similarly to the work developed in [9, 26, 30], where an advanced-delayed differential equation from nervous conduction was analyzed and solved numerically, we have performed a similar approach. We considered, like the author of [38], the adimensional version of (7.10) transforming the x variable into y = x/x0 , obtaining (7.11) y '' (t) + αy ' (t) + ω2 y(t) = p

y(t − τ ) − y(t + τ ) . 1 + y(t + τ )

(7.11)

and deducing the formula of an adapted method of steps so we could extend the solution on the intervals of interest supposing enough regularity. If we consider successive intervals with amplitude τ , the MS formula is given by formula (7.12) y(t + τ ) = − pn (t)(y '' (t) + αy ' ) + y(t − τ ) + g(y(t)), t ∈ R

(7.12)

where g(y(t)) = − pn (t)ω2 y(t) and pn (t) a polynomial function with order n ∈ N . To linearize the problem, we applied the Newton method obtaining a set of BVP with boundary functions defined by the Titze approximation. The iterative problem stops when some rule is satisfied. So, the nonlinear problem is simplified as sequence of linear boundary problems by means of the Newton iterative scheme. The last step consisted in the application of least squares algorithm, finite element method and collocation method to get the approximate solution. In [44] we have used the same parameters defined in [27], in the present work, we have considered the parameters displayed in the Table 7.1, the same that Drioli have used in [34].

Table 7.1 Considered parameters. Adapted from [34] Valiue Parameter ρ air density (kg·m−3 ) m mass (kg) r damping (N·s·m−3 ) k stiffness (N·m−1 ) L glottal length (m) Pl lung pressure (N·m−2 ) Sm fold surface (mm2 ) T fold thickness (mm) x01 , x02 rest positions (mm)

1.15 0.00017 0.069 34 0.014 300–3000 15–60 0.06–2 0.005–1

7 Revisiting a Model that Describes the Process of the Vocal Oscillation …

99

√ Table 7.2 Numerical results using different methods. ε = ||x − x˜ N ||2 / (k − 1) on k successive intervals, k = 3, 4, 5. N − 1Points k=3 k=4 k=5 N ε p ε p ε p Finite elements 32 8.14e − 6 64 6.92e − 7 128 5.97e − 8 Least squares 32 2.82e − 5 64 3.51e − 6 128 4.37e − 7 Collocation 32 2.14e − 4 64 5.12e − 5 128 1.25e − 5

3.57 3.56 3.54

2.77e − 6 3.36 2.58e − 7 3.43 2.3362e − 9 3.4624

1.36e − 6 1.24e − 7 1.17e − 9

3.36 3.38 3.43

3.00 3.01 3.01

1.27e − 5 1.57e − 6 1.95e − 7

2.99 3.01 3.01

4.92e − 6 6.08e − 7 7.67e − 8

2.97 3.01 2.99

2.12 2.07 2.03

6.69e − 5 1.55e − 5 3.73e − 6

2.19 2.11 2.06

1.97e − 5 4.41e − 6 1.04e − 6

2.25 2.16 2.09

7.5 Numerical Results The chosen parameters and initial values are displayed in [34] (see Table 7.1). We have used an uniform mesh. In Table 7.2 are presented the absolute error ε N (2-norm) and the estimated order of convergence p = log2 ε2N /log2 ε N of approximate solution of the equation (7.11) using a Least Squares Scheme, Finite Elements Method and Collocation with N points mesh. The approximate solution appear to have a good accuracy for the three methods. The estimate order of convergence is in agreement with the chosen method. For collocation, p ≈ 2; for FEM; p ≈ 3.5; for least squaes, p ≈ 3. The FEM promoted more accurate estimates as expeted. When the COL method is applied, the absolute error is of order 2 × 10−4 for both set of parameters, with a partition of 31 subintervals. The estimated order of convergence p is compatible with the expected one, p ≈ 2.

7.6 Conclusion Our initial proposal was to apply the more recent approaches and techniques using an adapted method of steps for the nonlinear case (7.11). The method introduced previously in [26] and extended from [9], using a numerical scheme based on an adapted method of steps, was rebuilt and re-adapted, using a uniform mesh. Using MS, FEM and collocation we compute the solution to any interval, imposing enough regularity to the initial functions.

100

M. F. Teodoro

Also, we needed to chose an adequate set of parameters. We have selected the set established in [34]. We have got adequate accuracy and the estimates of order of convergence were compatible with the 3 used methods, FEM, LS and collocation. COLL and by FEM were accurate. The use of Homotopic analytical method, some times used [24] to solve some delay differential equations is a suggestion for future. One possible issue as future work is to determine the threshold for x0 (the minimum value) so the wave can propagate. Following [45–48], the selection of distinct bases of functions is another point to perform in future. Acknowledgements This work was supported by Portuguese funds through the Center of Naval Research (CINAV), Portuguese Naval Academy, Portugal and The Portuguese Foundation for Science and Technology (FCT), through the Center for Computational and Stochastic Mathematics (CEMAT), University of Lisbon, Portugal, project UID/Multi/04621/2019.

References 1. Caruntu B, Bota C (2014) Analytical approximate solutions for a general class of nonlinear delay differential equations. Sci World J 2014 (2014) 2. Abell K, Elmer C, Humphries A, Vleck EV (2005) Computation of mixed type functional differential boundary value problems. SIADS - Siam J Appl Dyn Syst 4:755–781 3. Mallet-Paret J (1999) The global structure of traveling waves in spatially discrete dynamical systems. J Dyn Differ Equ 11:49–128 4. Pontryagin L, Boltyanskii V, Gamkrelidze R, Mishchenko E (1962) The mathematical theory of optimal process. Interscience, New York 5. Rustichini A (1989) Functional differential equations of mixed type: the linear autonomous case. J Dyn Differ Equ 1:121–143 6. Bell J (1984) Behaviour of some models of myelinated axons. IMA J Math Appl Med Biol 1:149–167 7. Bell J, Cosner C (1983) Threshold conditions for a diffusive model of a myelinated axon. J Math Biol 18:39–52 8. Chi H, Bell J, Hassard B (1986) Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction. J Math Biol 24:5583–601 9. Pinelas S, Chipot M, Dosla Z (eds) (2013) Analysis and computational approximation of a forward-backward equation arising in nerve conduction. Springer Proceedings in Math and Stat, vol 47. Springer Science+Business Media, New York 10. Alvarez-Rodriguez U, Perez-Leija A, Egusquiza I, Grafe M, Sanz M, Lamata L, Szameit A, Solano E (2017) Advanced-retarded differential equations in quantum photonic systems. Sci Rep 7 11. Rustichini A (1989) Hopf bifurcation for functional differential equations of mixed type. J Dyn Differ Equ 1:145–177 12. Titze I (1994) Principles of voice production. Prentice-Hall, Englewood Cliffs 13. Hirano M (1981) Clinical examination of voice. Springer, New York 14. Hopf bifurcation in an advanced-delay differential equation for the aeroelastic oscillation of some physiological systems. In: Proceeding series of the brazilian society of computational and applied mathematics. Sociedade Brasileira de Matemática Aplicada e Computacional. Melville, New York (2007). http://arquivo.sbmac.org.br/eventos/cnmac/xxx_cnmac/PDF/27. pdf

7 Revisiting a Model that Describes the Process of the Vocal Oscillation …

101

15. Sayce L, Zimmerman Z, Gartling G, Rousseau B, Branski R (2012) Epithelial response to vocal fold microflap injury in a preclinical model. The Laryngoscope. https://doi.org/10.1002/ lary.30169 16. Lee KH, Barton DAW, Renson L (2022) Modelling of physical systems with a hopf bifurcation using mechanistic models and machine learning. https://arxiv.org/pdf/2209.06910.pdf 17. Zhang Z, Neubauer J, Berry DA (2006) Aerodynamically and acoustically driven modes of vibration in a physical model of the vocal folds. J Acoust Soc Am 120(5):2841–2849 18. Ishizaka K, Matsudaira M (1972) Fluid mechanical considerations of vocal cord vibration. Speech communications research laboratory, monography, vol 8 19. Iakovleva V, Vanegas C (2005) On the solution of differential equations withe delayed and advanced arguments. Electron J Differ Equ Conf 13:57–63 20. Ford N, Lumb P (2009) Mixed-type functional differential equations: a numerical approach. J Comput Appl Math 229(2):471–479 21. Teodoro M, Lima P, Ford N, Lumb P (2008) Numerical modelling of a functional differential equation with deviating arguments using a collocation method. In: Simos T, Psihoyios G, Tsitouras C (eds) Numerical analysis and application mathematics. AIP conference proceedings, AIP, Melville, vol 1048, pp 553–557 22. Lima P, Teodoro M, Ford N, Lumb P (2010) Analytical and numerical investigation of mixed type functional differential equations. J Comput Appl Math 234:2732–2744 23. Teodoro M, Lima P, Ford N, Lumb P (2009) New approach to the numerical solution of forward-backward equations. Front Math China 4:155–168 24. Shakery F, Dehghan M (2008) Solution of delay differential equations via a homotopy perturbation method. Math Comput Model 48:486–498 25. Lima P, Teodoro M, Ford N, Lumb P (2010) Finite element solution of a linear mixed-type functional differential equation. Numer Algorithms 55:301–320 26. Teodoro M, Lima P, Ford NJ, Lumb P (2012) Numerical approximation of a nonlinear delayadvance functional differential equations by a finite element method. In: Simos T, Psihoyios G, Tsitouras C, Anastassi Z (eds) Numerical analysis and application mathematics. AIP conference proceedings, AIP, Melville, vol 1479, pp 406–409 27. Titze I (1988) The physics of small amplitude oscillation of the vocal folds. J Acoust Soc Am 83:1536–1552 28. Flanagan JL, Landgraf LL (1968) Self-oscillating source for vocal-tract synthesizers. IEEE Trans Audio Electroacoust 16:57–64 29. Teodoro M (2017) An issue about the existence of solutions for a linear non-autonomous mtfde. In: Pinelas S, a, Z.D, y, O.D, Kloeden P (eds) Differential and difference equations with applications. Springer proceedings in mathematics and statistics, vol 164. Springer, Springer International Publishing AG, Cham 30. Teodoro M (2017) Numerical solution of a delay-advanced differential equation from physiology. Appl Math Inf Sci 11(5):1287–1297 31. Ishizaka K, Flanagan J (1972) Synthesis of voiced sounds from a two-mass model of the vocal cords. Bell Labs Tech J 51(6):1233–1268. https://xueshu.baidu.com 32. Ishizaka K, Matsudaira M (1977) Speech physiology and acoustic phonetics. MacMillan, New York 33. Hirano M (1975) Phonosurgery: basic and clinical investigation. Oto-RhinoLaryngological Society of Japan, Fukuoka, Kioto University. Oficial Report of the 78th annual convention of the Oto-RhinoLaryngological Society of Japan 34. Drioli C (2005) A flow waveform-matched low-dimensional glottal model based on physical knowledge. J Acoust Soc Am 117:3184–3195 35. Pelorson X, Hirschberg A, Aure´gan Y (1994) Modelling of voiced sounds production using a modified two-mass model. J de Phys IV 4(C5), C5–453–C5–456. https://hal.archives-ouvertes. fr/jpa-00253089 36. Cisonni J, Hirtum AV, Pelorson X, Lucero J (2006) The influence of geometrical and mechanical input parameters on theoretical models of phonation. Acta Acust Acust 97:1–12

102

M. F. Teodoro

37. Miri AK, Barthelat F, Mongeau L (2012) Effects of dehydration on the viscoelastic properties of vocal folds in large deformations. J Voice 26(6):688–697 38. Lucero J (2008) Advanced-delay equations for aerolastics oscillations in physiology. Biophys Rev Lett 3(1):125–133 39. Lucero JJC, Koenig LL, Lourenco KG, Ruty N, Pelorson X (2011) A lumped mucosal wave model of vocal folds revisited: recent extensions and oscillation hysteresis. J Acoust Soc Am 129(3):1568–1579 40. Finkelhor BK, Titze IR, Durham PL (1988) The effect of viscosity changes in the vocal folds on the range of oscillation. J Voice 1(4):320–325 41. Hirtum AV, Cisonni J, Lucero J, Pelorson X (2006) Theoretical and experimental validation of a non-linear vocal fold model. https://www.researchgate.net/publication/228718103 42. Wu L, Zhang Z (2022) Computational study of the impact of dehydration-induced vocal fold stiffness changes on voice production. J Voice. https://www.sciencedirect.com/science/article/ pii/S0892199722000315 43. Yang S, Zhang Y, Mills RD, Jiang JJ (2017) Quantitative study of the effects of dehydration on the viscoelastic parameters in the vocal fold mucosa. J Voice 31(3):269–274. https://www. sciencedirect.com/science/article/pii/S0892199716300728 44. Teodoro M (2017) Numerical solution of a delay-advanced equation from acoustics. Int J Mech 11:107–114 45. Buhmann M (2003) Radial basis functions: theory and implementations. Cambridge University Press, Cambridge 46. Jafarian A, Measoomy Nia S, Golmankhaneh A, Baleanu D (2013) Numerical solution of linear integral equations system using the bernstein collocation method. Adv Contin Discret Models (123) 47. Yalçinba¸s S, Aynigül M, Sezer M (2011) A collocation method using hermite polynomials for approximate solution of pantograph equations. J Franklin Inst 348(6):1128–1139 48. Baishya C (2019) A new application of hermite collocation method. Int J Math, Eng Manag Sci 4(1):182–190

Chapter 8

Shear Stress and Temperature Analysis of Inconel 718 During the Backward Flow Forming Process Using the Finite Element Method Acar Can Kocabıçak and Magd Abdel Wahab Abstract Flow forming (tube spinning) is an incremental plastic deformation process that makes it possible to carry out forming operations of metal tubes over a mandrel using enormous forces. During the flow forming process, the rollers apply the radial force, and the machine’s headstock unit generates the axial force onto the workpiece. This present work compares the shear stress distribution in all directions. In this study, the finite element method (FEM) is operated to observe shear stress distributions on the workpiece during the flow forming operation. The FORGE® NxT 3.2 and Jmat Pro software are used to reach the most suitable material properties for the process simulation. The annealed condition Inconel 718 material is used for the FEM analyses. Two different reduction ratios, such as 37.5% and 50%, are applied to compare shear stress and process temperature differences in each direction. It is well understood that Arbitrary Lagrangian–Eulerian (ALE) method can be used effectively to predict temperature results of high plasticity processes such as the flow forming process. Furthermore, the re-meshing approach helps reduce simulation times and increase the accuracy during the operation. The results show that the temperature on the workpiece material with a 50% reduction ratio is higher than the 37.5% reduction ratio. Moreover, when the reduction ratio increases, the shear (tangential) stress on the workpiece also increases. In future work, the same forming parameters would be operated on experimental studies to improve and extend the conclusion. It is planned to perform experimental studies with a more detailed comparison by covering stress & strain reaction forces and damage criteria factors to validate FEM results. Keywords Flow forming · Inconel 718 · Finite element method · Metal forming · Arbitrary lagrangian eulerian A. C. Kocabıçak (B) Repkon Machine and Tool Industry and Trade Inc., Istanbul 34980, Turkey e-mail: [email protected] A. C. Kocabıçak · M. Abdel Wahab Faculty of Engineering and Architecture, Ghent University, Technologiepark Zwijnaarde 46, B-9052 Zwijnaarde, Belgium © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_8

103

104

A. C. Kocabıçak and M. Abdel Wahab

8.1 Introduction Flow forming is a metal-forming operation used to manufacture tubular rotational parts, mainly for aerospace and military industries. The starting material named preform loads on the mandrel at the beginning of the process. The part rotates with the main spindle, and the workpiece is exposed to the radial forces from the rollers. This way, while the workpiece’s wall thickness reduces, the workpiece’s length increases because of the volume constancy principle during the plasticity process. Various authors have investigated the flow forming process to understand the basis of this deformation technique [1–4]. Maj et al. studied the principle of the flow forming and heat treatment of Inconel 718 material by performing experimental trials [5]. They performed mechanical and microstructure tests to compare heat treatment and the flow forming process. It was observed that while mechanical properties increased after the flow forming due to work hardening phenomena, elongation reduced significantly. Standard aging heat treatment was carried out to increase the elongation after the forming. Finite element (FE) procedures are commonly used to predict experimental study results. Hua et al. [6] developed the backward flow forming process model of the Hastalloy workpiece using three forming rollers. The simulation was performed using ANSYS software. It was obtained that stress and strain distributions are multi-directional in the flow forming process. Similar results were found with Shinde et al. [7] developing the 3D thermo-mechanical FE model using Abaqus/Explicit software. In order to define the material model, the Johnson–Cook model has been performed to predict strain, strain rate, and temperature [8]. Parsa et al. [9] generated FEM using two forming rollers, and they investigated different process parameter effects such as feed rate and roller angles on the workpiece material properties. Xia et al. [10] used the MARC FE software to compare forming forces difference between the forward and backward flow forming processes. The simulation results show that radial forces are higher than the tangential forces for both processes. Li et al. [11] have carried out the FE model of the three rollers staggered tube spinning process. They found that when the axially staggering increases, the radial forces to tangential force reduces. Song et al. [12] worked on the FE optimization procedure about the diametric growth during the three roller tube spinning operation with DEFORM FEM software. They used two materials, the 6061 aluminum alloy and 304L stainless steel, to examine the diametric growth and residual stresses. It has been shown that the normal and shear stresses play a significant role in diametric growth results during the flow forming. Aghchai et al. [13] generated the FE model and response surface principle to investigate the diametric growth of an AISI 321 stainless steel tubular workpiece. It was designated that while reduction ratio is the most critical parameter on the diametric growth, feed rate effects are slight in this case. Roy et al. [14, 15] performed experimental and FE models to generate a correlation between plastic strain and micro-indentation hardness results. In this present work, the FE model is performed for the flow forming process of Inconel 718 material with two different reduction ratios, such as 37.5 and 50%. The

8 Shear Stress and Temperature Analysis of Inconel 718 During …

105

material properties for the simulations are defined using Jmat Pro software, and it is found that workpiece temperature during the 50% reduction ratio is greater than the 37.5% reduction ratio because of the plasticity energy. It is also well understood that shear stresses on the workpiece increase with higher reduction ratios. Furthermore, the roller forces play a significant role in operating flow forming operations successfully.

8.2 Material and Finite Element Procedure 8.2.1 Material Definition It is generally known to use ductile materials for all plasticity cases to perform the process on suitability. Furthermore, annealing and normalization operations are standard heat treatment processes to moderate residual stresses and obtain homogenous grain structure. Therefore, annealed heat treatment Inconel 718 material is preferred to improve the plasticity forming capacity before the flow forming. It is considered the starting material in isotropic conditions in the annealing heat treatment process. The chemical composition of the preform is presented in Table 8.1. Various authors used the JmatPro software to specify material properties data for their theoretical studies to predict plasticity and heat treatment results with different temperatures [16–18]. This framework contains thermal properties, density, coefficient of thermal expansion, specific heat, elasticity values, and stress–strain curves in different temperatures obtained for the FE analysis. The acquired flow-stress curves in different temperatures from 25 to 460 °C are given in Fig. 8.1. Thermal and physical properties of the Inconel 718 material are shown in Table 8.2, respectively. The material data obtained from the Jmat Pro was transferred to FORGE® NxT 3.2 software to perform FEM analysis.

8.2.2 Process Principle and Finite Element Method The backward flow forming technique was generated with the FORGE® NxT 3.2 software in this present work. This method consists of the tubular workpiece, mandrel, and three forming rollers. Before starting the process, the workpiece loads on the Table 8.1 Chemical composition of the Inconel 718 alloy Element type

Ni

Al

Co

Cr

Cu

Fe

Mn

Mo

Nb

Si

Ti

C

Weight (%)

53.2

0.55

0.05

18.41

0.02

18.4

0.046

3.04

5.24

0.05

0.94

0.02

106

A. C. Kocabıçak and M. Abdel Wahab

Fig. 8.1 Stress–Strain Curves of annealed Inconel 718 in different temperatures obtained by JmatPro Software

Table 8.2 Thermal and physical properties of the material Temp. (°C)

Density (kg/m3 )

Thermal conductivity (W/mK)

Coefficient of thermal expansion (10−5 /K)

Specific heat (J/kg K)

Poisson’s ratio (υ)

Young Modulus (Gpa)

25

8267.8

11.3

1.33

425

0.296

204.1

160

8219.8

12.5

1.44

453

0.295

196.5

260

8183.8

14.8

1.52

468

0.300

190.9

360

8157.7

16.4

1.60

484

0.311

184.7

460

8111.0

17.9

1.68

501

0.318

177.5

mandrel and rotates at the same speed as the mandrel. Then, while the tubular workpiece rotates in the same direction and speed as the mandrel, rollers start to apply forces on it from the outside surface of it. The tubular forming rollers move in the opposite direction. The preform is fixed using the headstock clamping tool and the rollers to move in the axial direction with a constant feed rate. The workpiece flows under the rollers in the opposite direction during the forming stages, as given in Fig. 8.2. The rollers keep free in the rotation direction and rotate only by friction from the workpiece contact surface. It is a general fact that three rollers can be located in the same symmetrical position or axially staggered around the tubular workpiece. The rollers place in the same symmetrical position at an angle of 120° from each other. In other words, the axial roller staggered is not used in this FE model; rollers and workpiece locate in a parallel direction.

8 Shear Stress and Temperature Analysis of Inconel 718 During …

107

Fig. 8.2 Backward flow forming principle

Even though the forming rollers and the mandrel are considered rigid bodies, the very-fıne mesh is generated on the contact points to improve the accuracy. Moreover, Arbitrary Lagrangian–Eulerian (ALE) re-meshing method was applied to the workpiece to reduce the failures in the deformation areas. The plate and rotate manipulator are used in FORGE® NxT 3.2 to define the movements of the workpiece during the process. This principle provides boundary conditions on the workpiece and helps to adjust the movement directions. The workpiece and rollers’ movement must be opposite in the backward flow forming. Therefore, the workpiece has been fixed using a plate manipulator to avoid the material flowing in the same direction as rollers. While the plate manipulator restricts the movement through material flow in the forward direction, the rotate manipulator lets the workpiece rotate at the same speed and direction as the mandrel. The generated process parameters for the FEM study are presented in Table 8.3. This study was performed with two different reduction ratios, such as 37.5% and 50%. The preform dimension is chosen the same for both simulations. The outside diameter of the preform is Ø49.50 mm, wall thickness is 4 mm, and length is 150 mm. The obtained wall thicknesses of the flow formed tubes are 2.5 mm and 2 mm, respectively, after completing the forming simulation. Table 8.3 Process parameters for FEM study

Feed (V)

37.5 mm/min.

Mandrel rotate (n)

125 rpm

Roller-preform btw friction coefficient

0.075–0.15

Mandrel–preform btw friction coefficient

0.075–0.15

Interaction roller-mandrel-preform

20,000 W/m2 .k

Ambient (cooling)

5500 W/m2 .k

108

A. C. Kocabıçak and M. Abdel Wahab

The flow forming process is an incrementally high plastic deformation process; hence it takes a very long time to obtain FEM results depending on the computer properties. The analysis duration was 40 s, and at least one week was needed to obtain results for each simulation. In order to obtain more accurate results, the 3D volumetric tetrahedron elements were generated for the preform meshing operation. The structured meshes were used for the workpiece, and 3D tetrahedral elements were generated. For the preform structure, 60,120 nodes, 34,768 triangles, and 323,135 tetrahedral mesh were applied. The maximum mesh size was set by 2 mm and reduced to 1 and 0.5 mm, respectively.

8.2.3 Heat Transfer Principle The following formula obtains the general term of heat flow: ρc

dT − div(kgrad(T )) − r σ :εϕ ˙c = 0 dt

(8.1)

where the first term is temperature evolution, the second term is internal conduction, and the last term is internal dissipation ρ = density of the material. c = specific heat capacity. k = thermal conductivity. T = temperature. t = time. r = mechanical energy to heat conversation coefficient. The radiation, conduction–convection, and constant fluxes occur during the plastic deformation processes such as flow forming operation,. It is predicted that plastic deformation causes an increase in the heat flux per unit volume during the flow forming process.

8.3 Results and Discussions All shear (tangential) stresses are investigated separately for the 37.5 and 50% reduction rates. Stresses in σx y , σ yz , σzx are given for both reduction rates, respectively, and it is found that shear stress increases by a higher reduction ratio. While the maximum shear stress in σx y direction with a 37.5% reduction ratio was obtained around 969 MPa; this value was found around 50 MPa higher with a 50% reduction ratio. Similar shear stress results are found after completing each process. It is well understood that when the reduction rate increases from 37.5 to 50%, shear stress in all directions are higher, as given in Figs. 8.3, 8.4, and 8.5, respectively.

8 Shear Stress and Temperature Analysis of Inconel 718 During …

Fig. 8.3 Shear stress distribution in XY direction for 37.5 and 50% reduction rates

Fig. 8.4 Shear stress distribution in YZ direction for 37.5% and 50% reduction rates

Fig. 8.5 Shear stress distribution in ZX direction for 37.5 and 50% reduction rates

109

110

A. C. Kocabıçak and M. Abdel Wahab

The maximum shear stress in σ yz direction is observed at 702 MPa with a 37.5% reduction ratio, and 841 MPa is obtained after the 50% reduction rate. Figure 8.4 shows that maximum shear stress in σ yz direction increased around 140 MPa after increasing the reduction ratio. Besides these results, shear stress in σzx is found around 601 MPa with 37.5%, and 922 MPa is found in 50% reduction rate. To summarize, the shear stress increased by approximately 320 MPa in σzx after applying the 50% reduction ratio with the flow forming process, this conclusion would be explained by higher forces and pressures with greater reduction rates [19, 20]. Furthermore, maximum stresses are found in σx y direction for both reduction rates. Because of these higher shear stress results during the 50% reduction rate process, mechanical properties are also expected to be greater due to the strain hardening mechanism. The process temperature difference between 37.5 and 50% reduction rates is also investigated during the FEM. It is found that the process temperature of 50% reduction ratio is around 86 ˚C greater than the 37.5% reduction ratio, as shown in Fig. 8.6. This difference would be explained by plasticity energy and friction energy from the rollers. The following picture shows that the process temperature is high in the local zone. The flow forming is a locally forming process, and while this area is under the rollers, the temperature is higher in this zone. Plasticity energy plays a significant role in the workpiece temperature during forming. Therefore, if the percentage reduction rate is very high, it may lead to excessive heat generation in the workpiece [21]. Besides the temperature and stress results, the mechanical properties of the workpiece would be higher with a 50% reduction rate compared to a 37.5% reduction rate. These higher temperature and stress distribution is expected to influence the mechanical properties of the workpiece.

8.4 Conclusion and Future Works In the Inconel 718 alloy flow forming process, the reduction ratio influences the process temperature and shear stress results. Therefore, the following conclusions can be deduced based on the FEM study’s 37.5% and 50% reduction rates. The reduction rate is a significant parameter in the heat generation on the workpiece during the flow forming process. When the reduction ratio increases, the shear stresses in all directions also increase because of higher forming forces requirement. The maximum shear stresses are found in σ yz direction for both operations (37.5 and 50%) as 960 and 1040 MPa, respectively. Besides, the minimum shear stress in σzx direction is found at 601 MPa after performing a 37.5% reduction rate. However, minimum shear stress is observed in σ yz direction while performing the higher reduction rate (50%) process. It was observed that the thermo-mechanical results mainly depend on preform properties and deformation rates. The workpiece temperature increases with applying higher reduction rates. Furthermore, process parameters such as feed rate, mandrel speed (rpm), and roller geometry would be

8 Shear Stress and Temperature Analysis of Inconel 718 During …

111

Fig. 8.6 Temperature difference between 37.5 and 50% reduction rates

other important values that affect the workpiece temperature and mechanical properties during the flow forming. In future work, mechanical properties, hardness, and grain structures will be compared with an experimental study. Furthermore, higher reduction rates would also be performed on the Inconel 718 alloys to find out the flow forming limits using suitable damage criteria. Acknowledgements The authors would like to thank Repkon Machine and Tool Industry and Trade Inc., which financially supported this study. The Repkon engineer prepared this work, and the authors wish to express gratitude to Ghent University, Belgium, for financial support for this research.

References 1. Kocabıçak AC, Karaka¸s A, Aydın G, Yalçınkaya S (2021) Lecture notes in mechanical engineering. Springer 2. Karaka¸s A, Kocabıçak AC, Yalçınkaya S, Sahin ¸ Y (2021) Lecture notes in mechanical engineering. 3. Sivanandini M, Dhami S, Pabla B (2012) Int J Sci Eng Res 3:1 4. Singh AK, Narasimhan K, Singh R (2018) Finite element modeling of backward flow forming of Ti6Al4V alloy

112

A. C. Kocabıçak and M. Abdel Wahab

5. Maj P, Błyskun P, Kut S, Romelczyk-Baishya B, Mrugała T, Adamczyk B, Mizera J (2018) J Mater Process Technol 253:64 6. Hua FA, Yang YS, Zhang YN, Guo MH, Guo DY, Tong WH, Hu ZQ (2005) J Mater Process Technol 168:68 7. Shinde H, Mahajan P, Singh AK, Singh R, Narasimhan K (2016) Int J Adv Manuf Technol 87:1851 8. Pantalé O, Gueye BJ (2013) Engineering (United States) 2013 9. Parsa MH, Pazooki AMA, Nili Ahmadabadi M (2009) Int J Adv Manuf Technol 42:463 10. Xia QX, Cheng XQ, Hu Y, Ruan F (2006)48:726 11. Li Y, Wang J, Lu G, Chen Q (2012) 227:1429 12. Song X, Fong KS, Oon SR, Tiong WR, Li PF, Korsunsky AM, Danno A (2014) Int J Adv Manuf Technol 71:207 13. Jalali Aghchai A, Razani NA, Mollaei Dariani B (2012) Proc Inst Mech Eng Part B J Eng Manuf 226:2002 14. Roy MJ, Klassen RJ, Wood JT (2009) J Mater Process Technol 209:1018 15. Roy MJ, Maijer DM, Klassen RJ, Wood JT, Schost E (2010) J Mater Process Technol 210:1976 16. Guo Z, Lasne P, Saunders N, Schillé JP (2018) Proc Manuf 15:372 17. Saunders N, Guo Z, Li X, Miodownik AP, Schillé Jom JP (2003) 55:60 18. Ebrahimi GR, Momeni A, Ezatpour HR (2018) Mater Sci Eng A 714:25 19. Tsivoulas D, Quinta da Fonseca J, Tuffs M, Preuss M (2015) Mater Sci Eng A 624:193 20. Gür CH, Arda EB (2003) Mater Sci Technol 19:1590 21. Singh AK, Narasimhan K, Singh R (2019)

Chapter 9

Mechanical Design and Optimization of Large-Scale Parabolic Trough Solar Collectors for Industrial Applications Ossama Mokhiamar, Mohammed Siddeq, and Osama Elsamni

Abstract Parabolic trough solar collectors have become effective and promising alternative to provide high temperatures in electricity generation and industrial applications. The challenge that is tracked in this present research is to provide a reliable tool to design a low-cost large scale parabolic trough collector that achieves the required thermal loads. The idea is to reduce the entire weight of the structure and keep the deflection within the tolerance allowed to reduce the heat losses due to the less scattering of the solar rays away from the absorber pipe. Two design scenarios are proposed and compared. Both designs resemble the spine or the backbone of humans, but they differ in the arrangement and number of the ribs and the longitudinal supporting ducts. The first design proposal is made up of more ribs and less longitudinal supporting ducts, while the second has less ribs and more supporting ducts. A square shaped duct is used as the main backbone at which all the ribs are assembled. Various optimization trials are conducted using SolidWorks software while changing the size of the backbone duct, the number of ribs, and the thickness and sizes of the longitudinal supporting duct. The optimization process targets the best combination of parameters that achieve the minimum weight and deflection with a reasonable factor of safety. The second scenario has shown much better results than the first, hence it has been used for further investigations via changing the rim angle and the backbone size to minimize the torque required. Keywords Solar energy · Solar concentrator · Parabolic trough · SolidWorks · Optimization

O. Mokhiamar · M. Siddeq · O. Elsamni (B) Faculty of Engineering, Mechanical Engineering Department, Alexandria University, El-Chatby, Alexandria 21544, Egypt e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_9

113

114

O. Mokhiamar et al.

9.1 Introduction With the implementation of environmental regulations in industrial countries aimed at reducing carbon emissions, demand for clean energy has risen dramatically. Solar energy is often recognized as the most important clean and renewable energy source, particularly in Middle Eastern nations such as Egypt, which has 3600 h/year of sunlight [1]. As a result, solar energy is incredibly valuable and enticing to research and the market. Despite the huge amount of agricultural waste in Egypt (30–35 million metric tons), 65% of this waste isn’t recycled [2] and is disposed of in an environmentally hazardous manner. Because organic wastes are high in soil nutrients, they are commonly used as a fertilizer source in many countries. Using solar energy to produce biochar is a novel and promising technique. Parabolic troughs can provide the necessary temperature for waste conversion into biochar. Giwa et al. [3] developed a system of parabolic trough fields for the conversion of date palm into biochar. The process’s economic and environmental metrics are far superior to those of conventional techniques. Parabolic troughs are used by around 96.3% of concentrating collectors [4]. There are different areas of applications of concentrating parabolic collectors in industrial and power generation applications. The mechanical design procedure for parabolic trough collectors is sophisticated and extensive, especially in the case of a large field of collectors. Design elements and optimization for the supporting structure, fixation, and transmission systems remain fertile ground for research and development to create a system without deflection hence minimize losses from reflecting rays. The influence of gravity loads on reflecting sheet deformation and focal point position modification using three different supporting systems was investigated in [5]. Macedo-Valencia et al. [6] created a single trough with a structure made up of lateral edge rips and an Ashape base supported by a polished aluminum reflecting sheet. A collector structure comprised of aluminum base supports and longitudinal supporting bars, as well as a mechanical gear/belt transmission system, were proposed and examined using a MATLAB GUI to optimize manufacturing cost and installation time [7]. Murtuza et al. [8] proposed a structure made up of truss elements, parabolic rips, and longitudinal support bars. A clamping method was used to secure the reflecting sheet to the framework. Schweitzer et al. [9] proposed a supporting structure design that lowered the fabrication costs by 20–25%. The design was based on a cantilever arm of welded trusses that is supported by a torque box backbone. A comprehensive review of different models of parabolic troughs was published by Fredriksson et al. [10]. The review discusses in detail the different structures for small and large-scale collectors. The influence of wind loads on parabolic trough stability is also an important research challenge. Collectors having a wide projected area are subjected to strong air drag forces. This force may have an impact on the system’s thermal performance as well as the likelihood of mechanical structure breakdown. As a result, researchers focused on giving a complete understanding of wind stress on parabolic troughs.

9 Mechanical Design and Optimization of Large-Scale Parabolic Trough …

115

Zhang et al. [11] calculated the effect of the gap size between reflective sheets on wind load reduction. It was fond that the optimal gab was found to be 0.06 m for axial gabs and 0.02 m for radial gabs. Fu et al. [12] investigated the connection between the mechanical structure of a parabolic trough and the wind load. In comparison to a basic model, the analysis resulted in an optimal deflection and collector weight. The weight and maximum deflection were reduced by around 4.6 and 5.8%, respectively. The present study is the first phase of a project that aimed at producing biochar from the agricultural wastes. These wastes need high temperatures ranged between 300 and 500 ◦ C in the absence of air. At this temperature, the pyrolysis process can take place directly inside the absorber tube of the parabolic trough or indirectly using a heating fluid. The relatively wide range of temperature necessitates the dimensions and the concentration ratio of the parabolic trough to change. Hence the aim of the present study is to find the optimum mechanical design of a large-sized parabolic trough concentrator to meet the specified temperature limit for each type of the agricultural waste. The objective will be minimizing the entire weight, deflection, and stress while getting a reasonable factor of safety. The large-scale parabolic trough is made up of series of the proposed modules to attain the maximum temperature needed to produce the biochar from the agriculture waste. Two different designs are proposed and analyzed under the effects of weight and wind loads. The optimization was carried out using the SolidWorks commercial software.

9.2 Trough Structural Design Aspects The two proposed designs share the same backbone carrying duct but differ in the number of ribs attached to the backbone and the longitudinal ducts those will be parallel to the backbone. In the first proposed design more ribs are connected to the main hollow backbone bar and less longitudinal supporting ducts. In the second scenario, more longitudinal square-shaped ducts are considered with less ribs attached to the main backbone. The square shaped longitudinal ducts help to increase the stiffness of the whole structure, hence lead to lower deflection. Both proposed designs are shown in Fig. 9.1. The parameters to be optimized are the sidewall thickness, the ribs thickness, the number of ribs, and the longitudinal squared-shaped supporting ducts and their thickness.

9.3 Model Preparation Before conducting the optimization process, the model should be setup and prepared in terms of drawing, meshing, and applying the body and external forces. The curved length of the module is set 5 m and the absorber pipe diameter is 1 inch so that the curvature ratio approaches 140. The module length is set at 6 m length. Accordingly, the meshing and the loads were applied to the proposed module.

116

O. Mokhiamar et al.

Ribs

Backbone duct

(a)

Sidewall

(b)

longitudinal duct

Fig. 9.1 The overall structure of; a the first proposed design; and b the second proposal design

9.3.1 Mesh Sizing The mesh type was chosen as blended curvature-based mesh as it is more suitable for structures containing many curvatures. The mesh quality was programmed to be as high as possible to maintain more accurate results. In the present simulations, the maximum element size was chosen as 10 mm, the minimum element size was 1 mm, and the number of Jacobian points was 16. As a result, the number of nodes was 5682131 and the number of elements was 2872390. It is worthy noted that the maximum aspect ratio was 9.44 with 96% of the elements being less than 3.

9.3.2 Static Configuration Bearing supports will be used to support the structure’s sides. The loads affecting the parabolic are the structure entire weight and the wind force. The wind force was calculated based on a flow simulation considering a wind speed of 30 km/h. The effect of this force was considered as a distributed load on the reflecting sheets.

9.3.3 The Optimization Process First Scenario The optimization technique implemented to meet the targeted requirements uses parametric cases conducted inside SolidWorks software. In the optimization process, the range of the parameters to be optimized should be assigned as well as the output constraints. The ranges of the parameters were taken as follows: – Sidewall thickness ranges from 10 to 20 mm. – The thickness of the support ranges from 5 to 10 mm in 1 mm increments.

9 Mechanical Design and Optimization of Large-Scale Parabolic Trough …

117

– Rib thickness ranges from 1 to 3 mm with a step of 1 mm. – The number of ribs with the values 6, 8, and 12. On the other hand, the output constraints are. – – – – –

Maximum deflection. Total mass. Center of gravity Maximum stress. Minimum factor of safety (based on strength).

Second Scenario The steps are like the ones that was explained the first model, except the parameters to be optimized. The parameters here are the number of longitudinal square-shaped ducts, their dimensions, and their thickness. In the optimization process, the ranges were as follows: – Number of ducts: 8, 10 and 12. – Duct dimension from 20 to 90 mm with a step of 10 mm. – Duct thickness from 1 to 3 mm with a step of 1 mm.

9.4 Results and Discussion 9.4.1 The Optimization of the First Scenario In this section, the results obtained from the simulations are presented and discussed. Figure 9.2 illustrates the effect of changing the sidewall thickness on the mass of the structure, the maximum stress produced, the minimum factor of safety and finally the maximum deflection. The simulation is conducted while fixing the rib number, support thickness and rib thickness. It is clear from the figure that as the sidewall thickness increases, the mass increases linearly and the factor of safety also increases. On the other hand, the deflection and stress decreased. In the next optimization process, the 10 mm sidewall thickness is fixed during seeking the optimum dimensions of the remaining parameters, since it gives a reasonable factor of safety and deflection, while producing the minimum mass. The objective of second round of the optimization process is to find the optimum dimensions of the support thickness, rib thickness and number of ribs that meet the design requirements. The range of the parameters is mentioned in Sect. 9.3.3. It is clear in Fig. 9.3 that the minimum factor of safety is greater than 2 and the maximum factor of safety is approximately 2.15. On the contrary, as the support thickness increases, the mass also increases, whereas the maximum deflection decreases. Going deep inside the results, one can finally select 5 mm for the support thickness, 1 mm for the rib thickness and 6 numbers of ribs. These dimensions give minimum mass and minimum possible deflection with a reasonable factor of safety.

118

O. Mokhiamar et al.

140

720

120

680

100

12 14 16 18 Sidewall thickness (mm)

20

80

Factor of safety

Mass (kg)

760

10

4

160

4.4

(b)

Factor of safety Deflection

4

3.6

3.6

3.2

3.2

2.8

2.8

2.4

2.4

2 1.6

Deflection (mm)

Mass Stress

800

640

4.4

180

(a)

Stress (MPa)

840

2 10

12 14 16 18 Sidewall thickness (mm)

20

1.6

Fig. 9.2 Effects of change the sidewall thickness; a Mass and stress; b Factor of safety and deflection

(a)

(b)

630

2.12

Factor of safety

Mass (kg)

620

2.16

610 600

2.08

2.04

590 580

2 5

6 7 8 9 Support thickness (mm)

(c)

10

5

10

4.2 4.15

t = 1 mm and N = 6 ribs t = 1 mm and N = 8 ribs t = 1 mm and N = 12 ribs t = 2 mm and N = 6 ribs t = 2 mm and N = 8 ribs t = 2 mm and N = 12 ribs t = 3 mm and N = 6 ribs t = 3 mm and N = 8 ribs t = 3 mm and N = 12 ribs

4.1

Deflection (mm)

6 7 8 9 Support thickness (mm)

4.05 4 3.95 3.9 3.85 3.8 3.75 3.7 5

6 7 8 9 Support thickness (mm)

Fig. 9.3 Optimization of the first scenario

10

9 Mechanical Design and Optimization of Large-Scale Parabolic Trough …

119

9.4.2 The Optimization of the Second Scenario Figure 9.4 illustrates the results obtained from the optimization process in the second proposed design. In Fig. 9.4a, the change of the mass versus the change of the dimension of the square duct at different square duct thicknesses and numbers is presented. The change of the factor of safety and deflection at the same conditions are shown in Fig. 9.4b and c, respectively. Comparing the results obtained from the previous two optimization processes, it’s clear that the second design gives better parameters regarding mass, deflection, factor of safety and finally stress. It’s worthily noted that in the two mentioned scenarios, the uncommon features have been studied in order to figure out which one is better. Therefore, after deciding the better design, more studies should be performed on the common features to find the remaining optimum dimensions of the module. They are backbone dimension, shape of sidewall and rim angle.

(a)

(b)

630

2.3

Factor of safety

620

Mass (kg)

2.4

610 600 590

2.2 2.1 2 1.9 1.8

580

1.7 5

6 7 8 9 Support thickness (mm)

(c)

10

20 30 40 50 60 70 80 90 Square shaped tube dimensions (mm)

3.6 t = 1 mm and N = 8 tubes t = 1 mm and N = 10 tubes t = 1 mm and N = 12 tubes t = 2 mm and N = 8 tubes t = 2 mm and N = 10 tubes t = 2 mm and N = 12 tubes t = 3 mm and N = 8 tubes t = 3 mm and N = 10 tubes t = 3 mm and N = 12 tubes

Deflection (mm)

3.2 2.8 2.4 2 1.6 1.2 0.8 20 30 40 50 60 70 80 90 Square shaped tube dimensions (mm)

Fig. 9.4 Optimization of the second scenario

120

O. Mokhiamar et al.

Fig. 9.5 Optimization of the backbone dimensions

9.4.3 Optimum Backbone Dimensions Using SolidWorks software, optimization study was performed to show the effect of changing the backbone dimensions on stress, deflection, minimum factor of safety and mass as shown in Fig. 9.5. The backbone dimension was varied from 120 to 180 mm. It is clear from figure that 120 mm dimension for the backbone is the optimum dimension. This dimension will be used in the further studies.

9.4.4 Shape of Sidewalls The shape of sidewall affects mainly the solidarity of the whole structure. In the second proposed design, the sidewall consists of three webs. However, while checking this design against the torque applied on the module, it was unsafe. Therefore, it’s needed to re-design the sidewall shape to find the proper number of webs. Figure 9.6 shows the relation between number of webs and both the stress induced and the factor of safety. It’s noted that the chosen width is 60 mm.

9.4.5 The Effect of the Rim Angle The rim angle is the angle between the line of aperture width and focal distance. The change of the rim angle affects the mass of the module, deflection of different loads and the torque need to operate the module. Therefore, in this section the effect of the rim angle has been investigated. Four angles were investigated in this study. They are 75°, 90°, 105° and finally 120°. The results are illustrated in Fig. 9.7. Form the figure it is obvious that as the torque,

9 Mechanical Design and Optimization of Large-Scale Parabolic Trough …

121

Fig. 9.6 A sidewall with 3 webs (Left); the effect of number of webs of the sidewall (Right)

Fig. 9.7 Effect of the rim angle

needed to drive the module, decreases the rim angle increases. The minimum torque and minimum mas occurred at 120°. On the other hand, the minimum deflection was found at 90° rim angle.

9.5 Conclusions The present work aims at designing and optimizing the dimensions of a large-scale parabolic trough with a minimum low deflection and minimum mass. The curved length of the parabola is 5 m and the length of the module in the longitudinal direction is 6 m. This module can be duplicated as needed in the axial direction to reach to the targeted temperature. Lowering the mass will reduce the cost, while lowering the deflection will preserve the thermal efficiency of the collector. Two different scenarios

122

O. Mokhiamar et al.

were proposed. The first scenario depends on installing more ribs and less longitudinal ducts, whereas the second scenario depends on more longitudinal ducts with less ribs. The two proposed designs were conducted using the SolidWorks software and compared. From the results, the following points can be drawn: • The second proposal offers a lower range of deflection compared to the first one at the same conditions, hence it can provide better thermal efficiency. It also reduces the weight of the structure by about 50% in some cases. Therefore, it can provide lower cost in terms of the used materials. • Using square-shaped ducts increases the stiffness of the whole structure and, consequently, causes minimum deflection. Moreover, it supports the reflecting sheets and can ensure the parabolic shape of the sheets. The availability of square ducts in a variety of sizes on the market gives the second design the upper hand in terms of cutting and machining expenses. • Increasing the backbone dimension has no significant effect on the stress, deflection, and factor of safety values. So, a dimension of 120 mm can be safely chosen as it has the minimum weight which leads to minimum cost and torque needed. • The number of sidewall webs affects the stability of the structure while applying the driving torque at the same sidewall thickness. Seven webs provide a considerable factor of safety while lowering the number of webs gives higher stresses and lower factors of safety which lead to unsafe structure. In future work, more parameters will be included in the optimization process, and a cost analysis will be conducted. Acknowledgements This research is supported by Science, Technology & Innovation Funding Authority (STIFA) under grant number 33541.

References 1. Ibrahim SMA (1985) Predicted and measured global solar radiation in Egypt. Sol Energy 35(2):185–188 2. Abou Hussein SD, Sawan OM (2010) The utilization of agricultural waste as one of the environmental issues in Egypt (A Case Study). J Appl Sci Res 6:1116–1124 3. Giwa A, Yusuf A, Ajumobi O, Dzidzienyo P (2019) Pyrolysis of date palm waste to biochar using concentrated solar thermal energy: economic and sustainability implications. Waste Manag 93:14–22 4. García IL, Álvarez JL, Blanco D (2011) Performance model for parabolic trough solar thermal power plants with thermal storage: comparison to operating plant data. Sol Energy 85:2443– 2460 5. Meiser S, Schneider S, Lüpfert E, Schiricke B, Pitz-Paal R (2017) Evaluation and assessment of gravity load on mirror shape and focusing quality of parabolic trough solar mirrors using finite-element analysis. Appl Energy 185:1210–1216 6. Macedo-Valencia J, Ramírez-Ávila J, Acosta R, Jaramillo OA, Aguilar JO (2014) Design, construction and evaluation of parabolic trough collector as demonstrative prototype. Energy Procedia 57:989–998

9 Mechanical Design and Optimization of Large-Scale Parabolic Trough …

123

7. Montes IEP, Benitez AM, Chavez OM, Herrera AEL (2014) Design and construction of a parabolic trough solar collector for process heat production. Energy Procedia 57:2149–2158 8. Murtuza SA, Byregowda HV, Ali MM, Imran M (2017) Experimental and simulation studies of parabolic trough collector design for obtaining solar energy. Resour Effic Technol 3:414–421 9. Schweitzer A., Schiel W., Birkle M., Nava P., Riffelmann K-J, Wohlfahrt A, Kuhlmann C (2014) ULTIMATE TROUGH®-Fabrication, erection, and commissioning of the world’s largest parabolic trough collector. Energy Procedia 49:1848–1857 10. Fredriksson J, Eickhoff M, Giese L, Herzog M (2021) A comparison and evaluation of innovative parabolic trough collector concepts for large-scale application. Sol Energy 215:266–310 11. Zhang L, Yang MC, Zhu YZ, Chen HJ (2015) Numerical study and optimization of mirror gap effect on wind load on parabolic trough solar collectors. Energy Procedia 69:233–241 12. Fu W, Yang MC, Zhu YZ, Yang L (2015) The wind-structure interaction analysis and optimization of parabolic trough collector. Energy Procedia 69:77–83

Chapter 10

Finite Element Modeling of Ultrasonic Nanocrystalline Surface Modification Process of Alloy 718 Chao Li, Ruslan Karimbaev, Auezhan Amanov, and Magd Abdel Wahab

Abstract As one of the emerging surface treatment technologies, ultrasonic nanocrystal surface modification (UNSM) tends to generate a plastic hardening layer to modify the mechanical and tribological properties of the workpiece near the surface. Among other surface modification methods, the lifting of mechanical performance near-surface region is mainly attributed to the presence of severe plastic deformation (SP2D). Thus, a more gradient distribution of residual stress is also introduced along the working process. The most notable advantage of UNSM technology is to make the surface optimization results precisely controllable through input parameters adjusting. Although the process and optimization results of UNSM on metals have been extensively studied experimentally, there are few studies on the effect of parameters on surface modification characteristics. This is due to the difficulty of conducting experiments to investigate the influence mechanism of the parameters and the procedure is costly and time-consuming. Nevertheless, this greatly limits the specific application studies of UNSM-treated specimens, such as accurate modeling before wear and fatigue testing, etc. In this work, a dynamic process of UNSM is built based on the finite element method (FEM). A displacement-controlled tip ball is modeled to simulate ultrasonic striking behaviors and a single-way process extracted from UNSM has been parametrically investigated carefully. The experimental data is used to verify and corroborate the FE results and it shows a good agreement. The depth and maximum value of residual stress are measured in the post-processing stage. As a result, the numerical results show that the UNSM-treated specimen exhibits a higher gradient of residual stress and plastic strain than its substrate. Keywords FEM · UNSM process · Residual stress · Severe plastic deformation C. Li · M. Abdel Wahab (B) Laboratory Soete, Ghent University, Ghent, Belgium e-mail: [email protected] R. Karimbaev · A. Amanov Department of Fusion Science and Technology, Sun Moon University, Asan-Si, Korea A. Amanov Department of Mechanical Engineering, Sun Moon University, Asan-Si, Korea © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Abdel Wahab (ed.), Proceedings of the 5th International Conference on Numerical Modelling in Engineering, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-99-0373-3_10

125

126

C. Li et al.

10.1 Introduction With the increasing demand for air transport and the fact that energy is scarce, the criterion for key components is also gradually raised [1]. For example, higher requirements are placed on the reliability and serving life of some critical gear such as aircraft engine blades and reaction rods in nuclear power plants [2]. As one of the most commercially dominant super alloys, 718 alloy is widely used in aerospace, and atomic energy industries [3]. Due to its extraordinary corrosion and oxidation resistance, and excellent mechanical strength whether in elevated high temperature, it stands out among other nickel-based super alloys. However, it is precisely because of its high hardness that 718 alloy is difficult to process and form. DED, known as a newly developed Additive Manufacturing (AM) method, is extensively applied in the manufacturing process. Compared with other conventional machining methods, such as casting and forging, directed energy deposition (DED) has the unique advantage of being able to directly form complex or even arbitrary shapes to meet practical needs [4]. However, DED tend to result in relatively poor surface quality which has become the main limitation preventing the wider application of 718 alloy. Hence, some post-processing options can be considered to improve near-surface properties. In this regard, how to improve the surface properties of 718 alloys made by DED has become an urgent issue related to the expansion of wider application fields. As one of most promising technique, UNSM treatment introduces strain hardening, gradient nano-crystalline structures, and especially residual stress into near surface of metals [5]. And they are beneficial to reduce the degree of surface deterioration, e.g., wear, corrosion, and fatigue features such as crack initiation and propagation. The most significant advantage of UNSM is that surface properties such as degree of plastic hardening and roughness can be precisely tuned with controllable static and dynamic forces. Compared with other surface treatment techniques which based on surface severe plastic deformation (S2PD) generation, such as laser penning (LP) and shot penning (SP), a more uniformly roughness can be produced and undesirable pile-up that occurs in laser penning can be avoided [6]. In terms of device, UNSM components comprised of ultrasound oscillator, air compressor, piezoelectric vibration transducer, booster, and a horn with a tip ball [6]. The ultrasound is generated by the oscillator, transmitted by the transducer, and then intensified as it passes through the booster. And static load is generated by an air compressor that provides constant pressure to bring the probe tip and workpiece into closer contact. Available studies have shown that the tribological, wear and fatigue properties of metallic samples are significantly enhanced due to the refinement of the martensitic lath and the formation of a plastic hardening layer by UNSM process [7]. Moreover, surface topography modified by surface striking of UNSM is important for both mechanical and tribological characteristics [8]. Regard of surface topography, the quality of surface morphology mainly refer to the roughness and waviness and form. The most used surface topography parameter in tribological studies is roughness [9]. Kim investigated the effect of the process variables of the UNSM on the surface quality characteristics of the treated specimens. It has been observed that the surface

10 Finite Element Modeling of Ultrasonic Nanocrystalline Surface …

127

waviness and roughness were mainly determined by static load and least affected by scanning speed. Compared with the untreated specimen, the surface roughness of 0.3297 mm and waviness of 1.8097 mm are produced in the optimal processing condition, which corresponds to an improvement of 80 and 72%, respectively [10]. However, there is no significant difference in surface roughness with increasing static load, suggesting that static load in UNSM has little effect on surface finish under some parameters. In addition, the smaller the path spacing, the lower the waviness and roughness generated on the surface, while the surface modification effect will be minimized by static load of out-of-range. Compared to the original TNTZ alloy, the UNSM-treated surface exhibited a higher hardness value which increased from 190 to 385 HV [11]. Similarly, a microhardness tester also shows that the surface hardness of 718 alloy subjected to UNSM at RT was increased by 35% compared with its substrate. Furthermore, a surface profilometer was used to examine the surface integrity and a fretting test rig based on a ball-on-disk contact condition was built to investigate the tribological effect induced by UNSM. At room temperature (RT), the wear test shows that friction coefficient of UNSM-treated 718 alloy decreased from 0.68 to 0.65, and especially reduced to 0.10 and 0.08 at high temperatures (HTs) of 400 and 600 °C, respectively. wear resistance of metal. UNSM can not only improve the surface integrity, but also wear resistance can be enhanced. The wear track was measured by a surface profilometer, and the results showed that the wear depth significantly decreased from 0.6 to 0.3 μm and the wear width narrowed from 341 to 275 μm compared with the control samples. Due to the plastic deformation, greater and deeper compressive residual stresses were induced in the UNSM-processed samples. The UNSM process has shown that the compressive residual stress of the subsurface will be significantly induced. The tensile residual stress exist in the top surface is converted to a compressive residual stress of magnitude 1094 MPa after the UNSM process. In addition, the magnitude of the residual compressive stress gradually decreases with increasing depth until it becomes a tensile residual stress at a depth of 130 μm [12]. Furthermore, it was found that the presence of tensile residual stress accelerates crack growth during cyclic loading and impairs component fatigue performance. The compressive residual stress produced by UNSM reduces the rate of crack propagation and leads to higher fatigue strength. The estimated crack growth rate of the UNSM-treated 718Plus alloy was 60% lower than that of the untreated alloy, indicating that fatigue life can be prolonged by fatigue crack nucleation and propagation suppressing [13]. Since the UNSM process does not involve chemical modification, UNSM-treated components exhibit excellent biocompatibility in orthodontic Treatment, dental implant and orthopedic devices [11]. In cell culture experiments, the UNSM-treated TNTZ alloy is applied to the culture of biological cells to investigate its effect on biological activity. Abrasion test results showed that UNSM-treated specimens exhibited more than 7 times higher abrasion resistance than untreated specimens. While the cell culture results showed that MC3T3 cells adhered and spread more easily on UNSM-treated samples than on substrate, demonstrating its enhanced biological activity, making it a strong candidate in applications of medical implantation. Instead of conventional experiments, some analytical models have emerged to simulate the machining process of UNSM to

128

C. Li et al.

achieve fast and robust prediction of residual stress distribution induced by UNSM. And linear regression analysis was used to study the effect of UNSM parameters such as vibration amplitude, tip size, static load, and temperature on residual stress distribution [14]. The analysis shows that the balance between residual stress and surface finish needs to be considered when selecting UNSM parameters. The increase in ultrasonic shock amplitude and static load resulted in an increase in the magnitude and depth of residual stress, while its potential to degrade in surface finish was also detected. At the same time, the smaller tip size and lower temperature can significantly increase the magnitude of residual stress, but also reduce the surface finish of the specimen. For a more intuitive understanding of the experiment, the process of UNSM technology can be carried on numerical simulation platform, especially using finite element method. A static and explicit dynamic finite element analysis (FEA) were performed to simulate quasi-static and dynamic indentation on copper workpieces [15]. Under various indentation conditions, the geometric evolution of the stacking and the corresponding stress and strain distributions in the surface area of the workpiece are obtained. While the processing velocity is fast enough, the strike frequency is as high as 20,000 Hz, keeping the interval space on the micron scale. Therefore, considering only 10 strikes does not reflect the truly UNSM processing, and only a roughly non-uniform simulation results will be produced which cannot be compared with experiments directly. Razi simulated a series of UNSM tests with different input parameters including static loads, striking amplitudes, ball materials, and scanning speed [6]. The maximum equivalent plastic strain and its depth were correlated with the response surface method to find the best parameter settings. A finite element model based on simplified physical theory and an equivalent static loading method were proposes by Wu [16]. Both the static and dynamic load were converted into an equivalent initial velocity which was then applied on workpiece to simulate the real loading process. The effective plastic strain is considered as an indicator to evaluate the degree of work hardening produced in the simulation. However, the linear processing path cannot truly reflect the complete processing mechanism of UNSM, especially when an entire surface of the specimen needs to be modified. In this work, a single way of UNSM process is simulated. Through a collaboration between experimental and numerical analysis, the aim of this study is to reveal the underlying mechanism by which dynamic striking significantly enhances.

10.2 Finite Element Simulations 10.2.1 Model Description To understand the working process of UNSM, a single way is considered in this study. A FE-based dynamic 3D model is built using ABAQUS® /Standard and ABAQUS® /Explicit FE codes. The material of the workpiece is alloy 718 and the

10 Finite Element Modeling of Ultrasonic Nanocrystalline Surface …

129

size built in finite element model is 6 × 6 × 3 mm. As shown in Figs. 10.1 and 10.2, the sample is considered as deformable and meshed with C3D8R elements. The mesh size of contact area is defined as 8 and 10 μm and the simulation will be compared. As shown in Table 10.1, the tip ball is chosen as tungsten carbide and can be considered as a rigid body in FEM with 2.4 mm diameter. To achieve the contact between tip and workpiece properly, the interaction between indenter and specimen defined as surface-to-surface contact. The master–slave algorithm is employed while the surface of tip ball and top surface of workpiece are defined as master and slave surface, respectively. Hence, to save analysis steps and time of moving the impact ball in FEM, 50 of tip ball models were built along the scanning path at the beginning stage for impact preparation. The scanning interval between two consecutive impacts d = v/f = 2.5 μm. The Due to the symmetry of FE model, only half model needs to be built to reduce the nodes which each one has 6 degrees-of-freedom (DOF).

10.2.2 Constitutive Model Since the ultrasonic generator brings an ultra-high vibration frequency to the tool tip, the influence of the strain rate cannot be ignored. In this paper, the Johnson-Cook model is used to describe the plasticity, and the parameters assigned to the material are as follows. [ ( pl )] [ ( pl )n ] ε˙ eq · 1 + Cln σeq = A + B εeq (10.1) ε˙ 0 pl

where σ is the equivalent stress, and εeq is the equivalent plastic strain. A is the yield stress. B, C and n are the material constants. In this study, the elastic modulus and

Fig. 10.1 Schematic illustration of fully scanning strategy of UNSM

130

C. Li et al.

Fig. 10.2 Schematic illustration of half 3D FE model

Table 10.1 Parameters used in UNSM treatment process Static force [N]

Ball material

Ball diameter [mm]

Vibration amplitude [μm]

Vibration frequency [Hz]

Scanning speed [mm/min]

100

WC

2.4

10

20,000

3,000

Table 10.2 Material properties and Johnson-Cook parameters for alloy 718 Sample

E [GPa]

A [MPa]

B

n

C

Alloy 718

200

900

1200

0.28

0.07

Johnson-Cook parameters used in finite element simulations are listed in Table 10.2 [17].

10.2.3 Loading Condition There are mainly two forces attributed to the effectiveness of the UNSM process as shown in Fig. 10.3a. The first one is static load Fs used to bring tip ball and workpiece into contact. Another one is dynamic load Fa sin(2π f t) with ultrasonic frequency f produced by the ultrasonic generator. It is used to produce severe plastic deformation (SPD) near the top surface of workpiece and Fa is the amplitude of dynamic force. The grey area in Fig. 10.3a stands for the acting period produced by total load Ftotal which combined with static load Fs and dynamic load Fd , i.e., Ftotal = Fs + Fd .

10 Finite Element Modeling of Ultrasonic Nanocrystalline Surface …

131

(a) Acting period

Xs (b)

Fig. 10.3 Diagrams of a force and b displacement

In this study, the displacement of tip ball X total is used to simulate the effectiveness of UNSM technique on workpiece. As shown in Fig. 10.3b, the acting period is consisted of static displacement X s and dynamic displacement X d , i.e., X total = X s + X d . To minimum the time consuming and calculation cost used in the finite element simulation process, the displacement data in tabular form is employed in this study. As shown in Fig. 10.3b, two kinds of simplified tabular data are defined as the boundary condition of tip ball. Firstly, each of the acting period is divided into 6 phases evenly which defined as Type 1. The refined tabular data is designed to consider the displacement region with large gradients which defined as Type 2. Then the simulation results between these two kinds of displacement forms will be compared (Fig. 10.4).

132

C. Li et al.

Fig. 10.4 Different forms of displacement: a roughly and b refined curves

10.3 Results and Discussion 10.3.1 Model Validation The static displacement X s can be validated with Hertzian solution, when Fs = 100 N, the X s calculated by FEM is 0.00639 mm using Abaqus/Standard. And the Hertzian contact solution of sphere-on-plane can be expressed as: ( Xs = 1 E

=

)1/3

9F 2 16R E

2

1 − νc2 1 − νs2 + Es Ec

(10.2)

(10.3)

In this way, the analytical solution based on Eqs. (10.2) and (10.3) is 0.00679 mm which has a difference of 5% compared with the finite element result.

10.3.2 Case Study The finite element model based on Johnson-Cook constitutive model is built and 3 cases are designed to investigate its accuracy, as shown in Table 10.3. Case 1 and Case 2 are used to compare the difference between the rough and refined displacement data. Case 2 and Case 3 are aimed to find the effect of the refined mesh size on model improvement. The distribution of residual stress can be a criterion to evaluate the consistency of finite element model with experimental result. The simulation results of residual stress distribution based on different displacement forms, mesh sizes and experimental measurements are represented in Fig. 10.5.

10 Finite Element Modeling of Ultrasonic Nanocrystalline Surface … Table 10.3 Case study for different displacement forms and mesh sizes

133

Case

Displacement

Mesh size

1

Tabular data

10

2

Refined tabular data

10

3

Refined tabular data

8

Fig. 10.5 Comparison of residual stress distributions for different displacement forms and mesh sizes obtained from FEM with experimental data

According to the figure, the maximum residual stress in tension and compression remains approximately the same for all cases. However, the location of the compressive residual stress in Case 3 is closer to the shallow layer, which is more approaching to the experimental measurements. It is indicated that Case 3 has a higher accuracy with experimental data, i.e., finer size of mesh and refined displacement data. The finer mesh size predicts the location of maximum compressive residual stress more accurately. However, no significant difference was observed in the distribution of compressive residual stress in Case 1 and Case 2 at the increasing stage. In the reducing stage of compressive residual stress, the residual stress of Case 2 is higher than that of Case 1 at the same position which is more approach to the experimental measurement. It suggests that the refined displacement form provides more accurate predictions at deeper layers.

10.4 Conclusions A finite element model based on Johnson-Cook constitutive model is built to simulate a single way process of UNSM technique. Only half of the finite element model need to be considered due to the symmetry of model and loading condition to save the calculation cost.

134

C. Li et al.

The displacement of the tip ball is used to simulate the effectiveness of its striking behavior of ultrasonic oscillation. Three cases are considered to investigate the influence of rough displacement form, refined displacement form and mesh size on finite element model, respectively. The distribution of residual stress can be used as a criterion to examine the consistency of finite element model with experimental measurement. The results show that finer mesh size and refined displacement form provide a highest consistency with experimental data. The refined displacement leads to a more accurate prediction at deeper layer, while there is no significant difference in the shallow layer. Moreover, the finer mesh size can predict the location of maximum compressive residual stress more closely. Acknowledgements This work was supported under the framework of an international cooperation program managed by the National Research Foundation of Korea and Het Fonds Wetenschappelijk Onderzoek – Vlaanderen (FWO) (FY2020K2A9A1A06103270) and funded by the Chinese Scholarship Council (CSC).

References 1. Kracke A (2016) Superalloys, the most successful alloy system of modern times-past, present, and future. Superalloy 718 and Derivat, 13–50 2. Klimko M (2017) Strength analysis of an aircraft turbo-compressor engine turbine discIn: In: AIP conference proceedings 1889, September 2017 3. Godec M (2021) Hybrid additive manufacturing of Inconel 718 for future space applications. Mater Charact 172 4. Saboori A (2019) Application of directed energy deposition-based additive manufacturing in repair. Appl Sci 9(16) 5. Miller M (2013) An integrated experimental and simulation study on ultrasonic nano-crystal surface modification 6. Razi M, Shahraki S, Teimouri R (2020) Numerical study of optimized processing condition in rolling strike ultrasonic nanocrystalline surface modification of copper. Int J Light Mater Manuf 3(2):160–171 7. Shen N (2018) Microstructure prediction of severe plastic deformation manufacturing processes for metals, 1–191 8. Amanov A, Pyun YS, Sasaki S (2014) Effects of ultrasonic nanocrystalline surface modification (UNSM) technique on the tribological behavior of sintered Cu-based alloy. Tribol Int 72:187– 197 9. Hirschmann ACO (2010) Surface modification of Inconel 718 superalloy by plasma immersion ion implantation. In: 7th International Symposium on Superalloy 718 and Derivatives, vol 2, pp 993–1001 10. Amanov A, Umarov R (2018) The effects of ultrasonic nanocrystal surface modification temperature on the mechanical properties and fretting wear resistance of Inconel 690 alloy. Appl Surf Sci 441:515–529 11. Acharya S, Suwas S, Chatterjee K (2021) Review of recent developments in surface nanocrystallization of metallic biomaterials. Nanoscale 13(4):2286–2301 12. Liu R (2021) Application of ultrasonic nanocrystal surface modification (UNSM) technique for surface strengthening of titanium and titanium alloys: a mini review. J Mater Res Technol 11:351–377

10 Finite Element Modeling of Ultrasonic Nanocrystalline Surface …

135

13. Kattoura M, Mannava SR, Qian D, Vasudevan VK (2018) Effect of ultrasonic nanocrystal surface modification on residual stress, microstructure and fatigue behavior of ATI 718 Plus alloy. Mater Sci Eng A 711(October 2017):364–377 14. Zhao J, Dong Y, Ye C (2021) Optimization of residual stresses generated by ultrasonic nanocrystalline surface modification through analytical modeling and data-driven prediction. Int J Mech Sci 197(July 2020):106307 15. Tarasov AF, Altukhov AV, Gribkov EP, Abdulov AR (2019) Development and FEM modeling of a new severe plastic deformation process according to the reverse shear scheme. Model Simul Eng 2019:1–10. Article ID 8563830. https://doi.org/10.1155/2019/8563830 16. Wu B, Zhang L, Zhang J, Murakami RI, Pyoun YS (2015) An investigation of ultrasonic nanocrystal surface modification machining process by numerical simulation. Adv Eng Softw 83:59–69 17. Khan MK, Fitzpatrick ME, Wang QY, Pyoun YS, Amanov A (2018) Effect of ultrasonic nanocrystal surface modification on residual stress and fatigue cracking in engineering alloys. Fat Fract Eng Mater Struct 41(4):844–855