Peridynamic Modeling, Numerical Techniques, and Applications 0128200693, 9780128200698

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Mechanics of Advanced Materials Series The Mechanics of Advanced Materials book series focuses on materials- and mechanics-related issues around the behavior of advanced materials, including the mechanical characterization, mathematical modeling, and numerical simulations of material response to mechanical loads, various environmental factors (temperature changes, electromagnetic fields, etc.), as well as novel applications of advanced materials and structures. Volumes in the series cover advanced materials topics and numerical analysis of their behavior, bringing together knowledge of material behavior and the tools of mechanics that can be used to better understand, and predict materials behavior. It presents new trends in experimental, theoretical, and numerical results concerning advanced materials and provides regular reviews to aid readers in identifying the main trends in researchin order to facilitate the adoption of these new and advanced materials in a broad range of applications.

Series editor-in-chief: Vadim V. Silberschmidt Vadim V. Silberschmidt is Chair of Mechanics of Materials and Head of the Mechanics of Advanced Materials Research Group, Loughborough University, United Kingdom. He was appointed to the Chair of Mechanics of Materials at the Wolfson School of Mechanical and Manufacturing Engineering at Loughborough University, United Kingdom in 2000. Prior to this, he was a Senior Researcher at the Institute A for Mechanics at Technische Universita¨t Mu¨nchen in Germany. Educated in the USSR, he worked at the Institute of Continuous Media Mechanics and Institute for Geosciences [bothdthe USSR (laterdRussian) Academy of Sciences]. In 1993e94, he worked as a visiting researcher, Fellow of the Alexander-von-Humboldt Foundation at Institute for Structure Mechanics DLR (German Aerospace Association), Braunschweig, Germany. In 2011e14, he was Associate Dean (Research). He is a Charted Engineer, Fellow of the Institution of Mechanical Engineers and Institute of Physics, where he also chaired Applied Mechanics Group in 2008e11. He serves as Editor-in-Chief (EiC) of the Elsevier book series on Mechanics of Advanced Materials. He is also EiC, associate editor, and/or serves on the board of a number of renowned journals. He has coauthored four research monographs and over 550 peer-reviewed scientific papers on mechanics and micromechanics of deformation, damage, and fracture in advanced materials under various conditions.

Series editor: Thomas Bo¨hlke Thomas Bo¨hlke is Professor and Chair of Continuum Mechanics at the Karlsruhe Institute of Technology (KIT), Germany. He previously held professorial positions at the University of Kassel and at the Otto-von-Guericke University, Magdeburg, Germany. His research interests include FE-based multiscale methods, homogenization of elastic, brittle-elastic, and visco-plastic material properties, mathematical description of microstructures, and localization and failure mechanisms. He has authored over 130 peerreviewed papers and has authored or coauthored two monographs.

Series editor: David L. McDowell David L. McDowell is Regents’ Professor and Carter N. Paden, Jr. Distinguished Chair in Metals Processing at Georgia Tech University, United States. He joined Georgia Tech in 1983 and holds a dual appointment in the GWW School of Mechanical Engineering and the School of Materials Science and Engineering. He served as the Director of the Mechanical Properties Research Laboratory from 1992 to 2012. In 2012 he was named Founding Director of the Institute for Materials (IMat), one of Georgia Tech’s Interdisciplinary Research Institutes charged with fostering an innovation ecosystem for research and education. He has served as Executive Director of IMat since 2013. His research focuses on nonlinear constitutive models for engineering materials, including cellular metallic materials, nonlinear and time-dependent fracture mechanics, finite strain inelasticity and defect field mechanics, distributed damage evolution, constitutive relations, and microstructure-sensitive computational approaches to deformation and damage of heterogeneous alloys, combined computational and experimental strategies for modeling high cycle fatigue in advanced engineering alloys, atomistic simulations of dislocation nucleation and mediation at grain boundaries, multiscale computational mechanics of materials ranging from atomistics to continuum, and system-based computational materials design. A Fellow of SES, ASM International, ASME, and AAM, he is the recipient of the 1997 ASME Materials Division Nadai Award for career achievement and the 2008 Khan International Medal for lifelong contributions to the field of metal plasticity. He currently serves on the editorial boards of several journals and is coeditor of the International Journal of Fatigue.

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ELSEVIER SERIES IN MECHANICS OF ADVANCED MATERIALS

PERIDYNAMIC MODELING, NUMERICAL TECHNIQUES, AND APPLICATIONS Edited by

ERKAN OTERKUS Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom

SELDA OTERKUS Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom

ERDOGAN MADENCI Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-820069-8 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Chiara Giglio Production Project Manager: Sojan P. Pazhayattil Cover Designer: Matthew Limbert Typeset by TNQ Technologies

Contributors Sundaram Vinod K. Anicode Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States Atila Barut Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States Tinh Quoc Bui Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Tokyo, Japan Cagan Diyaroglu Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom Mehmet Dorduncu Mechanical Engineering Department, Erciyes University, Kayseri, Turkey Yakubu Kasimu Galadima Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom Ugo Galvanetto Industrial Engineering Department, University of Padova, Padova, Italy; CISAS “G. Colombo”, University of Padova, Padova, Italy Xiaoqiao He Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong SAR, China Masaki Hojo Department of Mechanical Engineering and Science, Kyoto University, Kyoto, Japan Michiya Imachi Graduate School of Advanced Science and Engineering, Hiroshima University, Higashihiroshima, Japan Ali Javili Department of Mechanical Engineering, Bilkent University, Ankara, Turkey Lei Ju College of Ship Building Engineering, Harbin Engineering University, China Emma Lejeune Department of Mechanical Engineering, Boston University, Boston, MA, United States Christian Linder Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, United States Xuefeng Liu Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, Guangdong, China Chun Lu Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, Guangdong, China Erdogan Madenci Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States

xi

xii

Contributors

Naoki Matsuda Department of Mechanical Engineering and Science, Kyoto University, Kyoto, Japan Andrew McBride Glasgow Computational Engineering Centre, School of Engineering, University of Glasgow, Glasgow, United Kingdom Cody Mitts Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States Cong Tien Nguyen Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom Masaaki Nishikawa Department of Mechanical Engineering and Science, Kyoto University, Kyoto, Japan Erkan Oterkus Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom Selda Oterkus Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom Murat Ozdemir Department of Naval Architecture and Marine Engineering, Ordu University, Ordu, Turkey Anil Pathrikar Centre of Excellence in Advanced Mechanics of Materials, Indian Institute of Science, Bangalore, Karnataka, India Timon Rabczuk Institute of Structural Mechanics, Bauhaus University Weimar, Weimar, Germany Debasish Roy Centre of Excellence in Advanced Mechanics of Materials, Indian Institute of Science, Bangalore, Karnataka, India Pranesh Roy Centre of Excellence in Advanced Mechanics of Materials, Indian Institute of Science, Bangalore, Karnataka, India Arman Shojaei Institute of Material Systems Modeling, Helmholtz-Zentrum Geesthacht, Geesthacht, Germany Stewart A. Silling Sandia National Laboratories, Albuquerque, New Mexico, United States Paul Steinmann Glasgow Computational Engineering Centre, School of Engineering, University of Glasgow, Glasgow, United Kingdom; Institute of Applied Mechanics, Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg, Erlangen, Germany Satoyuki Tanaka Graduate School of Advanced Science and Engineering, Hiroshima University, Higashihiroshima, Japan Bozo Vazic Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom Qing Wang College of Ship Building Engineering, Harbin Engineering University, China Wenxuan Xia Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom Yanzhuo Xue College of Ship Building Engineering, Harbin Engineering University, China

Contributors

xiii

Zhenghao Yang Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom Mirco Zaccariotto Industrial Engineering Department, University of Padova, Padova, Italy; CISAS “G. Colombo”, University of Padova, Padova, Italy Xiaoying Zhuang Chair of Computational Science and Simulation Technology, Institute of Photonics, Leibniz University Hannover, Hannover, Germany; Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai, China

Preface Although peridynamics was originally introduced by Dr. Stewart A. Silling from Sandia National Laboratories in 2000 for predicting and simulating the failure response of structures, it has been extended and applied to many challenging problems from different disciplines. It has become a unique approach for multiphysics analysis with damage prediction capability across varying length scales. Since its inception, peridynamics is growing exponentially by contributions and publications of researchers from different parts of the world. This book brings together a wide range of recent contributions in the area of peridynamics. We hope that it offers new ideas and motivates peridynamic researchers to explore new applications. The book starts with an introductory chapter authored by Dr. Stewart Silling. The remaining nineteen chapters in the book are divided into two sections: new concepts in peridynamics and new applications in peridynamics. It presents new techniques such as dual horizon peridynamics, damage modeling using the phase-field approach, peridynamics for axisymmetric analysis, beam and plate models in peridynamics, coupled peridynamics and XFEM, fracture mechanics evaluation with peridynamics, and contact analysis of rigid and deformable bodies. Also, it presents cutting-edge applications of peridynamics such as ice modeling, composites delamination and damage in ceramics, modeling at nanoscale, and more. We profusely appreciate and thank the renowned authors for their contributions and commitment for different chapters of this book. We hope that this book further accelerates the growth of peridynamics by providing the beginner as well as the current peridynamic researchers with recent progress and novel applications of peridynamics. Lastly, we appreciate the encouragement and support of many colleagues in the field of peridynamics in the preparation of this book, program managers at AFOSR for the MURI Center for Material Failure Prediction through Peridynamics at the University of Arizona (AFOSR Grant No. FA9550-14-1-0073), and program managers at EOARD for the financial support to the University of Strathclyde (AFOSR Grant No. FA9550-18-1-7004). Erkan Oterkus, Selda Oterkus, and Erdogan Madenci

xv

C H A P T E R

1 Introduction Stewart A. Silling Sandia National Laboratories, Albuquerque, New Mexico, United States

O U T L I N E 1. What is peridynamics?

2

2. Peridynamics obtained from the smoothing of an atomic system

3

3. Material models 3.1 Linear microelastic model 3.2 Prototype microelastic brittle model 3.3 Microelastic nucleation and growth model 3.4 Nonlinear and rate-dependent bond-based models 3.5 Ordinary state-based material models 3.6 Non-ordinary state-based materials and the correspondence model

5 6 8 10 11 12 15

4. Relation to the local theory

18

5. Simple meshless discretization

19

6. Some research trends in the peridynamic theory 6.1 Special purpose material models 6.2 Wave dispersion 6.3 Material stability 6.4 Micropolar theories 6.5 Better meshless numerical techniques 6.6 Ductile material response 6.7 Multiple physical fields 6.8 Material variability

20 21 21 21 22 22 23 24 24

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00008-1

1

© 2021 Elsevier Inc. All rights reserved.

2

1. Introduction

7. Conclusions

25

Acknowledgments

26

References

26

1. What is peridynamics? The peridynamic theory is an alternative form of continuum mechanics that is more compatible than the standard (local) theory with discontinuities such as growing cracks. In the peridynamic theory, the equation of motion and material models use integrals rather than partial differential equations. This allows the peridynamic equations to be applied directly on the surface of a crack. The equation of motion replaces the term in the local theory that characterizes the internal forces within a material. Z V $ sðx; tÞ/ fðq; x; tÞdq (1.1) Hx

where s is the stress tensor and f(q,x,t) is a force density (per unit volume squared) that a neighboring point q exerts on x. This force density is determined by the deformation according to the material model. It is always required as a consequence of Newton’s laws that fðx; q; tÞ ¼  fðq; x; tÞ

(1.2)

for all x, q, t. The region of integration in Eq. (1.1) is the family of x, which is a neighborhood whose radius d > 0 is called the horizon (Fig. 1.1). The horizon is a cutoff distance for force interactions. Using Eq. (1.1), the peridynamic equation of motion is as follows: Z € tÞ ¼ rðxÞyðx; fðq; x; tÞdq þ bðx; tÞ (1.3) Hx

where r is the mass density, y is the deformation map, and b is the external body force density field. The attribute of the theory that allows interactions between points such as x and q directly across a finite distance is sometimes called strong nonlocality. In introducing peridynamics, the basic Eq. (1.3) is usually presented as an assumption that is elaborated upon to show that it has desirable properties, such as being able to sustain growing cracks, and not violating any laws of physics (Silling, 2000). This leaves some people wondering, “Where does this come from?” In particular, the strongly nonlocal nature of Eq. (1.3) is perceived as lacking motivation. With this in mind, I hope

3

2. Peridynamics obtained from the smoothing

Bond

ℋ = family of

FIGURE 1.1 The horizon and family of a material point x.

that the discussion in the next section helps to motivate the peridynamic continuum theory especially with regard to nonlocality.

2. Peridynamics obtained from the smoothing of an atomic system A fundamental concept in continuum mechanics is the representation of a material as a mathematically continuous field, even though in reality any material is made of atoms and molecules. One way to justify the assumption of a continuum is to apply a smoothing function to the system of particles. In the following, we define a continuous displacement field in this way and investigate the evolution equation that this continuous field obeys. The evolution equation turns out to be the peridynamic equation of motion. Consider an assembly of mutually interacting particles (point masses) in a crystal with mass Mk, k ¼ 1,2, .,N. Let the reference positions of these particles be denoted xk, and the displacement vectors uk(t). For simplicity, thermal oscillations will be neglected in the following discussion. Suppose any particle [ exerts a force Fk[ ðtÞ on any particle k. As a notational convenience, set Fkk ¼ 0 for any k. These forces are assumed to have the antisymmetry given by F[k ðtÞ ¼  Fk[ ðtÞ

(1.4)

4

1. Introduction

for all k, [, and t. It is also assumed that there is a cutoff distance d for the atomic interactions such that Fk[ ¼ 0 if jx[ xk j > d. Additionally, particle k is subject to a prescribed external force Bk(t). For any x˛R3, define a smoothing function U(x,$) such that the following normalization holds: Z Uðx; pÞdx ¼ 1 (1.5) for any point p. It is convenient to assume that at any x, U(x,$) vanishes outside a neighborhood of radius R, where R is a positive number. Define the smoothed mass density and body force density fields by X X rðxÞ ¼ Uðx; xk ÞMk ; bðx; tÞ ¼ Uðx; xk ÞBk ðtÞ: (1.6) k

k

Define the smoothed displacement field by 1 X uðx; tÞ ¼ Uðx; xk ÞMk uk ðtÞ rðxÞ k

(1.7)

Now we investigate the evolution equation for u. The particles obey Newton’s second law, X Mk u€k ðtÞ ¼ Fk[ ðtÞ þ Bt ðtÞ. (1.8) [

Differentiating Eq. (1.7) twice with respect to time yields X € tÞ ¼ Uðx; xk ÞMk u€k ðtÞ rðxÞuðx;

(1.9)

k

From Eqs. (1.6), (1.8), and (1.9), € tÞ ¼ rðxÞuðx;

X

X Uðx; xk Þ Fk[ ðtÞ þ Bk ðtÞ

k

¼

XX k

"

#

[

Uðx; xk ÞFk[ ðtÞ þ bðx; tÞ

(1.10)

[

From Eqs. (1.5) and (1.10), € tÞ ¼ rðxÞuðx;

Z  XX Uðq; x[ Þdq þ bðx; tÞ Uðx; xk ÞFk[ ðtÞ k

(1.11)

[

Eq. (1.11) can be rewritten as follows: Z € tÞ ¼ rðxÞuðx; fðq; x; tÞdq þ bðx; tÞ

(1.12)

3. Material models

where fðq; x; tÞ ¼

XX Uðx; xk Þ Uðq; x[ ÞFk[ ðtÞ k

5

(1.13)

[

and b is given by Eq. (1.6). It is easily shown from Eqs. (1.4) and (1.13) that f has the antisymmetry Eq. (1.2). The vector defined by x¼q  x

(1.14)

is called a bond. (When talking about bonds, it is always assumed that x s 0, without explicitly stating this.) The function f is called the pairwise bond force density and has dimensions of force/volume2. From Eq. (1.13), the points x and q interact only if jxj  d;

d ¼ 2R þ d

(1.15)

The length d is the horizon for the continuum model (Fig. 1.1). In summary, we defined a weighting function U and used it to define smoothed fields r, b, and u. With these definitions, and Newton’s second law applied to the particles, the smoothed displacement field was found to obey the peridynamic equation of motion Eq. (1.12). The peridynamic bond forces that appear in this equation of motion are defined by Eq. (1.13). Conceptually, the equation of motion is, and should be, nonlocal because changing the displacement of a single particle k directly affects the smoothed displacements at all the points x whose smoothing function U(x,$) have nonzero values at xk. A more complete derivation of the peridynamic equations was obtained from statistical mechanics by Lehoucq and Sears (2011). The definition of the pairwise bond force density Eq. (1.13) is not very practical as a material model because it does not directly relate the values of u near x to the bond forces f. More practical methods of determining f are discussed in Section 3.

3. Material models The purpose of a material model in peridynamics is to determine the values of the pairwise bond force density f(q,x,t) in terms of the smoothed displacements in the vicinity of x and q and any other physically relevant fields such as temperature. (The word “smoothed” will be omitted from now on, since we are no longer concerned with the underlying atomic system.) It is assumed that there is a horizon d, such that jq  xj > d 0 fðq; x; tÞ ¼ 0

(1.16)

for all x,q,t. The following discussion starts with the conceptually simplest material model and progresses to more advanced models.

6

1. Introduction

3.1 Linear microelastic model The simplest type of material model is called linear microelastic. In this model, each bond acts like a linear spring, and the body can be thought of as a network composed of an infinite number of these springs. To define the pairwise bond force density in a linear microelastic material, we first define the unit vector M in the direction of a deformed bond x ¼ q  x: Mðx; tÞ ¼

yðq; tÞ  yðx; tÞ . jyðq; tÞ  yðx; tÞj

(1.17)

Also define the bond elongation e and the bond strain s by eðx; tÞ ¼ jyðq; tÞ  yðx; tÞj  jxj;

sðx; tÞ ¼

eðx; tÞ . jxj

(1.18)

The bond elongation is therefore the change in length of a bond as it deforms. The pairwise bond force density is then given by fðq; x; tÞ ¼ bf ðsÞMðx; tÞ;

bf ðsÞ ¼ cðxÞsðx; tÞ

(1.19)

where c(x) is the spring constant for the bond x and bf is a scalar valued function that gives the magnitude of the force density. The linear microelastic material model is elastic in the usual sense of continuum mechanics. A body composed of linear microelastic material stores strain energy due to quasi-static loading from external forces. This energy storage is reversible, since it can all be recovered by reversing the external forces and unloading the body to its original condition. In other words, there is no energy dissipation. The stored energy can be identified with individual bonds. Each bond x has a micropotential w(s) such that bf ðsÞ ¼ dw ðsÞ ¼ 1 dw ðsÞ; de x ds

x¼ jxj

(1.20)

Hence, for the linear microelastic material, wðx; tÞ ¼

xcðxÞs2 . 2

(1.21)

The micropotential is related to the strain energy density at any point x by summing up the contributions from all the bonds connected to x: Z 1 Wðx; tÞ ¼ wðx; tÞdx (1.22) 2 Hx

The factor of 1/2 appears in (1.22) because each endpoint of a bond “owns” only half of the bonds’s micropotential.

3. Material models

7

The strain energy density given by (1.22) has the same meaning as in the local theory, since it represents the potential energy per unit volume that is stored at x due to deformation of the nearby material. If the material is isotropic as well as linear microelastic, this interpretation provides an easy way to calibrate c(x) if the general form of the dependence of c on bond length is given. For an isotropic material, c(x) depends only on the bond length, x ¼ |x|. By requiring the peridynamic strain energy density given by (1.22) to equal its value in the local theory for an isotropic expansion, it is easily shown that c is related to the bulk modulus k by the following expression (Silling and Askari, 2005): Z d 9k (1.23) x3 cðxÞdx ¼ . 2p 0 As a special case, if c(x) has the form cðxÞ ¼ c0

(1.24)

for any 0 < 0 > = > < VB > = C B6 7 Kt;t Kt;u 5 Vt  F A ¼ 0 (3.16) dVT @4 Kt;B > > > > ; : ; : Ku;B Ku;t Ku;u Vu 0 where the submatrix KB;B represents the unknown displacement in the internal region. The submatrices KB;t and KB;u represent the unknown displacements in the boundary region of the domain. The coupling and self-stiffness between the unknown vectors VB , Vt , and Vu are represented by the submatrices Kt;B ; Kt;t ; Kt;u ; Ku;B ; Ku;t , and Ku;u . The virtual work due to external tractions can be expressed as Z t $dudG ¼ dVT F (3.17) G

I. New concepts in peridynamics

66

3. Peridynamics for axisymmetric analysis

where F represents the applied load vector on the boundary. The explicit expression for the submatrices and the applied load vector are given by Mitts et al. (2020). For arbitrary virtual variations of the unknown vector V the weak form of PD governing is derived as

where

2

KB;B

6 H ¼ 4 Kt;B Ku;B

KB;t

KB;u

Kt;t Ku;t

Kt;u Ku;u

HV ¼ R

(3.18)

9 8 9 8 > > = < 0 > < VL > = 7 F and R ¼ ; V ¼ V 5 t > > > ; : > : ; Vu 0

(3.19)

3

For specified displacement constraints of Vu ¼ Vu , Eq. (3.19) can be reduced as

 ( K V ) KB;B KB;t B;u u VB ¼ (3.20) Kt;B Kt;t Vt Rt  Kt;u Vu The coefficient matrix H is nonsymmetric and sparsely populated. This linear system of equations can be solved using a preconditioned biconjugate gradient stabilization method (Bi-CGSTAB) (Van Der Vorst, 1992). This is an iterative technique and after solution is found the algorithm checks for damage. If damage occurs the coefficient matrix, H, changes and will continue to change as damage progresses. The incremental form of the PD equilibrium equations can be written as  tþDt  tþDt H VtþDt ¼ RtþDt  FtþDt (3.21a) K1 DVK K1 ¼ J with 

 tþDt

DVtþDt ¼ VtþDt  VtþDt K K K1

(3.21b)

where H VK1 represents the coefficient matrix at the t þ Dt load step at

the ðK  1Þth iteration. The residual is represented by JtþDt, and it is defined as the difference between external force, RtþDt , and the internal  tþDt  force, FtþDt K1 . Once damage occurs, H VK1 is updated as a result of the broken interactions between material points. When inertial effects are considered, the equation HV ¼ R can be rewritten as € þ HV ¼ R MV

(3.22)

€ is the acceleration in which M is the lumped diagonal mass matrix and V vector representing each material point. This transient system is solved

I. New concepts in peridynamics

67

5. Failure criteria

numerically using the finite difference method. Expressing the transient equation of motion in an incremental form results in € ¼ Rt  Ft MV

(3.23)

The external and internal force vectors are evaluated at time t. The internal force vector can be decomposed as Ft ¼ Ht Vt þ FtDt

(3.24)

The displacement vector at time step t þ Dt is calculated as _ Dt þ Dt2 M1 Jt VtþDt ¼ Vt þ V

(3.25)

Jt ¼ Rt  Ft .

(3.26)

t

where

To reduce the computational requirements, the solution procedure employs the Bi-CGSTAB method until failure is detected. To facilitate a smooth transition from the implicit to explicit solver, the displacement vector, VðÞ found during the failure load step, is used as an initial condition for the explicit algorithm. The initial condition of the velocity can be constructed as _ ¼V V

ðÞ

ðÞ

 VB Dt

(3.27) ðÞ

where Dt is the time step size. The displacement vector, VB , represents the implicit solution immediately prior to VðÞ .

5. Failure criteria In PD, damage is modeled by eliminating the interactions between the material points. This allows for natural crack initiation and propagation. The traditional failure criteria used in PD is based on the concept of critical stretch. When the stretch s between two material points exceeds the critical stretch value sc , the interaction (bond) between them is removed. Upon removal of a bond, damage initiates, and its growth continues naturally. The critical stretch parameter can be expressed in terms of the strain energy release rate Gc (Silling and Askari, 2005). The PD theory is not limited to critical stretch for bond breakage to reflect damage in the material. The critical stretch is usually related to the critical energy release rate from linear elastic fracture mechanics. If this parameter is not available, then it would be difficult to determine value of

I. New concepts in peridynamics

68

3. Peridynamics for axisymmetric analysis

FIGURE 3.4 Visibility of material point interactions across a crack surface.

critical stretch. Therefore, this study demonstrates the use of a stressbased criterion from classical continuum mechanics. It employs the PD representation of the stress components, and identifies the bond with the maximum stress. The weak form of PD enables the direct calculation of the stress components. Therefore, the method implemented for damage initiation and propagation is based on the maximum principle stress criteria. As shown in Fig. 3.4, the bond interaction is cut (red dotted line) at the center point and a crack surface is formed if the maximum principle stress between material point k and j exceeds the specified uniaxial strength. All other material points that have interactions (blue dotted lines) which extend through the newly generated crack surface (red dotted line) also have their interactions severed. These material points can no longer “see” the material points opposite the crack. This is known as the visibility criterion developed by Belytschko et al. (1994). Madenci et al. (2018) implemented the visibility criterion for a PD truss element in a finite element framework.

I. New concepts in peridynamics

5. Failure criteria

69

The k-th point is found by searching the material points for the largest stress n o ðkÞPD ð1ÞPD ð2ÞPD ðNÞPD smax ¼ max smax ; smax ; .smax (3.28) with

 ðNðkÞ ÞPD ð jÞPD ð1ÞPD ð2ÞPD smax ¼ max smax ; smax ; .smax

(3.29)

where N is the total number of material points in the discretization and NðkÞ is the number of points in the family of material point k. The ðkÞð jÞPD

maximum principle stress smax between material point k and j is defined as n o ðkÞð jÞPD PD PD ¼ max sPD smax (3.30) 1ðkÞð jÞ s2ðkÞð jÞ s3ðkÞð jÞ PD PD where sPD 1ðkÞð jÞ , s2ðkÞð jÞ , s3ðkÞð jÞ are the average of the principle stresses

between material points k and j. The average stress state between these points is defined as sPD ðkÞð jÞ ¼

PD sPD ðkÞ þ sð jÞ

2

(3.31)

PD in which sPD ðkÞ and sð jÞ are the stress states at material points k and j, ðkÞð jÞPD

respectively. If smax exceeds the critical stress, sult of the material, an incremental crack is introduced at the center cðkÞð jÞ of the bond point between points k and j. It is defined as   xðkÞ þ xð jÞ (3.32) cðkÞðjÞ ¼ 2 in which, xðkÞ and xð jÞ denote the locations of points k and j, respectively. The incremental crack indicated by red dotted line in Fig. 3.4 is defined by

D D cðkÞð jÞ þ tðkÞð jÞ ; cðkÞð jÞ  tðkÞð jÞ . (3.33) 2 2 The unit vector tðkÞð jÞ is normal to the direction of the maximum principle stress sPD maxðkÞð jÞ given by vector vðkÞð jÞ, i.e., tðkÞð jÞ $vðkÞð jÞ ¼ 0. The visibility of other material points is then checked, and their interactions (blue dotted lines) are removed if the crack (red dotted line) is situated between them (Madenci et al., 2018). In order to update the stiffness matrix, a failure parameter is introduced as ( ðkÞð jÞPD < sult or visible 1 if smax mðkÞðjÞ ¼ (3.34) 0 otherwise

I. New concepts in peridynamics

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3. Peridynamics for axisymmetric analysis

The local damage fðkÞ at a material point is defined as the ratio of number of broken bonds to the total number of bonds in the form (Silling and Askari, 2005) N ðkÞ P

fðkÞ ¼ 1 

j¼1

mðkÞð jÞ

NðkÞ

(3.35)

This ratio, ranging between 0 and 1, indicates possible crack formation when its values are about 1=2.

6. Numerical results Numerical results concern first the verification of the present approach by capturing the displacement and stress fields in a cylindrical body without a crack. Subsequently, crack propagation is simulated for an internal ring crack by employing combined implicit and explicit analyses solution methods. The implicit solution is performed until immediately before the onset of crack growth, and its propagation continues with an explicit analysis. During the numerical simulations, the weight function wðjxjÞ which indicates the degree of interaction between the material points is specified as wðjxjÞ ¼ d2 =jxj2

(3.36)

Also, a convergence study is performed for each configuration, and the global error measure is defined as (Mukherjee and Mukherjee, 1997) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u K h i2 u1X 1 ðrÞ ðcÞ t   (3.37) ε ¼  ðrÞ  u  ui u max K i¼1 i   where uðeÞ max is the absolute value of maximum displacement of the exact solution. The superscripts r and c denote the reference and computed solution, respectively. The parameter, K, represents the number of material points in the domain. The reference solution is obtained by using ANSYS, which is a commercially available finite element analysis software.

6.1 A cylindrical body with or without an internal ring crack under tension The radius and length of the cylinder are R ¼ 40 mm and L ¼ 40 mm, respectively. The Young’s modulus, density, and Poisson’s ratio are E ¼ 360 GPa, r ¼ 3100 kg/m3, and y ¼ 0:185, respectively. Fig. 3.5 shows

I. New concepts in peridynamics

6. Numerical results

71

FIGURE 3.5 Axisymmetric model of an elastic cylinder under prescribed displacement constraints.

the geometry, boundary conditions, and applied displacement constraints in the axisymmetric model of a cylinder. In the absence of a ring crack, the boundary conditions are enforced as wðr; z ¼ LÞ ¼ w0

(3.38a)

srz ðr; z ¼ LÞ ¼ 0

(3.38b)

uðr; z ¼ 0Þ ¼ 0

(3.38c)

wðr; z ¼ 0Þ ¼ 0

(3.38d)

uðr ¼ 0; zÞ ¼ 0

(3.38e)

srr ðr ¼ R; zÞ ¼ 0

(3.38f)

srz ðr ¼ R; zÞ ¼ 0

(3.38g)

The verification of the present approach is established by comparing the PD predictions with those of ANSYS with PLANE 182 elements. The PD discretization is achieved by using the center points of these elements. The mesh consists of 200  200 material points to maximize accuracy and minimize computational time. The horizon size is specified as d ¼ 4D with D ¼ Dr ¼ Dz ¼ 0:2 mm based on a convergence study.

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3. Peridynamics for axisymmetric analysis

(a)

( b)

Displacement u (r,z = 20)

m

Displacement u (r = 20, z) m

FIGURE 3.6 PD displacement predictions: (A) radial and (B) axial directions.

(a)

(b)

FIGURE 3.7 PD displacements along centerline of domain: (A) radial and (B) axial directions.

The PD displacement predictions are shown in Fig. 3.6. They are compared with the ANSYS results along the geometric center lines (r ¼ 20 mm and z ¼ 20 mm) in Fig. 3.7. Both the PD and ANSYS results are in good agreement. In the presence of an internal ring crack, the pre-existing crack is introduced by removing the interactions between the material points that cross through the crack plane rather than the direct imposition of traction-free conditions on the crack surfaces. The removal of the bonds is shown in Fig. 3.8. An internal ring crack has a length of 2a ¼ 10 mm. Its center is located d ¼ 15 mm away from the radial surface and 20 mm from the bottom edge. First, it focuses on the accuracy of the displacement field in the presence of a crack and the crack tip stress field. Second, it demonstrates crack growth path. Fig. 3.9 shows the PD displacement predictions for the

I. New concepts in peridynamics

6. Numerical results

73

FIGURE 3.8 Axisymmetric model of elastic cylinder with an internal ring crack.

FIGURE 3.9 PD discretization of axisymmetric model of elastic cylinder and removal of interactions across crack surfaces.

I. New concepts in peridynamics

74

3. Peridynamics for axisymmetric analysis

(a)

(b)

FIGURE 3.10 PD displacement predictions in the presence of an internal ring crack: (A) radial and (B) axial directions.

radial and axial directions. The PD prediction of axial displacement along the outer boundary is compared with the ANSYS results in Fig. 3.10. The PD and ANSYS results are in good agreement. The PD stress predictions shown in Fig. 3.11 correctly capture the stress concentration near the crack tip. Figs. 3.10 and 3.11 show the displacement and stress fields just before the crack begins to propagate. Once the critical stress value is reached and a bond is severed, the algorithm switches to explicit time integration to accurately capture crack growth. Fig. 3.12 shows damage (ratio of broken bonds to unbroken bonds) prior to crack growth, during growth from both ends, and the crack ending at the right edge of the cylinder.

7. Conclusions The axisymmetric PD model is validated by considering simple deformation analysis with and without a crack. The PD deformation and stress field compare well with those of ANSYS. The PD predictions capture the expected crack propagation path. The pre-existing internal ring crack starts growing simultaneously from both tips toward the center and free surface of the body until it reaches the free surface. The crack tip closer to the free edge grows faster. The present approach is accurate and effective for predicting damage initiation and growth in axisymmetric linear elastic solids.

I. New concepts in peridynamics

75

7. Conclusions

(a)

(b)

(c)

(d)

FIGURE 3.11 PD predictions of stress components in the presence of a crack: (A) srr , (B) szz , (C) sqq , and (D) srz .

(a)

(b)

(c)

FIGURE 3.12 Crack propagation as time progresses: (A) t ¼ t0 , (B) t ¼ t1 , and (C) t ¼ t2 .

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3. Peridynamics for axisymmetric analysis

References Belytschko, T., Lu, Y.Y., Gu, L., 1994. Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229e256. Gu, X., Madenci, E., Zhang, Q., 2018. Revisit of non-ordinary state based peridynamics. Eng. Fract. Mech. 190, 31e52. Madenci, E., Oterkus, E., 2014. Peridynamic Theory and its Applications. Springer, New York. Madenci, E., Barut, A., Futch, M., 2016. Peridynamic differential operator and its applications. Comput. Methods Appl. Mech. Eng. 304, 408e451. Madenci, E., Dorduncu, M., Barut, A., Futch, M., 2017. Numerical solution of linear and nonlinear partial differential equations using the peridynamic differential operator. Numer. Methods Part. Differ. Equ. 33, 1726e1753. Madenci, E., Dorduncu, M., Barut, A., Phan, N., 2018. Weak form of peridynamics for nonlocal essential and natural boundary conditions. Comput. Methods Appl. Mech. Eng. 337, 598e631. Madenci, Dorduncu, Barut, Phan, 2018. A state-based peridynamic analysis in a finite element framework. Eng. Fract. Mech. 195, 104e128. Madenci, E., Barut, A., Dorduncu, M., 2019. Peridynamic Differential Operator for Numerical Analysis. Springer, New York. Mitts, C., Naboulsi, S., Przybyla, C., Madenci, E., 2020. Axisymmetric peridynamic analysis of crack deflection in a single strand ceramic matrix composite. Eng. Fract. Mech. 235, 107074. Mukherjee, Y.X., Mukherjee, S., 1997. On boundary conditions in the element-free Galerkin method. Comput. Mech. 19, 264e270. Roy, P., 2018. Non-classical Continuum Models for Solids Using Peridynamics and Gauge Theory. Indian Institute of Science (Ph.D. thesis). Silling, S., Askari, E., 2005. A mesh-free method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526e1535. Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E., 2007. Peridynamic states and constitutive modeling. J. Elasticity 88, 151e184. Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solid. 48, 175e209. Van Der Vorst, H.A., 1992. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631e644. Wildman, R.A., Gazonas, G.A., 2016. Absorbing boundary conditions with verification. In: Handbook of Peridynamic Modeling, pp. 284e310. Zhang, Y., Qiao, P., 2018. An axisymmetric ordinary state-based peridynamic model for linear elastic solids. Comput. Methods Appl. Mech. Eng. 341, 517e550. Zhang, Y., Qiao, P., 2019. Peridynamic simulation of two-dimensional axisymmetric pull-out tests. Int. J. Solid Struct. 168, 41e57.

I. New concepts in peridynamics

C H A P T E R

4 Peridynamics damage model through phase field theory Pranesh Roy, Anil Pathrikar, Debasish Roy Centre of Excellence in Advanced Mechanics of Materials, Indian Institute of Science, Bangalore, Karnataka, India

O U T L I N E 1. Introduction

78

2. Phase field theory: A brief recap

79

3. PD 3.1 3.2 3.3 3.4 3.5

82 82 82 84 84 85

reformulation of phase field theory Kinematics Governing equations Kinematic correspondence Constitutive correspondence Equations in an explicit form

4. Criterion for bond breaking

86

5. Numerical illustrations 5.1 Dynamic crack branching 5.2 Simulation of Kalthoff-Winkler experiment

87 87 90

6. Concluding remarks

94

References

95

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00007-X

77

© 2021 Elsevier Inc. All rights reserved.

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4. Peridynamics damage model through phase field theory

1. Introduction Prediction of material fracture is of utmost importance for design of various structures, e.g., aerospace structures, dams, buildings, etc. Development of a proper computational model is challenging, as cracks being material discontinuities do not directly fall within the ambit of classical continuum mechanics which is based on partial differential equations. Despite such difficulties, many theories have indeed been developed, notably the linear elastic fracture mechanics (LEFM) and damage mechanics. Among the LEFM-based computational technique are extended finite element method (X-FEM) and virtual crack closure technique (VCCT), which treat a crack as a sharp discontinuity and use Griffith’s criterion, i.e., when the energy release rate exceeds a critical value, new surface develops in an irreversible manner. Damage mechanicsebased models are also available, viz. the cohesive zone model (CZM). These models require a predefined crack path and an additional crack propagation criterion and hence face difficulties in the case of more complex crack propagation. The phase field theory is attractive, as it combines fracture and damage mechanics within a single framework. Depending on the perspective, phase fieldebased damage models can be classified as physically based models derived from the Ginzburg-Landau type phase transition theory or as the ones originating from a regularization of the crack functional in Griffith’s theory (Schneider et al., 2016). In both viewpoints, discontinuities are given a smeared representation through an additional field variable called the phase field or order parameter. This approach can track spontaneous emergence and propagation of cracks, eliminating the need for ad hoc crack tracking algorithms as in conventional fracture mechanics. This in turn also makes the model computationally more tractable. Observing the phase field at different material points, the state of the material may be identified. In the phase field model, undamaged, fractured, and damaged state of the material correspond to s ¼ 1, s ¼ 0, and s˛ð0; 1Þ, respectively, where s denotes the phase field. Here, one must together solve the governing equation for the phase field coupled with the momentum balance equations. A large volume of work is available on these lines; for instance, a discrete time model for dynamic fracture based on crack regularization is proposed by Bourdin et al. (2011). Miehe et al. (2010a,b) developed a thermodynamically consistent phase field model. Ginzburg-Landau type evolution equation for the order parameter is considered by Kuhn and Mu¨ller (2010), which uses a mobility constant associated with the evolution of the phase field. Schlu¨ter et al. (2014) investigate dynamic crack propagation using a phase field model. Apart from these theoretical developments, numerical methods have also been

I. New concepts in peridynamics

2. Phase field theory: A brief recap

79

developed. Borden et al. (2012) have presented a local adaptive refinement strategy based on locally refined T-splines for spatial integration and a staggered numerical time integration scheme. Miehe et al. (2010a,b) have proposed a robust algorithm based on operator splitting for numerical time integration. Since all of these methods require solutions of partial differential equations, they cannot directly faithfully reproduce the fragmentation of the material body. In this chapter, we develop a phase fieldebased damage model within the peridynamics (PD) setting so as to combine the advantages of both the methods and ameliorate certain limitations of the stand-alone versions. A Ginzburg-Landau type phase field evolution is formulated in an integrodifferential form, which is coupled with the equation of motiondagain written in the PD form. Using the phase field, a rational bond breaking criterion is proposed, thereby eliminating the ad hoc bond stretch or bond energyebased criteria typically used in the literature on PD (Silling and Askari, 2005; Breitenfeld, 2014; Chowdhury et al., 2016). This makes physical fragmentation of the material body possible. Note that the phase field, being a regularized representation of discontinuity, cannot by itself display physical fragmentation. As the phase field evolution equation involves derivative terms of the phase field; they may induce numerical problems when sharp cracks need to be modeled. We derive the governing equation of the phase fieldebased PD theory from Hamilton’s principle and incorporate within it a Ginzburg-Landau type dissipative relaxation term. An irreversibility criterion is also applied externally to capture the irreversible nature of damage. We use the constitutive correspondence method to relate the PD force states with the corresponding classical objects, as strictly PD-based constitutive models are rare in the literature. By way of demonstrating the efficacy of our proposal, we furnish a few numerical simulations on the dynamic crack propagation/ branching problem and on the Kalthoff-Winkler experiment, while comparing them with the available experimental evidence. The rest of the chapter is organized as follows. In Section 2, a brief recap of the phase field theory is presented. The phase fieldebased PD equations are derived in Section 3. A criterion for bond breaking in proposed in Section 4. Numerical simulations are furnished in Section 5 for a dynamic crack branching problem and the Kalthoff-Winkler experiment. A few concluding remarks are offered in Section 6.

2. Phase field theory: A brief recap Phase field theory provides a framework enabling treatment of damage and fracture mechanics in a seamless manner. The phase field variable or order parameter s offers a regularized representation of sharp crack

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4. Peridynamics damage model through phase field theory

FIGURE 4.1 (A) Material body with a discontinuity and (B) representation of the discontinuity through the phase field parameter.

topology (see Fig. 4.1). The variable s takes on the value 1 for the undamaged state, 0 for the fractured state, and an intermediate value between 0 and 1 for the damaged state. In order that a surface area G3Rn1 may open up (n ¼ 2 or 3 for 2D and 3D domains), the amount of fracture energy required to be released is approximately written as (see Schlu¨ter et al., 2014) # Z Z Z " Z ð1  sÞ2 2 Gc dA z gGc dV ¼ þ ls jVsj Gc dV ¼ js dV (4.1) 4ls G V V V Here, Gc , g, and ls , respectively, denote the critical energy release rate, the crack functional and the length scale parameter. js denotes the fracture energy density. Assuming that the fracture/damage process involves no dissipation, we may state Hamilton’s principle (Eq. 4.2 below), the resulting Euler-Lagrange equations (Eqs. 4.3 and 4.4), and boundary conditions (Eqs. 4.5 and 4.6) as  Z t2  Z Z Z dLdV þ f $ dudV þ t $ dudV dt ¼ 0; t1 < t2 (4.2) t1

V

V

vV

  vL V$ f¼0 vVu   vL vL  V$ ¼0 vs vVs   vL T nþt¼0 vVu vL $n ¼ 0 vVs

I. New concepts in peridynamics

(4.3) (4.4) (4.5) (4.6)

2. Phase field theory: A brief recap

81

Here L denotes the Lagrange density (assumed to depend on Vu, s, and Vs), f and t are the body and surface force vectors, respectively. L can be expressed as L ¼ jk  je  js

(4.7)

where jk is the kinetic energy density and je the strain energy density. jk may be written as 1 _ u_ jk ¼ ru$ 2

(4.8)

It is assumed that the degradation of stress happens only in tension  þ  and not in compression. Using a decomposition of j into positive je and e   negative j e parts, Eq. (4.7) can be recast as   (4.9) L ¼ jk  gðsÞjþ e þ je  js Here gðsÞ denotes the degradation function which is typically taken  as gðsÞ ¼ s2 þ h. The explicit expressions for jþ e and je are given below. l 2 jþ e ¼ CtrðεÞDþ þ mεþ : εþ 2 l 2 j e ¼ CtrðεÞD þ mε : ε 2

(4.10) (4.11)

Here l and m the Lame parameters and ε is the linearized strain tensor T

given by ε ¼ VuþðVuÞ . ε is split using a spectral decomposition (Miehe et 2 al., 2010; Hofacker and Miehe, 2013). ε ¼ εþ þ ε

(4.12)

with εþ : ¼

m X

Cεi Dþ ni 5ni

(4.13)

Cεi D ni 5ni

(4.14)

i¼1

and ε : ¼

m X i¼1

εi and ni are the eigenvalues and eigenvectors of the strain tensor, respectively. m is the dimension of the space of independent spatial coordinates. The operators C,D are defined as CxDþ : ¼ 12 ðx þjxjÞ and CxD : ¼ 1 ðx jxjÞ where x is a scalar. Envisioning the phase field as an order 2

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4. Peridynamics damage model through phase field theory

parameter, one may also apply a correction for dissipative relaxation. The corrected phase field equation is given as   s_ 1s þ 2sjþ  G ¼0 (4.15) 2lDs þ c e M 2l Here M > 0 is the mobility constant. In order to account for irreversibility (no crack healing) in the damage process, the following constraint may be imposed: s_  0

(4.16)

3. PD reformulation of phase field theory The stage is now set to derive the governing equations of the PD phase field theory. We consider only small deformation in this work.

3.1 Kinematics The state of the body is specified through the relative displacement vector state and the relative phase field scalar state whose definitions are furnished below. Relative displacement vector state: U½xCxD ¼ u0  u

(4.17)

Relative phase field scalar state: S ½xCxD ¼ s0  s

(4.18)

In Eqs. (4.17) and (4.18), we have used the following abbreviations: u0 ¼ uðx0 Þ and s0 ¼ sðx0 Þ.

3.2 Governing equations In the absence of dissipation, the mechanics of fracture may be described by Hamilton’s principle stated below.  Z t2  Z Z dLdV þ f $ dudV dt ¼ 0; t1 < t2 (4.19) t1

U

U

Here L denotes the Lagrange density per unit volume, f the body force per unit volume, U3Rn the material body domain, and n the spatial dimension of the material body. t1 and t2 are arbitrary time instants with

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3. PD reformulation of phase field theory

_ U, S , and s, one may write its varit1 < t2 . Allowing L to depend on u, ation as dL ¼ Du_ L$du_ þ DU L∙dU þ DS L∙dS þ Ds L ds

(4.20)

where, Du_ , DU , DS , and Ds are Fre´chet derivative operators with respect _ U, S , and s, respectively. Variation of the Lagrange density may be to u, written as dL ¼ I$du_ þ ð TÞ∙dU þ ð P Þ∙dS þ ð pÞ ds

(4.21)

Here the following abbreviations are used: I ¼ Du_ L, T ¼ DU L, P ¼ DS L and p ¼ Ds L. Note also that we have abbreviated U½xCxD as U, S ½xCxD as S , T½xCxD as T, and P ½xCxD as P to avoid notational complexity.

 R The symbol ∙ denotes the inner product A∙B ¼ HðxÞ ACxD $BCxDdVx0 between vector states or scalar states as applicable. Using Eq. (4.21) in Eq. (4.19), we may write  Z t2 Z Z ðI$du_ þ ð TÞ∙dU þ ð P Þ∙dS þ ð pÞ dsÞdV þ f$dudV dt ¼ 0 t1

U

U

(4.22) Using the condition duðx; t1 Þ ¼ duðx; t2 Þ ¼ 0 and employing integration by parts, the first term on the left-hand side of Eq. (4.22) may be simplified as t2 Z t2 Z Z t2 Z Z t2 Z Z dI dI _ $dudVdt ¼  $dudVdt I$dudVdt ¼ I$dudV  t1 U U t1 U dt t1 U dt t1 (4.23) ðx 4x0 Þ,

Likewise, using a change of dummy variable the second and third terms on the left-hand side of Eq. (4.22) may be recast as Z t2 Z Z t2 Z Z  T∙dUdVdt ¼  T$dðu0  uÞdV 0 dVdt U

t1

Z ¼

Z 

t2 t1

Z U

Z Z

t1

Z

P ∙dS dVdt ¼  Z ¼

U U

t1

t2

U U t2

t1 t2

t1

  T  T0 $dudV 0 dVdt

Z Z U U

Z Z

U U



(4.24)

P dðs0  sÞdV 0 dVdt

 P  P0 dsdV 0 dVdt

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(4.25)

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4. Peridynamics damage model through phase field theory

Using Eq. (4.23) through (4.25) in Eq. (4.22) and noting that du are independent and arbitrary, one may obtain the following Lagrange equations. Z n o 0 0 dI T  T dV þ f ¼ dt H ð xÞ Z   P  P 0 dV 0  p ¼ 0

and ds Euler-

(4.26) (4.27)

HðxÞ

In the above equations, we assume that interactions occur only within a finite neighborhood H4U. Following the Ginzburg-Landau theory, we apply a correction for dissipative relaxation in Eq. (4.27) as Z   s_ (4.28) P  P 0 dV 0  p ¼ M HðxÞ

3.3 Kinematic correspondence Since constitutive equations strictly based on PD are scarce, we follow the constitutive correspondence proposed by Silling et al. (2007). The nonlocal measures GU and GS are defined as follows. Nonlocal displacement gradient: Z  1 0 GU ¼ wðjxjÞðUCxD5xÞdV K (4.29) H

Nonlocal gradient of phase field: Z  1 GS ¼ wðjxjÞS CxDx dV 0 K Here K ¼

R

(4.30)

H

H wðjxjÞðx5xÞdV

0

is the shape tensor.

3.4 Constitutive correspondence In order to relate the PD force states with their classical counterparts, we make use of equivalence of the variations of classical and PD Lagrange densities under homogeneous deformation conditions. Replacing the classical gradient terms with the corresponding nonlocal measures, i.e., Vu by GU and Vs by GS , we may write the variation of the Lagrange density as dL ¼

vL vL vL vL $du_ þ ds þ : dGU þ $dGS vu_ vs vGU vGS

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(4.31)

3. PD reformulation of phase field theory

85

One may rearrange the terms in Eq. (4.31) using tensor identities as ! Z vL vL 1 vL ds K x $dUdV 0 þ dL ¼ $du_ þ wðjxjÞ vu_ vs vGU H ! vL 1  þ wðjxjÞ $ xK dS dV 0 vGS H Z

(4.32)

One arrives at the relations between the PD states and the corresponding classical ones through a comparison of Eqs. (4.32) and (4.21). I¼

vL vu_

TCxD ¼  wðjxjÞ

vL 1 K x vGU

vL vs vL 1  P CxD ¼ wðjxjÞ $ xK vGS p¼ 

(4.33) (4.34) (4.35) (4.36)

3.5 Equations in an explicit form In this section, we present an explicit form of the equation considered in the previous section. The small strain tensor may be expressed as εnl ¼

GU þ ðGU ÞT 2

(4.37)

Note that, the subscript ðnlÞ stands for “nonlocal.” With the nonlocal measures, the Lagrange density per unit volume in Eq. (4.7) is given by L ¼ jk  jnle  jnls

(4.38)

Here, jnle represents the nonlocal strain energy density and is given by    jnle ¼ s2 þ h jþ (4.39) nle þ jnle  where, jþ nle and jnle are the positive and the negative parts of the nonlocal strain energy density. These are expressed as

l 2 jþ nle ¼ 2Ctrðεnl ÞDþ þ mεnlþ : εnlþ l 2 j nle ¼ Ctrðεnl ÞD þ mεnl : εnl 2

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(4.40) (4.41)

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4. Peridynamics damage model through phase field theory

One may obtain the nonlocal fracture energy employing the nonlocal gradient of the phase field ðGS Þ (see also Eq. 4.1) as " # ð1  sÞ2 2 þ ljGS j jnl s ¼ Gc (4.42) 4l vL vL vL vL in Eq. (4.33) through (4.36) may be , , and The terms , vu_ vGS vGU vs written explicitly as follows. vL ¼ ru_ (4.43) vu_      vL ¼  s2 þ h lCtrðεnl ÞDþI þ 2m εnlþ  lCtrðεnl ÞD I þ 2mεnl (4.44) vGU vL ¼ 2Gc l GS vGS

(4.45)

vL Gc ð1  sÞ ¼  2sjþ nle þ vs 2l

(4.46)

Note that Eq. (4.46) is obtained using the following identity:

R  1  R vGS 2 v 0 ¼ 0 and H wðjxjÞx dV K H wðjxjÞ vsjGS j ¼ 2GS , vs ¼ 2GS ,

xdV 0 ¼ 0.

4. Criterion for bond breaking As the phase field provides for a diffused representation of discontinuities through a smooth function, it cannot by itself model a body breaking into parts. On the other hand, PD achieves the physical fragmentation of material body based on critical values of bond stretch or bond energy (see Silling and Askari, 2005; Breitenfeld, 2014; Chowdhury et al., 2016). The critical bond stretch or bond energy are determined as functions of the critical energy release rate and the horizon radius in an ad hoc manner, and this method becomes difficult to apply for non-ordinary stateebased PD case. In circumventing the limitations of the stand-alone versions of both the phase field and PD, we develop a more rational criterion for bond breaking based on the phase field. We split of the influence function w as (see Tupek et al., 2013): b ðs; s0 Þ wðjxj; s; s0 Þ ¼ wðjxjÞ w

(4.47)

Here wðjxjÞ is the influence function for the undamaged material. Considering that a bond breaks in tension (i.e., when its stretch l is b ðs; s0 Þ is chosen such that: positive), w b ðs; s0 Þ ¼ 0 w

if

s þ s0 ¼ 0 and l > 0 2

I. New concepts in peridynamics

(4.48)

5. Numerical illustrations

87

b ðs; s0 Þ is a non-increasing function of both s and s0 . We choose Here w 0 b ðs; s Þ ¼ 1 whenever the criterion in Eq. (4.48) is not met. One may notice w that Eq. (4.48) describes the condition when a bond between two material points x and x0 breaks.

5. Numerical illustrations The PD damage model, augmented through phase field, will be validated through a few numerical simulations followed by a comparison against experimental observations. We consider a dynamic crack branching problem where a two-dimensional glass plate with a pre-notch under plane strain condition is subjected to tension. The crack initiation and propagation are tracked through the phase field and the crack branching is successfully captured. The crack branching angle is compared with the experimental results. Next, the Kalthoff-Winkler experiment is simulated assuming plane strain conditions on a steel specimen under low velocity which ensures brittle fracture. The angle of deflection of the crack is measured and the same is compared with experimental results. In order to carry out numerical simulation, the domain is discretized into a finite number of nodes, each of which is associated with a finite volume. Riemann-sum approximation is used to represent the integral terms. Boundary conditions are imposed through volume constraints using finite area patches extending beyond the domain. For the details of numerical implementation of PD equations, refer to Breitenfeld et al. (2014).

5.1 Dynamic crack branching We consider a glass plate with a pre-notch under tensile loading at the top and bottom boundaries (see Fig. 4.2). Plane strain conditions are assumed. The length (L) and the width (b) of the plate are L ¼ 0:1 m and b ¼ 0:04 m, respectively. The material parameters used in the simulation are: mass density r ¼ 2450 kg/m3, Young’s modulus E ¼ 32 GPa, Poisson’s ratio n ¼ 0:2, and critical energy release rate Gc ¼ 3 J/m2 10 L (see Borden et al., 2012). The mobility constant M is considered to be Gc T sffiffiffiffiffiffiffiffiffiffi 5rL2 E (see Schlu¨ter et al., 2014). The residwith m ¼ 2ð1 þ nÞ 2m ual stiffness parameter h is chosen as 1  109 . where T ¼

I. New concepts in peridynamics

88

4. Peridynamics damage model through phase field theory

FIGURE 4.2 Geometry and boundary conditions of the plate.

The contour plots of the phase field and the hydrostatic stress are furnished in Fig. 4.3 at different time instants. Crack initiates at the notch tip at 14 ms and propagates horizontally until 39 ms when the crack tip starts thickening and it leads to a branching of the crack in two paths. The crack branches at an angle z 28+ , which is in agreement with the experimentally observed results. Ramulu and Kobayashi (1985) experimentally observed the crack tip thickening phenomenon prior to branching which may be due to the crack surface roughness that develops when instability initiates. To cite a few works on the instability of an accelerating crack front, one may start with the theoretical prediction of a limiting velocity vc ¼ 0:6 VR (see Yoffe, 1951) with VR the Rayleigh wave speed above which the crack starts to branch. However, experimental results by Fineberg et al. (1992) show that the critical velocity for crack branching is approximately 0:36 VR. Sharon and Fineberg (1996) predict  that branching angles follow a Gaussian distribution with a mean of 30 . Molecular dynamic simulations by Abraham et al. (1994) also show the  branching angle to be 30 . Ramulu et al. (1983) have determined the crack branching angle in a polycarbonate material; theoretically, the branching    angle may vary between 26 and 31 (with an average value of 28.5 ) and  the experimentally observed crack branching angle varies between 22   and 34 (with an average value of 25 ). These studies show that the phase fieldebased PD theory can indeed predict the crack branching angle. We have also carried out a study for a pre-notch of length L=3 located at the center of the specimen as shown in Fig. 4.2. The material parameters and boundary conditions are the same as in the previous problem. The contour plots of the phase field and the hydrostatic stress are shown in Fig. 4.4 at different time instants. One may observe that the crack branches at an angle z 4:5+ , which is much smaller compared to the previous problem where the pre-notch was placed on left side of the specimen.

I. New concepts in peridynamics

5. Numerical illustrations

89

FIGURE 4.3 (A) Spatial variation of phase field s and (B) hydrostatic stress sH ¼ s11 þ s22 þ s33   2  N m at different times. 3

I. New concepts in peridynamics

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4. Peridynamics damage model through phase field theory

FIGURE 4.4 (A) Spatial variation of phase field s and (B) hydrostatic stress sH ¼ s11 þ s22 þ s33   2  N m at different times. 3

5.2 Simulation of Kalthoff-Winkler experiment The Kalthoff-Winkler experiment shows that the brittleness or ductility of a material not only depends on the material parameters but also on the loading conditions. A transition of failure mode from brittle to ductile happens upon increasing the impact velocity (Kalthoff, 2000). For low velocities, the crack propagates at an approximate angle of 70+ with respect to the horizontal direction. When the impact velocity reaches a critical value, a dynamic failure mode transition occurs and the material fails in plastic deformation instead of damage. Within the present scope of our work, we have not considered plasticity. Thus, results pertaining to only brittle damage will be furnished. A cylindrical projectile impacts a steel plate with two pre-notches at a velocity of 33 m/s (see Fig. 4.5). As the contact velocity increases initially followed by a decrease when the projectile rebounds, a velocity profile as depicted in Fig. 4.6 is applied, with a small rise in time and a maximum value of v ¼ 16:5 m/s (see Borden et al., 2012; Hofacker and Miehe, 2013).

I. New concepts in peridynamics

5. Numerical illustrations

91

FIGURE 4.5 Geometry and boundary conditions for the Kalthoff-Winkler experiment.

FIGURE 4.6 Application of the impact velocity.

I. New concepts in peridynamics

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4. Peridynamics damage model through phase field theory

Exploiting the symmetry in the geometry, we model only half of the structure. Displacement uy in the vertical direction at the bottom of the structure is constrained to zero through a boundary patch as it is a line of symmetry and all other boundaries are traction-free. We assume plane strain conditions for simulation. The model parameters chosen in the simulation are borrowed from Borden et al. (2012). The geometry of the specimen is specified through the length L ¼ 0:1 m (half of the total length of the specimen), width b ¼ 0:1 m and pre-notch length 0.05 m. The material parameter considered are: mass density r ¼ 8000 kg/m3, Young’s modulus E ¼ 190GPa, Poisson’s ratio n ¼ 0:3 and critical energy release rate Gc ¼ 2:213  104 J/m2. 10 L (see Schlu¨ter et al., 2014); T ¼ The mobility constant M is given by Gc T sffiffiffiffiffiffiffiffiffiffi

5rL2 E . The residual stiffness parameter h is taken as where m ¼ 2ð1 þ nÞ 2m 1  109 . From Fig. 4.7, one may observe that the crack propagates at an angle of  69 with respect to the horizontal direction which matches well with the experimental observations (see Kalthoff, 2000). The evolution of the hydrostatic pressure is shown in Fig. 4.7. It may be noted that the hydrostatic pressure is positive (tensile) at the crack tip. When the compression wave reflects back from the right boundary, a second crack emerges and propagates from the right side. This crack has been obtained in many simulations (see Belytschko et al., 2003; Zhou et al., 2016; Kosteski et al., 2012; Dipasquale et al., 2014) although not detected in experiments. We suspect that an improper assessment (underestimation) of the critical energy release rate Gc may be responsible for the second crack. We have also carried out a simulation considering only a single notch, i.e., by removing the bottom pre-notch in Fig. 4.5. The material parameters used are the same. However, as the problem no longer remains symmetric, we have analyzed the entire structure. The impact velocity is applied as shown in Fig. 4.5 and its variation with time is shown in Fig. 4.6. From Fig. 4.8, it may be observed that the crack propagates at an angle  of 58 along with two cracks emerging from the right surface. These cracks eventually branch after their straight propagation due to an increasing crack tip velocity.

I. New concepts in peridynamics

5. Numerical illustrations

93

FIGURE 4.7 (A) Spatial variation of phase field s and (B) hydrostatic stress sH ¼ s11 þ s22 þ s33   2  N m at different times. 3

I. New concepts in peridynamics

94

4. Peridynamics damage model through phase field theory

FIGURE 4.8 (A) Spatial variation of phase field s and (B) hydrostatic stress sH ¼ s11 þ s22 þ s33   2  N m at different times. 3

6. Concluding remarks The phase field equations are reformulated and coupled with the PDbased equations of motion. This results in a nonlocal damage model that offers the alleviation of a few limitations encountered in certain competing models of a similar genre. Specifically, based on the phase field, a new bond breaking criterion is developed in order to replace the

I. New concepts in peridynamics

References

95

ad hoc bond stretch or bond energy based criteria. Taking advantages from both PD and phase field, this combined setup improves the model’s efficacy in simulating fracture and fragmentation. This work sets the stage for a computationally simpler approach to delamination problems in composites especially under mixed mode loading. Extending the present model to include thermo-visco-plasticity for ductile fracture is also our future interest.

References Abraham, F.F., Brodbeck, D., Rafey, R.A., Rudge, W.E., 1994. Instability dynamics of fracture: a computer simulation investigation. Phys. Rev. Lett. 73 (2), 272. Belytschko, T., Chen, H., Xu, J., Zi, G., 2003. Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int. J. Numer. Methods Eng. 58 (12), 1873e1905. Borden, M.J., Verhoosel, C.V., Scott, M.A., Hughes, T.J., Landis, C.M., 2012. A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217, 77e95. Bourdin, B., Larsen, C.J., Richardson, C.L., 2011. A time-discrete model for dynamic fracture based on crack regularization. Int. J. Fract. 168 (2), 133e143. Breitenfeld, M., 2014. Quasi-static Non-ordinary State-Based Peridynamics for the Modeling of 3d Fracture. Doctoral dissertation, University of Illinois at Urbana-Champaign. Breitenfeld, M.S., Geubelle, P.H., Weckner, O., Silling, S.A., 2014. Non-ordinary state-based peridynamic analysis of stationary crack problems. Comput. Methods Appl. Mech. Eng. 272, 233e250. Chowdhury, S.R., Roy, P., Roy, D., Reddy, J.N., 2016. A peridynamic theory for linear elastic shells. Int. J. Solid Struct. 84, 110e132. Dipasquale, D., Zaccariotto, M., Galvanetto, U., 2014. Crack propagation with adaptive grid refinement in 2D peridynamics. Int. J. Fract. 190 (1e2), 1e22. Fineberg, J., Gross, S.P., Marder, M., Swinney, H.L., 1992. Instability in the propagation of fast cracks. Phys. Rev. B 45 (10), 5146. Hofacker, M., Miehe, C., 2013. A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns. Int. J. Numer. Methods Eng. 93 (3), 276e301. Kalthoff, J.F., 2000. Modes of dynamic shear failure in solids. Int. J. Fract. 101 (1e2), 1e31. Kosteski, L., D’Ambra, R.B., Iturrioz, I., 2012. Crack propagation in elastic solids using the truss-like discrete element method. Int. J. Fract. 174 (2), 139e161. Kuhn, C., Mu¨ller, R., 2010. A continuum phase field model for fracture. Eng. Fract. Mech. 77 (18), 3625e3634. Miehe, C., Hofacker, M., Welschinger, F., 2010a. A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. 199 (45), 2765e2778. Miehe, C., Welschinger, F., Hofacker, M., 2010b. Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J. Numer. Methods Eng. 83 (10), 1273e1311. Ramulu, M., Kobayashi, A.S., 1985. Mechanics of crack curving and branchingda dynamic fracture analysis. In: Dynamic Fracture. Springer Netherlands, pp. 61e75. Ramulu, M., Kobayashi, A.S., Kang, B.S.J., Barker, D.B., 1983. Further studies on dynamic crack branching. Exp. Mech. 23 (4), 431e437. Schlu¨ter, A., Willenbu¨cher, A., Kuhn, C., Mu¨ller, R., 2014. Phase field approximation of dynamic brittle fracture. Comput. Mech. 54 (5), 1141e1161.

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Schneider, D., Schoof, E., Huang, Y., Selzer, M., Nestler, B., 2016. Phase-field modeling of crack propagation in multiphase systems. Comput. Methods Appl. Mech. Eng. 312, 186e195. Sharon, E., Fineberg, J., 1996. Microbranching instability and the dynamic fracture of brittle materials. Phys. Rev. B 54 (10), 7128. Silling, S.A., Askari, E., 2005. A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83 (17), 1526e1535. Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E., 2007. Peridynamic states and constitutive modeling. J. Elasticity 88 (2), 151e184. Tupek, M.R., Rimoli, J.J., Radovitzky, R., 2013. An approach for incorporating classical continuum damage models in state-based peridynamics. Comput. Methods Appl. Mech. Eng. 263, 20e26. Yoffe, E.H., 1951. LXXV. The moving griffith crack. Lond., Edinb., Dublin Philos. Mag. J. Sci. 42 (330), 739e750. Zhou, X., Wang, Y., Qian, Q., 2016. Numerical simulation of crack curving and branching in brittle materials under dynamic loads using the extended non-ordinary state-based peridynamics. Eur. J. Mech. Solid. 60, 277e299.

I. New concepts in peridynamics

C H A P T E R

5 Beam and plate models in peridynamics Zhenghao Yang, Erkan Oterkus, Selda Oterkus Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom

O U T L I N E 1. Introduction

98

2. Peridynamic Timoshenko beam formulation 2.1 Classical Timoshenko beam formulation 2.2 Peridynamic Timoshenko beam formulation

98 98 99

3. Peridynamic Mindlin plate formulation 3.1 Classical Mindlin plate formulation 3.2 Peridynamic Mindlin plate formulation

102 102 104

4. Numerical results 4.1 Simply supported beam subjected to transverse loading 4.2 Mindlin plate subjected to simply supported boundary conditions

109 109 110

5. Conclusions

111

References

112

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00006-8

97

© 2021 Elsevier Inc. All rights reserved.

98

5. Beam and plate models in peridynamics

1. Introduction In the original peridynamic formulation, each material point has only translational degrees of freedom (DOF). Numerical solution of peridynamic equations are usually done by using meshless technique and uniform discretization. For certain types of structures, especially if one of the dimensions is very small or very large with respect to other dimensions, to obtain a solution can be computationally very expensive. For such structures such as beams, plates, and shells, additional rotational degrees of freedom should be introduced and the formulation should be modified accordingly. There are currently various peridynamic formulations available in the literature to represent beams, plates, and shells. Taylor and Steigmann (2015) developed a formulation suitable for analysis of thin plates. Diyaroglu et al. (2019) proposed a state-based formulation for Euler beams. This formulation was further extended by Yang et al. (2020) to model Kirchhoff plates. By utilizing non-ordinary state-based peridynamics, O’Grady and Foster (2014a, 2014b) developed Euler and Kirchhoff plate formulations suitable for thin plates. For relatively thick plates, transverse shear deformations become important. Therefore, Diyaroglu et al. (2015) introduced peridynamic Timoshenko beam and Mindlin plate formulations by taking into account transverse shear deformations. Recently, Chowdhury et al. (2016) developed a formulation for linear elastic shells. In addition, Nguyen and Oterkus (2019) investigated thermomechanical behavior of shell structures. In this chapter, peridynamic Timoshenko beam and Mindlin plate formulations are presented, since they are suitable for both thin and thick beams and plates. The governing equations are obtained by utilizing Euler-Lagrange equations.

2. Peridynamic Timoshenko beam formulation 2.1 Classical Timoshenko beam formulation According to the Timoshenko beam theory, the displacement field of a material point can be represented in terms of the displacement field of the material points on the neutral axis in xz plane as uðx; z; tÞ ¼ zqðx; tÞ

(5.1a)

wðx; z; tÞ ¼ wðx; tÞ

(5.1b)

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2. Peridynamic Timoshenko beam formulation

99

where qðx; tÞ and wðx; tÞ denote the rotation and transverse displacement of the material point on the neutral axis, respectively. Thus, the straindisplacement relationships can be written as vq vx vw gxz ¼ q þ vx εxx ¼ z

(5.2a) (5.2b)

According to the Hooke’s Law, the stress components can be written as vq sxx ¼ Eεxx ¼ Ez vx
 vw sxz ¼ Ggxz ¼ G q þ vx

(5.3a) (5.3b)

According to the classical continuum mechanics (CCM) theory, the average cross-sectional strain energy density of a particular material point on the neutral axis of the beam can be cast by integrating the strain energy density over the cross-section and divided by the cross-sectional area as Z 1 ðsxx εxx þ sxz gxz ÞdA (5.4) WCCM ¼ 2A A

with A being the cross-section area. Inserting Eq. (5.3) into Eq. (5.4) yields (

 ) 1 vq 2 vw 2 WCCM ¼ þ kGA q þ EI (5.5) 2A vx vx where k is shear coefficient, which depends on the geometry of the crosssection and I represents the second moment of cross-sectional area, which is defined as Z I ¼ z2 dA (5.6) A

2.2 Peridynamic Timoshenko beam formulation The PD equations of motion can be derived by utilizing Euler-Lagrange equation: d vL vL  ¼0 dt vu_ ðkÞ vuðkÞ

I. New concepts in peridynamics

(5.7)

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5. Beam and plate models in peridynamics

where L ¼ T  U is the Lagrangian and u represents the displacement vector, which contains the following components u¼ðq

w ÞT

(5.8)

If we assume the cross-section as constant, the total kinetic energy of the system, T, can be cast as
 1X I _2 2 T¼ r (5.9) q þ w_ ðkÞ VðkÞ 2 k ðkÞ A ðkÞ For a conservative system, the first term of the Lagrange’s equation becomes 8 9 I€ > > < qðkÞ = d vL V ¼ rðkÞ A (5.10) > > ðkÞ dt vu_ ðkÞ : w € ; ðkÞ

The PD strain energy density function has a nonlocal characteristic such that the strain energy of a particular material point k depends on both its displacement and all other material points in its family, which can be expressed as   ðkÞ ðkÞ WPD ¼ WPD uðkÞ ; uð1k Þ ; uð2k Þ ; uð3k Þ ; . (5.11) where uðkÞ represents the displacement vector of the material point k and uðik Þ ði ¼ 1; 2; 3; .Þ is the displacement vector of the i th material point within the horizon of the material point k. The total potential energy stored in the body can be obtained by summing potential energies of all material points including strain energy and energy due to external loads as  X ðkÞ  X U¼ WPD uðkÞ ; uð1k Þ ; uð2k Þ ; uð3k Þ ; . VðkÞ  bðkÞ $uðkÞ VðkÞ (5.12) k

k

where b is the generalized body force density vector, respectively, which in this study can be defined as b ¼ ð bq

b z ÞT

(5.13)

Here, the entries of the body load density vector bq and bz correspond to moment and transverse force, respectively.

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101

Thus, the second term of the Lagrange’s equation is  vL v X ðnÞ   ¼ WPD uðnÞ ; uð1n Þ ; uð2n Þ ; uð3n Þ ; . VðnÞ vuðkÞ vuðkÞ n 

¼

v X b $u V vuðkÞ n ðnÞ ðnÞ ðnÞ XvW ðnÞ

PD

n;i

vuðkÞ

v X ðdnk þ dnik ÞVðnÞ  b d V vuðkÞ n ðnÞ nk ðnÞ

(5.14)

1 ðkÞ XvW ð jÞ vW PD ¼ @ PD VðkÞ þ Vð jÞ A  bðkÞ VðkÞ vuðkÞ vu ðkÞ j 0

Inserting Eqs. (5.10) and (5.14) into the Euler-Lagrange equation gives 9 8 ðkÞ XvW ð jÞ > > vW > > PD PD > 8 9 > > VðkÞ þ Vð jÞ > > > > > ( ðkÞ ) vqðkÞ > > > j = < vqðkÞ < I €q > = bq ðkÞ þ rðkÞ A VðkÞ ¼  VðkÞ > > > > ðkÞ > > : w > > vW ðkÞ €ðkÞ ; bz XvW ð jÞ > > PD PD > > > V þ V > > ; : vwðkÞ ðkÞ vwðkÞ ð jÞ > j

(5.15) The strain energy density function, Eq. (5.5), can be transformed into the corresponding PD form for the material point k, and its family member j as 8  2 > > > < X qðik Þ  qðkÞ 1 1 ðkÞ   Vðik Þ þ kGA EI WPD ¼ 2   2 d A2 > > k ÞðkÞ  x i > ði : 9
2 qðik Þ þ qðkÞ > > > xðik ÞðkÞ = X wðik Þ  wðkÞ þ 2   V kÞ ði   > > xðik ÞðkÞ  i > ;

(5.16a)

8  2 > > > 1 1 < X qðij Þ  qð jÞ ð jÞ   Vðij Þ þ kGA EI WPD ¼ 2   2 d A2 > > j Þð jÞ  x i > ði : 9
2 qðij Þ þ qðjÞ > > > xðij Þð jÞ = X wðij Þ  wð jÞ þ 2   V jÞ ði   > > xðij Þð jÞ  i > ;

(5.16b)

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5. Beam and plate models in peridynamics

Substituting Eq. (5.16) into Eq. (5.15) and renaming the summation indices yield the final PD equations of motion for Timoshenko beam as Xqð jÞ  qðkÞ I   Vð jÞ rðkÞ €qðkÞ ¼ cb   A x  j ð jÞðkÞ


   qð jÞ þ qðkÞ 1 X ðkÞ xð jÞðkÞ sign xð jÞðkÞ Vð jÞ þ bq  cs wð jÞ  wðkÞ þ 2 j 2

!   X wð jÞ  wðkÞ qðkÞ þ qð jÞ ðkÞ   þ €ðkÞ ¼ cs rðkÞ w sign xð jÞðkÞ Vð jÞ þ bz   2 xð jÞðkÞ  j

(5.17a)

(5.17b)

where cb and cs are the PD material parameters associated with bending and transverse shear deformations, respectively, which are defined as cb ¼

2EI 2kG and cs ¼ 2 2 2 d A d A

(5.18a,b)

3. Peridynamic Mindlin plate formulation 3.1 Classical Mindlin plate formulation According to Mindlin plate theory, the displacement components of a material point can be represented in terms of mid-plane (xy plane) displacements and rotations as uðx; y; z; tÞ ¼ zqx ðx; y; tÞ

(5.19a)

nðx; y; z; tÞ ¼ zqy ðx; y; tÞ

(5.19b)

wðx; y; z; tÞ ¼ wðx; y; tÞ

(5.19c)

where qx and qy denote the mid-plane rotations about positive y- direction and negative x-direction, respectively. Moreover, wðx; y; tÞ denotes the mid-plane transverse displacements. The positive set of the degrees-offreedom is shown in Fig. 5.1. Thus, based on the assumptions given in Eq. (5.19), the straindisplacement relationships can be written as vqx vx vqy εyy ¼ z vy εxx ¼ z

I. New concepts in peridynamics

(5.20a) (5.20b)

3. Peridynamic Mindlin plate formulation

103

FIGURE 5.1 Positive set of the degrees-of-freedom in Mindlin plate formulation.


 1 vqx vqy þ εxy ¼ εyx ¼ z 2 vy vx 
vw gxz ¼ ks qx þ vx 
vw gyz ¼ ks qy þ vy which can also be expressed in indicial notation as
 1 vqI vqJ þ εIJ ¼ z 2 vxJ vxI
 vw gI3 ¼ ks qI þ vxI

(5.20c) (5.20d) (5.20e)

(5.21a) (5.21b)

where ks is introduced as shear coefficient. Note that the subscript indices, I; J; . ¼ 1ð ¼ xÞ; 2ð ¼ yÞ, and this convention will be applied throughout this study. The stress-strain relationships can be written for isotropic materials as: E ðεxx þ nεyy Þ 1  n2 E syy ¼ ðεyy þ nεxx Þ 1  n2 sxx ¼

(5.22a) (5.22b)

sxy ¼ Gðεxy þ εyx Þ

(5.22c)

sxz ¼ Ggxz

(5.22d)

syz ¼ Ggyz

(5.22e)

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5. Beam and plate models in peridynamics

Note that the transverse normal stress szz is considered to be negligible compared to in-plane stresses and this simplifies the 3D Hooke’s law into the 2D plane-stress material constitutive law. The stresses can also be written in indicial notation as:

where CIJKL ¼

sIJ ¼ CIJKL εKL

(5.23a)

sI3 ¼ GgI3

(5.23b)

  E 1n ðd d þ d d Þ þ nd d IL JK IK JL IJ KL 2 1  n2

(5.24)

The linear elastic strain energy density of the Mindlin plate can be expressed as 1 WCCM ¼ ðsIJ εIJ þ sI3 gI3 Þ 2

(5.25)

Inserting Eqs. (5.21), (5.23), and (5.24) into Eq. (5.25) and rearranging indices yields
   E 2 1  n vqI vqI vqI vqJ vqI vqJ 3n  1 vqI vqJ z þ þ WCCM ¼ þ 4 4 vxI vxJ vxJ
vxJ vxJ
vxI vxI vx 1  n2 J vw vw 2G þks qI þ qI þ (5.26a) 2 vxI vxI For a particular material point on the mid-plane, the average strain energy density can be obtained by integrating the strain energy density function, Eq. (5.26a), through the transverse direction and dividing by the thickness as
   E h2 1  n vqI vqI vqI vqJ vqI vqJ 3n  1 vqI vqJ þ þ WCCM ¼ þ 4 4 vxI vxJ vxJ
vxJ vx
vxI  vxJ 1  n2 12 J vxI vw vw 2G þks qI þ qI þ (5.26b) 2 vxI vxI

3.2 Peridynamic Mindlin plate formulation As for the Timoshenko beam, the PD equations of motion of a Mindlin plate can be obtained using Euler-Lagrange equation given in Eq. (5.7) where the generalized displacement vector, u, for this study can be defined as u ¼ ð qx

qy

w ÞT

I. New concepts in peridynamics

(5.27)

3. Peridynamic Mindlin plate formulation

105

The kinetic energy per unit area of the plate, T, can be expressed as

1 T¼ 2

Zh=2   r u_ 2 þ v_2 þ w_ 2 dz

(5.28a)

h=2

Inserting Eq. (5.19) into Eq. (5.28a) gives

1 T¼ 2

  Zh=2   1 h3  _ 2 _ 2  2 2 r z2 q_ x þ z2 q_ y þ w_ 2 dz ¼ r qx þ qy þ hw_ 2 2 12

(5.28b)

h=2

The total kinetic energy of the plate, T, can be cast by integrating Eq. (5.28b) over the mid-plane as Z T¼

TdA ¼ A

1 2

Z 

 h3  _ 2 _ 2  qx þ qy þ hw_ 2 dA 12

(5.29a)

A

which can be written in the discretized form as  2   1X h _ ðkÞ 2 h2  _ ðkÞ 2 2 r þ þ w_ ðkÞ VðkÞ T¼ q q 2 k ðkÞ 12 x 12 y

(5.29b)

The first term of Euler-Lagrange equation can be obtained by utilizing Eqs. (5.29b) and (5.27) as 8 2 9 > h €ðkÞ > > > > qx > > > > > 12 > > > > < = d vL d vT 2 ¼ ¼ rðkÞ h ðkÞ VðkÞ (5.30) € > > dt vu_ ðkÞ dt vu_ ðkÞ > > 12qy > > > > > > > > > > : € ; wðkÞ The total potential energy can be written as  X ðkÞ  X U¼ WPD uðkÞ ; uð1k Þ ; uð2k Þ ; uð3k Þ ; . VðkÞ  bðkÞ ,uðkÞ VðkÞ k

k

I. New concepts in peridynamics

(5.31)

106

5. Beam and plate models in peridynamics

Thus, the second term in Eq. (5.7) can be evaluated as  vL v X ðkÞ  ¼ WPD uðnÞ ; uð1n Þ ; uð2n Þ ; uð3n Þ ; . VðnÞ  vuðkÞ vuðkÞ n 

¼

v X b $u V vuðkÞ n ðnÞ ðnÞ ðnÞ XvW ðnÞ

PD

n;i

vuðkÞ

v X ðdnk þ dnik ÞVðnÞ  b d V vuðkÞ n ðnÞ nk ðnÞ

(5.32)

1 ðkÞ XvW ð jÞ vW PD ¼ @ PD VðkÞ þ Vðhskip1ptjÞ A  bðkÞ VðkÞ vuðkÞ vu ðkÞ j 0

where the generalized body force density vector b can be defined as b ¼ ð bqx

bqy

bz ÞT

(5.33)

Here, bq and bz correspond to moment and transverse force, respectively. Inserting Eqs. (5.30) and (5.33) into Euler-Lagrange equation yields 9 8 ðkÞ XvW ð jÞ vWPD > > > PD > > V > > ðkÞ VðkÞ þ ðkÞ ð jÞ > 8 2 9 > > > > vq vq > > j x x > > > h €ðkÞ > > > > > 8 bðkÞ 9 > > > > q > > > > x > > > > 12 > qx > > > > > > > > > ð jÞ ðkÞ < = > < vW < = = XvW PD PD ðkÞ V þ V þ V rðkÞ h2 ðkÞ VðkÞ ¼  b ðkÞ ð jÞ ðkÞ ðkÞ qy > ðkÞ €q > > > > > vqy > > > > ; y > j vqy : > > > > 12 > > > > > > > > ðkÞ > > > > > > > > bz > > : € ; > > ð jÞ ðkÞ wðkÞ > > XvW > > vWPD > > PD > Vð jÞ > > > ; : vw VðkÞ þ vw ðkÞ

j

ðkÞ

(5.34)

I. New concepts in peridynamics

107

3. Peridynamic Mindlin plate formulation

The strain energy densities of the material point k and j can be expressed in peridynamic form by transforming all differential terms in Eq. (5.26b) as 2 
 2 0 13 ðik Þ ðkÞ ðik ÞðkÞ 6 1  n B 12 X qI  qI nI C7 7 6 B Vðik Þ C 6 @ A7 3 x 7 6 4 k pd h ði ÞðkÞ i 7 6 7 26 E h 6 ðkÞ 7 WPD ¼ 7 6 2 12 6 7 1n 7 6 6 0 12 7 7 6 k 7 6 3n  1 Xqði Þ  qðkÞ ðik ÞðkÞ I I 5 4þ @ 2 A n V kÞ ði I 4 xðik ÞðkÞ pd2 h i 0

þk2s

3 X

G 2 pd3 h

@wðik Þ  wðkÞ þ

ð jÞ

12 ðkÞ k þ qI ði ÞðkÞ A xðik ÞðkÞ nI 2

xðik ÞðkÞ

i




2

WPD

ðik Þ

qI

 2 ð jÞ ðij Þð jÞ ðij Þ qI  qI nI

Vðik Þ (5.35a) 3

7 6 1  n 12 X 7 6 V jÞ ði 7 6 4 xðij Þð jÞ pd3 h i 7 6 7 26 E h 6 7 ¼ 7 6 7 1  n2 12 6 6 0 12 7 7 6 j ð jÞ ði Þ 7 6 3n  1 2 XqI  qI ðij Þð jÞ 5 4 @ A þ n V jÞ ði 4 xðij Þð jÞ I pd2 h i 0

ðij Þ

qI

12 ð jÞ j þ qI ði Þð jÞ A xðij Þð jÞ nI 2

@wðij Þ  wð jÞ þ X G 3 þk2s 2 pd3 h i xðij Þð jÞ

Vðij Þ (5.35b)

with n1 ¼ cos f, n2 ¼ sin f, and f is the orientation of peridynamic bond (interaction).

I. New concepts in peridynamics

108

5. Beam and plate models in peridynamics

Inserting Eq. (5.35a) and (5.35b) into Eq. (5.34) and renaming the summation indices yield the governing equations of PD Mindlin plate formulation as:   ð jÞ ðkÞ h2 €ðkÞ E h X qI  qI ð jÞðkÞ ð jÞðkÞ rðkÞ qL ¼ nI nL Vðik Þ þ 1 þ n pd3 j xð jÞðkÞ 12  3n  1 h2 2 XFðjÞ þ FðkÞ ð jÞðkÞ nL Vð jÞ 12 pd2 h j xð jÞðkÞ 1  n2 4




E

(5.36a)

! ð jÞ ðkÞ 6 X wð jÞ  wðkÞ qI þ qI ð jÞðkÞ ð jÞðkÞ ðkÞ nI þ nL Vð jÞ þ bqL 2 pd3 h j xðik ÞðkÞ 2 ! ð jÞ ðkÞ 6 X wð jÞ  wðkÞ qI þ qI ð jÞðkÞ ðkÞ 2 €ðkÞ ¼ ks G 3 nI rðkÞ w þ Vð jÞ þ bz (5.36b) xð jÞðkÞ 2 pd h j G k2s

where F(k) and F( j) in Eq. (5.36a) can be expressed as
 ðik Þ ðkÞ q  qI 2 X I ðik ÞðkÞ nI Vðik Þ FðkÞ ¼ 2 xðik ÞðkÞ pd h i
 ðjÞ ðij Þ  q q I 2 X I ðij Þð jÞ Fð jÞ ¼ 2 nI Vðij Þ x j pd h i ði Þð jÞ

(5.37a)

(5.37b)

In particular, Eq. (5.36a) and (5.36b) can be simplified for the Poisson’s ratio, n ¼ 1=3, as: ð jÞ

ðkÞ

Xq  q h2 €ðkÞ ð jÞðkÞ ð jÞðkÞ I I nI nL Vð jÞ qL ¼ cb 12 x ð jÞðkÞ j ! ð jÞ ðkÞ qI þ qI cs X ð jÞðkÞ ðjÞðkÞ ðkÞ xð jÞðkÞ nI  wð jÞ  wðkÞ þ nL Vð jÞ þ bqL 2 j 2 rðkÞ

! X wð jÞ  wðkÞ qðkÞ þ qð jÞ ð jÞðkÞ ðkÞ I I €ðkÞ ¼ cs nI rðkÞ w þ Vð jÞ þ bz x 2 ð jÞðkÞ j

(5.38a)

(5.38b)

where cb and cs represent PD material parameters related with bending and transverse shear deformations, respectively, which are defined as cb ¼

3Eh 4pd3

I. New concepts in peridynamics

(5.39a)

109

4. Numerical results

and cs ¼

9k2s E 4pd3 h

(5.39b)

4. Numerical results 4.1 Simply supported beam subjected to transverse loading A simply supported beam with a solid, circular cross-section is considered as shown in Fig. 5.2. The length is L ¼ 1 m and the radius of the cross-section is r ¼ 0:1 m. The Young’s modulus and Poisson’s ratio of the plate are E ¼ 200GPa and v ¼ 0:3, respectively. The second 4 moment of the area and shear coefficient are specified as I ¼ pr4 and k ¼ 9 10, respectively, according to the geometry of the cross-section. The model is discretized into one single row of material points along with the thickness direction and the distance between material points is Dx ¼ 0:002 m. The horizon size is chosen as d ¼ 3:015Dx. A fictitious region is introduced outside the left and right ends as the external boundaries with a width of d. The beam is subjected to a concentrated transverse force of Pz ¼ 1000N at the middle of beam. The load is conPz verted to a body load of bz ¼ 2DV ¼ 7:9577  106 N/m3 and it is imposed on two material points at the center of the beam, as shown in Fig. 5.3. The FE model of the simply supported beam is created by using BEAM188 element in ANSYS with 100 elements along the length. The PD solution of the transverse displacement w, and rotational displacement q,

FIGURE 5.2 Simply supported Timoshenko beam subjected to transverse load.

FIGURE 5.3 Numerical discretization of the Timoshenko beam.

I. New concepts in peridynamics

110

5. Beam and plate models in peridynamics 10

0

FE PD

3 2

Rotational Displacement, θ

Transverse Displacement, w(m)

10

4 FE PD

-0.5

-1

1 0 -1 -2 -3

-1.5

0

0.1

0.2

0.3

0.4 0.5 0.6 Location, x(m)

0.7

0.8

0.9

-4

1

0

0.1

0.2

(a)

0.3

0.4 0.5 0.6 Location, x(m)

0.7

0.8

0.9

1

(b)

FIGURE 5.4 Variation of (A) transverse displacement, (B) rotation along the beam.

are compared with the FE method results. As depicted in Fig. 5.4, the PD and the FE method results agree well with each other.

4.2 Mindlin plate subjected to simply supported boundary conditions In this second numerical example, a simply supported Mindlin plate is considered as shown in Fig. 5.5. The plate has a length and width of L ¼ W ¼ 1 m. The thickness of the plate is specified as h ¼ 0:15 m. The Young’s modulus is E ¼ 200GPa and Poisson’s ratio is n ¼ 0:3. The shear 2 coefficient is used as k2s ¼ p12. For the discretization of the model, 101 1 m. points are used in both directions. The discretization size is Dx ¼ 101 The horizon size is chosen as d ¼ 3:606Dx. A distributed transverse load of p ¼ 1000 N/m is applied along a row of material points through the pW

central line as a body load of bz ¼ 101DV ¼ 6:733  105 N/m3. The FE model is generated by using SHELL181 element of ANSYS with 50  50 elements. As depicted in Figs. 5.6 and 5.7, the PD solution of the

FIGURE 5.5 Simply supported Mindlin plate and loading condition.

I. New concepts in peridynamics

111

5. Conclusions 10

0

FE PD

-0.2

-0.4

Transverse Displacement, w(m)

Transverse Displacement, w(m)

-0.2

10

0

FE PD

-0.6 -0.8 -1 -1.2

-0.4 -0.6 -0.8 -1 -1.2

-1.4 -0.5

-0.4

-0.3

-0.2

-0.1 0 0.1 Location, x(m)

0.2

0.3

0.4

-1.4 -0.5

0.5

-0.4

-0.3

-0.2

(a)

FIGURE 5.6 (B) y  axis.

3

0.3

0.4

0.5

Variation of transverse displacements along the central (A)x  axis,

4 FE PD

10 FE PD

3

Rotational Displacement, θy (rad)

2

Rotational Displacement, θx (rad)

0.2

(b)

10

1

0

-1

-2

-3 -0.5

-0.1 0 0.1 Location, y(m)

2 1 0 -1 -2 -3

-0.4

-0.3

-0.2

-0.1 0 0.1 Location, x(m)

0.2

0.3

0.4

0.5

-4 -0.5

-0.4

-0.3

-0.2

(a)

-0.1 0 0.1 Location, y(m)

0.2

0.3

0.4

0.5

(b)

FIGURE 5.7 Variation of rotations along the central (A)x  axis, (B)y  axis.

transverse displacement, w, and rotations, qL , along the central x and y axes are compared with results from the FE method. According to this comparison, it can be concluded that the PD and the FE method results agree well with each other.

5. Conclusions In this chapter, peridynamic formulations for Timoshenko beam and Mindlin plate were presented. The governing equations were obtained by using Euler-Lagrange equations. The capability of formulations was demonstrated by considering two benchmark problems; simply supported Timoshenko beam and simply supported Mindlin plate subjected to transverse loading conditions. Peridynamic results were compared

I. New concepts in peridynamics

112

5. Beam and plate models in peridynamics

against finite element analysis results and a very good agreement was observed between the two solutions. Similar approach can be followed for other beam and plate formulations. The formulation is not limited to simply supported boundary conditions. Other types of boundary conditions can also be considered including clamped, mixed, and free boundary conditions. Moreover, Mindlin plate formulation does not have any limitation on material constants as experienced in bond-based peridynamic formulation.

References Chowdhury, S.R., Roy, P., Roy, D., Reddy, J.N., 2016. A peridynamic theory for linear elastic shells. Int. J. Solid Struct. 84, 110e132. Diyaroglu, C., Oterkus, E., Oterkus, S., Madenci, E., 2015. Peridynamics for bending of beams and plates with transverse shear deformation. Int. J. Solid Struct. 69, 152e168. Diyaroglu, C., Oterkus, E., Oterkus, S., 2019. An EulereBernoulli beam formulation in an ordinary state-based peridynamic framework. Math. Mech. Solid 24 (2), 361e376. Nguyen, C.T., Oterkus, S., 2019. Peridynamics for the thermomechanical behavior of shell structures. Eng. Fract. Mech. 219, 106623. O’Grady, J., Foster, J., 2014a. Peridynamic beams: a non-ordinary, state-based model. Int. J. Solid Struct. 51 (18), 3177e3183. O’Grady, J., Foster, J., 2014b. Peridynamic plates and flat shells: a non-ordinary, state-based model. Int. J. Solid Struct. 51 (25e26), 4572e4579. Taylor, M., Steigmann, D.J., 2015. A two-dimensional peridynamic model for thin plates. Math. Mech. Solid 20 (8), 998e1010. Yang, Z., Vazic, B., Diyaroglu, C., Oterkus, E., Oterkus, S., 2020. A Kirchhoff plate formulation in a state-based peridynamic framework. Math. Mech. Solid 25 (3), 727e738.

I. New concepts in peridynamics

C H A P T E R

6 Coupling of CCM and PD in a meshless way Mirco Zaccariotto1, 2, Arman Shojaei3, Ugo Galvanetto1, 2 1

Industrial Engineering Department, University of Padova, Padova, Italy; CISAS “G. Colombo”, University of Padova, Padova, Italy; 3 Institute of Material Systems Modeling, Helmholtz-Zentrum Geesthacht, Geesthacht, Germany

2

O U T L I N E 1. Introduction

114

2. The splice method, at a continuum level 2.1 Peridynamics formulation 2.2 Splice between a PD region and a CCM region

115 115 118

3. A meshless discretisation of CCM: the finite point method

120

4. A meshless discretisation of PD

127

5. Details on the discretised version of the coupling

129

6. Numerical examples 6.1 Example 1: pre-cracked plate subjected to traction 6.2 Example 2: Kalthoff-Winkler experiment

130 131 133

7. Conclusions

135

Acknowledgments

136

References

136

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00014-7

113

© 2021 Elsevier Inc. All rights reserved.

114

6. Coupling of CCM and PD in a meshless way

1. Introduction Crack propagation in solid bodies is an important problem regarding many practical applications, but the accurate modeling of damage and fracture phenomena is still an open issue. The classical continuum mechanics (CCM) theory assumes that as a body deforms it remains continuous, the theory is formulated using partial differential equations, the spatial derivatives of which are not defined in a discontinuity (singularity), such as a crack. In the last 30 years several numerical approaches have been proposed to deal with discontinuities, equipping the classical theory with the capability to describe crack propagation. The most prominent methods are: the interface elements endowed with Cohesive Zone Models (Mi et al., 1998; Mun˜oz et al., 2006; Salih et al., 2016), the extended finite element method (Zi and Belytschko, 2003; Belytschko et al., 2003), the phase field method (Francfort and Marigo, 1998; Gibaud et al., 2018), the element erosion technique (Unosson et al., 2006), meshless methods (Rabczuk and Belytschko, 2004; Li et al., 2002), and FEM re-meshing techniques (Hou, 2017; Bouchard et al., 2000). Most of the mentioned methods use ad hoc modifications of CCM and by simplifying assumptions, but each one presents some drawbacks (Bobaru et al., 2016; Madenci and Oterkus, 2014). Peridynamics (PD) is a nonlocal theory proposed by Silling (2000) and Silling et al. (2007) that is an alternative and promising nonlocal theory of solid mechanics that addresses discontinuous problems. PD is based on an integral formulation with no use of spatial derivatives. For this reason, computational methods based on PD can solve the problem of crack propagation (Kilic et al., 2009; Ha and Bobaru, 2011; Agwai et al., 2011; Oterkus et al., 2012; Madenci et al., 2020). The main advantage of peridynamics is that no a priori knowledge about the crack initiation and propagation is required: the crack is free to arise and grow in every part of the structure, only according to physical and geometrical constraints. Unfortunately, the PD-based methods are not computationally efficient, due to the nonlocal nature of the approach. In order to reduce the computational cost, the domain of the problem can be divided in two zones: the PD-based methods are used only around cracks or in regions where cracks are likely to develop or propagate and CCM-based methods elsewhere. Such an approach requires an effective coupling, between the two parts of the model, that should not introduce inaccuracies or spurious effects and, hopefully, the method should be simple and easy to implement. We proposed a coupling strategy in Shojaei et al. (2017b) and Shojaei et al. (2016) where PD is coupled with meshless methods. Then we

I. New concepts in peridynamics

2. The splice method, at a continuum level

115

applied a similar approach to equip FEM models with the capability to simulate crack propagation (Galvanetto et al., 2016; Zaccariotto et al., 2017a,b, 2018; Ni et al., 2019), to study adaptive multi-grid methods (Shojaei et al., 2018) and to investigate multi-physics phenomena (Bazazzadeh et al., 2020). The proposed method is a possible implementation of the splice model described in Silling et al. (2015). In this chapter, we will focus on the coupling of a meshless discretization of CCM with the discretized formulation of PD using the approach presented in Shojaei et al. (2017b). The contents are organized as follows. In Section 2 a short summary of peridynamics and the splice model is given, Section 3 and 4 show a meshless discretization of CCM and PD respectively. Section 5 provides details of the discretized version of the coupling, Section 6 presents several examples dealing with linear and nonlinear problems. Section 7 closes the chapter.

2. The splice method, at a continuum level 2.1 Peridynamics formulation The peridynamic theory is a strongly nonlocal formulation of solid mechanics used for the study of continuous bodies with evolving discontinuities, including cracks and long-range forces. Each material point x in the reference configuration of a body B interacts through the material model with other material points within a distance d. The maximum interaction distance d is called the horizon, and the material within the horizon of x is called the family (Hx) of x. The peridynamic equation of motion of point x is defined by the following integro-differential equation, Z € tÞ ¼ ruðx; (6.1) fT½x; tCxD  T½x0 ; tCxDgdVx0 þ bðx; tÞ Hx

and consequently the internal force density at point x can be defined as Z Lðx; tÞ ¼ (6.2) fT½x; tCxD  T½x0 ; tCxDgdVx0 Hx

€ is the In the previous equations r is the material mass density, u acceleration, and b is the external body force density. T½x; tCxD is the force vector state which indicates the force per volume square vector that point x0 exerts on point x. However, in order to maintain the global equilibrium of the bond, the reaction T½x; tCxD acts on the other point of the bond at position x0 , as a consequence of Newton’s third law. The vector x from x to any neighboring material point x0 ˛ Hx is called bond, x ¼ x0  x

x0 ˛Hx

I. New concepts in peridynamics

(6.3)

116

6. Coupling of CCM and PD in a meshless way

Furthermore, the relative displacement vector is defined as h ¼ uðx0 ; tÞ  uðx; tÞ

(6.4)

The relative position in the deformed configuration of a bond is equal to x þ h. The state-valued functions, related to the previous vectors, are respectively the reference vector state and the deformation vector state XCxD ¼ x

(6.5)

YCxD ¼ x þ h

(6.6)

The unit vector state in the direction of the deformed bond YCxD is the deformed direction vector state, MCxD ¼

YCxD jYCxDj

(6.7)

the norms of reference vector state and deformation vector state are, respectively, the bond lengths in initial and deformed configurations. Therefore, the reference position scalar state and the deformation scalar state are given by x ¼ jXCxDj

(6.8)

y ¼ jYCxDj

(6.9)

The extension scalar state represents the axial elongation of the bond, e¼y  x

(6.10)

In Ordinary State-Based Peridynamics (OSB-PD) (Silling et al., 2007), if the material is taken to be homogenous and the force state T depends only on the deformation state Y the force vector state is aligned with the corresponding deformed bond, in this case the material is called ordinary. The force vector state can be written as T½x; tCxD ¼ t½x; tM½x; tCxD

(6.11)

where t is the modulus state the value of which depends on the constitutive law of the material; in the case of linear solids t can be evaluated based on classical constants (Le et al., 2014; Sarego et al., 2016) and for a 2D plane model it can be obtained by the following formula, 8  > 2ð2n  1Þ 0 1þn wx > > k  am q þ awed plane stress; > < n1 9ð1  nÞ m t½x; tCxD ¼   > 1 wx > > > q þ awed plane strain 2k  am : 9 m (6.12)

I. New concepts in peridynamics

2. The splice method, at a continuum level

117

in which ed is the scalar deviatoric state component of the bond e obtained by ed ¼ e 

qx 3

(6.13)

q is the peridynamic version of the volume dilatation that indicates the relative volume change DV/V of the nonlocal region Hx, for a 2D case q can be obtained by Z 8 2ð1  2nÞ > > 0 > < mð1  nÞ H ðwxe ÞdVx plane stress; x q¼ (6.14) Z > 2 > > ðwxe ÞdVx0 plane strain. : mÞ Hx where m is the weighted volume, given by Z wjxjdVx0 m¼

(6.15)

Hx

the influence function w weighs the effect of the force vector state, depending on the initial length of the bond (Silling et al., 2007; Littlewood, 2015). Finally, a and k0 in Eq. (6.12) are parameters related to classical material properties such as the bulk modulus k and the shear modulus m. The relevant formulas for the 2D cases are, 8   > m 1þn 2 > >

m > > :k þ plane strain. 9 a¼

8m m

(6.17)

The Bond-Based version of Peridynamics (BB-PD) can be considered as a particular case of the OSB-PD in which the interaction between two material points is independent of other bonds, the force vector state, that point x0 applies on point xðT½x; tCxDÞ and that point x applies on point x0 ðT½x0 ; tÞCxD have the same magnitude but opposite sign. Consequently, in BB-PD the force density of a bond x is called the pairwise force function f defined as   1 1 (6.18) fðu0  u; x0  x; tÞ ¼ T½x; tCxD  T½x0 ; tCxD ¼ f   f 2 2 All the constitutive information of the material is included in f. In this way the formulation can be simplified, but BB-PD is restricted to a fixed

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118

6. Coupling of CCM and PD in a meshless way

value of Poisson’s ratio which is v ¼ 1/4 for 3D and plane strain and v ¼ 1/3 for plane stress. In order to describe the discontinuities in the material, a failure criterion needs to be introduced into PD-based models. In Dipasquale et al. (2017) different failure criteria for OSB-PD are investigated. For simplicity in the following sections the bond stretch criterion will be used. Accordingly, each bond is associated with a history-dependent damage function mðx; tÞ that, based on the bond status, can take either the value of 0 (broken bond) or 1 (active bond). In this way the damage level 4 at point x and time instant t can be defined as the ratio between the number of broken bonds and the initial total number of bonds connected to the point x R 0 H mðx; tÞdVx 4ðx; tÞ ¼ 1  xR (6.19) 0 Hx dVx The damage level 4 can take values in the range [0, 1], 4 ¼ 0 represents an undamaged status while 4 ¼ 1 is the case in which there is the complete separation of the material point x from all its family points.

2.2 Splice between a PD region and a CCM region The original peridynamics formulation is based on the implicit assumption that for homogeneous bodies the horizon is constant in the whole domain. Possible methods for allowing variations of the horizon size as a function of position are described in Silling et al. (2015). The same paper defines a strategy of changing the horizon in a material model such that the bulk properties are invariant to this change defining in this way a variable scale homogeneous (VSH) body. However, in the absence of body forces, a uniform deformation of a VSH body is not necessarily in equilibrium. The lack of equilibrium is an artifact due to the position dependence of the horizon. Reference Silling et al. (2015) shows that the out of balance forces are proportional to the second derivative of d (x) consequently if d (x) is a linear function of position then the equilibrium is recovered. Cases in which d (x) is a nonlinear function of x require special methods to guarantee the equilibrium when a uniform deformation is applied. The partial stress method and the splice method are proposed as two approaches able to address this issue (Silling et al., 2015). The first method introduces a modified form of the momentum balance using a new field called the partial stress tensor that depends on the nonlocal deformation state: it is demonstrated as in a VSH body under uniform deformation the equilibrium is guaranteed. The second approach, the splice method, uses a different strategy to solve the issue: the body B is divided in two regions Bþ and B associated with two values of the horizon radius respectively dþ and d (the body B is called a splice of the two regions Bþ and B).

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2. The splice method, at a continuum level

119

FIGURE 6.1 The body B is divided in two regions Bþ and B associated with two values of the horizon radius, respectively, dþ and d. B is called a splice of the two regions Bþ and B.

The internal force density at any point x ˛ B is evaluated by (see Eq. 6.1) (Fig. 6.1).  8Z  0 > T D ½x; tCxD  T D ½x ; tCxD dVx0 if x˛B D ; <  LðxÞ ¼ Lsplice ðxÞ ¼ ZB  > : T L ½x; tCxD  T L ½x0 ; tCxD dVx0 if x˛B L B

(6.20) In which the two force state vector fields are obtained using the following formulae, 8 1 b > > Tþ ½xCxD ¼ 1þD T > 1 ðY1 ½xÞCx=dþ D > < dþ (6.21) > > > T ½xCxD ¼ 1 T b ðY ½xÞCx=d D >  :  1 1 d1þD  where D is the number of dimensions, dþ and d are the horizon radius b 1 is the reference material model given for d ¼ 1 (Silling et al., values and T 2015). In this way the internal force density at each material point x is obtained using the horizon radius assigned at that material point, even if, for the force state evaluation, family points x0 belonging to a region of the body associated with a different horizon radius value are involved. The idea of the splice method can be applied also to the special case in which one region of the body is modeled using a horizon value which tends to 0, corresponding to the local approach of classical continuum mechanics (CCM). In this way the splice method can be adopted as a strategy to implement the local-nonlocal coupling. Eq. (6.22) defines the internal force density at a material point x for a coupled local-nonlocal model studied using the splice approach. This formula can be obtained from Eq. (6.20) assuming the horizon value is

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reducing to zero (Silling and Lehoucq, 2008) for the region modeled by CCM, T(x,t), adopting the assumption of small displacements, is the Cauchy stress tensor.  (Z  Tþ ½x; tCxD  Tþ ½x0 ; tCxD dVx0 if x˛BPD LðxÞ ¼ (6.22) B divTðx; tÞ if x˛BCCM The internal force at a certain material point x is then evaluated depending only on the region where the material point x is located; this apply also for material points of the BPD region near the interface (see Fig. 6.2), in this case, they will have family points located in the BCCM region.

3. A meshless discretisation of CCM: the finite point method In the traditional FEM the most popular computational method for structural mechanics, crack is defined as a boundary with free traction. Therefore, solving problems with crack propagation requires the redefinition of the body boundary in time. One of the main viable options to deal with moving cracks using conventional FEM (equipped with some strategies to define the crack propagation direction and the increase of the crack length) is to update the mesh during each step of analysis in such a way that the element edges coincide with crack borders during all steps (Wawrzynek and Ingraffea, 1989). Moreover, such re-meshing processes are affected by numerical difficulties, complexity in computer programming and often lead to degradation of solution accuracy (Olson et al., 1991). With the aim of eliminating re-meshing techniques, over the past decades, meshless methods have attracted the attention of many researchers.

FIGURE 6.2 The body B is divided in two regions BCCM and BPD: the first is described by classical continuum mechanics (local theory) and the other by peridynamics (nonlocal theory). The dashed line is the interface between the two regions. A material point x of the BPD region can have family nodes located in the BCCM region.

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3. A meshless discretisation of CCM

In classical continuum mechanics for a 2D elastic body B, the elastodynamic problem in a homogeneous and isotropic medium is described using the following equation: € tÞ ¼ divTðx; tÞ þ bðx; tÞ ¼ ST DSuðx; tÞ þ bðx; tÞ ruðx;

(6.23)

In which u ¼ [u, v] is the displacement vector whose components u and v are, respectively, along x and y directions of the global coordinate system (hereinafter we will adopt the assumption of small displacements); x ¼ [x, y] ˛ B is the position vector of a material point of the body B, t is time, r is the material density, b is the body force vector and S is the differential operator defined through the following formula 2 3 v=vx 0 6 7 v=vy 5 S¼4 0 (6.24) v=vy v=vx T

and D is the matrix containing the material constants, for the plane-strain case is 2 3 1v v 0 E 6 7 D¼ 1v 0 (6.25) 4 v 5 ð1 þ vÞð1  2vÞ 0 0 ð1  2vÞ=2 while for the plane-stress case we have 2 1 v E 6 D¼ 4v 1 ð1 þ vÞð1  2vÞ 0 0

0

3

7 0 5 ð1  vÞ=2

(6.26)

where E is the Young’s modulus and v the Poisson’s ratio. To solve Eq. (6.23) we have to consider the given boundary conditions, displacements are imposed on the portion of the boundary Gu whereas tractions are imposed on the boundary portion Gt (see Fig. 6.3), the relevant equations are uðx; tÞ ¼ uðx; tÞ x˛Gu

(6.27)

FIGURE 6.3 The body B is bounded by a boundary G, in Gu displacement boundary conditions u are applied and in Gt stress boundary conditions t are applied.

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nT DST uðx; tÞ ¼ tðx; tÞ x˛Gt

(6.28)

Finally, the initial displacement and velocity conditions can be summarized in the following equations, uðx; 0Þ ¼ u0 ðxÞ

x˛B

(6.29)

_ 0Þ ¼ u_ 0 ðxÞ uðx;

x˛B

(6.30)

In Eq. (6.28) the matrix n is defined as follows: 3 2 nx 0 7 6 n ¼ 4 0 ny 5 ny nx

(6.31)

where nx and ny are the components of the outward unit vector normal at the boundary. Many meshless methods could be used to study a CCM elastodynamic problem, depending on how equations are discretized. Meshless methods can be classified into two major categories: meshless methods based on weak-form, such as the element-free Galerkin method (Belytschko et al., 1994), and meshless methods based on the strong-form such as the finite point method (FPM) (On˜ate et al., 1996). The methods belonging to the second category do not need any background mesh, they are truly meshless, simpler to implement, and computationally less expensive than meshless methods based on weak-form. However, FPM, in comparison with weak form-based methods, is most often less stable and accurate for problems governed by partial differential equations with derivativebased boundary conditions (Neumann) such as solid mechanics problems with stress boundary conditions. Neumann boundary conditions should be imposed directly through a series of independent equations, which are different from the governing equations in the problem domain (On˜ate et al., 2001). The basis of the success of the FPM for solid mechanics applications is the stabilization of the discrete differential equations reached using the finite calculus method or simpler modifications (On˜ate et al., 2001; On˜ate, 1998; Boroomand et al., 2005). Let Xt ˛ B, i¼ 1, 2, . a set of nodes (not necessarily regularly distributed) covering the domain B and the boundary G, around each of these nodes it is possible to define a subdomain Ci called cloud (see Fig. 6.4). Each cloud has a local coordinate system with origin on the node Xi (the central node) and it contains the neighboring nodes (the family nodes) of Xi as Xj, j ¼ 1, 2, ., n. The unknown function u(X,t) has two components u and v that can be expressed, adopting the local coordinate system, as [u(x, y,t), v(x, y, t)] where x and y are the components of the local coordinate system. u(X,t) will be approximated at Xi for every Ci using the nodal values associated to the family nodes.

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123

FIGURE 6.4 The image shows the schematic definitions of cloud, cloud center, family node, local coordinate system in the case of a regular node distribution. Clouds are shown as circles in the figure, but this is not a requirement.

The approach can be rooted on truncated Taylor series (Boroomand et al., 2005). Considering a function u(x, y) continuously differentiable up to the required order, its expansion in a two-dimensional space, using the local coordinate system x ¼ [x, y], is uðxÞ ¼ uð0Þ þ x

vu vu 1 v2 u v2 u ð0Þ þ y ð0Þ þ x2 2 ð0Þ þ xy ð0Þ vx vy 2 vx vxvy

1 1 v2 u þ y2 y2 2 ð0Þ þ . 2 2 vy

(6.32)

Using only a finite number of terms we can define the function b u ðxÞ that approximates the function u(x, y) uðxÞ y b u ðxÞ ¼ uð0Þ þ x

  vu vu ð0Þ þ y ð0Þ þ . þ O hpþ1 vx vy

(6.33)

where p is the order of the approximation and h is a measure of average spacing between nodes. In general, the unknown function can be approximated by a complete polynomial as in Eq. (6.34). uðxÞ y b u ðxÞ ¼ a0 þ a1 x þ a2 y þ a3 x2 þ a4 xy þ a5 y2 þ .

(6.34)

where the coefficients a0, a1, a2,. in the case of a truncated series can be defined for a finite region (the cloud) around the central node imposing the following condition b u ðxij Þ ¼ uðxij Þ

j ¼ 1; .; n

I. New concepts in peridynamics

(6.35)

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6. Coupling of CCM and PD in a meshless way

where xij ¼ Xi  Xj and Xj ˛ Ci (see Fig. 6.4). The number n of equations must be at least equal to the number of monomials used in Eq. (6.34). Unfortunately, depending on the family node distribution, following the previous approach a singular or ill-conditioned system of equations could be obtained (Boroomand et al., 2005). This issue can be solved by adopting a least square procedure using a number of nodal unknowns greater than the number of monomials. Therefore, a displacement component, for example, u at time step tn, can be approximated by the n function b u ðxi j Þ as n b u ðxij Þ ¼

nb X

pj ðxij Þanj ¼ pT ðxij Þan

(6.36)

j¼1

where pðxij Þ is a vector of nb monomial bases, chosen ensuring that the bases are complete, and a is the vector of unknown coefficients to be defined in term of nodal values. For a 2D elasticity problem possible base vectors are (On˜ate, 1998) P ¼ ½1; x; yT

for nb ¼ 3

(6.37)

and  T p ¼ 1; x; y; x2 ; xy; y2

for nb ¼ 6

(6.38)

The local coordinates of the family nodes associated to a central node Xi can be collected in a vector 0 1 « Bx C B i3 C B C C xR ¼ B (6.39) B xi4 C xij ¼ Xi  Xj where Xj ˛Ci Bx C @ i5 A « Consequently, evaluating Eq. (6.36) 2 0 n 1 pT b u1 6 6 pT Bu b n2 C C 6 B n b zB u C¼6 @« A 6 « 4 n b u ni pT

for all family nodes in Ci we have 3 ðxi1 Þ 7 ðxi2 Þ 7 7 n (6.40) 7a ¼ Can 7 « 5 ðxini Þ

The matrix C depends on the coordinates of the family nodes X1, X2, . b n within Xni and is associated to the local approximation of the function u Ci. To proceed with the approximation, u can be sampled at the ni family

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125

nodes of Ci but assuming (as usual) ni > nb, then the matrix C will be not a square matrix. Consequently the approximation cannot fit all the values uðxij Þ. This problem is overcome by using a weighted least squares scheme for approximating the unknown field function that results in a minimization of the weighted sum of the square error at each point, as follows J¼

nR X

2 b nj  pT ðxij Þan wðrj Þ u

(6.41)

j¼1

where rj ¼ jxij j is the distance between the family node j and the central node i and wðrj Þ is a weight function that should be taken suitably for Ci, we use as suggested in Boroomand et al. (2005), Mossaiby et al. (2020), and Shojaei et al., 2017a the function defined by 8  2 2 2 2 rj =d >  eðrm =d Þ >

> : 0 rj > rm Furthermore, d and rm are two parameters proportional to the distance between the central node and the most remote node in the cloud rrmax. We adopt d ¼ 0.25rrmax and rm ¼ 2rrmax as suggested in Boroomand et al. (2005). The weight function takes a value equal to 1 at the central node where the unknown function has to be estimated. The minimization of the norm of J in Eq. (6.41) with respect to an provides the following system of equations, n

Aan ¼ Bb uR

(6.43)

where A¼

ni X

wðrj Þpðxij ÞpT ðxij Þ

(6.44)

j¼1

B ¼ ½ wðr1 Þpðxi1 Þ

wðr2 Þpðxi2 Þ /

wðrini Þpðxni Þ 

(6.45)

It should be noted as matrices A and B are only dependent on the nodes’ position into the cloud. Finally, an can be evaluated from Eq. (6.43) as follows an ¼ A1 Bb uR n

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(6.46)

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6. Coupling of CCM and PD in a meshless way

b n ðxÞ can be obtained by substituting an The approximation function u from Eq. (6.46) into Eq. (6.36) which gives, n n b u ðxÞ ¼ pT ðxÞA1 Bb uR ¼

ni X

n

Nj ðxÞ b uj

(6.47)

j¼1

Where Nj is the shape function (On˜ate, 1998) of the node Xj in the local approximation of the cloud. Shape functions can be evaluated on the basis of the initial position of the nodes; furthermore, the functions Nj are not depending from the time instant tn so they can be estimated once at the beginning of the analysis. The discretized system of equations of the finite point method can be obtained by substituting the displacements un(x) ¼ (un(x), vn(x)) in Eq. (6.23) with the approximated functions defined in Eq. (6.47), which results, for each node, the following equation ru€ni ¼ ST DSN Xi uni þ bni (6.48) where u€ni ¼ u€n ðXi Þ. The vector unR contains the nodal values of displacement components of the family nodes T  (6.49) unR ¼ un1 ; vn1 ; un2 ; vn2 ; .unni ; vnni Finally, N is the matrix of shape functions defined as follows,

 N1 0 N2 0 N3 0 / N¼ 0 N1 0 N2 0 N3 /

(6.50)

Furthermore, by applying the same approach to the boundary conditions defined in Eqs. (6.28) and (6.27), the discretized form of traction boundary condition equation becomes   tðXi tn Þ ¼ nDSNT jxi unR Xi ˛Gt (6.51) where tn ¼ nDt and Dt is the time step. The imposed displacement boundary conditions can be obtained by applying to the node in the boundary part Gu the given value of the displacement. In the case of a static problem the discretized equations of the finite point method can be re-arranged (On˜ate et al., 2001) to obtain a system of algebraic equations with the following form Ku ¼ f

I. New concepts in peridynamics

(6.52)

4. A meshless discretisation of PD

127

where K is the stiffness matrix, u is the unknown nodal displacements vector, and f is a vector of known forces. As pointed out in On˜ate et al. (2001) the collocation procedure leads to a nonsymmetric stiffness matrix.

4. A meshless discretisation of PD The peridynamic continuum formulation can be discretized by using a mesh-free scheme as suggested in Silling and Askari (2005), therefore the solution domain is modeled with an array of nodes (see Fig. 6.5), for sake of simplicity we adopt, as the majority of works in the literature do, a uniform distribution of nodes. The distance between two nearest neighboring nodes Dx ¼ Dy ¼ D is called the grid spacing. A cube (or a square cell in 2D) of material with a side length equal to one grid spacing is associated to each node and is called node volume. The union of all volumes should cover the problem domain and represent well the border. Time is discretized into time instants as t1,2,., tn. The discretized PD equations of motion are h i o 8X n h n i n n n n n n < j T xj Cxj  xi D T xj Cxi  xj D bðxÞVj þ bi for OSB  PD n ru€i ¼ X

 : n n f u  u ; x  x bðxÞVj þ bni for BB  PD j i j i j (6.53)

FIGURE 6.5 A spatial discretization in a peridynamic model.

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6. Coupling of CCM and PD in a meshless way

In which n is the time step and subscripts are associated with the node number uni ¼ uðxi ; tn Þ. The factor bðxÞ is the volume correction factor used to evaluate the portion of the volume Vj within the neighborhood (see Fig. 6.5) of the central node xi. The time integration approach adopted for all the dynamic simulations in this study, involving both discrete Eqs. (6.48) and (6.53), is an explicit Velocity-Verlet time integration scheme. Given displacement, velocity and acceleration of each node at the time instant tn namely uni ; u_ ni ; u€ni, the simulation can proceed at tnþ1 ¼ tn þ Dt as follows: -

nþ1=2 € ni Dt ¼ u_ ni þ u evaluate u_ i

-

¼ unþ1 þ u_ i Dt evaluate unþ1 i i nþ1 € i using the relevant equation of motion Eq. (6.48) or Eq. evaluate u (6.53)

-

nþ1=2 € nþ1 evaluate u_ nþ1 ¼ u_ i þ 12 u Dt i i

nþ1=2

The time step Dt must be smaller than the critical time step defined by Silling and Askari (2005) (Fig. 6.6): Dtc ¼

Dmin ck

(6.54)

where Dmin is the minimum nodal distance in the discretized domain and Ck is the speed of sound in the material.

FIGURE 6.6 (a) a domain B, divided in two regions modeled respectively using PD and FPM. The rectangular area is enlarged in image b). (b) PD nodes (circle markers) and FPM nodes (diamond markers), the dashed line divides the two portions of the model.

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5. Details on the discretised version of the coupling

129

5. Details on the discretised version of the coupling In the adopted coupling technique, the domain B is partially described with a CCM model discretized using the FPM. The remaining part of the domain is studied with a PD model discretized with the mesh-free approach, in this way the coupling has been achieved in a completely meshless scheme. Nodes are either of FPM type or of PD type but no overlapping region exists in terms of nature of the nodes. The coupling method can be used with both BB-PD and SB-PD, as shown in the references (Zaccariotto et al., 2018; Ni et al., 2019). However, we limit the presentation to the BB case. The coupling uses a local/nonlocal framework that benefits from the advantages of both methods while avoiding their limitations. The parts of the domain where cracks either exist or are likely to propagate will be modeled by PD; the remaining part of the domain will be described by the meshless method FPM that requires less computational effort. For every node of the domain B the relevant equation of motion, either Eq. (6.48) or Eq. (6.53) will be adopted. We define the coupling zone the set of PD nodes, the neighborhood of which contains one or more FPM nodes, united to the set of FPM nodes the clouds of which contain one or more PD nodes. If a PD node xPD belongs to the coupling zone in its neighborhood there will be FPM nodes (see Fig. 6.7), but they will play the role of PD nodes when Eq. (6.53) will be applied to the node xPD . The same strategy, using Eq. (6.48), will be adopted for an FPM node having in its cloud some PD nodes (see Fig. 6.7). Node a (see Fig. 6.7) is a PD node, but among its family nodes there are some FPM nodes (b1, b2); by adopting the proposed coupling strategy the acceleration of node a can be evaluated, using Eq. (6.53), assuming all family nodes play the role of PD nodes,         r€ una ¼ . þ f una1  una ; xa1  xa b xa1 a Va1 þ f una2  una ; xa2  xa b xa2 a Va2 þ         . þ f unb1  una ; xb1  xa b xb1 a Vb1 þ f unb2  una ; xb2  xa b xb2 a Vb2 þ . þ bna (6.55) In the same manner of the acceleration of node c, an FPM node can be evaluated using Eq. (6.48) assuming all family nodes play the role of FPM nodes even if some of them (d1, d2,.) are PD nodes, ru€nc ¼ . þ fcc1 þ fcc2 þ fcc þ .fcd1 þ fcd2 þ . þ bnc in which fij ¼ ST DS Nj jxi unj .

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(6.56)

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6. Coupling of CCM and PD in a meshless way

FIGURE 6.7 The PD node a has in its neighborhood some FPM nodes (b1 and b2); similarly the FPM central node c has in its cloud some PD nodes (d1, d2, .). In this figure the cloud radius is smaller than the horizon d.

As a consequence of the proposed coupling approach all PD nodes in the coupling zone will have a complete horizon, solving the issue of the softening effect at interfaces between two regions modeled using nonlocal and local methods. Moreover, since no arbitrary choice or tuning of parameters or blending functions are required the practical implementation of the method is simple. An additional improvement is the possibility to adopt an adaptive partitioning of the solution domain switching the type of a node from FPM to PD, in this way the PD region can anticipate the crack tip during the crack propagation (Shojaei et al., 2016) so that cracks propagate always within the PD region.

6. Numerical examples In this section the performance of the proposed coupling approach in terms of accuracy and efficiency in the solution of two dynamic crack propagation problems is investigated. In both examples, for the FPM nodes circular clouds with radius 2Dx are chosen, which leads to only 13 nodes on average in the clouds. For the local approximation of clouds quadratic polynomial basis functions are employed and for the nodes at the boundaries the clouds are a bit enlarged to ensure that at least nine nodes are included. The simulations are performed with an in-house research code and the run times reported later are measured on an Intel 393 Core i7-3770 3.40 GHz CPU, on a 64 bit Windows 10 Enterprise system.

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131

6.1 Example 1: pre-cracked plate subjected to traction The aim of this example is to show the capability of the present method in the solution of dynamic problems involving crack branching. The setup of the problem is shown in Fig. 6.8: a rectangular plate with a notch at its center parallel to the longer side. A uniform traction with magnitude 19.2 N/mm2 is applied to the horizontal sides of the plate. The plate is made of Duran 50 glass and the mechanical properties are: E ¼ 65,000 N/ mm2, r ¼ 2235  109 kg/mm2, v ¼ 1/3, and G0 ¼ 204  106 J/mm2. The solution of the problem for a time duration of 3  105 s is computed. Two models for the solution of this example are considered. The first model is referred to as PD-only, which corresponds to a model discretized purely with PD nodes. The solution obtained by this model is a reference for evaluating the performance of the present coupling approach in terms of accuracy and efficiency. The second model, referred to as PD-FPM, is a coupled model that employs both FPM and PD nodes in its discretization. As discussed earlier, the coupling model makes use of a switching technique (Shojaei et al., 2016) that switches the FPM nodes to PD nodes adaptively. This switching algorithm is triggered when the stretch between FPM nodes reaches a critical value, here taken 90% of the critical stretch parameter s0 of the PD portion. In this way, the use of the PD nodes will be restricted only to the necessary regions of the problem domain. The interested reader may refer to Shojaei et al. (2016) for more details. In both models the domains are discretized with a grid spacing Dx ¼ 0.25 mm which leads to 64,562 nodes for each model and the time is discretized with a constant time step Dt ¼ 1  108 s; the solver runs over 3000 instants in both models. The peridynamic horizon in both models is taken constant and equal to d ¼ 1 mm. The initial discretization of the FPM-PD model is shown in Fig. 6.9 (first row-third column). In the

FIGURE 6.8 Example 1: problem domain, boundary and load conditions.

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PD-only

PD-FPM

PD-FPM Discr.

30 × 10−6

25 × 10−6

20 × 10−6

10 × 10−6

0.0

[s]

FIGURE 6.9 Example 1: contour plot of damage for (left) PD-only and (middle) PD-FPM as well as (right) PD region in PD-FPM at different time instants.

PD-FPM model, at the beginning of the simulation, a very small portion of the whole discretization (0.94%) is allocated to the PD nodes. This portion just includes a very small region around the notch at the middle of the plate. The thickness of this region is determined so that the visibility criterion for the FPM nodes is met. Fig. 6.9 reports the contour plots of damage evolution produced by both models as well as the variation of the discretization in the PD-FPM model at different time instants. The PD-FPM model is capable of capturing a crack pattern very similar to that of the PD-only model. In both models, the propagating cracks start to propagate and branch at a similar instant in time. In the PD-FPM model the FPM nodes are changing to PD nodes suitably in time. In this sense, at the last time instant the PD portion of the discretization (which is only 5.8% of the whole discretization) follows the critical zones around the propagating cracks. This is an indication that the switching algorithm is performing well in time. The computational run time of both models are reported in Table 6.1. The results confirm that the PD-FPM model can reproduce the solution of the PD-only using less computational resources.

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TABLE 6.1 Comparison of the computational resources used by the models in example 1. Model

Init. PD portion [%]

Final PD portion [%]

CPU time [s]

PD-only

100

100

3561.76

PD-FPM

0.94

5.8

1007.85

6.2 Example 2: Kalthoff-Winkler experiment This example examines the performance of the present method in simulating the well-known Kalthoff’s experiment. The solution of this benchmark problem has been the subject of several studies in the literature for evaluating the accuracy of models in dynamic fracture analysis, for example, Belytschko et al. (2003), Rabczuk et al. (2010), Song et al. (2006). The schematic setup and geometry of the problem are depicted in Fig. 6.10. In the experiment, a rigid projectile at the speed of 32  103 mm/s hits laterally a plate made of 18Ni1900, with two parallel notches. The impact takes place in the region between the notches and, as discussed in Kalthoff (2000), in the experiment a brittle fracturing phenomenon mainly

FIGURE 6.10

Example 2: Kalthoff-Winkler’s experimental setup.

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in mode I is observed. Once the impact occurs, compressive stress waves start to propagate in the plate and immediately cracks nucleate and propagate from the tips of the notches. The cracks propagate straightly with an angle of almost 70 with respect to the horizontal direction. Some studies in the literature, such as Dipasquale et al. (2014) and Gu et al. (2017), have already proven that standard PD models can successfully capture this angle accurately. In the present study, the mechanical properties and the loading conditions are set similar to those reported in Dipasquale et al. (2014) which are: E ¼ 190  103 N/mm2, r ¼ 8000  109 kg/mm3, v ¼ 1/4 (planestrain condition is assumed), and G0 ¼ 0.022170 J/mm2. It is assumed that the projectile and the plate have similar amount of mass. The impacting load is imposed to the model by prescribing a constant speed of v ¼ 16:5  103 mm/s to the boundary region between the notches, and the prescribed velocity is kept constant in the whole simulation. In fact, this way of modeling is logical, since one can assume that in such a short time, the projectile and the plate are attached and they move together. A time duration of 104 s is considered for the problem. The results of the present PD-FPM model are compared with those of a reference PD-only model. In both models, the domain is discretized with a grid spacing Dx ¼ 0.5 mm, and this results in the total number of 80,602 nodes. The peridynamic horizon in both models is taken constant and equal to d ¼ 2 mm. Moreover, by taking Dt ¼ 2  109 s the time is discretized in 5000 instants. Fig. 6.11 (first row-third column) illustrates the initial configuration of the discretized PD-FPM model at the beginning of the simulation. The PD nodes are placed only in a restricted rectangular region (about 15% of the whole domain) that includes the notches as well as the region in the exposure of impact. We assume that the switching technique is triggered as soon as the stretch between the FPM nodes exceeds 90% of the critical stretch s0. Fig. 6.11 shows the contour plots of damage in both models as well as the adaptive change of FPM nodes to PD nodes in the discretized PD-FPM model. The results reveal that the PD-FPM model can suitably produce a solution in excellent agreement with that of the PD-only model. Moreover, the results show that the adaptive switching algorithm in the PD-FPM model performs well in time. The portions of the discretization in which FPM nodes switch to PD nodes are restricted almost to the critical parts of the solution domain. As shown in Fig. 6.11 the obtained crack path inclination angles are 66.89 and 67.80 , respectively, in the PD-only and the PD-FPM models, which are in very good agreement with the angle obtained in the experiment. To get an insight into the computational efficiency of the PD-FPM model compared with the PD-only model, the run times required by both models are reported in Table 6.2. Briefly, the PD-FPM model can reproduce the solution of the PD-only at a much smaller computational cost. I. New concepts in peridynamics

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7. Conclusions

PD-only

PD-FPM

PD-FPM Discr.

100 × 10−6

40 × 10−6

0.0

[s]

FIGURE 6.11 Example 2: contour plot of damage for (left) PD-only and (middle) PDFPM as well as (right) PD region in PD-FPM at different time instants. TABLE 6.2 Comparison of the computational resources used by the models in example 2. Model

Init. PD portion [%]

Final PD portion [%]

CPU time [s]

PD-only

100

100

7331.9

PD-FPM

15.001

17.59

3220.7

7. Conclusions The chapter presents an effective way to couple a meshless discretization of classical continuum mechanics, as the finite point method, to peridynamic grids. The coupling approach is very simple, since there is no need to introduce neither blending function nor tuning of parameters. The PD region of the model can adaptively follow the crack during its

I. New concepts in peridynamics

136

6. Coupling of CCM and PD in a meshless way

propagations whereas the remaining part of the model is studied using the more computationally efficient finite point method. Numerical examples show the capabilities of the proposed approach: models using the described coupling method reproduce the same results of the PD-only models but with a significant reduction of the computational time.

Acknowledgments Ugo Galvanetto and Mirco Zaccariotto would like to acknowledge the support they received from MIUR under the PRIN 2017 research project “DEVISU” (2017ZX9X4K) and from University of Padua under the research projects BIRD2018 NR.183703/18 and BIRD2020 NR.202824/20.

References Agwai, A., Guven, I., Madenci, E., 2011. Predicting crack propagation with peridynamics: a comparative study. Int. J. Fract. 171 (1), 65e78. Bazazzadeh, S., Mossaiby, F., Shojaei, A., 2020. An adaptive thermo-mechanical peridynamic model for fracture analysis in ceramics. Eng. Fract. Mech. 223 nr 106708. Belytschko, T., Lu, Y.Y., Gu, L., 1994. Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229e256. Belytschko, T., Chen, H., Xu, J.X., Zi, G., 2003. Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int. J. Numer. Methods Eng. 58, 18731905. Bobaru, F., Foster, J.T., Geubelle, P.H., Silling, S.A., 2016. Handbook of Peridynamic Modeling. CRC press, Boca Raton, FL. Boroomand, B., Tabatabaei, A.A., On˜ate, E., 2005. Simple modifications for stabilization of the finite point method. Int. J. Numer. Methods Eng. 63, 351e379. Bouchard, P., Bay, F., Chastel, Y., Tovena, I., 2000. Crack propagation modelling using an advanced remeshing technique. Comput. Methods Appl. Mech. Eng. 189 (3), 723e742. Dipasquale, D., Zaccariotto, M., Galvanetto, U., 2014. Crack propagation with adaptive grid refinement in 2D peridynamics. Int. J. Fract. 190 (1e2), 1e22. Dipasquale, D., Sarego, G., Zaccariotto, M., Galvanetto, U., 2017. A discussion on failure criteria for ordinary state-based peridynamics. Eng. Fract. Mech. 186, 378e398. Francfort, G.A., Marigo, J.-J., 1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solid. 46 (8), 1319e1342. Galvanetto, U., Mudric, T., Shojaei, A., Zaccariotto, M., 2016. An effective way to couple FEM meshes and Peridynamics grids for the solution of static equilibrium problems. Mech. Res. Commun. 76, 41e47. ˜ .L., Lhuissier, P., Salvo, L., 2018. Modeling large viscoplastic strain in Gibaud, R., Guesnet, A multi-material with the discrete element method. Int. J. Mech. Sci. 136, 349e359. Gu, X., Zhang, Q., Xia, X., 2017. Voronoi-based peridynamics and cracking analysis with adaptive refinement. Int. J. Numer. Methods Eng. 112 (13), 2087e2109. Ha, Y.D., Bobaru, F., 2011. Characteristics of dynamic brittle fracture captured with peridynamics. Eng. Fract. Mech. 78, 1156e1168. Hou, C.-Y., 2017. Various remeshing arrangements for two-dimensional finite element crack closure analysis. Eng. Fract. Mech. 170 (Suppl. C), 59e76. Kalthoff, J.F., 2000. Modes of dynamic shear failure in solids. Int. J. Fract. 101 (1e2), 1e31.

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Kilic, B., Agwai, A., Madenci, E., 2009. Peridynamic theory for progressive damage prediction in center-cracked composite laminates. Compos. Struct. 90 (2), 141e151. Le, Q., Chan, W., Schwartz, J., 2014. A two-dimensional ordinary, state-based peridynamic model for linearly elastic solids. Int. J. Numer. Methods Eng. 98 (8), 547e561. Li, S., Liu, W.K., Rosakis, A.J., Belytschko, T., Hao, W., 2002. Mesh-free Galerkin simulations of dynamic shear band propagation and failure mode transition. Int. J. Solid Struct. 39 (5), 1213e1240. Littlewood, D.J., 2015. Roadmap for Peridynamic Software Implementation. SANDIA Report. Sandia National Laboratories, Albuquerque. Madenci, E., Oterkus, E., 2014. Peridynamic Theory and its Applications. Springer. Madenci, E., Barut, A., Purohit, P.K., 2020. A peridynamic approach to computation of elastic and entropic interactions of inclusions on a lipid membrane. J. Mech. Phys. Solid. 143 nr. 104046. Mi, Y., Crisfield, M.A., Gao, D., Hellweg, H.B., 1998. Progressive delamination using interface elements. J. Compos. Mater. 32 (14), 1246e1272. Mossaiby, F., Shojaei, A., Boroomand, B., Zaccariotto, M., Galvanetto, U., 2020. Local Dirichlet type absorbing boundary conditions for transient elastic wave propagation problems. Comput. Methods Appl. Mech. Eng. 362 nr.112856. 2020. Mun˜oz, J., Galvanetto, U., Robinson, P., 2006. On the numerical simulation of fatigue driven delamination with interface elements. Int. J. Fatig. 28 (10), 1136e1146 (The Third International Conference on Fatigue of Composites). Ni, T., Zaccariotto, M., Zhu, Q.-Z., Galvanetto, U., 2019. Coupling of FEM and Ordinary Statebased Peridynamics for Brittle Failure Analysis in 3D. Mechanics of Advanced Materials and Structures nr 1602237. Olson, L.G., Georgiou, G.C., Schultz, W.W., 1991. An efficient finite element method for treating singularities in Laplace’s equation. J. Comput. Phys. 96, 391e410. On˜ate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L., 1996. A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int. J. Numer. Methods Eng. 39, 3839e3866. On˜ate, E., Perazzo, F., Miquel, J., 2001. A finite point method for elasticity problems. Comput. Struct. 79, 2151e2163. On˜ate, E., 1998. Derivation of stabilized equations for advective-diffusive transport and fluid flow problems. Comput. Methods Appl. Mech. Eng. 151 (1e2), 233e267. Oterkus, E., Madenci, E., Weckner, O., Silling, S., Bogert, P., Tessler, A., 2012. Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot. Compos. Struct. 94 (3), 839e850. Rabczuk, T., Belytschko, T., 2004. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Methods Eng. 61 (13), 2316e2343. Rabczuk, T., Zi, G., Bordas, S., Nguyen-Xuan, H., 2010. A, simple and robust threedimensional cracking-particle method without enrichment. Comput. Methods Appl. Mech. Eng. 199 (37e40), 2437e2455. Salih, S., Davey, K., Zou, Z., 2016. Rate-dependent elastic and elasto-plastic cohesive zone models for dynamic crack propagation. Int. J. Solid Struct. 90, 95e115. Sarego, G., Le, Q.V., Bobaru, F., Zaccariotto, M., Galvanetto, U., 2016. Linearized state-based peridynamics for 2-D problems. Int. J. Numer. Methods Eng. 108 (10), 1174e1197. Shojaei, A., Mudric, T., Zaccariotto, M., Galvanetto, U., 2016. A coupled meshless finite point/Peridynamic method for 2D dynamic fracture analysis. Int. J. Mech. Sci. 119, 419e431. Shojaei, A., Mossaiby, F., Zaccariotto, M., Galvanetto, U., 2017a. The meshless finite point method for transient elastodynamic problems. Acta Mech. 228 (10), 3581e3593. Shojaei, A., Zaccariotto, M., Galvanetto, U., 2017b. Coupling of 2D discretized peridynamics with a meshless method based on classical elasticity using switching of nodal behaviour. Engin. Comp. 34 (5), 1334e1366.

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Shojaei, A., Mossaiby, F., Zaccariotto, M., Galvanetto, U., 2018. An adaptive multi-grid peridynamic method for dynamic fracture analysis. Int. J. Mech. Sci. 144, 600e617. Silling, S.A., Askari, E., 2005. A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526e1535. Silling, S.A., Lehoucq, R.B., 2008. Convergence of peridynamics to classical elasticity theory. J. Elasticity 93, 13e37. Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E., 2007. Peridynamic states and constitutive modeling. J. Elasticity 88 (2), 151e184. Silling, S.A., Littlewood, D.J., Seleson, P., 2015. Variable horizon in a peridynamic medium. J. Mech. Mater. Struct. 10 (5), 591e612. Silling, S., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solid. 48 (1), 175e209. Song, J.H., Areias, P.M., Belytschko, T., 2006. A method for dynamic crack and shear band propagation with phantom nodes. Int. J. Numer. Methods Eng. 67 (6), 868e893. Unosson, M., Olovsson, L., Simonsson, K., 2006. Failure modelling in finite element analyses: element erosion with crack-tip enhancement. Finite Elem. Anal. Des. 42 (4), 283e297. Wawrzynek, P.A., Ingraffea, A.R., 1989. An interactive approach to local remeshing around a propagating crack. Finite Elem. Anal. Des. 5, 87e96. Zaccariotto, M., Sarego, G., Dipasquale, D., Shojaei, A., Bazazzadeh, S., Mudric, T., et al., 2017a. Discontinuous mechanical problems studied with a peridynamics-based approach. Aerot Missili Spazio 96 (1), 44e55. Zaccariotto, M., Tomasi, D., Galvanetto, U., 2017b. An enhanced coupling of PD grids to FE meshes. Mech. Res. Commun. 84, 125e35. Zaccariotto, M., Mudric, T., Tomasi, D., Shojaei, A., Galvanetto, U., 2018. Coupling of fem meshes with peridynamic grids. Comput. Methods Appl. Mech. Eng. 330, 471e497. Zi, G., Belytschko, T., 2003. New crack-tip elements for XFEM and applications to cohesive cracks. Int. J. Numer. Methods Eng. 57, 2221e2240.

I. New concepts in peridynamics

C H A P T E R

7 Coupled peridynamics and XFEM Mehmet Dorduncu1, Erdogan Madenci2, Atila Barut2 1

Mechanical Engineering Department, Erciyes University, Kayseri, Turkey; Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States

2

O U T L I N E 1. Introduction

139

2. Peridynamic differential operator

141

3. XFEM in conjunction with peridynamics 3.1 Displacements at peridynamic material points 3.2 Principle of virtual work

143 144 148

4. Activation of enrichment functions

150

5. Numerical results 5.1 Plate with a straight crack under tension 5.2 Plate with an inclined crack under tension

151 151 153

6. Conclusions

157

References

157

1. Introduction The concept of eXtended Finite Element Method (XFEM) was introduced as a technique to model cracks and crack growth within the realm of finite elements without remeshing (Belytschko and Black, 1999; Moe¨s

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00013-5

139

© 2021 Elsevier Inc. All rights reserved.

140

7. Coupled peridynamics and XFEM

et al., 1999). It is based on the partition of unity property of finite elements (Melenk and Babuska, 1996). It permits cracks to propagate through any surface within an element; it removes the limitations of the cohesive zone elements. While being successfully used to predict crack path in isotropic materials, the XFEM suffers from the requirement of different external criteria for crack initiation (Cohesive Zone Element-CZE), crack surface generation (level set functions), crack propagation (CZE and Virtual Crack Closure technique-VCCT), and crack propagation path (Maximum Stress Criteria-MSC). Application of such criteria poses challenges to guide the crack path especially in the prediction of the evolution of an arbitrary crack shape. Also, there are often multiple cracks exhibiting complex patterns forming nonplanar 3D surfaces. Furthermore, the existing numerical methods for calculating fracture parameters encounter challenges due to this topological evolution. As an alternative to the classical continuum mechanics, the peridynamic (PD) theory introduced by Silling (2000) converts the existing governing field equations from their local to nonlocal form by introducing an internal length parameter. Peridynamics (PD) is extremely suitable to model discontinuities such as cracks because its governing equation does not include any spatial derivatives; thus, remaining valid regardless of discontinuities. It has been proven powerful for failure prediction because of its intrinsic failure criteria. It continuously monitors stretch between the material points and removes the interaction when the stretch reaches its critical value. It permits nucleation of cracks and propagation at multiple sites with arbitrary paths inside the material. Crack growth is unguided, and the critical stretch value can be expressed in terms of the Linear Elastic Fracture Mechanics (LEFM) fracture parameters as derived by Silling and Askari (2005). The PD differential operator introduced by Madenci et al. (2016, 2019) converts the local spatial differentiation to its nonlocal integral form in a unified manner regardless of the presence of jump discontinuities or singularities. The PD differential operator enables the determination of the derivatives by only performing integration while restoring the nonlocal interactions. However, these interactions are not required to remain intact; they can be broken. The broken interactions between the material points may align themselves along surfaces that form cracks. Although the deformation is discontinuous across such a crack, yet the PD differential operator provides nonsingular derivatives. This study employs the PD differential operator to augment the XFEM formulation to eliminate the aforementioned requirements without introducing additional external degrees of freedom (DoF). The PD augmented XFEM does not require a priori knowledge of crack path information for crack nucleation, and complex algorithms to track crack path(s). This approach simply activates the appropriate interpolation

I. New concepts in peridynamics

2. Peridynamic differential operator

141

functions in the XFEM formulation depending on the crack propagation path prior to the next incremental load step. This chapter is organized as follows: Section 2 describes the peridynamic differential operator. Section 3 presents the basic coupling concept and the derivation of the governing equations based on the principle of virtual work. Section 4 describes the approach for activating the enrichment functions, and Section 5 presents the numerical results for selfsimilar and mixed mode crack growth in a plate under tension.

2. Peridynamic differential operator Peridynamics provides the nonlocal representation of a scalar field f ¼ f ðxÞ at point x by accounting for the effect of its interactions with the other points, x0 ; in the domain, as shown in Fig. 7.1. Each point has its own family members and occupies an infinitesimally small entity. The points x and x0 only interact with the other points in their own families Hx and Hx0 respectively. The strength of the interaction between the material points in each family is specified by a nondimensional weight function, wðjxjÞ. Although not a limitation, the degree of interaction with the family

FIGURE 7.1 Interaction of peridynamic points x and x0 with arbitrary family size and shape.

I. New concepts in peridynamics

142

7. Coupled peridynamics and XFEM

members (weight function) can be specified as a Gaussian distribution in the form of 2

wðjxjÞ ¼ eð2jxj=dÞ .

(7.1)

in which the parameter d usually defined in terms of the spacing between the material points specifies the extent of family. Each point requires the construction of the PD functions. For a 2D spatial function f ðx þxÞ where x ¼ fx; ygT , x0 ¼ fx0 ; y0 gT , x ¼ 0 x  x. The vectors x and x0 specify the positions of two points in the undeformed configuration, respectively. The partial derivatives of up to second order are recast in their integral form as 9 8 > vf ðxÞ > > > > > > > > > > > vx > > > > 8 9 > > > > > > vf ðxÞ > > > > > > g10 > ðxÞ > > > > > 2 > > > > > > > vy > > > > > > > > > 01 > > > > g ðxÞ > > > > 2 > > > > < v2 f ðxÞ = Z < = 20 ¼ ð f ðx þ xÞ  f ðxÞÞ g2 ðxÞ dA (7.2) 2 vx > > > > > > > > > > > > > Hx > > > > > > g02 > > > v2 f ðxÞ > 2 ðxÞ > > > > > > > > > > > > > > > > 2 > 11 > > : vy > > g2 ðxÞ ; > > > > > > 2 > > > > > v f ðxÞ > > > > > > ; : vxvy > p p

in which gN1 2 ðxÞ are the PD functions introduced by Madenci et al. (2016, 2019). The superscripts, p1 and p2 denote the order of differentiation with respect to the variables x and y, respectively. The subscript N represents the order of Taylor series expansion in the construction of the PD functions. The use of PD differential operator requires numerical integration. Therefore, the integration is performed by employing a meshless scheme due to its simplicity. The domain is discretized into a finite number of collocation points, each with a specific entity such as volume or area. Associated with a particular point, the integration involves the summation of the entities within each family. As shown in Fig. 7.2, the material point xj with an incremental area of Aj interacts with other points in a family of square shape. The material point xj can interact with material points located in other elements. Therefore, the kinematics of material point xj is dependent on the nodal DoF of those elements whose nodes are indicated with dark solid circles.

I. New concepts in peridynamics

3. XFEM in conjunction with peridynamics

FIGURE 7.2

143

Peridynamic material point xj and its family that includes points from other

elements.

3. XFEM in conjunction with peridynamics A plate with a crack shown in Fig. 7.3 can be discretized with traditional and enriched finite elements. The elements away from the crack are traditional constant strain triangular (CST) elements. The elements that are completely or partially cut by a crack are known as extended finite elements whose displacement field is enhanced by enriched shape functions (Belytschko and Black, 1999; Moe¨s et al., 1999; Melenk and Babuska, 1996). The equilibrium equations for a plate are obtained by employing the peridynamic (nonlocal) representation of the strain energy at material points. Therefore, the plate is also discretized through a uniform grid to perform the PD differentiation of the displacement components at material point xj with ð j ¼ 1; .; KÞ and K indicates the total number of material points. Each grid point is referred to as the PD material point. The derivatives of displacements, strains, and stresses are expressed by employing the PD differential operator. Also, the stress and strain components at each material point are derived in terms of the nodal unknowns associated with the standard XFEM formulation. The total DoF remains the same, and the PD calculations do not require the solution of any additional equations. By monitoring the stretch between the PD material points, the interactions can be removed to nucleate the crack and guide the crack. Depending on the position of the crack, the appropriate enrichment functions are activated.

I. New concepts in peridynamics

144

7. Coupled peridynamics and XFEM

FIGURE 7.3 Finite element and peridynamic discretization of a plate with a crack.

3.1 Displacements at peridynamic material points In the standard XFEM formulation, the displacement vector can be approximated by uðxj Þ ¼

3 X n¼1

þ

3 X

Nn ðxj Þvn þ

3 X

Nn ðxj ÞHðxj Þan

n¼1

(7.3)

Nn ðxj Þfaðxj Þbn þ bðxj Þcn þ 4ðxj Þdn þ cðxj Þen g.

n¼1

where uj ¼ uðxj Þ ¼ fuxj ; uyj gT is the displacement vector at material point xj in a triangular element with three nodes. The interpolation functions j

j

Nn ðxj Þ ¼ xn defined in terms of the area coordinates xn form a partition of unity at the n-th node. The unknown displacement vector at the n-th node is denoted by vTn ¼ fvxn ; vyn g. The unknown coefficient vector, aTn ¼ faxn ; ayn g is associated with the discontinuous function, Hðxj Þ. This function is defined as  1 if xj ˛ þ R . (7.4) Hj ¼ Hðxj Þ ¼ 1 if xj ˛  R in which Rþ and R denote the regions above and below the cut (crack) separating the element as shown in Fig. 7.4. The unknown coefficient vectors associated with the singular enrichment functions are bTn ¼ fbxn ; byn g; cTn ¼ fcxn ; cyn g; dTn ¼ fdxn ; dyn g, and eTn ¼ fexn ; eyn g.

I. New concepts in peridynamics

3. XFEM in conjunction with peridynamics

145

The enrichment functions are defined as qj pffiffiffiffi rj cos ; 2 pffiffiffiffi qj bj ¼ bðxj Þ ¼ rj sin ; 2 pffiffiffiffi qj 4j ¼ 4ðxj Þ ¼ rj sin sin qj ; 2 qj pffiffiffiffi cj ¼ cðxj Þ ¼ rj cos cos qj 2 aj ¼ aðxj Þ ¼

with

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2  rð jÞ ¼ x0 ð jÞ þ y0 ð jÞ

and qð jÞ ¼ tan

1

y0 ð jÞ

(7.5a) (7.5b) (7.5c) (7.5d)

(7.6a)

!

x0 ð jÞ

(7.6b)

in which x0ð jÞ and y0ð jÞ are the relative positions of material xj with respect to the crack tip coordinates, as shown in Fig. 7.4. The details of these enrichment functions are given by Moe¨s et al. (1999) and Fleming et al. (1997). In the absence of a crack in the CST element, the displacement vector at material point xj can be written as uj ¼ Tj vej

(7.7)

where Tj is the transformation matrix and vej is the nodal unknown vector of element e. They are defined as 3 2 j j j x1 x2 x3 0 0 0 5 (7.8a) Tj ¼ 4 j j j 0 0 0 x1 x2 x3 and

e  veT j ¼ vx1 ; vx2 ; vx3 ; vy1 ; vy2 ; vy3

I. New concepts in peridynamics

(7.8b)

146 7. Coupled peridynamics and XFEM

I. New concepts in peridynamics

FIGURE 7.4 Position of material point xj with respect to a crack configuration in an element.

147

3. XFEM in conjunction with peridynamics

In the case of a crack cutting the element into two parts, the transformation matrix Tj and the nodal unknown vector of element vej are defined as 2 j # jÞ j j j j x1 x2 x3 H j x1 H j x2 H j x3 0 0 0 0 0 0 4 Tj ¼ j j j j j j 0 0 0 0 0 0 x1 x2 x3 Hj x1 Hj x2 Hj x3 (7.9a) and

e  veT j ¼ vx1 ; vx2 ; vx3 ; ax1 ; ax2 ; ax3 ; vy1 ; vy2 ; vy3 ; ay1 ; ay2 ; ay3

(7.9b)

In the case of a crack tip embedded in the element, the transformation matrix Tj and the nodal unknown vector of element vej are defined as 2

j

x1

6 60 6 Tj ¼6 60 4

j x1

j

j

j

0

0

0

0

0

j x2

j x3

ðj

j

j

j

j

j

j

0

0

0

0

0

0

0

0

0

0

0

0

0

0

j aj x1

j aj x2

j bj x3

j 4j x 1

j

j

j

0

0

0

0

0

0

0

j 4j x 2

j 4 j x3

j cj x1

x2 x3 aj x1 aj x2 aj x3 bj x1 bj x2 bj x3 4j x1 4j x2 4j x3 cj x1 cj x2

j aj x3

j bj x1

j b j x2

j

cj x2

and  veT ; j ¼ vx1 ; vx2 ; vx3 ; bx1 ; bx2 ; bx3 ; cx1 ; cx2 ; cx3 ; dx1 ; dx2 ; dx3 ; ex1 ; ex2 ; ex3 e vy1 ; vy2 ; vy3 ; by1 ; by2 ; by3 ; cy1 ; cy2 ; cy3 ; dy1 ; dy2 ; dy3 ; ey1 ; ey2 ; ey3

(7.10b)

The displacement vector of all material points that are in the family of material point xj can be defined as n o b Tj ¼ uTj1 uTj2 . uTjN u with j ¼ 1; .; K (7.11) where N indicates the number of family members associated with xj . Similarly, the unknowns of finite element nodes which are associated with the material points that are in the family of xj can be assembled as n o T eT eT b v . v v j ¼ Assembly veT (7.12) j1 j2 jN Therefore, the displacement vector of material point xj can be expressed in terms of the nodal displacement vector as bjb bj ¼ T u vj where the transformation matrix is defined by h i b T ¼ TT TT . TT T j j j j 1 2 N

I. New concepts in peridynamics

(7.13a)

(7.13b)

j

cj x3

3

7 7 0 7 7 7 0 7 5 j cj x3

148

7. Coupled peridynamics and XFEM

3.2 Principle of virtual work The internal virtual work for the domain is expressed as Z dWI ¼ sab dεab dA

(7.14)

A

where sab and εab with ða; b ¼ x; yÞ are the stress and strain components, and A is the area of the entire domain. The virtual quantity is expressed by dðÞ. The integration can be performed by finite summation over the area occupied by each PD material point as dWI ¼

K X

sTj dεj Aj

(7.15)

j¼1

in which sj and εj are the stress and strain vectors at material point xj , and they are defined in the form n oT j j j sj ¼ sxx ; syy ; sxy (7.16a) and n  oT n oT j j j j j j j εj ¼ εxx ; εyy ; gxy ¼ ux;x ; uy;y ; ux;y þ uy;x

(7.16b)

The parameter K denotes the total number of material points in the domain, and Aj is the area occupied by each material point. The stress and strain vectors are related as sj ¼ Cεj

(7.17)

where C is the material property matrix defined as 2 3 C11 C12 C16 6 7 C ¼ 4 C12 C26 C26 5 C16

C26

(7.18)

C66

The derivatives appearing in the strain components are obtained by employing the PD differential operator introduced by Madenci et al. (2016, 2019). Thus bj εj ¼ B j u

I. New concepts in peridynamics

(7.19)

149

3. XFEM in conjunction with peridynamics

b j is the displacement vector of all material points that are in the in which u family of xj . The matrix of PD differentiation Bj is defined as 2

j

Bx1

6 6 Bj ¼ 6 6 0 4 j By1

Bx2

j

0

j By1 j

By2

0

Bx1

Bx3

j

0

0

j By2

0

j By3

j

Bx2

j

By3

j

Bx3

j

j

0

0

j ByN

j

BxN

. BxN .

. ByN

j

3 7 7 7 7 5

(7.20)

where each coefficient performs differentiation with respect x or y as described in detail by Madenci et al. (2016). They are defined as j

vð$Þ ¼ g10 2 ðxk ÞAk vx

(7.21a)

j

vð$Þ ¼ g01 2 ðxk ÞAk vy

(7.21b)

Bxk ¼ and Byk ¼ pq

where g2 are the PD functions given explicitly by Madenci et al. (2016, 2019), and the subscript ðk ¼ 1; .; NÞ with N being the total number of PD points in the family of material point xj . b j from Eq. (7.13) and in conjunction with Eqs. (7.17) Substituting for u and (7.19), the internal virtual work is rewritten as dWIk ¼

K X

Tb bj db vj K jv

(7.22a)

j¼1

with bj ¼ T b T BT CBj T bj K j j

(7.22b)

The internal virtual work can be recast as dWI ¼ dvT Kv where

(7.23) 

 vT ¼ Assembly vT1

vT2

.

vTK

K ¼ Assemblyf K1

K2

.

KK g

(7.24a) (7.24b)

The virtual work by external loads acting on the finite element nodes is expressed as dWE ¼ dvT P where P is the vector of lumped loads applied at the nodes.

I. New concepts in peridynamics

(7.25)

150

7. Coupled peridynamics and XFEM

The principle of virtual work requires that dWI  dWE ¼ 0

(7.26)

dvT ðKv  PÞ ¼ 0

(7.27)

It leads to

Requiring the first variation of the nodal unknowns to vanish leads to the governing equation as Kv ¼ P

(7.28)

In the case of crack propagation, the solution of the problem is obtained by linearizing Eq. (7.28) in the form Kn vnþ1 ¼ Pnþ1

(7.29)

where Kn is updated as more interactions are broken due to crack propagation, and Pnþ1 denotes the next load step vector. The boundary conditions on displacements and the applied external loads are imposed on the nodes in a traditional way.

4. Activation of enrichment functions Peridynamic theory enables the simulation of crack propagation by progressively removing the interactions between the material points. When the failure criterion is satisfied, the interactions are removed, and enrichment functions are activated accordingly in the elements where new crack surfaces emerge. It is assumed that when the maximum principal bond strain Eprin between two material points defining a bond exceeds its critical value Ecritical , the onset of damage occurs and its growth continuous in an autonomous fashion. The maximum principal bond strain Eprin is defined as 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9

 = < E þE Eaa þ Ebb 2 aa bb þ with a; b ¼ x; y þ E2ab Eprin ¼ max ; : 2 2 (7.30) in which Eij is the average of strain field, εi and εj at material points, xi and xj , respectively.

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5. Numerical results

  The local damage f xðiÞ at a material point xðiÞ is defined as the ratio of the number of broken interactions to the total number of initial interactions as (Silling and Askari, 2005) N P

  j¼1 f xðiÞ ¼ 1 

mðiÞðjÞ N

.

(7.31)

The local damage ranges from zero to one. When the local damage is one, all the interactions initially associated with the point have been eliminated, while a local damage of zero means that all interactions are intact. The measure of local damage is an indicator of possible crack formation within a body. For example, initially a material point interacts with all materials in its horizon; thus, the local damage has a value of zero. However, the creation of a crack terminates half of the interactions within its horizon, resulting in a local damage value of one-half. In the past decade, PD has been widely adopted and applied to various engineering problems. An extensive literature survey on PD is given by Madenci and Oterkus (2014).

5. Numerical results The applicability of the coupled PD and XFEM approach is demonstrated by considering a plate with either a straight or an inclined crack under tension. The plate dimensions are specified as W ¼ L ¼ 1 m. The Young’s modulus and Poisson’s ratio of the isotropic plate are E ¼ 70GPa of n ¼ 0:3, respectively. The maximum critical bond strain values for tension and compression are specified as ETcritical ¼ 0.0163 and ECcritical ¼ 0.0271, respectively.

5.1 Plate with a straight crack under tension The plate is subjected to a uniform resultant tension load of N0 ¼ 1000 N/m along the right edge while it is fixed along the left edge as shown in Fig. 7.5. The pre-existing crack is located at the center of the plate and has a length of 2a ¼ 0:255 m. The finite element discretization is achieved by using 2704 nodal points and 5202 triangular elements, and PD domain is divided into 300 grid points in x and y directions, leading to 90,000 PD material points. The details of the material points near the crack surface and the XFEM mesh are shown in Fig. 7.6. The crack (shown with red color) cuts through the elements. In this figure, the nodes whose shape functions are enriched by the discontinuous functions and corresponding discontinuous DoF are

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FIGURE 7.5 Description of the geometry, boundary, and loading conditions of a plate with a straight crack.

FIGURE 7.6

Details of (A) XFEM and (B) PD material points near the crack surface.

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designated by the solid circles, while the nodes whose shape functions are enriched by the crack tip singular functions and their corresponding DoF are denoted by the hollow circles. Note that the interactions of the material points that are cut through the crack do not exist due to the presence of crack. The deformed configurations of the plate and crack opening are shown in Fig. 7.7. Based on the deformed configurations, the displacement of PD material points shows the elliptical opening of crack surfaces. It is apparent that the PD predictions capture the expected crack opening accurately. Fig. 7.8 shows the variations of normal stresses in the x and y directions as well as shear stress. Due to the geometrical discontinuities around the crack tips, stress concentrations are obvious, and probable failures may initiate at these locations. Therefore, it is crucial to acquire accurate stress variations for the failure analysis. As seen in Fig. 7.8, PD stress predictions successfully capture the expected stress concentrations near the crack tips. The onset and propagation of crack growth are monitored by using an implicit algorithm as described in Section 3. Fig. 7.9 shows the PD predictions of damage pattern. It is evident that damage initiates at the tip of the pre-existing crack and propagates toward the top and bottom edges.

5.2 Plate with an inclined crack under tension The plate geometry, XFEM and PD discretizations, and its properties are the same as those specified previously. The left and right edges of the plate are subjected to a uniform resultant tension load while the horizontal edges are free of traction as shown in Fig. 7.10. The pre-existing inclined crack located at the center has a length of 2a ¼ 0:396 m with an inclination angle of q ¼ 45+ .

FIGURE 7.7 Deformed plate configuration (left) and deformation of crack surfaces (right).

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FIGURE 7.8 Stress field variations: (A) normal stress in the x-direction, (B) normal stress in the y-direction, and (C) shear stress.

FIGURE 7.9 Crack propagation path in the plate with a pre-existing straight crack.

The details of XFEM discretization near the crack surface is shown in Fig. 7.11. The crack tips give rise to enriched shape functions with singular functions at the nodes denoted by the hollow circles. The nodes represented by solid circles have discontinuous functions in their shape functions.

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FIGURE 7.10 Description of geometry of a plate with an inclined crack.

FIGURE 7.11

Details of XFEM mesh with enriched nodes near the crack surface.

Fig. 7.12 shows the PD material points after the plate is deformed. As previously described, the crack is introduced through the removal of interactions between the PD material points in the PD domain. It is obvious that the inclined crack opening can be successfully captured by the present approach. Fig. 7.13 shows the variations of normal and shear stresses. The stress concentrations are apparent at the crack tips. Also, the normal and shear stress components shown in Fig. 7.13 capture the expected stress concentrations.

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FIGURE 7.12 Deformed plate configuration (left) and deformation of crack surfaces (right).

FIGURE 7.13 Stress field variations: (A) normal stress in the x-direction, (B) normal stress in the y-direction, and (C) shear stress.

The damage initiation and its growth are monitored as shown in Fig. 7.14. Once the cracks start propagating, they extend toward the lateral edges of the plate as expected.

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References

FIGURE 7.14

157

Crack propagation path in the plate with an inclined crack.

6. Conclusions The present approach employs the PD differential operator to augment the XFEM formulation without introducing additional external degrees of freedom. The PD augmented XFEM does not require a priori knowledge of crack path information for crack nucleation, and complex algorithms to track crack path(s) by simply activating the appropriate interpolation functions in the XFEM formulation depending on the crack propagation path prior to the next incremental load step. It is based on the principle of virtual work while employing the nonlocal stress and strain components. The kinematics of PD material points are controlled by the nodal unknowns of the finite elements. The use of PD differential operator always yields nonsingular strain field even in the presence of a through crack or a partial crack in an element. When the failure criteria is satisfied, the interactions are removed to nucleate and guide the crack propagation path. The broken interactions between the material points may align themselves along surfaces that form cracks. This approach may be very valuable especially in the presence of multiple cracks exhibiting complex patterns forming nonplanar 3D surfaces. It removes the challenges associated with the existing numerical methods for predicting complex crack paths.

References Belytschko, T., Black, T., 1999. Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601e620. Fleming, M., Chu, Y.A., Moran, B., Belytschko, T., 1997. Enriched element-free galerkin methods for crack tip fields. Int. J. Numer. Methods Eng. 40, 1483e1504. Madenci, E., Oterkus, E., 2014. Peridynamic Theory and its Applications. Springer, New York. Madenci, E., Barut, A., Futch, M., 2016. Peridynamic differential operator and its applications. Comput. Methods Appl. Mech. Eng. 304, 408e451. Madenci, E., Barut, A., Dorduncu, M., 2019. Peridynamic Differential Operator for Numerical Analysis. Springer, New York.

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Melenk, J.M., Babuska, I., 1996. The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139, 289e314. Moe¨s, N., Dolbow, J., Belytschko, T., 1999. A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131. Silling, S.A., Askari, E., 2005. A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 526e1535. Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solid. 48, 175e209.

I. New concepts in peridynamics

C H A P T E R

8 Peridynamics in dynamic fracture modeling Michiya Imachi1, Satoyuki Tanaka1, Murat Ozdemir2, Tinh Quoc Bui3, Selda Oterkus4, Erkan Oterkus4 1

Graduate School of Advanced Science and Engineering, Hiroshima University, Higashihiroshima, Japan; 2 Department of Naval Architecture and Marine Engineering, Ordu University, Ordu, Turkey; 3 Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Tokyo, Japan; 4 Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom

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2. Ordinary state-based peridynamics 2.1 Discretization of peridynamic formulation

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3. Fracture modeling 3.1 Interaction integrals 3.1.1 Interaction integral for stationary cracks 3.1.2 Interaction integral for propagating cracks 3.2 MLS approximation

165 166 166 168 169

4. Evaluation of mixed-mode DSIFs for stationary cracks

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5. Dynamic crack propagation and arrest modeling 5.1 Transition bond modeling 5.2 Crack arrest modeling with application phase

173 173 174

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00019-6

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© 2021 Elsevier Inc. All rights reserved.

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5.3 Numerical studies

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6. Concluding remarks

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References

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1. Introduction Cracks may cause undesired circumstances in engineering structures, e.g., a small crack in a welded part of a ship/offshore structure may propagate and result in a catastrophic failure. The determination of cracks and fracture parameters is therefore essential from the viewpoint of structural safety. Evaluation of fracture parameters, e.g., stress intensity factors (SIFs) for a static loading may give insight to some extent; however, such static results are essentially inadequate as most structures and their components suffer sophisticated dynamic loading conditions. In addition, a crack may propagate that causes variable strain energies and crack velocities. In this respect, examination of cracks along with the dynamic effects becomes necessary and cannot be omitted (Hahn et al., 1973; Kalthoff et al., 1976). The basic theory for the dynamic fracture has been presented by textbooks which are partially (Anderson, 2005) and completely (Freund, 1990; Ravi-Chandar, 2004) dedicated to dynamic fracture phenomena. Modeling and analysis of cracks utilizing conventional continuum mechanics approach inherently contain several challenges because of its nature. The Finite Element Method (FEM) in conventional manner (Hughes, 2000) suffers from representing discontinuities and singular fields in the vicinity of cracks. Re-meshing task to achieve the crack propagation notably reduces the computational efficiency in FEM. Although the conventional FEM has drawbacks in fracture simulation, some prominent works can be found. Node release technique in FEM framework has been considered for simulating crack propagation as reported by Kanninen (1978). Kobayashi (1979) analyzed the crack arrest problem by dynamic FEM. Nishioka et al. (1981) carried out fast fracture analysis by moving singular elements in which the crack tip is assumed to be fixed while the surrounding elements are moving. Considering the deficiencies in the conventional FEM for fracture analyses, eXtended FEM (XFEM) (Belytschko and Black, 1999; Moe¨s et al., 1999) was proposed as a numerical procedure for efficient simulation of crack-related problems. On the other hand, extending XFEM for threedimensional (3D) problems, in which many possible crack patterns exist, is still a challenging task.

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Beside the mesh-based methods, particle-based mesh-free methods might be considered as alternative tools for handling cracks and evaluating fracture parameters. Element-free Galerkin method (Belytschko et al., 1994), reproducing kernel particle method (RKPM) (Liu et al., 1995), and wavelet Galerkin method (Amaratunga et al., 1994; Tanaka et al., 2013) can be utilized for fracture problems. Tanaka et al. (2015, 2016) evaluated static fracture parameters employing RKPM. Even though it is a well-known fact that the mesh-free methods hold notable advantages in modeling of cracked solids, there are still problems in dealing with fracture phenomenon, crack branching, fragmentation, and multiple cracks. Over the past several decades, Silling (2000) reformulated continuum elasticity for long range forces. This novel method is called as Peridynamics (PD). The PD formulation can be considered as continuum version of molecular dynamics. The horizon concept enables nonlocal interactions in PD formulation, which can be also regarded as length scale parameter. The interactions between the particles (material points) inside the horizon are established by bonds transmitting the interaction forces to each other. The PD formulation has been divided into three major subcategories, namely, bond-based (Silling and Askari, 2005), ordinary state-based (OSB), and nonordinary state-based PD (Silling et al., 2007) formulations considering the definition of force interactions between the particles. The PD equation of motion contains integral terms rather than the differentiation of physical values. Thanks to its integration property, the PD promises great potential for simulating discontinuity problems such as cracks or material interfaces. The cracks are treated as part of the problems and special techniques are not required for inserting cracks. Simply releasing the bond forces between the material points is sufficient to maintain a damage/crack in a solid. The PD method therefore becomes suitable for handling crack propagation, crack branching, and multiple cracks. Owing to the capabilities of PD theory, it has been applied for many fracture-related problems. Kilic and Madenci (2009) examined the dynamic crack propagation path in quenched glass plates. Oterkus and Madenci (2012) presented a detailed PD model for the analysis of laminated composite materials, in which the PD bonds are separately defined for matrix and fiber directions rather than a homogenization technique. The failure models considered in Refs (Silling and Askari, 2005; Kilic and Madenci, 2009; Oterkus and Madenci, 2012) are based on the critical stretch criterion. Foster et al. (2011) proposed an energy-based failure criterion so that a PD bond will be irreversibly broken if the strain energy density stored in a bond exceeds a critical value. Dipasquale et al. (2017) discussed the several failure criteria in OSB-PD framework. They examined the critical stretchebased and critical energy densityebased criteria in both continuum and discrete form. On the other hand, the classical

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fracture mechanics parameters can be utilized for assessing crack propagation in PD framework. Hu et al. (2012) obtained nonlocal J-integral using bond-based PD formulation. The derivatives of the physical quantities were obtained by the central difference algorithm. Panchadhara and Gordon (2016) then utilized PD nonlocal SIFs for assessing crack initiation and propagation. Recently, Imachi et al. (2018, 2019, 2020) evaluated displacement fields and collapsed stress tensor (Lehoucq and Silling, 2008) within OSB-PD framework while derivatives of the physical quantities are approximated by moving least squares (MLS) (Lancaster and Salkauskas, 1981) approximation for dynamic stress intensity factors (DSIFs) computation. The DSIFs were then utilized in crack propagation and arrest modeling. The content of the chapter is established as follows. Firstly, the basic formulation for OSB-PD and its discrete form will be presented in Section 2. Then, fracture modeling for DSIFs will be described in Section 3. Section 4 covers the evaluation of mixed-mode DSIFs for stationary cracks. Then, the present formulation will be extended for crack propagation and arrest modeling along with the transition bond concept in Section 5. The concluding remarks will be summarized in Section 6.

2. Ordinary state-based peridynamics The basic theory is summarized in this section. The force interactions between the particles are described within OSB-PD framework, in which the force state depends on not only the bond deformation but also the dilatation of the particles. The equation of motion for a particle with unit volume can be written as (Silling et al., 2007): Z h i v2 uðx; tÞ 0 0 rðxÞ ¼ Tðx; tÞ  T ðx ; tÞ dVx0 þ bðx; tÞ: (8.1) vt2 H

The horizon is represented by H, and it has a circular shape whose radii is d at initial configuration of 2D problem domain. Eq. (8.1) is a general form and can be adopted in problems with variable density rðxÞ, e.g., functionally graded materials (Ozdemir et al., 2020). Displacement field and body force density for the particle at x are defined as uðx; tÞ and bðx; tÞ, respectively. Vx0 stands for the volume of the particle at x0 . Tðx; tÞ and T0 ðx0 ; tÞ are the force states interacting with each other. The force state vectors are aligned with the bond vector in the deformed configuration, and they can be written in OSB-PD framework as: T ¼ tm;

(8.2)

where m ¼ ðy0 yÞ=jy0 yj denotes unit bond vector. y0  y stands for relative position vector in the deformed configuration. I. New concepts in peridynamics

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An OSB-PD formulation for 2D solids was developed by Le et al. (2014). This formulation is utilized throughout the present work. The force state is derived from the strain energy density function, which is decomposed for dilatation and deviatoric parts of the deformation as given in Eq. (8.3).    1   W q; ed ¼ kq2 þ a uðxÞed $ ed ; (8.3) 2 where q is the dilatation, while ed stands for the deviatoric part of the extension state e. uðxÞ denotes the arbitrary scalar influence state. By representing initial relative position vector between the particles as x and the relative displacement vector by h, the extension state becomes e ¼ jx þhj .jxj. The deviatoric part of the extension state is expressed as ed ¼

e  qjxj 3. jxj is the initial distance between the particles. The dilatations for 2D plane stress and plane strain conditions are, respectively, given in Eqs. (8.4) and (8.5) (Le et al., 2014). q¼

2ð2n  1Þ ðujxjÞ$e  ; n  1 ðujxjÞ$x q¼2

ðujxjÞ$e  : ðujxjÞ$x

(8.4) (8.5)

Similarly, the PD material parameters are defined for 2D plane stress condition as: a¼

8G  ; ðujxjÞ$x

k¼K þ

Gðn þ 1Þ2 9ð2n  1Þ2

(8.6) ;

(8.7)

and for the plane strain condition: a¼

8G  ; ðujxjÞ$x

G k¼K þ : 9

(8.8) (8.9)

In Eqs. (8.6)e(8.9), the elastic bulk and shear moduli are represented by K and G, respectively. Finally, the magnitude of the force state vector in Eq. (8.2) is expressed for the plane stress condition as: 2ð2n  1Þ  a  d    ujxj   þ au ed ; t¼ ue $ x (8.10) kq  n1 3 ðujxjÞ$x

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and for the plane strain condition:    a ujxj   þ au ed : t ¼ 2 kq  u ed $ x 3 ðujxjÞ$x

(8.11)

2.1 Discretization of peridynamic formulation The PD problem domain can be discretized as similar to conventional particle methods (Silling and Askari, 2005). Then, the general form of PD equation of motion in Eq. (8.1) is discretized for each particle (i) in 2D domain by replacing the volume expression Vx0 by the area Að jÞ as: rðiÞ u€ðiÞ ¼

NP  X

 TðiÞð jÞ  Tð jÞðiÞ Að jÞ þ bðiÞ ;

(8.12)

j

where the acceleration term is denoted as u€ðiÞ . NP is the number of material points inside the horizon of particle (i). The acceleration term in Eq. (8.12) is simply obtained by 2 3 NP   X 1 4 u€nðiÞ ¼ (8.13) TðiÞð jÞ  Tð jÞðiÞ Að jÞ þ bðiÞ 5: rðiÞ j Then, the central difference scheme is adopted for evaluating displacement fields as: 2 3 NP  2 X  ðDtÞ 4 unþ1 TnðiÞð jÞ  Tnð jÞðiÞ Að jÞ þ bnðiÞ 5 þ 2unðiÞ  un1 (8.14) ðiÞ ; ðiÞ ¼ r ðiÞ j where the superscripts (n þ 1), (n), (n  1) stand for the physical quantities at the next, present, and previous time steps, respectively. The time step size is denoted by Dt. The magnitude of the force density vector for the plane stress condition can be written in discretized form as: 0 1   NP aðiÞ X 2ð2n  1Þ @   kqðiÞ  tðiÞð jÞ ¼ u ed x A A n1 3 j ðiÞð jÞ ðiÞð jÞ ðiÞð jÞ ð jÞ     uðiÞð jÞ xðiÞð jÞ   þ aðiÞ uðiÞðjÞ edðiÞðjÞ ; (8.15)    NP P    u x  x A j ðiÞð jÞ ðiÞð jÞ ðiÞð jÞ ðjÞ

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and the discrete form of the force density vector magnitude for the plane strain condition is expressed as: 1 0   NP aðiÞ X   tðiÞð jÞ ¼2@kðiÞ qðiÞ  u ed x A A 3 j ðiÞð jÞ ðiÞð jÞ ðiÞð jÞ ð jÞ   (8.16)   uðiÞð jÞ xðiÞð jÞ      þ aðiÞ uðiÞð jÞ edðiÞð jÞ : P    NP j uðiÞð jÞ xðiÞð jÞ xðiÞð jÞ Að jÞ The discretized form of dilatation terms at particle (i) for the plane stress and plane strain conditions are expressed in Eqs. (8.17) and (8.18), respectively.   N PP   uðiÞð jÞ xðiÞðjÞ edðiÞð jÞ Að jÞ 2ð2n  1Þ j qðiÞ ¼ ; (8.17)    n1 N PP    uðiÞð jÞ xðiÞð jÞ xðiÞð jÞ Að jÞ j

2 qðiÞ ¼

NP P j

NP P j

    uðiÞð jÞ xðiÞð jÞ edðiÞð jÞ Að jÞ

:       uðiÞð jÞ xðiÞð jÞ xðiÞð jÞ Að jÞ

(8.18)

3. Fracture modeling Introduction of cracks is relatively easier in PD framework compared to other numerical techniques. If a pre-existing crack cuts the PD bonds, those bonds are assumed to be broken throughout the PD simulation. The broken bond condition can be simply imposed by setting the scalar state influence function as zero, i.e., u ¼ 0 as shown in Fig. 8.1A. In a similar fashion, if a certain failure criterion for a PD bond is met, the bond is assumed to be irreversibly broken and new crack surfaces will be generated, see Fig. 8.1B. A stationary crack is simply modeled by eliminating the PD bonds, then the DSIFs for the crack are obtained by the interaction integrals. As for a propagating crack, a critical value of DSIFs is considered for the estimation of crack propagation, then the DSIFs extracted by the interaction integral are compared with the critical value.

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(a)

(b)

Propagating crack

Pre-existing crack

Broken bonds

Unbroken bonds Broken bonds

FIGURE 8.1 Schematic of crack modeling in PD: (A) pre-existing crack, (B) propagating crack.

3.1 Interaction integrals To evaluate the DSIFs, the interaction integral is employed for both stationary (Chen and Shield, 1977) and propagating cracks (Re´thore´ et al., 2005). For stationary cracks, the interaction integral contains derivatives of the displacement and stress fields near the crack tip zone, while the derivatives of the velocity field have to be calculated in addition to stress and displacement fields for the interaction integral of propagating cracks. 3.1.1 Interaction integral for stationary cracks Chen and Shield (1977) proposed the interaction integral for stationary cracks. To compute the interaction integral, actual and auxiliary components of physical quantities are considered. The equivalent domain integral for the J-integral is defined as: Z     1  aux act aux act aux I¼ þ s þ u þ s s u  q1;j sact ij ij i;1 i;1 ij 2 ij U       aux act aux act aux þ ε þ s þ u  εact d s u dU þ q 1 1j ij ij ij;j ij;j i;1 i;1 ¼ J act þ J aux þ M:

(8.19)

In Eq. (8.19), J act and J aux stand for the J-integrals of actual and auxiliary quantities. The M-integral is denoted by M, which is the interaction term of actual and auxiliary quantities and can be written as: Z     1  act aux act act aux aux act u þ s u  ε þ s ε M¼ s q1;j saux d1j ij i;1 ij i;1 ij ij 2 ij ij U  (8.20)   aux act act aux þ q1 sij;j ui;1 þ sij;j ui;1 dU:

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In Eqs. (8.19) and (8.20), sij and ui denote, respectively, stress and displacement components in a local coordinate system located at the crack tip. The “actual” and “auxiliary” field variables are represented by superscripts“act” and “aux,” respectively. d1j denotes the Kronecker delta and U is the integration domain. For evaluating the domain integral, a weight function q1 is defined as: 8 rG < rG1 >

: 0 otherwise where rG is distance from the crack tip. By representing inner and outer domain boundaries by G1 and G2 ; the distances to the crack tip from the inner and outer domain boundaries are denoted by rG1 and rG2 , respectively. The global axes and crack tip details with local axes are schematically given in Fig. 8.2A and B, respectively. In Fig. 8.2B, the domains are illustrated with circular shape; however, an arbitrary domain shape can also be adopted. Once the integrals in Eqs. (8.19) and (8.20) are computed, the actual and auxiliary J-integrals can be obtained. The relationship between the DSIFs and J-integrals is expressed as: ðKI Þ2 þ ðKII Þ2 ; E0 aux 2 aux 2 þ KII K aux J ¼ I ; E0 J act ¼

where 0

E ¼ (a)



E for plane stress

: E= 1  n2 for plane strain

(8.22) (8.23)

(8.24)

(b)

Crack

FIGURE 8.2 Crack tip definition with integration domains: (A) global axes, (B) crack tip details with local axes.

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Auxiliary DSIFs are denoted by KIaux and KIIaux in Eq. (8.23). The interaction integral in Eq. (8.19) can be written in terms of the actual and auxiliary DSIFs as:

2

2 KI þ KIaux þ KII þ KIIaux ; (8.25) I¼ E0 and the M-integral is expressed as: M¼

KI KIaux þ KII KIIaux : E0

(8.26)

Using the relations given by Eqs. (8.25) and (8.26), the actual DSIFs, KI and KII for mode-I and mode-II can be obtained by imposing KIaux ¼ 1, KIIaux ¼ 0 and KIaux ¼ 0, KIIaux ¼ 1, respectively. Finally, the actual DSIFs are written as: KI ¼

E0 M for 2

KIaux ¼ 1;

KIIaux ¼ 0;

(8.27)

KII ¼

E0 M for 2

KIaux ¼ 0;

KIIaux ¼ 1:

(8.28)

3.1.2 Interaction integral for propagating cracks The interaction integral proposed by Re´thore´ et al. (2005) is adopted for propagating cracks. The velocity components in the vicinity of the crack tip are utilized for crack propagation. These velocity components are associated with the damping effects. The interaction integral for propagating cracks is defined as: Z h    i act aux act act aux act I ¼ q1;j saux  saux dkj ij ui;k þ sij ui;k ml um;l  ru_ l u_ l U

þ q1

h

  i act aux act act aux u_ act saux þ r u_ aux dU; i i;k þ u_ i u_ i;k ij;j ui;k þ sij;j ui;k

(8.29)

where the velocity field is denoted as u_ i . The notation for the interaction integral of propagating cracks is same as the interaction integral of stationary cracks as previously defined in Eq. (8.19). The relationship between DSIFs of propagating cracks and the interaction integral can be written as: I¼

2 _ I KIaux þ AII ðaÞK _ II KIIaux : AI ðaÞK 0 E

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(8.30)

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_ and AII ðaÞ _ are universal functions depending on The parameters AI ðaÞ _ The universal functions are adopted from the crack propagation speed a. (Freund (1990) as:

4ad 1  a2s _ ¼ ; (8.31) AI ðaÞ ðk þ 1ÞD

4as 1  a2s _ ¼ AII ðaÞ ; (8.32) ðk þ 1ÞD where k is a material parameter, which is defined for the plane stress condition as k ¼ ð3 nÞ=ð1 þnÞ, and for the plane strain condition as k ¼ 3  4n. On the other hand, ad , as , and D are the functions of crack speed rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ffi 2 and wave speeds in solids, which can be defined as ad ¼ 1  a_ c2d , rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ffi

2 2 _ as ¼ 1  a c2s , and D ¼ 4ad as  1 þ a2s . cd and cs are the longitudinal and transverse wave speeds in solids, respectively. To calculate the actual DSIFs, KI and KII for propagating cracks, the auxiliary SIFs are considered as KIaux ¼ 1, KIIaux ¼ 0 for pure mode-I and KIaux ¼ 0, KIIaux ¼ 1 for pure mode-II conditions, respectively. Then, the relationship between actual DSIFs and interaction integrals can be written as: KI ¼

E0 E0 II ; KII ¼ I : _ _ II 2AI ðaÞ 2AII ðaÞ

(8.33)

In Eq. (8.33), II and III are the interaction integrals assuming the auxiliary fields as pure mode-I and mode-II, respectively. These expressions have been provided by Imachi et al. (2019). In calculating the interaction integrals, the spatial derivatives of the physical quantities are evaluated by the MLS approximation. On the other hand, the stress components are obtained within the OSB-PD framework as (Lehoucq and Silling, 2008): Z sact ¼ T5xdV: (8.34) H

3.2 MLS approximation Displacement gradients have to be evaluated in the interaction integrals. Even tough, some researchers, e.g., Hu et al. (2012), have obtained these gradients within PD framework, MLS (Lancaster and Salkauskas, 1981) can be considered as an efficient method for approximating the spatial derivatives of the physical components.

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8. Peridynamics in dynamic fracture modeling

Using MLS, a function located at x can be approximated as: ef ðxÞ ¼ pT ðxÞcðxÞ;

(8.35)

where pðxÞ is the basis function. A quadratic basis is usually adopted as 

pT ðxÞ ¼ 1 x1 x2 x21 x1 x2 x22 . cðxÞ is the coefficient vector minimizing the weighted L2 -norm R as: R¼

NP X

2 wðx  xi Þ pT ðxi ÞcðxÞ  f ðxi Þ

i

(8.36)

¼ ðPc  fÞT WðxÞðPc  fÞ;

where NP represents the number of WðxÞ ¼ diag½wðx x1 Þ/wðx xNP Þ. P function values as: 2 p1 ðx1 Þ p2 ðx1 Þ 6 p ðx Þ p ðx Þ 2 2 6 1 2 P¼6 4 « « p1 ðxn Þ p2 ðxNP Þ

particles for the approximation. is a matrix including the basis / / 1 /

pm ðxNP Þ

3

pm ðxNP Þ 7 7 7: 5 «

(8.37)

pm ðxNP Þ

Here, m is number of monomials in the basis function vector. Hence, the approximated function ef ðxÞ can be written as: ef ðxÞ ¼ pðxÞT G1 ðxÞHðxÞf ¼ 4ðxÞf;

(8.38)

where GðxÞ ¼ WðxÞPPT and HðxÞ ¼ WðxÞP. Cubic spline is adopted as weight function as given in Eq. (8.39) 8 > 3 3 > > 1  s2 þ s3 ð0  s  1Þ > > 2 4 > < 1 wðx  xi Þ ¼ ; (8.39) ð2  sÞ3 ð1  s  2Þ > > > 4 > > > : 0 ð2  sÞ where s is the normalized distance between the particles as s ¼ jx xi j= d. The shape function 4ðxÞ has to be modified when the horizon of a particle cuts the crack segment to consider the discontinuity. Diffraction method (Organ et al., 1996) is utilized for this purpose.

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4. Evaluation of mixed-mode DSIFs for stationary cracks

171

The displacement and velocity fields are utilized for calculating displacement and velocity gradients by MLS approximation as: act uact i;k ¼ 4i;k ui ; act u_ act i;k ¼ 4i;k u_ i ;

(8.40)

where 4i;k is the spatial derivative of MLS shape functions.

4. Evaluation of mixed-mode DSIFs for stationary cracks So far, evaluation of DSIFs and fracture modeling have been described. In this section, these procedures will be validated through numerical examples from the literature. A rectangular specimen with a slanted edge crack is considered. This problem has been investigated by many researchers, e.g., Fedelinski et al. (1996) and Liu et al. (2012). Therefore, it is seen as a well-known numerical example. Fedelinski et al. (1996) employed the dual boundary element method (DBEM) to calculate the DSIFs of stationary cracks. Liu et al. (2012) then solved several stationary crack problems under dynamic loading utilizing the singular edgeebased smoothed FEM. In the present work, the numerical results obtained by the OSB-PD method, which has been described so far, will be compared with the results from Fedelinski et al. (1996); Liu et al. (2012). The geometrical and material properties are adopted from the reference works (Fedelinski et al., 1996; Liu et al., 2012). The representative model of a rectangular plate with a slanted edge crack is given in Fig. 8.3. The geometric parameters shown in Fig. 8.3 are as follows. The plate width is B ¼ 44 mm, height is H ¼ 32 mm. The lower crack tip is located 6 mm far from the left edge, i.e., B1 ¼ 6 mm. The crack length and orientation angle are a ¼ 22:63 mm and a ¼ 45+ , respectively. The shear modulus G ¼ 29:4 GPa, Poisson’s ratio n ¼ 0:286, and the mass density r ¼ 2:45  106 kg/mm3 values are adopted. Other solution parameters are defined with reference to Imachi et al. (2018).

FIGURE 8.3 A rectangular plate model with a slanted edge crack. I. New concepts in peridynamics

172

FIGURE 8.4

8. Peridynamics in dynamic fracture modeling

Comparison of normalized mode-I DSIFs for a slanted edge crack.

FIGURE 8.5 Comparison of normalized mode-II DSIFs for a slanted edge crack.

pffiffiffiffiffiffi The obtained DSIFs are normalized by K0 ¼ s0 pa and compared with the reference results. Total simulation time is considered as t ¼ 25 ms. Normalized DSIFs for mode-I and mode-II are, respectively, represented in Figs. 8.4 and 8.5. As it is obvious from both the figures, the stress wave reaches the crack tip around 4 ms Then, the SIFs start increasing. The present results in general show good agreement with the reference results obtained by different numerical techniques, confirming the accuracy of the developed

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5. Dynamic crack propagation and arrest modeling

173

PD method. In particular, the present approach offers remarkable results in comparison with the DBEM by Fedelinski et al. (1996) and present results perfectly fit each other, while a slight difference can be observed between the present and reference results by Liu et al. (2012) in Fig. 8.5. A reflection of stress wave from the boundaries has more significant effects on the DSIFs of mode-I compared to those of mode-II. This can be inferred from the fluctuations of DSIFs in Fig. 8.4. On the other hand, the fluctuations become less in DSIFs of mode-II and almost monotonic increase of DSIFs is observed.

5. Dynamic crack propagation and arrest modeling In this section, the crack propagation modeling with the transition bond concept is described. The transition bond concept is also utilized for the crack arrest modeling with application phase. For our numerical experiment, we particularly consider the double cantilever beam (DCB) specimen (Kalthoff et al., 1976). In the application (prediction) phase, both the DSIFs and final crack lengths are estimated by the proposed OSB-PD modeling. Crack initiation and propagation are assessed by the conventional fracture mechanics framework. DSIFs are evaluated to compare with the critical SIFs. Imachi et al. (2019) reported that the DSIFs get significant oscillations when the bond force is suddenly released for the propagating cracks that disturbs the solution accuracy when the crack direction is assessed by maximum circumferential stress criterion. To suppress the oscillations, the transition bond concept was proposed and implemented in evaluation of DSIFs for the propagating cracks (Imachi et al., 2019) and the crack arrest modeling (Imachi et al., 2020).

5.1 Transition bond modeling Imachi et al. (2019) clearly showed influence of the transition bond concept on the DSIFs of propagation cracks. Furthermore, the sudden release of the PD forces caused very inaccurate results in the crack arrest modeling (Imachi et al., 2020) so that the cracks stopped immediately after the onset of crack propagation for lower crack initiation toughness values. This is because of significant oscillations in DSIFs, which sometimes become zero and the crack stops. Some works can be found in literature addressing the numerical oscillations in the crack propagation. Aoki et al. (1987) and Core´ et al. (2018) introduced damping effects near the crack tip zone. The nodal forces in FEM were gradually reduced near the crack tip elements in (Aoki et al., 1987), while the damping effects were introduced in the discrete element method by Core´ et al. (2018).

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Initial crack

Propagating crack

Broken bonds Transition bonds

Damping length

FIGURE 8.6 Transition bond concept with damping length.

To eliminate the oscillations in DSIFs, reducing the bond forces seems to be a reasonable approach once the crack propagation condition is met in the PD modeling. In this case, a transition bond is defined so that a bond continues to withstand some amount of force depending on the crack size increment, da, and a predefined damping length, Da. Transition bond concept is schematically illustrated in Fig. 8.6. An intact PD bond is assumed as a transition bond when a propagating crack segment cuts the bond. Then, the influence function u for the bond b ¼ f ,u. f stands for the gradual reduction of the bond is modified as u b b force as: 8 intact bond >

: 0 broken bond where a_ is the crack speed and t denotes time. fb is defined by   da n fb ¼ 1  : Da

(8.42)

In Eq. (8.42), n determines the characteristics of the reduction, e.g., n ¼ 1 states the linear reduction of the bond force. Imachi et al. (2019) carried out a series of simulations to examine the influence of parameters n and Da. In the present work, we utilize these parameters as Da ¼ d=2 and n ¼ 4.

5.2 Crack arrest modeling with application phase In the crack arrest modeling, two methodologies, namely, the generation and application phases (Nishioka and Atluri, 1982), are common practices. Generation phase can be considered as a validation phase in

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5. Dynamic crack propagation and arrest modeling

which the experimental crack data (crack path history) is introduced as input and DSIFs are evaluated. Implementation of the generation phase is limited, since the experimentally evaluated crack data is required. On the other hand, the application phase is capable of evaluating both the crack path histories and DSIFs assuming a relation between the crack speed and the DSIFs. The accuracy of the results in the application phase is strongly related with an expression defining the relation between DSIFs and crack velocity. This expression however needs to be determined based on some previous crack data. In this respect, an expression based on the crack data given by Kalthoff et al. (1976), similar to ones in Kanninen and Popelar (1985); Prabel et al. (2007), is defined as: Kd ¼

Ka   ; a_ m 1 vl

(8.43)

Dynamic stress intensity factor K d [ MN m− 3/ 2 ]

where Kd represents DSIF. The crack arrest toughness is utilized as Ka ¼ 0:024 MN$m3=2 , the limiting crack velocity, vl is taken as 470 m/s, and the arbitrary parameter is m ¼ 2:2. The prediction curve for DSIFs is given in Fig. 8.7 by comparing to the experimental crack data of (Kalthoff et al., 1976). The flowchart of the application phase has been illustrated by the authors in Imachi et al. (2020). The procedure therefore will be briefly described here. At first, the force states and stress components are evaluated under prescribed loading and boundary conditions within the OSB-PD framework. Then, it is assessed whether the crack propagation occurs or not. If the crack is stationary, the DSIFs are obtained for the stationary cracks by Eqs. (8.27) or (8.28) as described in Section 3. Then, the DSIFs for a stationary crack are compared with the critical SIFs. 2.0 Prediction curve Kalthoff’s exp.

1.8 1.5 1.2 1.0 0.8 0.5 0.2 0.0 0

50

100

150

200

250

300

350

Velocity [m/s]

FIGURE 8.7 The prediction curve for DSIFs versus crack speed.

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400

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8. Peridynamics in dynamic fracture modeling

If the crack propagation still does not exist, the same procedure is repeated. If the crack propagation occurs, the DSIFs for moving cracks are calculated by Eq. (8.33) as expressed in Section 3. Then, the crack speed by Eq. (8.43) is evaluated and the position of the crack tip is updated. At the final step, the bond condition with or without the transition bond concept is computed, see Eqs. (8.41) and (8.42).

5.3 Numerical studies The procedure described in Section 5.2 is validated through demonstrative numerical studies. DCB specimens are adopted from Kalthoff et al. (1976). A DCB specimen is schematically given in Fig. 8.8. The distance between the particles is set as dx ¼ 1:0 mm for all the specimens, while the horizon size is utilized as d ¼ 4dx. Wedge loading condition in the experiments is simulated by the forced displacement as shown in Fig. 8.8. A similar procedure has been implemented by the authors in our previous study, see Imachi et al. (2020). The parameters of the estimation curve in Eq. (8.43) are set for different values, which would generate slightly different results from Imachi et al. (2020). Four specimens are adopted from Kalthoff et al. (1976), which are no. 4, 8, 17, and 24. Then the final crack lengths when the crack is arrested as well as the DSIFs are compared with those given by Kalthoff et al. (1976) considering with or without transition bond conditions. Fig. 8.9 shows the DSIFs and final crack lengths for specimen no. 4 with and without transition bond conditions. Dashed line stands for the present results while the reference results from Kalthoff et al. (1976) are demonstrated by red and blue marks. The reference results for specimen no. 4 and 21 had been evaluated under practically same conditions by Kalthoff et al. (1976). As it is obvious in Fig. 8.9, the crack is arrested earlier compared to the experimental data when the transition bond condition is not considered. Furthermore, the numerical oscillations of DSIFs have significantly reduced the accuracy. On the other hand, U

a

U

FIGURE 8.8 Representative model of a DCB specimen.

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5. Dynamic crack propagation and arrest modeling

With transition bond

Without transition bond 2.0

1.8

Present Specimen no. 4 Specimen no. 21

1.5

Crack arrest

1.2

1.0

0.8

0.5

Crack arrest 1.5

1.2

1.0

0.8

0.5

0.2

0.2

0.0

Present Specimen no. 4 Specimen no. 21

1.8

Dynamic stress intensity factor K d [ MN m− 3/ 2 ]

Dynamic stress intensity factor K d [ MN m− 3/ 2 ]

2.0

80

100

120 140 160 Crack length a [mm]

180

200

0.0

80

100

120 140 160 Crack length a [mm]

180

200

FIGURE 8.9 DSIFs versus crack tip positions for specimens no. 4 and 21.

the numerical damping effects provided by the transition bond concept have dramatically increased the solution accuracy. The final crack length obtained by the transition bond condition is also in good agreement with the experiments. The crack initiation toughness for the specimen no. 4 is much higher compared to the other specimens, which enables to store more strain energy in the specimens prior to the onset of the crack propagation. In this case, the final crack lengths become much higher for the specimen no. 4. It is also observed that the difference between the final crack lengths of with and without transition bond conditions is relatively small. This could be explained by the higher crack initiation toughness of the specimens. Similar comments can be made for the specimen no. 8 as the DSIFs and final crack lengths are presented in Fig. 8.10 with and without transition bond conditions. The crack is arrested much earlier for this specimen. The crack initiation toughness of specimen no. 8 is smaller compared to that of specimen no. 4, as a result of which the final crack lengths become smaller. Moreover, the difference between the final crack lengths of with and without transition bond conditions becomes more visible. The results for specimens no. 17 and 24 are presented in Figs. 8.11 and 8.12, respectively. These specimens have much smaller crack initiation toughness values compared to those of specimens no. 4 and 8. The lower

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8. Peridynamics in dynamic fracture modeling

With transition bond

Without transition bond 2.0

2.0

Present Specimen no. 8

Present Specimen no. 8 1.8

1.5

Crack arrest

1.2

1.0

0.8

0.5

Dynamic stress intensity factor K d [ MN m− 3/ 2 ]

Dynamic stress intensity factor K d [ MN m− 3/ 2 ]

1.8

Crack arrest 1.2

1.0

0.8

0.5

0.2

0.2

0.0

1.5

80

100

120 140 160 Crack length a [mm]

FIGURE 8.10

180

0.0

200

100

120 140 160 Crack length a [mm]

Present Specimen no. 17

1.8

1.8

1.5

1.2

Crack arrest 1.0

0.8

0.5

0.2

Dynamic stress intensity factor Kd [ MN m− 3/ 2 ]

Dynamic stress intensity factor Kd [ MN m− 3/ 2 ]

200

Without transition bond

2.0 Present Specimen no. 17

0.0

180

DSIFs versus crack tip positions for specimen no. 8.

With transition bond

2.0

80

1.5

Crack arrest

1.2

1.0

0.8

0.5

0.2

80

100

120 140 160 Crack length a [mm]

180

200

0.0

80

100

120 140 160 Crack length a [mm]

180

200

FIGURE 8.11 DSIFs versus crack tip positions for specimen no. 17.

crack initiation toughness represents a sharper crack tip. For these specimens, the strain energies stored in the bodies prior to the onset of crack propagation would be much smaller, as a result of which the final crack lengths become small. It is clear from Figs. 8.11 and 8.12 that the numerical oscillations have caused very earlier arrest of cracks when the I. New concepts in peridynamics

179

6. Concluding remarks

With transition bond

Without transition bond 2.0

2.0

Present Specimen no. 24

Present Specimen no. 24 1.8

Dynamic stress intensity factor K d [ MN m− 3/ 2 ]

Dynamic stress intensity factor K d [ MN m− 3/ 2 ]

1.8

1.5

1.2

Crack arrest 1.0

0.8

0.5

1.2

Crack arrest 1.0

0.8

0.5

0.2

0.2

0.0

1.5

80

100

120 140 160 Crack length a [mm]

FIGURE 8.12

180

200

0.0

80

100

120 140 160 Crack length a [mm]

180

200

DSIFs versus crack tip positions for specimen no. 24.

transition bond condition is not taken into account. The difference between the final crack lengths of with and without transition bond cases becomes more pronounced for the specimens with lower crack initiation toughness values.

6. Concluding remarks In this contribution, we have presented the application of OSB-PD approach to 2D dynamic fracture problems. Obtained results confirm the high accuracy of the developed method as the computed solutions are compared with reference results derived from analytical, experimental, and other numerical methods. As a first case, the DSIFs of a stationary crack have been computed and compared with the analytical and numerical works. In calculating the DSIFs, MLS approach has been employed to obtain the displacement and velocity gradients. It was shown that this procedure is efficient. The dynamic crack propagation with arrest modeling has been described and several problems have been solved. Both crack path histories and DSIFs were compared with the reference works. The influence of the transition bond modeling has been discussed and demonstrated. It was clearly observed that the transition bond concept has significantly reduced the numerical oscillations caused by sudden releasing of the bond force. The transition bond concept also increases the accuracy of the crack arrest modeling so that a pre-mature crack arrest has been avoided by suppressing the numerical oscillations. I. New concepts in peridynamics

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C H A P T E R

9 Contact analysis of rigid and deformable bodies with peridynamics Sundaram Vinod K. Anicode, Erdogan Madenci Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States

O U T L I N E 1. Introduction

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2. Approach 2.1 Bond-based peridynamic model 2.2 Rigid impactor model

185 185 187

3. Contact model between the impactor and target

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4. Numerical results 4.1 Normal impact of a rigid sphere (single sub-volume) on a simply supported plate 4.2 Normal impact of a rigid sphere (multiple sub-volume) on a fully clamped plate

198

5. Conclusions

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References

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Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00001-9

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1. Introduction This chapter focuses on a generalized particle-based contact model for peridynamic impact simulations. A review of various contact detection methods and the existing algorithms for both meshless and mesh-based approaches can be found in Bourago and Kukudzhanov (2005). The commonly accepted contact models in the framework of peridynamics involve either rigid impactor or short-range force algorithms (which can also be named as penalty method) as explained in detail by Madenci and Oterkus (2014). If the impactor is rigid and the target is flexible and governed by PD equations of motion (EOM), the material points inside the rigid impactor are moved to the closest region outside the impactor surface. The reaction forces are then calculated in the next time step for the rigid body motion of the impactor. However, this formulation suffers especially if the rigid target has sharp corners and edges which lead to uncertainties for the PD material point location outside the impactor surface. On the other hand, the short-range force approach is commonly used if both the impactor and target are flexible. Several numerical studies by Madenci and Oterkus (2014) verified the capability of both approaches. Macek and Silling (2007) incorporated the short-range force contact algorithm into the finite element (FE) software, ABAQUS, to solve very complex plate perforation and projectile penetration problems. Lee et al. (2016) modeled a rigid impactor and flexible target with FE and PD nodes, respectively. The node-to-surface contact algorithm is achieved by inverse parametric mapping technique between the contacting elements and PD nodes. The penalty stiffness is enforced by using the nodal mass of PD nodes and the segment mass of the FE target surface. Lai et al. (2018) used the average stiffness of the particles for penalty parameter and solved the impact problem using non-ordinary state-based PD. The aforementioned methods provide verifications for very complex impact phenomena, but the contact force/moment evolution is not realistic to that of an actual model as they strongly depend on the penalty stiffness or short-range force constants (which are chosen arbitrarily in most of the studies) and the instantaneous number of contacting points. Besides, the stick-slip with friction contact model is unavoidable in order to provide reaction results that are realistic. Silling (2016) developed a bond-based friction model where the forces are applied parallel to the bonds. Although this model is computationally efficient, it introduces errors in the approximation of friction for pure-sliding case. Recently, Kamensky et al. (2019) improved this approximation by formulating a nonlocal friction for meshless approach and extended it to state-based peridynamics. There is no rigorous PD study in the literature that investigates the reaction forces and moments between the impactor and

I. New concepts in peridynamics

2. Approach

185

target sites. This chapter presents a generalized particle-based contact model similar to the one used in discrete element modeling (DEM) (Cundall and Strack, 1979) and SPH (Campbell et al., 2000; Bui et al., 2014). It avoids arbitrary selection of the penalty parameter by a direct estimation using contact mechanics. Furthermore, an incremental approach is used for the contact force update in order to accurately represent the conditions during the contact of an impactor with arbitrary shape. Section 2 describes the approach for modeling the governing equation of motion of the target and the impactor. Section 3 describes a generalized contact model during an impact and its numerical implementation. Finally, Section 4 provides numerical results for the validation of the present approach.

2. Approach In the peridynamic theory, material points interact with each other directly through the prescribed response function, which contains all the constitutive information associated with the material. The response function includes a length parameter called internal length (horizon), d: The locality of interactions depends on the horizon, and interactions become more local with a decreasing horizon. As the interactions between material points cease, cracks may initiate and align themselves along surfaces that form cracks, yet the integral equations continue to remain valid.

2.1 Bond-based peridynamic model As introduced by Silling (2000), the bond-based PD concerns the physics of a material point x that interacts with another material point x0 within a certain range, as shown in Fig. 9.1. The interaction domain Hx of material point x is defined by its horizon d. Material points x0 located within the interaction domain Hx are called the family members of x. The interactions of a material point x are governed by PD equation of motion as Z € ðx; tÞ ¼ fðu; u0 ; x; x0 ; tÞdVx0 þ bðx; tÞ rðxÞ u (9.1) Hx

where rðxÞ and x represent the density and position vector of the main material point x in the undeformed configuration, respectively, and t

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FIGURE 9.1 PD Interactions of material point x with the other points x0 within its family.

represents the time. The displacement vector is u and the body load on the main material point is given by body force density vector, bðx; tÞ. The interaction or peridynamic force, which is exerted upon the material point x by material point x0 , is denoted by fðx0 x; u0 uÞ and is expressed as fðu; u0 ; x; x0 Þ ¼ mcs

y0  y jy0  yj

(9.2a)

with s¼

jy0  yj  jx0  xj jx0  xj

(9.2b)

where the deformed position of material points y, y0 are defined as y ¼ x þ u and y0 ¼ x0 þ u0 . The parameter s represents the stretch of a bond and c is the bond constant. The bond constant is obtained by equating strain energy densities of classical continuum mechanics (CCM) and PD theories under simple loading conditions. It can be derived as c¼

15E pd ð1 þ nÞ 4

(9.3)

in which E and n denote the Young’s modulus and Poisson’s ratio of the material. The elastic perfectly plastic constitutive model for the bonds (interactions) is shown in Fig. 9.2. The force-density relationship can be written as fðsÞ ¼ cs; if syc < sðtÞ < sot and fðsÞ ¼ csyc ; if sðtÞ < syc for elastic region and plastic regions, respectively. Local damage at a point is defined as the weighted ratio of the number of eliminated interactions to the total number of initial interactions of a

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2. Approach

FIGURE 9.2 Constitutive model of a bond.

material point with its family members. The local damage at a point can be quantified as (Silling and Askari, 2005) which ranges from 0 to 1. R mðx0  x; tÞdV 0 H R . (9.4) fðx; tÞ ¼ 1  dV 0 H

When the local damage is 1, all the interactions initially associated with the point have been eliminated, while a local damage of 0 means that all interactions are still intact. The measure of local damage is an indicator of possible crack formation within a body. For example, initially a material point interacts with all materials in its horizon; thus, the local damage has a value of 0. However, the creation of a crack terminates half of the interactions within its horizon resulting in a local damage value of ½.

2.2 Rigid impactor model The impactor is not deformable at any instant, and it moves with its own velocity governed by the rigid-body dynamics. As shown in Fig. 9.3, the impactor is composed of many rigid infinitesimal sub-volumes of DV. The relative position of the centroid of each sub-volume is given by vector r with respect to the body-fixed (local) coordinate system of the impactor. The density of the impactor is denoted by r. Therefore, the virtual work of the impactor with an arbitrary shape can be expressed as Z Z Z Z T€ T T rDR RdV ¼ rDR fb dV þ DR tn dG þ DRT fR dV (9.5) V

V

A

I. New concepts in peridynamics

VR

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9. Contact analysis of rigid and deformable bodies with peridynamics

FIGURE 9.3

Volume discretization of the impactor.

where R is the global position vector of each point representing the centroid of each sub-volume. The body force vector of each point is denoted by fb, and the external traction force acting normal to the surface A of the impactor is denoted by tn. The reaction force fR between the impactor and target prevents the interpenetration of PD material points inside the impactor. The volume of the PD material points interpenetrated into the impactor is denoted by VR. The global position vector of each subvolume can be expressed in terms of the Rigid-Body (RB) motion of the impactor as R ¼ R0 þ T0 r0

(9.6)

where R0 and T0 denote the position vector and orientation matrix of the impactor, respectively. The vector r0 represents the position vector of each sub-volume with respect to the local (body-fixed) coordinates of the impactor. Hence, the velocity of each sub-volume R_ becomes e 0 T0 r0 R_ ¼ R_ 0 þ T_ 0 r0 ¼ R_ 0 þ u

(9.7)

e 0 T0 T_ 0 ¼ u

(9.8)

with

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2. Approach

in which u0 denotes the angular velocity vector of the impactor, and the skew-symmetric matrix representation of the angular velocity vector e 0. Note that is denoted by u e 0 rhu0  r ¼ r  u0 h  eru0 e 0 Tr0 ¼ u u

(9.9)

Tr0

is the position vector of each sub-volume in global coorwhere r ¼ dinate system. Hence, Eq. (9.7) can be rewritten as R_ ¼ R_ 0  eru0

(9.10)

€ becomes The acceleration of each sub-volume R €¼R € 0  eru _0þu e 0u e 0r R

(9.11)

The virtual displacement vector of each sub-volume in the impactor, DR, can be approximated as _ DR ¼ RDt

(9.12)

where Dt is an infinitesimally small-time increment. Substituting from Eq. (9.10) for R_ in Eq. (9.12) yields DR ¼ R_ 0 Dt  eru0 Dt ¼ DR0  erDq0

(9.13)

DR0 ¼ R_ 0 Dt

(9.14a)

Dq0 ¼ u0 Dt

(9.14b)

with

and in which Dq0 represents infinitesimally small (i.e., jDq0 j 0. The

ðkÞ

tangential overlap dðiÞ;t is evaluated by using the relative tangential veðkÞ

ðkÞ

ðkÞ

locity d_ ðiÞ;t ¼ vðiÞ  d_ ðiÞ;n over Ds time increment as ðkÞ ðkÞ ðkÞ dðiÞ;t ¼ dðiÞ;t1 þ d_ ðiÞ;t Ds

(9.31)

The relative velocity of the point of contact as computed by Luding ðkÞ ðkÞ (2008) is vðiÞ ¼ vG þ u0  xk xG þak nðiÞ  vi , where ak ¼ rðkÞ  0:5d. The force-displacement relation in the normal direction is based on Hertzian contact law and can be extended for the multiparticle interactions (Bui et al., 2014). Accordingly, the normal contact force FðiÞ;n , at point i due to interaction with its neighbors, NðiÞ becomes FðiÞ;n ¼

NðiÞ 1 X ðkÞ ðkÞ Kn dðiÞ;n þ Cn d_ ðiÞ;n NðiÞ k¼1

(9.32)

where the normal direction stiffness Kn and damping Cn constants are defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kn ¼ 2E

ðkÞ

RdðiÞ;n

(9.33a)

FIGURE 9.5 Contact model between the impactor and peridynamic points.

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9. Contact analysis of rigid and deformable bodies with peridynamics

and Cn ¼ wn

pffiffiffiffiffiffiffiffiffiffi mKn

in which wn is the damping ratio in the normal direction and 1n2ðkÞ 1 EðkÞ , R

(9.33b) 1 E

¼

1n2ðiÞ EðiÞ þ

¼ R1ðiÞ þ R1ðkÞ and m1 ¼ m1ðiÞ þ m1ðkÞ denote the effective Young’s modulus,

radius, and mass of contacting particles i and k. For the tangential direction, the force-displacement relation is obtained using the elastic no-slip Mindlin law. The effective tangential force at the point i due to interaction with its neighbors (Bui et al., 2014) becomes FðiÞ;t ¼

NðiÞ 1 X ðkÞ ðkÞ Kt dðiÞ;t þ Ct d_ ðiÞ;t NðiÞ k¼1

(9.34)

where the tangential direction stiffness Kt and damping Ct constants are defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ

Kt ¼ 8G and Ct ¼ wt 2nðiÞ GðiÞ

RdðiÞ;n pffiffiffiffiffiffiffiffiffi mKt

(9.35a)

(9.35b)

2nðkÞ GðkÞ

þ and wt indicate the effective shear modulus and in which ¼ tangential direction damping ratio, respectively. The effect of friction is included if the tangential force FðiÞ;t exceeds mFðiÞ;n and Eq. (9.34) takes the form 1 G

FðiÞ;t ¼ mFðiÞ;n

(9.36)

Fig. 9.6 describes the contact setup for the peridynamic particle interacting with the impactor sub-volume during an impact. The normal and tangential contact forces acting on PD particle i are invoked in Eq. (9.1) as a body force in the form bn ðxi ; sÞ ¼

1 F VðiÞ ðiÞ;n

(9.37a)

bt ðxi ; sÞ ¼

1 F VðiÞ ðiÞ;t

(9.37b)

and

I. New concepts in peridynamics

3. Contact model between the impactor and target

I. New concepts in peridynamics

FIGURE 9.6 Contact model for peridynamics.

195

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9. Contact analysis of rigid and deformable bodies with peridynamics

They are invoked in Eq. (9.27a) as a force density vector due to contact as ! FðiÞ;n þ FðiÞ;t R fcf ¼  (9.38) VðiÞ NðiÞ The moment, mRðkÞ appearing in Eq. (9.27b) due to the tangential force acting on the impactor is evaluated as !  FðiÞ;t ðkÞ R mðkÞ ¼  xk  xG þ ak nðiÞ  (9.39) VðiÞ NðiÞ Since the impactor is discretized using multiple sub-volumes, the contacting time of each sub-volume with the PD particle can be different, resulting in the multiple contact force history (Kruggel-Emden et al., 2008; Hohner et al., 2011; Thorton et al., 2013). To capture the contact force evolution due to multiple contacts, which is more realistic to that of acting on the impactor, Eqs. (9.30)e(9.35) are used in the incremental form. At the instant of time s, compute the number of PD particles Npd in contact with the impactor. Also for each PD particle i compute the number of contacting impactor sub-volumes NðiÞ . The algorithm for the incremental approach is outlined as (1) Set i ¼ 1 • If i  Npd and damage flag 0 go to step (4), else go to step (7) (4) For the normal direction • Compute the incremental overlap and velocity ðkÞ;s

ðkÞ;s

ðkÞ;s1

DdðiÞ;n ¼ dðiÞ;n  dðiÞ;n

ðkÞ;s ðkÞ;s ðkÞ;s1 ; Dd_ ðiÞ;n ¼ d_ ðiÞ;n  d_ ðiÞ;n ;

• Using the above kinematic variables, the incremental normal force is  ðkÞ;s ðkÞ;s ðkÞ;s DFðiÞ;n ¼ an Kns DdðiÞ;n þCsn Dd_ ðiÞ;n ; an is a user-defined tuning parameter for the normal force evolution.

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3. Contact model between the impactor and target

(5) Similarly, for the tangential direction. • Compute the incremental overlap and velocity ðkÞ;s

ðkÞ;s1

ðkÞ;s

b DdðiÞ;t ¼ dðiÞ;t  d ðiÞ;t

ðkÞ;s

ðkÞ;s

ðkÞ;s1

; Dd_ ðiÞ;t ¼ d_ ðiÞ;t  d_ ðiÞ;t

;

where.  b ðkÞ;s1 ¼ dðkÞ;s1  dðkÞ;s1 $nðkÞ nðkÞ is the previous tangential d ðiÞ;t ðiÞ;t ðiÞ;t ðiÞ ðiÞ overlap projected onto the current tangential plane and ðkÞ;s b ðkÞ;s1 þ 1 d_ ðkÞ;s Ds d ¼d ðiÞ;t

ðiÞ;t

NðiÞ ðiÞ;t

ðkÞ;s

• The sticking component DFðiÞ;stick , based on Eq. (9.34) is computed as  ðkÞ;s ðkÞ;s ðkÞ ðkÞ;s1 ðkÞ;s DFðiÞ;stick ¼ lðiÞ DFðiÞ;stick þ Kts DdðiÞ;t þ Cst Dd_ ðiÞ;t ; where ðkÞ

lðiÞ ¼

8 > > >
t > > : Ks1 ; t

9 DFsðiÞ;n  0 > > > = DFsðiÞ;n < 0

> > > ;

• The average tangential slip component, based on Eq. (9.36) is ðkÞ;s DFðiÞ;slip ¼ N1i mFsðiÞ;n . • Take the minimum of the above two forces, i.e.,  ðkÞ;s ðkÞ;s ðkÞ;s DFðiÞ;t ¼ min DFðiÞ;stick ; DFðiÞ;slip (6) Collect the forces computed at steps (4) and (5) and sum for each k. (7) k ¼ k þ 1. If k > NðiÞ exit inner loop and go to step (8), else go to step (2). (8) For each PD point i, • contact force is FsðiÞ ¼ FsðiÞ;n þ FsðiÞ;t where 1 • FsðiÞ;n ¼ Fs1 ðiÞ;n þ NðiÞ

N ðiÞ P k¼1

ðkÞ;s

DFðiÞ;n and FsðiÞ;t ¼ at

N ðiÞ P k¼1

ðkÞ;s

DFðiÞ;t ; at is a

user-defined tuning parameter for the tangential force evolution. • The force density at PD point i and impactor point k are calculated using Eqs. (9.37) and (9.38). (9) i ¼ i þ 1 If i > Npd exit outer loop else go to step (1).

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9. Contact analysis of rigid and deformable bodies with peridynamics

4. Numerical results The numerical results concern the validation of the contact algorithm using a single and multiple particle interactions under normal impact loading without allowing damage in the material.

4.1 Normal impact of a rigid sphere (single sub-volume) on a simply supported plate As shown in Fig. 9.7A, the spherical impactor of radius R ¼ 0:01 m has a normal impact with the plate at a velocity of Vz ¼ 1 m/s. The interactions between target material points are governed by the PD equations of motion. The plate is simply supported, and its geometry is defined by its length, width, and the total thickness as L ¼ W ¼ 120 mm and H ¼ 80 mm, respectively. The target is made of steel with Young’s modulus E ¼ 206GPa, Poisson’s ratio n ¼ 0:28, and density r ¼ 7833 kg/ m3. The plate is discretized into a 100  100  4 grid with the grid spacing specified as Dx ¼ 2:0 mm. Fig. 9.7b shows the PD discretization of a quarter model of the plate with the 25  25  4 grid. The time step size

FIGURE 9.7 Normal impact of rigid sphere on a simply supported plate: (A) Geometry (B) PD discretization.

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4. Numerical results

199

FIGURE 9.8 Variation of impactor parameters during the contact period: (A) Normal displacement, (B) Normal velocity, and (C) Normal force.

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9. Contact analysis of rigid and deformable bodies with peridynamics

is Dt ¼ 1:8  108 s with a total solution time of t ¼ 100 ms. Fig. 9.8 shows the variation of impactor parameters during the contact duration. The contact terminates at about 80 ms. The transverse (normal) displacement of the plate just after rigid impactor rebounds is shown in Fig. 9.9. Table 9.1 presents the comparison of the PD results with theoretical results (Chun and Lam, 1998).

4.2 Normal impact of a rigid sphere (multiple sub-volume) on a fully clamped plate The effect of changing the discretization size of the impactor mesh is investigated and the results of the rebound kinematics are compared with FE simulations from ANSYS. Fig. 9.10A depicts the impact of the rigid sphere with R ¼ 25 mm on the square plate with an impact velocity V ¼  80 m/s at angle q ¼ 0 normal to the surface. The plate geometry is defined by its length, width, and the total thickness as L ¼ W ¼ 120 mm

FIGURE 9.9

PD prediction of normal displacement in the plate at time t ¼ 80 ms.

TABLE 9.1 Comparison of the results. PD simulations

Kruggel-Emden et al. (2008)

Uz;max ðmmÞ

0.027

0.025

Vz;max =Vz;min

0.75/1.0

0.75/1.0

Fn;max ðkNÞ

1.2335

1.25

Uz;max ðmmÞ

0.00781

0.00781

Parameters Impactor

Plate

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5. Conclusions

FIGURE 9.10 Impact of rigid sphere on a fully clamped plate: (A) Geometry and (B) Discretized model. TABLE 9.2 Material parameters for normal impact of sphere and plate. Property

Sphere

Plate

Young’s modulus, E ðGPaÞ

2.26

210

Poisson’s ratio, n

0.28

0.25

1380

7850

3

Density, r ðkg=m

and H ¼ 80 mm, respectively. The plate is discretized with 10 10 10 grid spacing of Dx ¼ 8 mm. The impactor sub-volumes for contact detection are created with a grid-spacing of Dx0 . Fig. 9.10B shows the discretized model of the rigid sphere and the plate. Their material properties are given in Table 9.2. The explicit dynamic analysis is performed for impactor grid spacing of Dx0 ¼ Dx; Dx=3 and Dx=5 for a time step of the Dt ¼ 1:8  108 s and total time t ¼ 200 ms. Fig. 9.11 shows the kinematics of the impactor which matches closely with the finite element analysis.

5. Conclusions This study presents new generalized particle-based contact model in the framework of peridynamics. It includes the capability to simulate the realistic impact conditions of an arbitrarily shaped impactor using an incremental contact force update scheme and also model the frictional

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FIGURE 9.11 Comparison of impactor kinematics during the impact for various impactor mesh sizes: (A) Displacement and (B) Velocity.

stick-slip-based contact. This contact algorithm is validated for the contact parameters such as maximum penetration, rebound velocity, contact duration, and force reaction by performing impact on the target material without allowing failure. The effect of impactor mesh discretization size was also investigated. It was found that the contact results do not change significantly and it correlates well with FEM.

References Bourago, N.G., Kukudzhanov, V.N., 2005. A review of contact algorithms. Mech. Solid. 40, 35e71.

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References

203

Bui, H.H., Kodikara, J.K., Bouazza, A., Haque, A., Ranjith, P.G., 2014. A novel computational approach for large deformation and post-failure analyses of segmental retaining wall systems. Int. J. Numer. Anal. Methods GeoMech. 38, 1321e1340. Campbell, J., Vignjevic, R., Libersky, L., 2000. A contact algorithm for smoothed particle hydrodynamics. Comput. Methods Appl. Mech. Eng. 184, 49e65. Chun, L., Lam, K.Y., 1998. Dynamic response of fully-clamped laminated composite plates subjected to low-velocity impact of a mass. Int. J. Solid Struct. 35, 963e979. Cundall, P.A., Strack, O.D., 1979. A discrete numerical model for granular assemblies. Geotechnique 29, 47e65. Hohner, D., Wirtz, S., Kruggel, H., Scherer, V., 2011. Comparison of the multi-sphere and polyhedral approach to simulate non-spherical particles within the discrete element method: influence on temporal force evolution for multiple contacts. Powder Technol. 208, 643e656. Kamensky, D., Behzadinasab, M., Foster, J.T., Bazilevs, Y., 2019. Peridynamic modeling of frictional contact. J. Peridynamics Nonlocal Model. 1, 1e15. Kruggel-Emden, H., Rickelt, S., Wirtz, S., Scherer, V., 2008. A study on the validity of the multi-sphere Discrete Element Method. Powder Technol. 188, 153e165. Lai, X., Liu, L.S., Li, S.F., Zeleke, M., Liu, Q., Wang, Z., 2018. A non-ordinary state-based peridynamics modeling of fractures in quasi-brittle materials. Int. J. Impact Eng. 111, 130e146. Lee, J., Liu, W., Hong, J.W., 2016. Impact fracture analysis enhanced by contact of peridynamic and finite element formulations. Int. J. Impact Eng. 87, 108e119. Luding, S., 2008. “Introduction to discrete element methods: basic of contact force models and how to perform the microemacro transition to continuum theory. Eur. J. Environ. Civil Eng. 12, 785e826. Macek, R.W., Silling, S.A., 2007. Peridynamics via finite element analysis. Finite Elem. Anal. Des. 43, 1169e1178. Madenci, E., Oterkus, E., 2014. Peridynamic Theory and its Applications. Springer, New York. Silling, S.A., Askari, E., 2005. A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526e1535. Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solid. 48, 175e209. Silling, S.A., 2016. Meshfree Peridynamics for Soft Materials. SAND2016-4586 C. Thorton, C., Cummins, S.J., Cleary, P.W., 2013. An investigation of the comparative behavior of alternative contact force models during in-elastic collisions. Powder Technol. 233, 30e46.

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C H A P T E R

10 Modeling inelasticity in peridynamics Selda Oterkus1, Erdogan Madenci2 1

Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom; 2 Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States

O U T L I N E 1. Introduction

206

2. Peridynamic plasticity formulation

207

3. Peridynamic viscoelasticity formulation

214

4. Numerical results 4.1 Plate under tensile loading 4.2 Plate with a pre-existing crack under tensile loading

217 217 218

5. Conclusions

219

References

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Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00011-1

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© 2021 Elsevier Inc. All rights reserved.

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10. Modeling inelasticity in peridynamics

1. Introduction Peridynamic (PD) theory is not limited to elastic deformations. Various peridynamic formulations are available in the literature to represent inelastic deformations for plastic, viscoelastic, and viscoplastic material behavior. Among the previous studies concerning plastic deformation, Mitchell (2011a) developed ordinary state-based plasticity model for peridynamics. Madenci and Oterkus (2016) introduced an ordinary state-based peridynamic model to consider plastic deformations according to von Mises yield criterion with isotropic hardening. More recently, Liu et al. (2020) presented a new ordinary state-based peridynamic model to consider nonlinear hardening plasticity material behavior. Sun and Sundararaghavan (2014), Luo et al. (2018), and Gu et al. (2019) presented the peridynamic crystal plasticity formulations. Rahaman et al. (2017) developed a peridynamic model for plasticity by incorporating microinertia-based flow rule, entropy equivalence, and localization residuals. Lammi and Vogler (2014) introduced peridynamic plasticity model for the dynamic flow and fracture of the concrete. Chen et al. (2018) presented fracture animation for elastoplastic solids by employing peridynamics. Among the previous studies concerning viscoelastic deformation, Mitchell (2011b) developed an ordinary state-based viscoelasticity model for peridynamics. Madenci and Oterkus (2017) introduced another model within ordinary state-based framework suitable for thermoviscoelastic deformations. Delorme et al. (2017) presented the generalization of the ordinary state-based peridynamic model for linear viscoelasticity. Silling (2019) demonstrated the attenuation of waves in a viscoelastic peridynamic medium. Dorduncu et al. (2016) developed a peridynamic truss element for viscoelastic deformations. Weckner and Mohamed (2013) applied the Green’s function approach along with Fourier and Laplace transforms to derive integral-representation formulas for viscoelastic material behavior. Peridynamic formulations for viscoplasticity are also available such as Foster et al. (2010), Pathrikar et al. (2019), and Amani et al. (2016). This chapter presents the peridynamic formulations for plastic and viscoelastic deformations developed by Madenci and Oterkus (2016) and Madenci and Oterkus (2017), respectively. Numerical examples concerning a metallic plate under tension with and without a pre-existing crack demonstrate the capability of these formulations.

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2. Peridynamic plasticity formulation

2. Peridynamic plasticity formulation Peridynamics is a nonlocal continuum mechanics formulation with a governing equation in the form of integro-differential equation which can be written as Z € tÞ ¼ ðtðu0  u; x0  x; tÞ  t0 ðu0  u; x0  x; tÞÞdV 0 þ bðx; tÞ (10.1) rðxÞuðx; H

where r is the density, H is the horizon, u€ and u are the acceleration and displacement of a material point, b is the body load, and t is the peridynamic force density between two material points, x and x0 . This equation can be written in discretized form as rðkÞ u€ðkÞ ¼

N h X

  tðkÞð jÞ uð jÞ  uð jÞ ; xð jÞ  xðkÞ ; t

j¼1



 tð jÞðkÞ uð jÞ  uð jÞ ; xðkÞ  xð jÞ ; t

i

(10.2)

Vð jÞ þ bðkÞ

where N is the number of material points, j inside the horizon of the material point k. Strain energy density WðkÞ of the material point k can be k , and distortional, W m , parts as split into dilatational, WðkÞ ðkÞ m k WðkÞ ¼ WðkÞ þ WðkÞ ;

(10.3)

with k WðkÞ ¼ ak q2ðkÞ

and

8 < X N

m WðkÞ ¼ b :

j¼1

9    2 =      wðkÞðjÞ yð jÞ  yðkÞ   xð jÞ  xðkÞ  Vð jÞ  am q2ðkÞ ; ;

where peridynamic parameters ak , 8 > < k ak ¼ 1 > : k 2 8 > < 2m am ¼ 5m > : 6

(10.4a)

(10.4b)

am , and b can be defined as for ð2  DÞ for ð3  DÞ

(10.5a)

for ð2  DÞ for ð3  DÞ

I. New concepts in peridynamics

(10.5b)

208

10. Modeling inelasticity in peridynamics

and

8 6m > > > < phd4 b¼ > 15m > > : 2pd5

for ð2  DÞ (10.5c) for ð3  DÞ

The influence function wðkÞðjÞ can be defined as d . wðkÞð jÞ ¼   xð jÞ  xðkÞ 

(10.6)

where d is the size of the horizon. The peridynamic dilatation term qðkÞ can be expressed as qðkÞ ¼ d

N X

        wðkÞð jÞ yð jÞ  yðkÞ   xð jÞ  xðkÞ  LðkÞð jÞ Vð jÞ ;

(10.7)

j¼1

with

8 2 > > > < phd3 d¼ > 9 > > : 4pd4

for ð2  DÞ (10.8) for ð3  DÞ

and yð jÞ  yðkÞ xð jÞ  xðkÞ $ . LðkÞð jÞ ¼     yð jÞ  yðkÞ  xð jÞ  xðkÞ 

(10.9)

where y is the position of the material point in the deformed configuration. The peridynamic force density between two material points k and j can be obtained from the strain energy density given in Eq. (10.3) as tðkÞð jÞ ¼

vWðkÞ 1   Vð jÞ v y  y  ð jÞ ðkÞ

y  yðkÞ y  yðkÞ  ð jÞ  ¼ tðkÞðjÞ  ð jÞ      yð jÞ  yðkÞ  yð jÞ  yðkÞ 

(10.10)

where Vð jÞ is the volume of the material point j and  LðkÞð jÞ  qðkÞ þ 2dbsðkÞð jÞ tðkÞð jÞ ¼ ak  am 2dd   xð jÞ  xðkÞ 

I. New concepts in peridynamics

(10.11)

209

2. Peridynamic plasticity formulation

with the stretch sðkÞð jÞ being defined as         yð jÞ  yðkÞ   xð jÞ  xðkÞ    sðkÞð jÞ ¼   xð jÞ  xðkÞ 

(10.12)

Similar to the strain energy density, the peridynamic force density can be split into dilatational, tkðkÞð jÞ , and distortional, tmðkÞð jÞ , parts as tðkÞð jÞ ¼ tkðkÞð jÞ þ tmðkÞð jÞ ;

(10.13)

y  yðkÞ 2dak d LðkÞð jÞ qðkÞ  ð jÞ  tkðkÞð jÞ ¼     yð jÞ  yðkÞ  xð jÞ  xðkÞ 

(10.14a)

with

and

! tmðkÞð jÞ ¼

2dam d LðkÞð jÞ qðkÞ 2dbsðkÞð jÞ    xð jÞ  xðkÞ 

y  yðkÞ  ð jÞ .   yð jÞ  yðkÞ 

(10.14b)

Based on the relationships given in Eq. (10.14a) and (10.14b), the dilatation and stretch expressions can be rewritten as     xð jÞ  xðkÞ  k qðkÞ ¼ tðkÞð jÞ (10.15a) 2dak dLðkÞð jÞ and

"

LðkÞð jÞ 1 hm i d  q sðkÞð jÞ ¼ tðkÞð jÞ þ am 2db b x  x  ðkÞ ð jÞ ðkÞ

# (10.15b)

The stretch expression given in Eq. (10.15b) can also be split into dilatational, skðkÞð jÞ , and distortional, smðkÞð jÞ , components as sðkÞð jÞ ¼ skðkÞð jÞ þ smðkÞð jÞ with

" skðkÞðjÞ ¼

(10.16)

#

LðkÞðjÞ am d  q .  b x  x  ðkÞ ðjÞ ðkÞ

I. New concepts in peridynamics

(10.17a)

210

10. Modeling inelasticity in peridynamics

and smðkÞð jÞ ¼

1 hm i t 2db ðkÞð jÞ

(10.17b)

Similarly, the distortional part of the strain energy density given in Eq. (10.4b) can be rewritten as

m WðkÞ ¼b

N X

    ds2ðkÞð jÞ xð jÞ  xðkÞ Vð jÞ  am q2ðkÞ .

(10.18a)

j¼1

or m ¼b WðkÞ

N X j¼1

1 m d LðkÞð jÞ q t d þ am  2db ðkÞð jÞ b x  x  ðkÞ ð jÞ ðkÞ

!2

    xð jÞ  xðkÞ Vð jÞ  am q2ðkÞ . (10.18b)

Since plastic deformation is a nonlinear process, the solution requires an incremental approach. Therefore, the peridynamic force and stretch can be written in their incremental forms as DtðkÞð jÞ and DsðkÞð jÞ , respectively. Incremental stretch can be split into elastic, DseðkÞð jÞ , and plastic, p

DsðkÞð jÞ , components as p

DsðkÞð jÞ ¼ DseðkÞð jÞ þ DsðkÞð jÞ ;

(10.19)

Similarly, incremental form of dilatation can be expressed as DqðkÞ ¼ DqeðkÞ ¼ dd

N X

DseðkÞð jÞ LðkÞð jÞ Vð jÞ

(10.20)

j¼1 p

with the plastic component of the dilatation being zero, i.e., DqðkÞ ¼ 0. p Plastic stretch, sðkÞð jÞ , at any time instant can be written as p p sðkÞð jÞ ¼ sðkÞð jÞ ðt ¼ 0Þ þ

Z 0

t

p

s_ðkÞð jÞ ðtÞdt.

p

(10.21)

where s_ðkÞð jÞ is the rate of plastic stretch. Based on von Mises yield criteria, yield stress can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sY ¼ 6mW mðkÞ ; (10.22)

I. New concepts in peridynamics

2. Peridynamic plasticity formulation

211

By using the distortional part of the strain energy density expression given in Eqs. (10.18a) and (10.22) can be rewritten as b

N X j¼1

  s2   ds2ðkÞðjÞ xð jÞ  xðkÞ Vð jÞ  am q2ðkÞ ¼ Y ; 6m

(10.23)

By considering strain hardening material behavior, the yield function can be expressed as   p m FðkÞ ¼ WðkÞ  G sðkÞ ; (10.24)   p p where G sðkÞ is the final state of strain hardening and sðkÞ is the equivalent plastic stretch. For the plasticity tooccur,the yield function given in Eq. (10.24) should

be equal to 0, i.e., FðkÞ tðkÞð jÞ ¼ 0. For an increment of peridynamic force, DtðkÞð jÞ , depending on the rate of change of the yield function and its value, i.e., ! N vFðkÞ vFðkÞ 1X DFðkÞ ¼ Dt þ Dt (10.25) 4 j¼1 vtðkÞð jÞ ðkÞð jÞ vtð jÞðkÞ ð jÞðkÞ The state of the loading can be described as DFðkÞ < 0/unloading DFðkÞ > 0/loading

(10.26)

DFðkÞ ¼ 0/neutral loading The incremental strain energy density from the plastic state of tðkÞð jÞ and sðkÞð jÞ to a new plastic state of tðkÞð jÞ þ DtðkÞð jÞ and sðkÞðjÞ þ DsðkÞð jÞ can be expressed as p

e þ DWðkÞ ; DWðkÞ ¼ DWðkÞ

(10.27)

where e DWðkÞ ¼

N   1X DtðkÞð jÞ DseðkÞð jÞ þ Dtð jÞðkÞ Dseð jÞðkÞ Vð jÞ 4 j¼1

(10.28a)

p

N   1X p p DtðkÞð jÞ DsðkÞð jÞ þ Dtð jÞðkÞ Dsð jÞðkÞ Vð jÞ . 4 j¼1

(10.28b)

DWðkÞ ¼

I. New concepts in peridynamics

212

10. Modeling inelasticity in peridynamics

Eqs. (10.21) and (10.24) can also be written in a different form as 8 9 9 1 08 vFðkÞ > vFðkÞ > > > > > > > > > 9 > 9C > > > 8 > 8 B> > vtðkÞð1Þ > vtð1ÞðkÞ > > > > > C B> Dt Dtð1ÞðkÞ > > > > > > > > ðkÞð1Þ < < < < = = = =C B 1B C « « « « $ $ þ DFðkÞ ¼ B C ¼ 0; > > > > > > > 4 B> : Dt : Dt > > ; > ;C > > > > C B> > vFðkÞ > vFðkÞ > ðkÞðNÞ ðNÞðkÞ A > > > > @> > > > > > > > > : vt : vt ; ; ðkÞðNÞ

ðNÞðkÞ

(10.29a) 08 Dsp 9 8 Dsp V 9 8 Vð1Þ 9 ðkÞð1Þ ð1Þ > > Dt > > ðkÞð1Þ > < = = > < < ð1ÞðkÞ = 1B p « « « DWðkÞ ¼ B $ þ $ > > ; 4 @> > ; > : Dt ; > : p : p ðkÞðNÞ DsðNÞðkÞ VðNÞ DsðkÞðNÞ VðNÞ (10.29b) 91 8 Dt > > ð1ÞðkÞ > > =C < C ¼ 0: « A > > > > ; : Dt ðNÞðkÞ

According to Eq. (10.29a) and (10.29b), it can be concluded that vectors in Eq. (10.29a) and (10.29b) are parallel to each other. Therefore, it leads to the following relationships 8 9 vFðkÞ > > 8 p 9 > > > > > > Ds Vð1Þ > > > > > > > < ðkÞð1Þ < vtðkÞð1Þ > = = « « (10.30a) ¼ CðkÞ > > > > > > > > : Dsp > > ; vF ðkÞ > > V > > ðkÞðNÞ ðNÞ > > : vt ; ðkÞðNÞ

8 9 vFðkÞ > > > > > > > > > > vt > > ð1ÞðkÞ < = « . ¼ CðkÞ > > > > > > > > ; vFðkÞ > > > > ðNÞ > > : vt ; ðNÞðkÞ

8 p Ds Vð1Þ > > < ð1ÞðkÞ « > > : Dsp V ðNÞðkÞ

9 > > =

(10.30b)

From, Eq. (10.30a) and (10.30b), the incremental plastic stretch of the bond between material points xðkÞ and xð jÞ can be written as p

DsðkÞð jÞ ¼

vFðkÞ 1 CðkÞ ; Vð jÞ vtðkÞð jÞ

I. New concepts in peridynamics

(10.31)

2. Peridynamic plasticity formulation

213

Since distortional deformation causes plastic deformation, Eq. (10.31) can be rewritten as p

DsðkÞð jÞ ¼

vFðkÞ 1 CðkÞ m . Vð jÞ vtðkÞð jÞ

(10.32)

where CðkÞ is the proportionality constant. Note that the yield surface defined in Eq. (10.22) is valid for uni-axial loading condition. For a general loading condition, equivalent stress and equivalent plastic stretch can be defined as 2 31=2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   N  ffi pffiffiffiffiffiffi X   seðkÞ ¼ 6mW mðkÞ sðkÞð jÞ ¼ 6m4bd s2ðkÞð jÞ xð jÞ  xðkÞ Vð jÞ  am q2ðkÞ 5 j¼1

(10.33a) and p

DsðkÞ ¼ A0 with

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi p W mðkÞ DsðkÞð jÞ

8 pffiffiffi > 3 > > > < pffiffiffiffiffiffiffiffiffiffiffiffi4 for ð2  DÞ pbhd A0 ¼ rffiffiffiffiffiffiffiffiffiffi > > 5 > > : for ð3  DÞ pbd5

(10.33b)

(10.34)

After substituting Eq. (10.18a) in Eq. (10.33b), the equivalent plastic stretch can be obtained as seen in Fig. 10.1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X  N  2  u   p p DsðkÞ ¼ A0 tbd DsðkÞð jÞ xð jÞ  xðkÞ Vð jÞ .

(10.35)

j¼1

Failure definition in peridynamic plasticity formulation is different than widely used critical stretch criterion since the peridynamic forcestretch relationship is nonlinear. Therefore, the peridynamic bond breakage can be based on an energy-based criterion of the form JðkÞð jÞ  1; c JðkÞð jÞ

I. New concepts in peridynamics

(10.36)

214

10. Modeling inelasticity in peridynamics

FIGURE 10.1

Force-stretch relationship between two material points.

with c Jðk þ Þð j Þ ¼

Jc . Nc

(10.37)

where Jc is the critical energy release rate and Nc represents the number of interactions crossing a crack surface. The energy of the bond between material points xðkÞ and xð jÞ , JðkÞð jÞ , can be calculated as JðkÞð jÞ ¼

1 w V V ðDxÞh ðkÞð jÞ ðkÞ ð jÞ

(10.38)

where Dx is the grid size, h is the thickness, VðkÞ and Vð jÞ represent the volumes of the material points xðkÞ and xð jÞ , respectively, and the micropotential, wðkÞð jÞ , can be obtained as sðkÞðjÞ Z

wðkÞð jÞ ¼

tðkÞð jÞ jxj  xk jdsðkÞð jÞ .

(10.39)

0

3. Peridynamic viscoelasticity formulation For a linear viscoelastic isotropic material, peridynamic force density can be expressed in terms of Boltzmann hereditary superposition integral as tðkÞð jÞ ¼ tkðkÞð jÞ þ tmðkÞð jÞ

I. New concepts in peridynamics

(10.40)

215

3. Peridynamic viscoelasticity formulation

with 2dd

  xð jÞ  xðkÞ 

tkðkÞð jÞ ¼ 

Zt

ak ðt  t0 Þ

(10.41a)

0

Zt tmðkÞð jÞ ¼ 2d

vqðkÞ 0 dt vt0

0

bðt  t Þ

vsmðkÞð jÞ vt0

dt0

(10.41b)

0

The time-dependent peridynamic parameters ak ðtÞ, am ðtÞ, and bðtÞ depend on time-dependent bulk modulus kðtÞ and shear modulus mðtÞ which can be expressed in the form of Prony series as kðtÞ ¼ kN þ

N X

ki et=si

(10.42a)

mi et=si

(10.42b)

i¼1

and mðtÞ ¼ mN þ

N X i¼1

where kN and mN represent asymptotic values of bulk and shear moduli, N is the number of terms in Prony series, and si is the relaxation time. If the bulk modulus is independent of time, the dilatational and distortional components of the peridynamic force can be expressed as 2dak d qðkÞ tkðkÞð jÞ ¼   xð jÞ  xðkÞ 

(10.43a)

and Zt tmðkÞð jÞ ¼

2dbN 0

vsmðkÞð jÞ vt0

0

dt þ

N X

Zt bi

i¼1

0

2deðtt Þ=si

vsmðkÞð jÞ vt0

dt0

(10.43b)

0

Eq. (10.43b) can also be written as tmðkÞð jÞ ¼ 2db0 smðkÞð jÞ 

N X i¼1

ðkÞð jÞ

2dbi gi

I. New concepts in peridynamics

ðtÞ.

(10.44)

216

10. Modeling inelasticity in peridynamics

ðkÞð jÞ

where gi

ðtÞ is the viscous stretch, ðkÞð jÞ gi ðtÞ ¼

Zt h i vsm ðkÞð jÞ 0 ðtt0 Þ=si dt 1e vt0

(10.45a)

0

and b0 ¼ bN þ

N X

bi

(10.45b)

i¼1

To perform numerical time integration, viscous stretch can be written in discrete form as h i ðkÞð jÞ ðkÞð jÞ gi ðtnþ1 Þ ¼ 1  eDt=si smðkÞð jÞ þ DsmðkÞð jÞ þ eDt=si gi ðtn Þ (10.46) h i Dsm ðkÞð jÞ  si 1  eDt=si Dt where Dt represents the time increment. As Dt=si /0, the viscous stretch term given in Eq. (10.46) can be rewritten as     Dsm Dt ðkÞð jÞ Dt ðkÞð jÞ ðkÞð jÞ ðkÞð jÞ m gi ðtnþ1 Þ ¼ gi ðtn Þ þ ðtn Þ þ sðkÞð jÞ  gi si si 2 (10.47) Failure criterion for an interaction between two material points xðkÞ and xð jÞ can be expressed as GðkÞð jÞ  GcðkÞð jÞ

(10.48)

with GcðkÞð jÞ ¼

Gc Nc

(10.49)

where Gc represents the critical strain energy release rate and Nc is the total number of interactions creating a unit crack surface. Energy release rate of the interaction can be calculated as GðkÞð jÞ ¼

1 w V V ðDxÞh ðkÞð jÞ ðkÞ ð jÞ

I. New concepts in peridynamics

(10.50)

217

4. Numerical results

where Dx is the grid size, h is the thickness, VðkÞ and Vð jÞ represent the volumes of the material points xðkÞ and xð jÞ , respectively. The viscoelastic micropotential, wðkÞð jÞ , can be obtained as þ sðkÞð Z jÞ ðt¼0 Þ

wðkÞð jÞ ðtÞ ¼

sðkÞð Z jÞ ðtÞ

t0þ ðkÞð jÞ jxj xk jdsðkÞð jÞ þ 0

tðkÞð jÞ ðtÞjxj xk jdsðkÞð jÞ sðkÞð jÞ

ðt¼0þ Þ

(10.51) where t0þ ðkÞð jÞ is the instantaneous elastic force density.

4. Numerical results 4.1 Plate under tensile loading In order to demonstrate the applicability of PD theory for plastic material behavior, a plate under tension is considered as shown in Fig. 10.2. The length and width of the plate are specified as L ¼ 1 m and W ¼ 1 m, respectively. The plate is made of AISI 1020 steel with Young’s Modulus of E ¼ 209:322GPa, Poisson’s ratio of n ¼ 0:3, and Yield Stress of sy ¼ 264MPa. The isotropic nonlinear hardening model is used to characterize the stress-strain relationship in the peridynamic model.

FIGURE 10.2

Plate under tension.

I. New concepts in peridynamics

218

10. Modeling inelasticity in peridynamics

5.E+08

Stress (Pa)

4.E+08 3.E+08 2.E+08 experiment

1.E+08 0.E+00

PD 0

0.05

0.1 0.15 Strain (m/m)

0.2

0.25

FIGURE 10.3 Comparison of peridynamic and experimental results for stress-strain curve of AISI 1020 steel.

The model is created by using a discretization size of Dx ¼ 0:02 m and the horizon size of d ¼ 3Dx. The boundary conditions are specified as: uy ðx; y ¼ LÞ ¼ U

(10.52a)

uy ðx; y ¼ 0Þ ¼ 0

(10.52b)

In order to implement these boundary conditions in PD theory, a fictitious region is created along the boundary with a size of horizon d (Madenci and Oterkus, 2016). As shown in Fig. 10.3, there is a very good agreement between the peridynamic predictions and experimental measurements by Netto et al. (2005). The peridynamic model successfully captures the nonlinear material behavior.

4.2 Plate with a pre-existing crack under tensile loading A plate with a pre-existing central crack under tension is considered in order to demonstrate the applicability of peridynamic viscoelastic formulation. As shown in Fig. 10.4, the length and width of the plate are specified as L ¼ 0:3 m and W ¼ 0:15 m, respectively. Its thickness is specified as h ¼ 0:003 m and the crack length is 2a ¼ 0:03 m. The bulk response KðtÞ of the plate is assumed as elastic and the shear response GðtÞ is considered as viscoelastic. They are specified as KðtÞ ¼ K

(10.53a)

and GðtÞ ¼ GN þ

N X

Gi e

st

i

i¼1

I. New concepts in peridynamics

(10.53b)

219

5. Conclusions

FIGURE 10.4

A plate with a pre-existing crack under tension.

TABLE 10.1 Prony coefficients. i

si

1

1.00E-04

3.51

2

1.00E-02

0.05

3

3.16E-01

14.40

EN

Ei (GPa)

182.04

The coefficients of the Prony series are given in Table 10.1. It has a Poisson’s ratio of n ¼ 0:4. The peridynamic model is created by using a discretization size of Dx ¼ 0:003 m and horizon size of d ¼ 3Dx. The time step size is chosen as Dt ¼ 0:1 s. The boundary conditions are specified as: sxx ðx ¼ L = 2; yÞ ¼ so ¼ 2.2222  105 Pa

(10.54a)

uy ðx ¼  L = 2; yÞ ¼ 0

(10.54b)

The peridynamic predictions are compared against finite element analysis results obtained using ANSYS, a commercial finite element software. As shown in Figs. 10.5 and 10.6, they agree well with ANSYS results for both horizontal and vertical displacements at t ¼ 5 s.

5. Conclusions In this chapter, peridynamic plasticity and viscoelasticity formulations are presented. Both formulations accurately capture the nonlinear material behavior. Their accuracy was established by considering a metallic plate under tension with and without a pre-existing crack. In the case of

I. New concepts in peridynamics

220

10. Modeling inelasticity in peridynamics

FIGURE 10.5

PD results at t ¼ 5 s (A) horizontal (B) vertical displacements.

FIGURE 10.6 ANSYS results at t ¼ 5 s (A) horizontal (B) vertical displacements.

plastic deformation, the peridynamic predictions are compared with the published experimental measurements. In the case of viscoelastic deformation, the peridynamic predictions are compared with the finite element results.

References Amani, J., Oterkus, E., Areias, P., Zi, G., Nguyen-Thoi, T., Rabczuk, T., 2016. A non-ordinary state-based peridynamics formulation for thermoplastic fracture. Int. J. Impact Eng. 87, 83e94. Chen, W., Zhu, F., Zhao, J., Li, S., Wang, G., 2018, February. Peridynamics-based fracture animation for elastoplastic solids. Comput. Graph. Forum 37 (No. 1), 112e124. Delorme, R., Tabiai, I., Lebel, L.L., Le´vesque, M., 2017. Generalization of the ordinary statebased peridynamic model for isotropic linear viscoelasticity. Mech. Time-Dependent Mater. 21 (4), 549e575.

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References

221

Dorduncu, M., Barut, A., Madenci, E., 2016. Peridynamic truss element for viscoelastic deformation. In: 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 1721. Foster, J.T., Silling, S.A., Chen, W.W., 2010. Viscoplasticity using peridynamics. Int. J. Numer. Methods Eng. 81 (10), 1242e1258. Gu, X., Zhang, Q., Madenci, E., 2019. Non-ordinary state-based peridynamic simulation of elastoplastic deformation and dynamic cracking of polycrystal. Eng. Fract. Mech. 218, 106568. Lammi, C.J., Vogler, T.J., 2014. A Nonlocal Peridynamic Plasticity Model for the Dynamic Flow and Fracture of Concrete. Report SAND2014-18257, Sandia National Laboratories. Liu, Z., Bie, Y., Cui, Z., Cui, X., 2020. Ordinary state-based peridynamics for nonlinear hardening plastic materials’ deformation and its fracture process. Eng. Fract. Mech. 223, 106782. Luo, J., Ramazani, A., Sundararaghavan, V., 2018. Simulation of micro-scale shear bands using peridynamics with an adaptive dynamic relaxation method. Int. J. Solid Struct. 130, 36e48. Madenci, E., Oterkus, S., 2016. Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening. J. Mech. Phys. Solid. 86, 192e219. Madenci, E., Oterkus, S., 2017. Ordinary state-based peridynamics for thermoviscoelastic deformation. Eng. Fract. Mech. 175, 31e45. Mitchell, J.A., 2011a. A Nonlocal Ordinary State-Based Plasticity Model for Peridynamics (No. SAND2011-4974C). Sandia National Lab.(SNL-NM), Albuquerque, NM (United States). Mitchell, J.A., 2011b. A non-local, ordinary-state-based viscoelasticity model for peridynamics. Sandia Nat. Lab Rep. 8064, 1e28. Netto, T.A., Ferraz, U.S., Estefen, S.F., 2005. The effect of corrosion defects on the burst pressure of pipelines. J. Constr. Steel Res. 61 (8), 1185e1204. Pathrikar, A., Rahaman, M.M., Roy, D., 2019. A thermodynamically consistent peridynamics model for visco-plasticity and damage. Comput. Methods Appl. Mech. Eng. 348, 29e63. Rahaman, M.M., Roy, P., Roy, D., Reddy, J.N., 2017. A peridynamic model for plasticity: micro-inertia based flow rule, entropy equivalence and localization residuals. Comput. Methods Appl. Mech. Eng. 327, 369e391. Silling, S.A., 2019. Attenuation of waves in a viscoelastic peridynamic medium. Math. Mech. Solid 24 (11), 3597e3613. Sun, S., Sundararaghavan, V., 2014. A peridynamic implementation of crystal plasticity. Int. J. Solid Struct. 51 (19e20), 3350e3360. Weckner, O., Mohamed, N.A.N., 2013. Viscoelastic material models in peridynamics. Appl. Math. Comput. 219 (11), 6039e6043.

I. New concepts in peridynamics

C H A P T E R

11 Kinematically exact peridynamics Ali Javili1, Andrew McBride2, Paul Steinmann2, 3 1

Department of Mechanical Engineering, Bilkent University, Ankara, Turkey; 2 Glasgow Computational Engineering Centre, School of Engineering, University of Glasgow, Glasgow, United Kingdom; 3 Institute of Applied Mechanics, Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg, Erlangen, Germany

O U T L I N E 1. Introduction

224

2. Kinematics

225

3. Governing equations 3.1 Internal potential energy 3.1.1 One-neighbor interactions 3.1.2 Two-neighbor interactions 3.1.3 Three-neighbor interactions 3.2 External potential energy 3.3 Equilibrium

227 227 228 230 232 235 236

4. Computational implementation

237

5. Harmonic potentials

238

6. Examples

240

7. Conclusion

243

References

243

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00016-0

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© 2021 Elsevier Inc. All rights reserved.

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1. Introduction Peridynamics (PD) is an alternative approach to formulate continuum mechanics (Silling, 2000), the roots of which can be, in part, traced back to the pioneering works of Piola (Dell’Isola et al., 2015). As a nonlocal theory, the behavior of each material point is dictated by its interactions with other material points in its vicinity. In contrast to classical continuum mechanics, the integro-differential governing equations of PD are appropriate for problems involving discontinuities such as cracks. Since PD inherently accounts for geometrical discontinuities, it is readily employed in fracture mechanics and related problems (Kilic and Madenci, 2009; Foster et al., 2011; Silling et al., 2010; Ren et al., 2016; Agwai et al., 2011; Dipasquale et al., 2014; Chen and Bobaru, 2015; Diyaroglu et al., 2016). However, the range of PD applications is broad and not limited to fracture mechanics. Various applications and extensions of PD other than those dealing exclusively with material failure include (Ostoja-Starzewski et al., 2013; Breitenfeld et al., 2014; Oterkus et al., 2014; Bobaru et al., 2009; Rahman and Foster, 2014; Zaccariotto et al., 2018; Silling and Bobaru, 2005; O’Grady and Foster, 2014; Taylor and Steigmann, 2015; Aguiar and Fosdick, 2014; Tupek and Radovitzky, 2014; Madenci and Oterkus, 2017; Silling, 2017; Rahaman et al., 2017; Lejeune and Linder, 2017; Butt et al., 2017), among many others. For a brief description of PD together with a review of its applications and related studies in different fields to date, see (Javili et al., 2019b). The original PD theory of Silling (2000) was restricted to bond-based interactions. The bond-based assumption of PD limited its applicability for material modeling, including the inability to account for Poisson’s ratio other than 1/4 for isotropic materials. This shortcoming was addressed in various contributions and rectified in Silling et al. (2007) via the introduction of the notion of state and categorizing interactions as bond-based and state-based, see also Silling and Lehoucq (2010). The goal here is to elaborate on the kinematically exact formulation of continuum kinematicse inspired peridynamics (CPD) proposed by Javili et al. (2019a). CPD provides an alternative PD formulation whose underlying concepts are analogous to classical continuum mechanics. In particular, the interaction potential is composed of three parts accounting for one-neighbor interactions, two-neighbor interactions, and three-neighbor interactions within the horizon. In contrast to Javili et al. (2019a), the formulation and derivations are presented here in a discretized form from the onset, thereby providing an accessible framework that can be readily adopted for computational implementation.

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2. Kinematics

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2. Kinematics Central to our formulation is the geometrically exact description of kinematics inspired by classical continuum mechanics. Consider deformation of a continuum body, as illustrated in Fig. 11.1. The continuum body occupies the material configuration B 0 3 R3 at time t ¼ 0 that is mapped to the spatial configuration Bt 3R3 via the nonlinear deformation map y as x ¼ yðX; tÞ: B 0  Rþ /B t ; with X and x identifying points in the material and spatial configurations, respectively. In contrast to standard local continuum mechanics, the nonlocality assumption dictates that any point X in the material configuration can interact with other points within its finite neighborhood H 0 ðXÞ : The neighborhood H 0 is referred to as the material horizon. The measure of the horizon in the material configuration is denoted d and is generally the radius of a spherical neighborhood at X. In Fig. 11.1 the material and spatial horizons H 0 and H t , respectively, are shown as spheres for simplicity. The horizons H 0 and H t coincide with the points X and x in the limit of an infinitesimal neighborhood. Another key

FIGURE 11.1 Motion of a continuum body within the CPD formulation. The continuum body that occupies the material configuration B0 3 R3 at time t ¼ 0 is mapped to the spatial configuration Bt 3 R3 via the nonlinear deformation map y. The neighborhood of X is mapped to the neighborhood of x.

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characteristic of CPD is that the kinematics of the local continuum mechanics formalism are recovered for infinitesimal neighborhood ( Javili et al., 2019a). Recall the three key elements of classical continuum mechanics that describe the kinematics, i.e., the local kinematics measures of relative deformation: deformation gradient : F:¼ Grad y; cofactor of F: K:¼ Cof F; determinant of F:

(11.1)

J:¼ Det F;

mapping differential line elements, area elements, and volume elements, respectively. Inspired by these local measures, we introduce three nonlocal kinematic measures of relative deformation that underpin CPD. We derive and satisfy the governing equations at each collocation point identified with its coordinates X 0 and x0 in the material and spatial configurations, respectively. Of particular importance in our formulation is
 the neighbor set X 1 ; X 2 ; X 3 of the collocation point X0 identified as
1 2 3       X ;X ;X cX 1 ˛ H 0 X 0 ; X 2 ˛H 0 X 0 ; X 3 ˛H 0 X 0 . Note that the neighbor set is not a set of only three neighbors, but it is a set of all the possible triplets of neighbors within the horizon. The neighbors X1, X2, and X3 are referred to as the first, second, and third neighbors of the collocation point X 0 , respectively, and are mapped to their spatial counterparts via the nonlinear deformation map y. That is       x1 ¼ y X 1 ; t ; x2 ¼ y X 2 ; t ; x3 ¼ y X 3 ; t . The relative positions between a point and its neighbors are denoted by X{,} and x{,} in the material and spatial configurations, respectively, where the superscript {,} identifies the pair. That is X01 : ¼ X 1  X 0 ;

x01 : ¼ x1  x0 ;

X02 : ¼ X 2  X 0 ;

x02 : ¼ x2  x0 ;

03

3

0

X : ¼X X ;

x03 : ¼ x3  x0 .

Next, in the spirit of the local measures Eq. (11.1), we introduce three nonlocal kinematic measures of relative deformation. The first relative deformation measure x01 mimics the linear map F of the infinitesimal line element dX in the material configuration to its spatial counterpart dx. In view of our proposed formalism, the relative measure x01 ¼ x1  x0 ;

(11.2)

is the main ingredient to describe one-neighbor interactions and in the infinitesimal limit the relation x01 ¼ F$X01 holds (Javili et al., 2019a). The

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3. Governing equations

second relative deformation measure a012 is reminiscent of the linear map K from the infinitesimal vectorial area element dA in the material configuration to its spatial counterpart da. In our proposed framework, the relative (area) measure  
 a012 ¼ x1  x0  x2  x0 ; (11.3) is the main ingredient to describe two-neighbor interactions. In the infinitesimal limit, the relation a012 ¼ K$A012 is recovered ( Javili et al., 2019a). The third relative deformation measure v0123 mimics the linear map J of the infinitesimal volume element dV in the material configuration to its spatial counterpart dv. The relative (volume) measure       v0123 ¼ x1  x0  x2  x0 $ x3  x0 ; (11.4) is the main ingredient to describe three-neighbor interactions. Similarly, the relation v0123 ¼ J V 0123 is obtained in the infinitesimal limit ( Javili et al., 2019a).

3. Governing equations In order to obtain the governing equations, the total potential energy functional is minimized. This is done by setting the first variation of the total potential energy functional to zero. The total potential energy functional P that we seek to minimize with respect to all admissible (spatial) variations dy at a fixed material placement is composed of internal and external contributions J and Y, respectively. That is P¼JþY

0 dP ¼ 0

cdy;

(11.5)

We emphasize that the current discussion on the variational setting is for nondissipative processes.

3.1 Internal potential energy The internal potential energy functional J itself is composed of the internal potential energy due to one-neighbor interactions J1, twoneighbor interactions J2, and three-neighbor interactions J3. That is j ¼ j1 þ j2 þ j3 ; where the subscript indicates the type of interaction. Furthermore, we define the point-wise internal energy densities per volume in the material

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configuration j01 ; j02 and j03 associated with one-neighbor, two-neighbor, and three-neighbor interaction energies, respectively. That is Z Z Z 0 0 0 0 (11.6) J1 ¼ j1 dV ; J2 ¼ j2 dV ; J3 ¼ j03 dV 0 . B0

B0

B0

Equipped with the definitions of the point-wise internal energy densities, their contributions to the internal potential energy, or rather their variations, due to one-, two-, and three-neighbor interaction energies are explored next in Sections 3.1.1e3.1.3, respectively. 3.1.1 One-neighbor interactions The energy density per volume in the material configuration due to oneneighbor interactions J01 is a point-wise quantity. That is J01 is obtained via the integration of neighbor-wise energy densities. Let J01 1 denote the oneneighbor interaction energy density per volume squared in the material configuration. Note that while J01 is an energy density per volume, J01 1 is an energy density per volume squared. The number of superscripts identifies the nature of the interaction densities. One superscript corresponds to the familiar energy density per volume and two superscripts are associated with energy density per volume squared. As will be seen in the following sections, this notation avoids possible confusion. Consequently, the one-neighbor interaction energy density per volume 0 J1 reads Z 1 0 j1 ¼ j01 dV 1 . 2 H0

The factor one-half prevents double-counting since the combinations of {0,1} will eventually contribute twice to J1, see Eq. (11.6). Next, the variation of J1 can be expressed as Z dJ1 ¼ d j01 dV 0 B0

Z ¼d B0

1 2

Z 1 0 j01 1 dV dV

(11.7)

H0

Z Z 1 0 j01 1 dV dV ;

¼ B0 H 0

where the factor one-half disappears due to the variation rules on multiple integrals.

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The one-neighbor interaction energy density J01 1 is expressed in terms of the first relative deformation measure x01, see Eq. (11.2). That is   01 01 j01 . (11.8) 1 ¼ j1 x The description (11.8) does not fulfill material frame indifference in its current form. That is, the energy density is not invariant with respect to superposed rigid body motions. To sufficiently satisfy material frame indifference, we express j01 1 in terms of the objective scalar measure |x01| as   01  01  . j01 x 1 ¼ j1 The variation of J1 thus yields Z Z 01 1 0 dJ1 ¼ p01 1 $dx dV dV ;

(11.9)

B0 H 0

where we define p01 1 as the force density per volume squared due to oneneighbor interaction energies by p01 1 :¼

vj01 1 vx01

.

Note, p01 1 has two superscripts and is therefore a (force) density per volume squared. Since x01 ¼ y1  y0, the identity dx01 ¼ dy1  dy0 ; holds and therefore Z Z Z 1 1 0 0 0 dJ1 ¼ p01 $dy dV dV  bint 01 1 $dy dV 1 B0 H 0

(11.10)

B0

0

with int b1 the internal force density per volume in the material configuration due to one-neighbour interaction energies defined by Z int 0 1 b1 : ¼ p01 1 dV H0 0

Note that we identify int b1 as an internal force density, since it is an energy conjugated quantity to dy0. As will be demonstrated shortly, the counterparts of Eq. (11.10) for two- and three-neighbor interaction energies are obtained via a formally similar process. The intermediate steps for two- and three-neighbor interactions, however, are slightly more involved.

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3.1.2 Two-neighbor interactions The energy density per volume in the material configuration due to two-neighbor interactions j02 is a point-wise quantity composed of the denote the twointegration of neighbor-wise energy densities. Let j012 2 neighbor interaction energy density per volume cubed in the material configuration. According to our convention, the number of superscripts identifies the nature of the interaction densities. That is, one superscript corresponds to the familiar energy density per volume and three superscripts are associated with energy density per volume cubed. Consequently, the two-neighbor interaction energy density per volume j02 reads Z Z 1 2 1 j012 j02 ¼ 2 dV dV ; 3 H0 H0

where the factor one-third prevents triple-counting since the combination of {0,1,2} will eventually contribute three times to J2, see Eq. (11.6). Next, the variation of J2 is expressed as Z Z Z Z 1 0 0 2 1 0 j012 dJ2 ¼ j2 dV ¼ d 2 dV dV dV . 3 B0

B0

H0 H0

Due to the variation rules on multiple integrals, the factor one-third disappears and eventually we obtain Z Z Z 2 1 0 dj2 ¼ j012 (11.11) 2 dV dV dV . B0 H 0 H 0

The two-neighbor interaction energy density j012 2 is expressed in terms of the second relative deformation measure a012, see Eq. (11.3), as  012  j012 ¼ j012 a . (11.12) 2 2 Again, the description Eq. (11.12) does not fulfill material frame indifference. To sufficiently satisfy material frame indifference, we 012 | instead of a012. That express j012 2 in terms of the objective measure |a is  012  a  . j012 ¼ j012 (11.13) 2 2 The variation of Eq. (11.12) reads dj012 2 ¼

vj012 2 $da012 ; va012

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3. Governing equations

231

and since a012 ¼ x01  x02, it follows that dj012 2 ¼

 01   vj012  vj012 02 2 2 $ dx  x $ x01  dx02 . þ 012 va012 va

(11.14)

Inserting Eq. (11.14) into Eq. (11.11) yields Z Z Z   vj012 2 $ dx01  x02 dV 2 dV 1 dV 0 þ dJ2 ¼ 012 va B0 H 0 H 0

Z Z Z B0 H 0 H 0

  vj012 2 $ x01  dx02 dV 2 dV 1 dV 0 . 012 va

To proceed, we exchange the order of integrations and the relative measure of deformation in the second term and relabel the quantities, which yields Z Z Z   vj012 2 dJ2 ¼ $ dx01  x02 dV 2 dV 1 dV 0  012 va B0 H 0 H 0

Z Z Z B0 H 0 H 0

  vj012 2 $ x01  dx02 dV 2 dV 1 dV 0 . 012 va

  012 a012  in Eq. (11.13), the property Due to the definition j012 2 ¼ j2 vj012 vj012 2 2 ¼  021 ; 012 va va holds leading to Z Z Z dj2 ¼

2 B0 H 0 H 0

or alternatively

  vj012 2 $ dx01  x02 dV 2 dV 1 dV 0 ; 012 va

Z Z 01 1 0 p01 2 $dx dV dV ;

dJ2 ¼

(11.15)

B0 H 0

where the force density per volume squared due to two-neighbor interactions p01 2 is defined by Z vj012 2 p01 ¼ 2x02  012 dV 2 . 2 va H0

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Since dx01 ¼ dy1  dy0, Eq. (11.15) can be expressed as Z Z Z int 0 1 1 0 dJ2 ¼ p01 $dy dV dV  b2 $dy0 dV 0 ; 2 B0 H 0

(11.16)

B0

0

with int b2 the internal force density per volume in the material configuration due to two-neighbor interactions defined by Z int 0 1 b2 : ¼ p01 2 dV H0 0

0

Note that, similar to int b1 , we identify int b2 as an internal force density, since it is an energy-conjugated quantity to dy0. 3.1.3 Three-neighbor interactions The point-wise energy density per volume in the material configuration due to three-neighbor interactions j03 is composed of the integration denote the three-neighbor of neighbor-wise energy densities. Let j0123 3 interaction energy density per volume to the fourth power in the material configuration. Recall our convention that one superscript corresponds to the familiar energy density per volume and four superscripts are associated with an energy density per volume to the fourth power. Consequently, the three-neighbor interaction energy density per volume j03 reads Z Z Z 1 j0123 dV 3 dV 2 dV 1 ; j03 ¼ 3 4 H0 H0 H0

where the factor one-fourth is introduced to prevent quadruple-counting due to the combinations of {0,1,2,3} that contribute four times to J3, see Eq. (11.6). Next, the variation of J3 is expressed as Z dJ3 ¼ d j03 dV 0 Z ¼d B0

1 4

Z Z Z

B0

j0123 dV 3 dV 2 dV 1 dV 0 . 3 H0 H0 H0

Due to the variation rules on multiple integrals, the factor one-fourth disappears and yields Z Z Z Z dJ3 ¼ dj0123 dV 3 dV 2 dV 1 dV 0 . (11.17) 3 B0 H 0 H 0 H 0

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3. Governing equations

233

The three-neighbor interaction energy density j0123 is expressed in 3 0123 , see Eq. (11.4). That is, terms of the third relative deformation measure v  0123  v . j0123 ¼ j0123 3 3

(11.18)

As before, since Eq. (11.18) does not fulfill material frame indifference, we express j0123 in terms of the objective measure |v0123| instead of v0123. 3 That is,  0123   . v (11.19) j0123 ¼ j0123 3 3 The variation of Eq. (11.18) reads

dj0123 ¼ 3 and the definition

dj0123 3 dv0123 ; dv0123

  v0123 ¼ x01  x02 $x03 furnishes dj0123 ¼ 3

 01  dj0123 3 dx  x02 $x03 0123 dv

þ

 01  dj0123 3 x  dx02 $x03 0123 dv

þ

 dj0123 3 dv

0123

 x  x02 $dx03 .

Inserting Eq. (11.20) into Eq. (11.17) yields Z Z Z Z dj3 ¼ B0 H 0 H 0 H 0

Z Z Z Z þ B0 H 0 H 0 H 0

Z Z Z Z þ B0 H 0 H 0 H 0

 01  vj0123 3 dx  x02 $x03 dV 3 dV 2 dV 1 dV 0 0123 vv  01  vj0123 3 x  dx02 $x03 dV 3 dV 2 dV 1 dV 0 0123 vv  01  vj0123 3 x  x02 $dx03 dV 3 dV 2 dV 1 dV 0 . 0123 vv

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(11.20)

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To proceed, we exchange the order of integrations and the relative measure of deformation on the second and third terms, and relabel the quantities accordingly, to obtain Z Z Z Z  01  vj0123 3 dJ3 ¼ dx  x02 $x03 dV 3 dV 2 dV 1 dV 0 0123 vv B0 H 0 H 0 H 0

Z Z Z Z þ B0 H 0 H 0 H 0

Z Z Z Z þ B0 H 0 H 0 H 0

 02  vj0123 3 x  dx01 $x03 dV 3 dV 2 dV 1 dV 0 0213 vv  03  vj0123 3 x  x02 $dx01 dV 3 dV 2 dV 1 dV 0 ; 0321 vv

which can be expressed alternatively as Z Z Z Z  01  vj0123 3 dJ3 ¼ dx  x02 $x03 dV 3 dV 2 dV 1 dV 0 0123 vv B0 H 0 H 0 H 0

Z Z Z Z  B0 H 0 H 0 H 0

Z Z Z Z  B0 H 0 H 0 H 0

 01  vj0123 3 dx  x02 $x03 dV 3 dV 2 dV 1 dV 0 0213 vv  01  vj0123 3 dx  x02 $x03 dV 3 dV 2 dV 1 dV 0 . 0321 vv

 0123   , the property v ¼ j0123 Due to the requirement j0123 3 3 vj0123 vj0123 vj0123 3 3 3 ¼  0213 ¼  0321 ; 0123 vv vv vv holds leading to Z Z Z Z  01  vj0123 3 dx  x02 $x03 dV 3 dV 2 dV 1 dV 0 ; dJ3 ¼ 3 0123 vv B0 H 0 H 0 H 0

or alternatively

Z Z 01 1 0 p01 3 $dx dV dV ;

dJ3 ¼ B0 H 0

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(11.21)

3. Governing equations

235

with the force density per volume squared due to three-neighbor interactions Z Z  vj0123  01 3 p3 : ¼ 3 x02  x03 dV 2 dV 3 . vv0123 H0 H0

¼ dy  dy0, Eq. (11.21) can be expressed as Z Z Z int 0 1 1 0 p01 $dy dV dV  b3 $dy0 dV 0 ; dJ3 ¼ 3

Since dx

01

1

B0 H 0

(11.22)

B0

0

with int b3 the internal force density per volume in the material configuration due to three-neighbor interactions defined by Z int 0 1 b3 : ¼ p01 3 dV . H0 0

Again, we identify int b3 as an internal force density, since it is an energy-conjugated quantity to dy0.

3.2 External potential energy The external potential energy consists of the contributions due to external loading applied to the system. The external loading can be divided into externally prescribed forces within the bulk and tractions on the surface of the body. We emphasize that externally prescribed tractions on the boundary of the body vB 0 are accommodated naturally in CPD. The variation of the external potential energy Y reads Z Z dY ¼  bext 0 $dy0 dV 0  t ext 0 $dy0 dA0 . (11.23) B0

vB 0

0

where ext b denotes the point-wise external force density per volume in the material configuration with dimension N/m3. The point-wise external traction on the boundary in the material configuration with dimension 0 N/m2 is denoted ext t . This format of the external potential energy follows from a more general treatment applicable to higher gradient and nonlocal continua, see (Javili et al., 2013; Auffray et al., 2015).

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3.3 Equilibrium To derive the equations governing equilibrium, we set the variation of the total potential energy P with respect to all admissible (spatial) variations dy at a fixed material placement to zero, see Eq. (11.5). That is dJ þ dY ¼ 0

cdy;

(11.24)

where dJ ¼ dJ1 þ dJ2 þ dJ3 is constructed from Eqs. (11.10), (11.16) and (11.22) as Z Z Z dj ¼ p01 $dy1 dV 1 dV 0  bint 0 $dy0 dV 0 ; With (11.25) B0 H 0 B0 0 0 0 0 01 01 p01 : ¼ p01 1 þ p2 þ p3 ; bint :¼ bint 1 þ bint 2 þ bint 3 ;

where p01 is the total internal force density per volume squared in the 0

material int b configuration. Similarly, b is the total internal force density 0

per volume in the int b material configuration. Note that b can alternatively be expressed as Z int 0 b ¼ p01 dV 1 . H0

Inserting the variation of internal and external potential energies, Eq. (11.25) and Eq. (11.23), respectively, into the equilibrium Eq. (11.24), yields Z Z Z Z Z 0 0 01 1 1 0 0 0 0 0 p $dy dV dV  bint $dy dV  bext $dy dV  t ext 0 $dy0 dA0 ¼ 0. B0 H 0

B0

B0

vB 0

(11.26) Due to the arbitrariness of dy ; the equilibrium Eq. (11.26) can be expressed as a balance of forces bint 0 þ bext 0 ¼ 0

cX 0 ;

together with the virtual power equivalence Z Z Z p01 $dy1 dV 1 dV 0  t ext 0 $dy0 dA0 ¼ 0: B0 H 0

(11.27)

(11.28)

vB 0

The continuous counterparts in classical continuum mechanics to the force balance Eq. (11.27) and the virtual power equivalence Eq. (11.28)

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4. Computational implementation

237

read Div P þ bext ¼ 0 cX˛B 0 and Z Z Divðdy$PÞdV  t ext $dydA ¼ 0; B0

vB 0

respectively, with P the Piola stress tensor. Note that the virtual power equivalence Eq. (11.28) is an underlying postulate of CPD since PD lacks the equivalent of Cauchy’s fundamental theorem that is used in combination with the Gauss theorem. If the external boundary is traction-free or if only displacement-type boundary conditions are prescribed, the virtual power equivalence Eq. (11.28) reduces to Z Z p01 $dy1 dV 1 dV 0 ¼ 0: B0

4. Computational implementation The underlying governing equation of CPD is the linear momentum balance Eq. (11.27). The angular momentum balance is a priori fulfilled if the internal potential energy densities satisfy material frame indifference or more precisely if  01  012  012  0123  0123  01 012 j01 ja j ; j3 ¼ j0123 jv j 1 ¼ j1 jx j ; j2 ¼ j2 3

(11.29)

For the force balance Eq. (11.27), externally prescribed body forces are omitted henceforth, since their incorporation into the framework is fairly straightforward and standard. Thus, our point-wise nonlocal equilibrium equation and the point of departure for this section reads Z 0 0 R ¼ 0 with R :¼ p01 dV 1 ; H0

where the point-wise residual vector for collocation point X0 is denoted as R0 and is composed of the contributions due to one-neighbor,

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two-neighbor, and three-neighbor interactions defined by R01 ; R02 ; and R03, respectively. That is R0 ¼ R01 þ R02 þ R03 ; with Z R01 :¼

Z

1 0 p01 1 dV ; R2 :¼ H0

Z 1 0 p01 2 dV ; R3 :¼

H0

1 p01 3 dV H0

After computing the point-wise residual vectors R0 at each collocation point X0, we assemble them into a global residual vector R. For nonlinear problems and large deformations, the deformation is computed incrementally. The nonlinear system of governing equations at each increment is expressed as R ¼ O for which an approximate solution is achieved via an interactive NewtoneRaphson scheme. To do so, we establish a consistent linearization of the resulting system of equations at iteration k. That is !

Rkþ1 ¼ O with

Rkþ1 ¼ Rk þ Kk $DXk ;

(11.30)

where Kk is the tangent at iteration k defined by  vR  ; Kk :¼ vXk k and X is the global deformation vector that consists of the pointwise deformation vectors of all collocation points. The update of deformation DX at iteration k is computed from Eq. (11.30). Then the global deformation vector X is updated at each iteration as Xkþ1 ¼ Xk þ DXk . This iterative process is continued until the norm of the residual vector R is sufficiently small. Further details on computational aspects of the proposed framework are given in (Javili et al., 2020a,b). The last step toward numerical implementation is to define interaction potentials. We particularize the generic form of interaction potentials Eq. (11.29) next.

5. Harmonic potentials In this section, we provide specific examples of harmonic potentials for one-neighbor, two-neighbor, and three-neighbor interactions such that the requirement of material frame indifference is fulfilled. To do so, the 012 and j0123 are expressed in terms of |x01|, interaction potentials j01 1 , j2 3 012 0123 |, respectively. Recall that we adopt a continuum |a |, and |v kinematicseinspired approach, where the deformation measures are common continuum descriptors, namely changes in length, area, and

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volume. This leads to an intuitive description of the interaction energy densities. To aid understanding, one-neighbor interaction energies can be viewed as the resistance against the change of length between a point and its neighbors, reminiscent of the elastic modulus in classical continuum mechanics. Two-neighbor interaction energies can be interpreted as the resistance against the change in the area of the triangle formed by a point and a pair of neighbors, analogous to Poisson-like effects of classical continuum mechanics in two dimensions. Finally, three-neighbor interaction energies are essentially the resistance against the change in the volume of the tetrahedron formed by each point and its triplet of neighbors, similar to Poisson-like effects in classical continuum mechanics in three dimensions. We define the elastic coefficients C1, C2, and C3 to characterize the resistance to the change of length, area, and volume, respectively. The proposed harmonic potentials read  

2 1  01  x01  01  01   1 ; J1 ¼ C1 X X  2 

2  1  012  a012  1 ; ¼ C2 A 2 jA012 j  

2 1  0123  v0123  0123 ; J3 ¼ C3 V  1 2 jV 0123 j J012 2

and hence

01  01  x vJ01 x 1 ¼ C1  01   1  01  ; 01    vx X x  

012   012  a012  vJ012 a 2    ;  1 ¼ C A 2 va012 ja012  jA012  

0123   0123  v0123  vJ0123 v 3     1 0123  . ¼ C3 V 0123 0123   vv jv jV In particular, the force density per volume squared due to oneneighbor interactions read

01  01  x vj01 x 01 1 p1 :¼ 01 ¼ C1  01   1  01  ; (11.31) X  x  vx

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11. Kinematically exact peridynamics

Which is nothing other than the force density vector of bond-based PD. This result shows that one-neighbor interactions in CPD recover exactly the bond-based PD formalism. However, two- and three-neighbor interactions are fundamentally different to state-based PD.

6. Examples The main goal of this section is to demonstrate the theory via a series of numerical examples. The numerical investigation is devised such that the key features of the framework are illustrated. The numerical examples to demonstrate elasticity at large deformations are carried out on a unit square domain with and without a square hole at its center subject to uniaxial extension, as shown in Fig. 11.2. For the three-dimensional examples, we consider a slab of a similar cross-section and thickness of 0.1 subject to plane deformations. The uniaxial extension is imposed on the domain via prescribed displacements on its left and right edges, and the upper and lower edges remain free of external loadings. An extension of 100% is prescribed in the horizontal direction. The extension is imposed by prescribing horizontal displacements on the left and right edges while vertical displacements are not allowed. The parameter C1 is fixed throughout and the ratios C2/C1 and C3/C1 are varied such that the influence of two- and three-neighbor interactions is clearly observed. The domain is discretized uniformly with grid spacing of D ¼ 0.02; that is 50 grid points on the edges. The collocation points coincide with the grid points and the measure of the horizon is d z 3D. Fig. 11.3 shows the deformation of the domain in the absence of twoneighbor interactions, with and without a square hole in the middle of the specimen. As we have only one-neighbor interactions, the deformation corresponds to Poisson’s ratio of 1/3 and thus the material is compressible. This situation coincides with bond-based peridynamics formulation. The color bars show the displacement in the vertical

FIGURE 11.2

Unit square without and with a square hole subject to uniaxial extension.

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FIGURE 11.3 Extension of a unit square without (left) and with (right) a hole with oneneighbor interactions. The material behavior is compressible and associated with Poisson’s ratio of 1/3.

direction. Furthermore, the convergence behavior associated with the NewtoneRaphson scheme at various intermediate increments is given. The numbers in the convergence boxes are the L2-norm of the normalized residual at a NewtoneRaphson iteration. It can be seen that the quadratic rate of convergence associated with a NewtoneRaphson scheme is obtained. To account for Poisson’s ratio other than 1/3, two-neighbor interactions are introduced next by prescribing C2/C1, 0. A larger value. Figs. 11.5 and 11.6 illustrate the evolution of the deformation for a slab without and with a hole at its center, respectively. In both examples, displacements in thickness direction are prevented. The computational simulations are carried out for both compressible and nearly incompressible materials with C3/C1 ¼ 0 and C3/C1 ¼ 107, respectively. The nearly incompressible material shown in the lower half of the figure resists considerably the change of volume and thus contracts significantly in the lateral direction. In addition, in Fig. 11.6, the hole expands far more than that for the compressible material shown above.

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FIGURE 11.4 Extension of a unit square without (left) and with (right) a hole with oneneighbor and pronounced two-neighbor interactions. The material behavior is nearly incompressible and associated with Poisson’s ratio of nearly 1 of C2/C1 leads to a higher resistance to area change. That is, in the limit of C2/C1 / N the material behavior will be incompressible. Fig. 11.4 shows the deformation of the domain in the presence of pronounced two-neighbor interactions corresponding to C2/C1 ¼ 107. That is, Poisson’s ratio is nearly 1 and thus the material is nearly incompressible resulting in a more pronounced lateral contraction so as to maintain the initial area. Again, the color bars show the displacement in the vertical direction and the quadratic rate of convergence associated with a NewtoneRaphson is obtained. All the computational simulations are carried out using our in-house code.

FIGURE 11.5

Square slab subject to uniaxial extension at large deformations for compressible (top) and nearly incompressible (bottom) material. The compressible and nearly incompressible deformations are associated with C3/C1 ¼ 0 and C3/C1 ¼ 107, respectively.

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FIGURE 11.6 Perforated square slab subject to uniaxial extension at large deformations for compressible (top) and nearly incompressible (bottom) material. The compressible and nearly incompressible deformations are associated with C3/C1 ¼ 0 and C3/C1 ¼ 107, respectively.

7. Conclusion Continuum kinematicseinspired peridynamics (CPD) was recently proposed by Javili et al. (Javili et al., 2020b) as a kinematically exact alternative to state-based peridynamics to formulate nonlocal continuum mechanics at finite deformations. In this manuscript the problem, formulation, and derivation of CPD have been presented together with numerical examples to illustrate the key features of the framework. The proposed methodology is fully implicit and the solution procedure shows the asymptotically quadratic rate of convergence associated with a NewtoneRaphson scheme. In summary, this contribution presents a kinematically exact formulation of peridynamics in a variationally consistent framework. We believe that this generic approach is broadly applicable to enhance understanding of material behavior for a large variety of applications in multi-field problems accounting for geometrical discontinuities.

References Aguiar, A.R., Fosdick, R., 2014. A constitutive model for a linearly elastic peridynamic body. Math. Mech. Solid 19 (5), 502e523. Agwai, A., Guven, I., Madenci, E., 2011. Predicting crack propagation with peridynamics: a comparative study. Int. J. Fract. 171 (1), 65e78. Auffray, N., Dell’Isola, F., Eremeyev, V.A., Madeo, A., Rosi, G., 2015. Analytical continuum mechanics a´ la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solid 20 (4), 375e417.

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Bobaru, F., Yang, M., Alves, F., Silling, S.A., Askari, E., Xu, J., 2009. Convergence, adaptive refinement, and scaling in 1D peridynamics. Int. J. Numer. Methods Eng. 77 (6), 852e877. Breitenfeld, M.S., Geubelle, P.H., Weckner, O., Silling, S.A., 2014. Non-ordinary state-based peridynamic analysis of stationary crack problems. Comput. Methods Appl. Mech. Eng. 272, 233e250. Butt, S.N., Timothy, J.J., Meschke, G., 2017. Wave dispersion and propagation in state-based peridynamics. Comput. Mech. 60 (5), 1e14. Chen, Z., Bobaru, F., 2015. Peridynamic modeling of pitting corrosion damage. J. Mech. Phys. Solid. 78, 352e381. Dell’Isola, F., Andreaus, U., Placidi, L., 2015. At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solid 20 (8), 887e928. Dipasquale, D., Zaccariotto, M., Galvanetto, U., 2014. Crack propagation with adaptive grid refinement in 2D peridynamics. Int. J. Fract. 190 (1e2), 1e22. Diyaroglu, C., Oterkus, E., Madenci, E., Rabczuk, T., Siddiq, A., 2016. Peridynamic modeling of composite laminates under explosive loading. Compos. Struct. 144, 14e23. Foster, J., Silling, S.A., Chen, W., 2011. An energy based failure criterion for use with peridynamic states. Int. J. Multiscale Comput. Eng. 9 (6), 675e688. Javili, A., Dell’Isola, F., Steinmann, P., 2013. Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solid. 61 (12), 2381e2401. Javili, A., McBride, A.T., Steinmann, P., 2019a. Continuum-kinematics-inspired peridynamics. Mechanical problems. J. Mech. Phys. Solid. 131, 125e146. Javili, A., Morasata, R., Oterkus, E., Oterkus, S., 2019b. Peridynamics review. Math. Mech. Solids 24, 3714e3739. Javili, A., McBride, A.T., Steinmann, P., 2020a. A Geometrically Exact Formulation of Peridynamics, Theoretical and Applied Fracture Mechanics. Javili, A., Firooz, S., McBride, A.T., Steinmann, P., 2020b. The computational frame work for continuum kinematics-inspired peridynamics. Comput. Mech. https://doi.org/10.1007/ s00466e020e01885e3. Kilic, B., Madenci, E., 2009. Prediction of crack paths in a quenched glass plate by using peridynamic theory. Int. J. Fract. 156 (2), 165e177. Lejeune, E., Linder, C., 2017. Modeling tumor growth with peridynamics. Biomech. Model. Mechanobiol. 16 (4), 1141e1157. Madenci, E., Oterkus, S., 2017. Ordinary state-based peridynamics for thermos viscoelastic deformation. Eng. Fract. Mech. 175, 31e45. O’Grady, J., Foster, J., 2014. Peridynamic beams: a non-ordinary, state-based model. Int. J. Solid Struct. 51 (18), 3177e3183. Ostoja-Starzewski, M., Demmie, P.N., Zubelewicz, A., 2013. On thermodynamic restrictions in peridynamics. J. Appl. Mech. 80 (1), 014502. Oterkus, S., Madenci, E., Agwai, A., 2014. Fully coupled peridynamic thermomechanics. J. Mech. Phys. Solid. 64 (1), 1e23. Rahaman, M.M., Roy, P., Roy, D., Reddy, J.N., 2017. A peridynamic model for plasticity: micro-inertia based flow rule, entropy equivalence and localization residuals. Comput. Methods Appl. Mech. Eng. 327, 369e391. Rahman, R., Foster, J.T., 2014. Bridging the length scales through nonlocal hierarchical multiscale modeling scheme. Comput. Mater. Sci. 92, 401e415. Ren, H., Zhuang, X., Cai, Y., Rabczuk, T., 2016. Dual-horizon peridynamics. Int. J. Numer. Methods Eng. 108, 1541e1476. Silling, S.A., Bobaru, F., 2005. Peridynamic modeling of membranes and fibers. Int. J. Non Lin. Mech. 40 (2e3), 395e409. Silling, S.A., Lehoucq, R.B., 2010. Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73e168.

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Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E., 2007. Peridynamic states and constitutive modeling. J. Elasticity 88 (2), 151e184. Silling, S.A., Weckner, O., Askari, E., Bobaru, F., 2010. Crack nucleation in a peridynamic solid. Int. J. Fract. 162 (1e2), 219e227. Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solid. 48 (1), 175e209. Silling, S.A., 2017. Stability of peridynamic correspondence material models and their particle discretizations. Comput. Methods Appl. Mech. Eng. 322, 42e57. Taylor, M., Steigmann, D.J., 2015. A two-dimensional peridynamic model for thin plates. Math. Mech. Solid 20 (8), 998e1010. Tupek, M.R., Radovitzky, R., 2014. An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures. J. Mech. Phys. Solid. 65 (1), 82e92. Zaccariotto, M., Mudric, T., Tomasi, D., Shojaei, A., Galvanetto, U., 2018. Coupling of FEM meshes with peridynamic grids. Comput. Methods Appl. Mech. Eng. 330, 471e497.

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C H A P T E R

12 Modeling biological materials with peridynamics Emma Lejeune1, Christian Linder2 1

Department of Mechanical Engineering, Boston University, Boston, MA, United States; 2 Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, United States

O U T L I N E 1. Introduction

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2. Methodology 2.1 Background and notation 2.2 Implementing growth and remodeling 2.3 Note on emergent behavior

252 252 256 258

3. Example applications 3.1 Fracture in biological materials 3.2 Tissue growth and shrinkage 3.2.1 Cell division and tissue growth 3.2.2 Cell death and tissue shrinkage 3.3 Connecting emergent behavior across scales

264 264 264 265 265 266

4. Conclusion and outlook

268

Acknowledgments

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Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00005-6

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© 2021 Elsevier Inc. All rights reserved.

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1. Introduction From a mechanical modeling perspective, capturing the behavior of biological materials presents a compelling challenge for three main reasons. First, biologically derived materials often have a complex microstructure that can lead to mechanical response that is quite different from standard engineering materials (Meador et al., 2020). For example, shell composites such as nacre have been shown to have unprecedented fracture toughness in relation to their yield strength (Shao et al., 2012). Second, biological materials can adapt in response to their surroundings (Ambrosi et al., 2011). For example, heart muscle can grow and remodel in response to changes in blood pressure (Rausch et al., 2011). Third, both the material microstructure and adaptive response influence mechanical behavior on multiple scales (Ambrosi et al., 2016). Therefore, even when a phenomena is well understood on one scale, it is not necessarily clear how that behavior will link across scales or how mechanical information will traverse scales and trigger adaptation (Lejeune et al., 2019). The inherently interesting nature of these challenges combined with the potential benefits of understanding, predicting, and ultimately controlling the mechanical behavior of biological materials has motivated researchers to develop novel numerical methods, and extend modeling frameworks originally formulated for engineered materials (Rodriguez et al., 1994). Peridynamics, a theoretical and computational framework that is designed to unify the mechanics of discrete and continuous media, is a technique originally developed for fracture mechanics applications (Silling and Lehoucq, 2010). Rather than using partial differential equations to formulate the equations of motion, peridynamics uses integral equations which exist on crack surfaces (Silling, 2000; Silling et al., 2007). The first paper introducing peridynamics was published in 2000 and presents peridynamics as a methodology for modeling discontinuities and long range forces using a constitutive relation based on bond-based (i.e., pair-wise) interactions between particles (Silling, 2000). Since this original work, peridynamic theory has been developed numerically (Bobaru and Ha, 2011), extended to include more complex constitutive models (Silling et al., 2007; Warren et al., 2009), and applied to model a variety of engineered systems (Kilic and Madenci, 2010b). In the context of biological materials, peridynamics is a compelling method for capturing material fracture, for example, bone fracture (Deng et al., 2008; Ghajari et al., 2014), and for capturing material behavior where longrange forces are important, for example, in lipid membranes (Madenci et al., 2020). And, because the peridynamic framework deals comfortably with both continuous and discrete media, it is a compelling method for modeling biological tissue on the cell population scale where the material is, in reality, somewhere in-between (Lejeune and Linder, 2017a).

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Over the course of the past decade, peridynamics has gained traction as a method for modeling biological materials. The range of this recent research is shown in Fig. 12.1. Peridynamics has been used to model diverse phenomena ranging from fracture to aspects of growth and remodeling (Javili et al., 2019). In this chapter, we will discuss some of these recent advances with an emphasis on adapting the peridynamic framework to capture mechanical behavior on the cell population scale. We begin in Section 2 with a brief methodological background; we highlight some notable applications in Section 3, and then we conclude in Section 4. We anticipate that in the coming decade the flexible nature of the peridynamics framework will lead to further adoption in modeling biological materials.

FIGURE 12.1 Examples adapted from the literature of models of biological materials based on peridynamics. Clockwise from the upper left: agent-based cell modeling (Lejeune et al., 2019), understanding emergent behavior in the cerebellum (Lejeune et al., 2019), modeling inclusions in lipid membranes (Madenci et al., 2020), modeling cortical bone fracture (Deng et al., 2008), modeling rupture in lipid membranes (Taylor et al., 2016), modeling tumor growth (Lejeune and Linder, 2017a), modeling mechanical inhomogeneities in growing spheroids (Lejeune and Linder, 2018a), and modeling tumor shrinkage (Lejeune and Linder, 2020).

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2. Methodology Here, we cover basic methodological advances for modeling biological materials with peridynamics. First, in Section 2.1, we briefly review the fundamental equations of peridynamics. Then, in Section 2.2, with a consistent notation, we introduce extensions to the framework that enable modeling biological mechanisms such as cell growth, division, and death, and present the fundamental equations in their corresponding discrete form. Finally, in Section 2.3, we cover the concept of emergent behavior in biological systems and provide additional context for understanding the results demonstrated in Section 3.

2.1 Background and notation As stated in Section 1, peridynamics is a theoretical and computational framework where the classical balance equations are integrals rather than partial differential equations (Silling, 2000). Here we introduce the basic notation and equations, with key terms illustrated in Fig. 12.2. First, we introduce the concept of a horizon. In the peridynamic formulation, a given point x interacts with other points within its horizon H x where H x is a line (1D), circle (2D), or sphere (3D) defined by horizon size dx and written as H x ¼ fx0 j kx0  xk < dx g.

(12.1)

Physically, H x is defined as the domain where any particle will experience force exerted by x. The two-dimensional case is illustrated in Fig. 12.2. In addition to the horizon, we introduce the dual-horizon, which will allow for nonuniformity in horizon size across different points (Ren et al., 2016), specifically the case where some point x is within the horizon of x0 but x0 is not within the horizon of x (Bobaru et al., 2009; Bobaru and Ha, 2011). The dual horizon is defined as the union of points whose horizons include x, written as 0 
 (12.2) H x ¼ x 0  x ˛ H x0 . For all points x within H x0 , x0 acts on x. And, unlike the horizon, the dual horizon is not necessarily a circle or sphere. However, if the horizon 0 size dx does not vary across points, then H x ¼ H x and the dual-horizon formulation will be identical to that of conventional peridynamics. The equation of motion is formulated as an integral of interaction forces between points on the body U. Here we introduce the terminology required to define the balance of linear momentum. Material points in the initial configuration U0 are illustrated in Fig. 12.2A as x and x0 . The bond vector between x and x0 in the initial configuration is defined by the term

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FIGURE 12.2 Illustration of key notation: (A) illustration of the reference configuration mapped to the deformed configuration; (B) illustration of model implementation in the mesh-free setting.

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x ¼ x0  x. In the discrete setting, and in numerical implementation, point x is referred to as node j, point x0 is referred to as node k, and the bond between them is defined as xjk ¼ xk  xj . The displacement vector uðx; tÞ and position vector yðx; tÞ ¼ x þ uðx; tÞ, illustrated in Fig. 12.2A, are defined in the current configuration. The relative displacement of a bond is then defined as h ¼ u0  u, and described in the discrete setting as hjk ¼ uk  uj . Here we adopt the typical state notation where we define a state of order m as a function AC ,D that maps the vector in angle brackets C ,D to a tensor of order m (Silling et al., 2007). For example, the relative position of bond x can be written as yCxD ¼ yðx0 ; tÞ  yðx; tÞ ¼ x þ h. In this chapter, states are written with an underline, and angle brackets are used to indicate the quantity which the state function is acting on. In dual-horizon peridynamics, the force between points x and x0 is defined in two distinct steps, first using the dual horizon and then using the horizon. We define force density vector f xx0 ðh; xÞ as the force per volume acting on particle x due to particle x0 . Point x is the location of the force and x0 is the source of the force. Likewise, f x0 x ðh; xÞ is the force density vector acting on particle x0 due to particle x. Each force density, f xx0 acting on x, is then accompanied by a reaction force density, f xx0 acting on x0 . The direct force density at x is computed from points in the dual horizon of x and the reaction force density at x is computed from points in the horizon of x. Therefore, the net force density acting on point x due to bond xx0 is a sum of the direct force density and reaction force density written as f xx0 ðh; xÞ  f x0 x ðh;  xÞ.

(12.3)

And, the net force density acting on a point x0 due to bond xx0 is f x0 x ð  h; xÞ  f xx0 ðh; xÞ.

(12.4) 0 x

By accounting for the contributions from H and H x (and subsequently direct and reaction forces) separately, the antisymmetry of net force density is preserved in the case of variable horizon size. Given the kinematic description and definition of force density, we can define the balance of linear momentum (Silling, 2000). The inertial force, body force, and internal force terms at point x and time t are equated as Z Z € tÞ ¼ ruðx; f xx0 ðh; xÞdVx0  f x0 x ðh;  xÞdVx0 þ bðx; tÞ (12.5) 0

x0 ˛H x

x0 ˛H x

where r is density, u€ is acceleration, and b is body force (Ren et al., 2016). 0 Integrating over the dual horizon H x contributes the direct force term acting on point x, and integrating over the horizon H x contributes the reaction

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force term. The discrete form of the balance of linear momentum is then written as X  X    € j ; tÞ ¼ ruðx f jk hjk ; xjk DVk  f kj  hjk ;  xjk DVk þ bðxj ; tÞ 0

k˛H j

k˛H j

(12.6) where the integral in Eq. (12.5) is simply replaced by a summation. We note that in the numerical implementation of dual-horizon peridynamics it is not necessary to compute the dual horizon explicitly. This equation can be assembled by looping through the horizon of each node and subsequently inferring each node’s dual horizon (Ren et al., 2016). Next, we define the equation for force density f xx0 with a constitutive law based on ordinary state-based peridynamics (Silling et al., 2007). The qualification “state-based” comes from the state notation defined previously. In state-based peridynamics, bond force is a function of the collective deformation of all bonds that act on the same points as the bond in question. Peridynamics is a nonlocal theory meaning that nonadjacent points can interact. The degree to which nonlocal forces come into play is controlled by two parameters: the horizon size d and the influence function uCxD. The influence function can be chosen to weight the effect of certain bonds more heavily or it can be set to a constant. For example, uCxD can be equal to ! jjxk2 uCxD ¼ exp  2 (12.7) or uCxD ¼ 1 d or another appropriate function (Ren et al., 2016; Littlewood, 2015). Given a chosen influence function, we can then compute the influence function weighted volume of the horizon at point x, mx , by integrating over H x as Z mx ¼ uCxDx$xdVx . (12.8) Hx

In addition, we define extension state eCxD as eCxD ¼ jjx þ hjj  jjxjj

(12.9)

based on the bonds deformation. Using mx defined at point x and eCxD defined for each bond associated with x, we then compute the dilation at x: Z n qx ¼ uCxDjjxjjeCxDdVx (12.10) mx Hx

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where n is the dimension number, n˛f2; 3g. The deviatoric extension state for bond x from the perspective of point x is then computed as ed CxD ¼ eCxD 

qx jjxjj . n

(12.11)

With these terms, the scalar force state that defines the linear elastic ordinary state-based constitutive law for each bond x from the perspective of point x is written as t x0 x ¼

nkqx nðn þ 2Þm uCxDkxk þ uCxDed CxD mx mx

(12.12)

where k and m are the Lame´ parameters bulk modulus and shear modulus, respectively (Littlewood, 2015). Given the scalar force state t, the force density vectors corresponding with each bond are computed as f x0 x ðh; xÞ ¼ tx0 x

h  x jjh þ xjj

(12.13)

which is the action force applied at point x0 , and f x0 x ðh; xÞ ¼ tx0 x

hþx jjh þ xjj

(12.14)

which is the reaction force applied at point x. To compute the total force at each point, force density vectors are summed over all bonds in the horizon. For more detail, we direct the reader to the peridynamics literature (Littlewood, 2015; Madenci and Oterkus, 2014; Oterkus, 2015; Silling and Lehoucq, 2010).

2.2 Implementing growth and remodeling Starting from the background given in Section 2.1, we now highlight one strategy for adding growth and remodeling to the peridynamic framework. Specifically, we discuss using peridynamics to model cellular behavior on the microscale. We note that a description of additional cases, for example, adding macroscale growth alone, can be found in the literature (Lejeune and Linder, 2017a). In this example, we treat each cell as an individual node and use the peridynamic equation of motion to maintain mechanical equilibrium (Lejeune and Linder, 2017a). With this treatment, it is possible to implement an algorithm, illustrated in Fig. 12.3, where the peridynamic framework interacts with a biological algorithm and is used to maintain mechanical equilibrium in a defined system (Lejeune and Linder, 2017a). Essentially, after each simulation step of algorithmically defined cell behavior, the entire system is relaxed back to mechanical equilibrium via an adaptive dynamic relaxation procedure (Kilic and

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2. Methodology

initial conditions no

for

load steps remain? yes

apply load step

apply bio algorithm

yes mechanical equilibrium? while no adaptive dynamic relaxation

simulation results

FIGURE 12.3 An example of an algorithm where a biological mechanism such as cell death or cell division is implemented in conjunction with the peridynamic equation of motion.

Madenci, 2010a). Here we redefine the constitutive relations in a manner that makes this possible and introduce the equations from Section 2.1 in their discrete form. First, we redefine the stretch-free separation distance between nodes as     xjk  ¼ 1 þ gj rj þ ð1 þ gk Þrk (12.15) where r is the initial radius associated with each node (cell), and g is the radial growth (or shrinkage) associated with each node where  1 < g < gmax . Given jjxjk jj, we define the stretch between node j and node k as     y  y   xjk  k j   (12.16) sjk ¼ xjk  which is used to determine bond damage gjk following  1 if s < smax gjk ¼ 0 otherwise

(12.17)

with smax defined as the maximum allowable stretch between bonds. Depending on the application, bond damage can be either reversible or irreversible (Lejeune and Linder, 2017a). We then define influence function u as simply ujk ¼ gjk .

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(12.18)

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The discrete form of the horizon weighted volume m is written as X  2 ujk xjk  DVk (12.19) mj ¼ k˛H j

where DVk is a function of r and g, and bond elongation e is written as     ejk ¼ yk  yj   xjk . (12.20) The discrete form of dilation q is written as   n X qj ¼ ujk xjk ejk DVk mj

(12.21)

k˛H j

and deviatoric bond elongation ed is defined as   qj xjk  d ejk ¼ ejk  n

(12.22)

where n ¼ 2 for a two-dimensional problem and n ¼ 3 for a threedimensional problem. With these terms, the equation for the magnitude of force density that arises at node k due to node j is tkj ¼

  nðn þ 2Þm nkqj ujk xjk  þ ujk edjk mj mj

(12.23)

where k and m are Lame´ parameters. Building on these definitions, force density is defined as yk  yj    f jk yj ; yk ¼ tjk $  y  y  k j    yk  yj    fkj yj ; yk ¼ tkj $  y  y  k

(12.24)

j

where only position in the current configuration y defines the direction of force density. In Table 12.1 we list standard ranges for the parameters required to implement these equations.

2.3 Note on emergent behavior In computational modeling of biological materials, there is an inherent trade-off between continuous and discrete approaches. Continuum modeling is typically used to understanding materials on the macroscale, where biological materials are treated as either growing surfaces (Rudraraju et al., 2019), volumetrically growing solids (Javili et al., 2015), or as constituents in a mixture theory approach (Byrne, 2003; Preziosi and

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2. Methodology

TABLE 12.1 This table summarizes typical parameter ranges for agent-based cell simulations. We note that a framework for running a global sensitivity analysis on these and other parameters has been presented in the literature (Lejeune and Linder, 2018b, 2020). Parameter

Value

Source

d0

1:01  1:15

Plausible values (Lejeune and Linder, 2017a)

smax

1:0  2:0

Plausible values (Lejeune and Linder, 2017a)

E

1 kPa

Plausible value (Xu et al., 2012)

n

0.45

Nearly incompressible material

r0

5 mm

Approximate cell size (Drasdo and Hohme, 2005)

Tosin, 2008). Discrete modeling, on the other hand, views biological materials as a collection of cellular and/or subcellular components (Drasdo et al., 2007; Sandersius and Newman, 2008). Discrete modeling, which is often favored by the biophysics community (Norton et al., 2010), allows a mechanistic description of cell behavior, but is computationally intractable on the macroscale, which severely limits many potential applications in the clinical setting such as modeling the macroscale mechanical interactions between tumors and healthy tissue or organs (Frieboes et al., 2007; Lowengrub et al., 2010). To capture the benefits of both discrete and continuum modeling, hybrid modeling approaches have been proposed (Stolarska et al., 2009). For example, one approach treats active cells on the perimeter of a growing system as discrete particles and the inactive cells at the center of the system as a continuum captured by a finite element mesh (Kim et al., 2007). In addition, there has been significant effort to formulate continuum models that phenomenologically reflect cellular scale behavior (Ambrosi et al., 2012; Araujo and McElwain, 2004). And, more recently, researchers have explored multiscale modeling frameworks for biological materials (Khang et al., 2020). In several of the results presented in Section 3, the goal is to understand how phenomena observed on the cellular scale will ultimately influence macroscale tissue behavior. Essentially, given some known cellular-scale mechanism such as cell division or cell death, what macroscale growth and remodeling-related behavior will emerge? Here we briefly define two

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tools for summarizing the results of agent-based cell models that will allow us to quantify emergent behavior. First, we define the average growth-induced deformation gradient F. Examples of this are illustrated in Figs. 12.4e12.6. To compute the average growth-induced deformation F, we define an array of initial stretch vectors L0 as h i p p p L0 ¼ l101 l201 .l011 l102 l202 .l022 .l10n l20n .l0nn . (12.25) Then, we define an array of current stretch vectors Lt which reflects the initial stretch vectors tracked into the current configuration: h i p p p Lt ¼ l1t1 l2t1 .lt11 l1t2 l2t2 .lt22 .l1tn l2tn .ltnn (12.26) where p reflects initial vector repeats due to cell splitting as illustrated in Fig. 12.5C. In the case of cell death, as in Fig. 12.6, only vectors that are present in both configurations are considered. Given these arrays, we then define average deformation in the current configuration F with the equation FL0 ¼ Lt .

(12.27)

To solve this over-determined system of equations, we use the normal  1 equation F ¼ Lt LT0 L0 LT0 . We can also define change in volume with respect to deformation as J ¼ det F. In addition to summarizing average population deformation, we define average population connectivity C. To compute C, we treat the population of cells as a mathematical graph structure G (Newman, 2010). Each cell is treated as a node in graph G. For every cell pair ðj; kÞ where the physical distance between node j and node k is less than or equal to rj þ rk , there is a corresponding edge in G between nodes j and k. This is illustrated in Fig. 12.6C. Average population connectivity C is then defined as 1=d

C¼

NG NSG  mSG NG

(12.28)

where NG is the number of nodes in G, NSG is the number of nodes in the largest connected subgroup, d ¼ 2 or d ¼ 3 is the dimension of the problem, and mSG is the dimensionless mean shortest path in the largest connected subgroup. The results in Section 3 show the ratio of connectivity at the end of the simulation to connectivity at the start of the simulation Cf =C0 . We note that both F and C have the added benefit of being a convenient way to summarize complicated model results with potential stochastic variation (Lejeune and Linder, 2020).

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Deng et al. 2009

Ghajari et al. 2014

c)

d)

Perré et al. 2016

e)

Taylor et al. 2016

Chen et al. 2019

2. Methodology

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b)

a)

FIGURE 12.4 Examples from the literature relevant to modeling fracture in biological materials with peridynamics: (A) the relationship between fracture and healing in cortical bone (Deng et al., 2008); (B) fracture in anisotropic cortical bone (Ghajari et al., 2014); (C) fracture in wood (Perre´ et al., 2016); (D) rupture of biological membranes (Taylor et al., 2016); (E) fracture in a porous material (Chen et al., 2019). All figures are adapted from the original manuscripts cited.

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d)

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FIGURE 12.5 Investigating cell divisionedriven tissue growth with a peridynamics-based model: (A) illustration of cell division; (B) different probability density functions for cell division angle in 2D; (C) method for computing an approximate growth-induced deformation gradient from an agent-based cell model (see Section 2.3 for additional details); (D) components of the growth-induced deformation gradient as a function of underlying division angle distribution where 4 ¼ bN ð0; 1Þ. Information is adapted from Lejeune, E., Linder, C., 2017b. Quantifying the relationship between cell division angle and morphogenesis through computational modeling. J. Theor. Biol. 418, 1e7.

d)

2. Methodology

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FIGURE 12.6

Investigating cell deathedriven tissue shrinkage with a peridynamics-based model: (A) illustration of cell death; (B) method for computing an approximate growth-induced deformation gradient from an agent-based cell model; (C) method for computing cell population connectivity based on graph theory (see Section 2.3 for additional details); (D) components of the growth-induced displacement-based volume change and population connectivity as a function of cell shrinkage during death. Information is adapted from Lejeune, E., Linder, C., 2020. Interpreting stochastic agent-based models of cell death. Comput. Methods Appl. Mech. Eng. 360, 112700.

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3. Example applications In this section, we discuss three broad examples of peridynamics applied to modeling biological materials. First, in Section 3.1, we highlight several examples of fracture in biological materials. Then, in Section 3.2, we show how peridynamics can be used to better understand tissue growth and shrinkage. Finally, in Section 3.3, we show an example where peridynamic simulation is a part of a multiscale modeling framework designed to better understand mechanical contributions to cerebellar morphogenesis.

3.1 Fracture in biological materials Peridynamics is best known as a method for modeling fracture (Madenci and Oterkus, 2014). Here we briefly note some examples of peridynamics being used to model fracture in biological materials. In Fig. 12.4A, we show an example of simulating fracture experiments in cortical bone (Deng et al., 2008). Notably, these simulations investigated the influence of weakened regions due to previous fracture and healing. Fig. 12.4B shows an example of simulating fracture experiments in cortical bone where the crack path depends on material anisotropy (Ghajari et al., 2014). We note that several recent advances in implementing anisotropic, porous, and spatially heterogeneous material behavior could potentially be relevant to simulating bone fracture (Karpenko et al., 2020). One such advance, simulating fracture in porous media, is shown in Fig. 12.4E (Chen et al., 2019). We also show an example of fracture in a complex wood microstructure in Fig. 12.4C (Perre´ et al., 2016), and rupture in a micron scale biological membrane in Fig. 12.4D (Taylor et al., 2016). Notably, there has been compelling recent work on simulating the complex behavior of inclusions in lipid membranes with peridynamics (Madenci et al., 2020). And, hyperelastic constitutive modeling, relevant to modeling soft tissue, has also been implemented in the peridynamic framework (Huang et al., 2019).

3.2 Tissue growth and shrinkage Here we show two examples where peridynamics is used to better understand tissue growth and shrinkage. We note that in both cases, the peridynamic framework is used to model biological materials, populations of cells, that from a mechanics perspective are not strictly classified as either continuous or discrete.

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3.2.1 Cell division and tissue growth Significant effort has been made toward defining rules for predicting division angle in an individual cell (Gillies and Cabernard, 2011). And, external cues such as peptide gradients (Lamb et al., 2014), applied stretch (Wyatt et al., 2015), and applied force (Nestor-Bergmann et al., 2014) have all been shown to influence cell division angle. In studies of cell division orientation, division angle will vary even between tightly controlled experiments. Based on this experimentally observed variability, it follows that cell division angle is best understood as a random variable (Akanuma et al., 2016; Bosveld et al., 2016; Corrigan et al., 2015; Juschke et al., 2013). At present, it is unknown how the stochastic cell division angle influences morphogenesis on the population and tissue scales (Matamoroaˆ-Vidal et al., 2015; Minc and Piel, 2012). To better understand how the distribution of division angle orientations will influence tissue scale growth, we implement a mechanics-based model of a population of cells where individual cells are represented with the peridynamic framework outlined in Section 2.2. We apply volumetric growth to each cell, and when cells exceed a threshold size they divide according to some probability distribution defined by angle 4, illustrated in Fig. 12.5B. The main quantitative simulation result is visualized in Fig. 12.5D, where the ellipses are visualizations of the average growth tensor, defined in Section 2.3, for a simulated cell population with respect to division angle probability distribution 4 (Lejeune and Linder, 2017b). The peridynamic framework allows us to demonstrate that in certain systems the degree of anisotropy in population scale growtheinduced deformation is directly connected to the underlying probability distribution of division angle 4. 3.2.2 Cell death and tissue shrinkage The ability to robustly model cell death has important applications ranging from understanding anomalous organ development (Yamaguchi et al., 2011) to neurodegeneration (Weickenmeier et al., 2019). A particularly compelling example where an enhanced understanding of cell death on the organ scale would help guide clinical decision making is the case where tumors are located in a high stakes regions and can mechanically damage the surrounding tissue if they continue to grow, illustrated in Fig. 12.1 (Bellomo et al., 2008; Clatz et al., 2005; Deisboeck et al., 2011). In these cases, the relevant medical interventions such as radiation therapy and chemotherapy largely function by inducing cell death (Baskar et al., 2012; Cohen-Jonathan et al., 1999). Computational modeling is relevant because even when the response of individual cells to treatment is well understood, it is not necessarily straightforward how the cellular-scale

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process of cell death will manifest on the macroscale. On the macroscale, many different fundamental modeling choices are potentially justifiable for capturing cell death. For example, macroscale tissue models may represent cell death as volumetric shrinkage, mechanical damage, a decrease in species concentration, or some combination of these mechanisms (Harris et al., 2018; Jain et al., 2014; Taber, 1995). The most applicable macroscale modeling choice will vary based on the specific type of cell death, illustrated in Fig. 12.6A (Majno and Joris, 1995), and the system in question (Ambrosi et al., 2016; Suzanne and Steller, 2013; Stylianopoulos, 2017). With the peridynamic framework described in Section 2.2 and the analysis tools summarized in Section 2.3, we are able to take a bottom-up approach to show how differences in cell death on the cellular scale are linked to different interpretations of cell death on larger length scales. Specifically, we show that in some cases cell death leads to gaps between cells, while in other cases it lead to tissue shrinkage, illustrated in Fig. 12.6B and C. In Fig. 12.6D, we plot the change in population shrinkage J and average connectivity C, the two quantities of interest defined in Section 2.3 with respect to degree of radial cell shrinkage a ¼ rmin =r0 with all other parameters fixed. Clearly, the amount that a cell shrinks before it stops exerting force on its neighbors is important to population scale tissue behavior. When a is small (more shrinkage) cell death predominantly manifests as a volumetric change. When a is large, cell death predominantly manifests as a change in porosity and/or as an increase in material damage. Notably, Fig. 12.6D also shows that the results for both the two-dimensional and three-dimensional cases are quantitatively different. Unlike with a standard continuum model, the model based on peridynamics allows these differences to naturally emerge.

3.3 Connecting emergent behavior across scales The cerebellum is a tightly folded structure located at the back of the head where the folds of the cerebellum are aligned such that the external surface appears to be covered in parallel grooves (Leto et al., 2015). Experiments have shown that a series of interconnected mechanisms drive cerebellar foliation (Sudarov and Joyner, 2007). However, the mechanism guiding the initial location of these folds, and subsequently cerebellar morphology, remains poorly understood (Leto et al., 2015). Critically, there is no definitive mechanistic explanation for the preferential emergence of parallel folds instead of the irregular folding pattern seen in the cerebral cortex (Lawton et al., 2019). With the framework defined in Sections 2.2 and 2.3, we are able to implement a multiscale model that connects the anisotropic cell division experimentally observed during cerebellar development (Legue´ et al., 2015) to anisotropic fold formation on the tissue scale (Lejeune et al., 2016). As shown in Fig. 12.7, II. New applications in peridynamics

3. Example applications

II. New applications in peridynamics

FIGURE 12.7 Investigating cerebellar morphogenesis with a peridynamics-based model: (A) illustration of cerebellar morphology and a schematic of cerebellar morphogenesis; (B) illustrated results of a single agent-based model simulation where the division angle is drawn from a probability distribution chosen to appear similar to the experimental results (Legue´ et al., 2015); (C) A plot of the components of F generated with the method described in Section 2.3; (D) The results of a tissue scale isogeometric analysis simulation where the growth-induced deformation gradient shown in (C) is applied to the outer layer of a cylindrically curved domain. Information is adapted from Lejeune, E., Dortdivanlioglu, B., Kuhl, E., Linder, C., 2019. Understanding the mechanical link between oriented cell division and cerebellar morphogenesis. Soft Matter 15 (10), 2204e2215.

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we implement an agent-based model of simulated cell clones, propagate information from our in silico cell clones to a tissue-scale model, and use the framework to understand how differential growth between the cerebellar layers drives geometric instability in three-dimensional space. In the macroscale model, we study the influence of physically realistic anisotropic growth on surface wrinkling through both incremental stability analysis (Biot, 1965) and numerical modeling with isogeometric analysis (Hughes et al., 2005). By approaching this problem with fully three-dimensional models on both scales, we are able to better understand how the symmetry of the cerebellum emerges, rather than treating this symmetry as inherent and only presenting a model in two-dimensional space. Looking forward, the framework implemented here with a cellular-scale peridynamics-based model coupled to a macroscale continuum model is a powerful tool for understanding emergent behavior across scales.

4. Conclusion and outlook In this chapter, we begin in Section 1 by introducing several reasons why modeling biological materials is a compelling research challenge. Then, we reviewed the basic equations for peridynamics in Section 2.1, described key details of adapting the peridynamic framework for biological materials in Section 2.2, and briefly describe methods for summarizing mechanically relevant results of agent-based model simulations in Section 2.3. In Section 3, we highlight three main applications for modeling biological materials with peridynamics: material fracture in Section 3.1, tissue growth and shrinkage in Section 3.2, and understanding emergent behavior across scales in Section 3.3. Practically, these applications cover bone fracture, lipid membrane rupture, tumor growth and shrinkage, and cerebellar morphogenesis. Looking forward, we anticipate multiple avenues for future research in modeling biological materials with peridynamics. For example, there are numerous ways in which the peridynamic framework can be extended for modeling of biological systems ranging from implementing unique cell types and additional cellular-scale mechanisms to adding growth components to multiphysics constitutive laws. Furthermore, we anticipate that additional potential of the peridynamic framework could be realized through a class of models referred to as “hybrid models.” In the broader literature of agent-based cell modeling, “hybrid models,” where select locations contain discrete representations of cells and subcellular components while other locations are represented as a continuum, are gaining traction (Van Liedekerke et al., 2015, 2018). In particular, hybrid models

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References

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are being used to mitigate the prohibitive computational cost of implementing large agent-based models. Peridynamics, by nature, is a promising tool for hybrid modeling because it is specifically designed to unify the mechanics of continuous and discontinuous media. With the peridynamic framework, both “continuous” approximate regions and “discrete” highly resolved regions can be implemented with the same set of equations. Thus, the flexibility of the peridynamic framework could enable substantial further advances in computational modeling of biological materials beyond what we show here.

Acknowledgments This work was supported by the National Science Foundation, United States of America Graduate Research Fellowship Grant No. DGE-114747 to EL and National Science Foundation, United States of America CAREER Grant No. CMMI-1553638 to CL.

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skin: a detailed histomechanical characterization, residual strain analysis, and constitutive model. Acta Biomater. 101, 403e413. Minc, N., Piel, M., 2012. Predicting division plane position and orientation. Trends Cell Biol. 22 (4), 193e200. Nestor-Bergmann, A., Goddard, G., Woolner, S., 2014. Force and the spindle: mechanical cues in mitotic spindle orientation. Semin. Cell Dev. Biol. 34, 133e139. Newman, M., 2010. Networks: An Introduction. Oxford university press. Norton, K.-A., Wininger, M., Bhanot, G., Ganesan, S., Barnard, N., Shinbrot, T., 2010. A 2d mechanistic model of breast ductal carcinoma in situ (dcis) morphology and progression. J. Theor. Biol. 263 (4), 393e406. https://doi.org/10.1016/j.jtbi.2009. 11.024. Oterkus, S., 2015. Peridynamics for the Solution of Multiphysics Problems. Ph.D. thesis. The University of Arizona. Perre´, P., Almeida, G., Ayouz, M., Frank, X., 2016. New modelling approaches to predict wood properties from its cellular structure: image-based representation and meshless methods. Ann. For. Sci. 73 (1), 147e162. Preziosi, L., Tosin, A., 2008. Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J. Math. Biol. 58 (4), 625e656. Rausch, M., Dam, A., Go¨ktepe, S., Abilez, O., Kuhl, E., 2011. Computational modeling of growth: systemic and pulmonary hypertension in the heart. Biomech. Model. Mechanobiol. 10 (6), 799e811. Ren, H., Zhuang, X., Cai, Y., Rabczuk, T., 2016. Dual-horizon peridynamics. Int. J. Numer. Methods Eng. 108 (12), 1451e1476. Rodriguez, E., Hoger, A., McCulloch, A., 1994. Stress-dependent finite growth in soft elastic tissue. J. Biomech. 27 (4), 455e467. Rudraraju, S., Moulton, D.E., Chirat, R., Goriely, A., Garikipati, K., 2019. A computational framework for the morpho-elastic development of molluskan shells by surface and volume growth. PLoS Comput. Biol. 15 (7) e1007213. Sandersius, S., Newman, T., 2008. Modeling cell rheology with the subcellular element model. Phys. Biol. 5 (1), 015002. Shao, Y., Zhao, H.-P., Feng, X.-Q., Gao, H., 2012. Discontinuous crack-bridging model for fracture toughness analysis of nacre. J. Mech. Phys. Solid. 60 (8), 1400e1419. Silling, S.A., Lehoucq, R.B., 2010. Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73e168. Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E., 2007. Peridynamic states and constitutive modeling. J. Elasticity 88 (2), 151e184. Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solid. 48, 175e209. Stolarska, M.A., Kim, Y., Othmer, H.G., 2009. Multi-scale models of cell and tissue dynamics. Phil. Trans. Math. Phys. Eng. Sci. 367 (1902), 3525e3553. https://doi.org/10.1098/ rsta.2009.0095. Stylianopoulos, T., 2017. The solid mechanics of cancer and strategies for improved therapy. J. Biomech. Eng. 139 (2), 021004. Sudarov, A., Joyner, A.L., 2007. Cerebellum morphogenesis: the foliation pattern is orchestrated by multi-cellular anchoring centers. Neural Dev. 2 (December), 26. Suzanne, M., Steller, H., 2013. Shaping organisms with apoptosis. Cell Death Differ. 20 (5), 669. Taber, L., 1995. Biomechanics of growth, remodeling, and morphogenesis. Appl. Mech. Rev. 48 (8), 487e545. Taylor, M., Go¨zen, I., Patel, S., Jesorka, A., Bertoldi, K., 2016. Peridynamic modeling of ruptures in biomembranes. PloS One 11 (11) e0165947.

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Van Liedekerke, P., Palm, M., Jagiella, N., Drasdo, D., 2015. Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results. Comput. Part. Mech. 2 (4), 401e444. Van Liedekerke, P., Buttenscho¨n, A., Drasdo, D., 2018. Off-lattice agent-based models for cell and tumor growth: numerical methods implementation, and applications. In: In Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes. Elsevier, pp. 245e267. Warren, T.L., Silling, S.A., Askari, A., Weckner, O., Epton, M.A., Xu, J., 2009. A non-ordinary state-based peridynamic method to model solid material deformation and fracture. Int. J. Solid Struct. 46 (5), 1186e1195. https://doi.org/10.1016/j.ijsolstr.2008.10.029. Weickenmeier, J., Jucker, M., Goriely, A., Kuhl, E., 2019. A physics-based model explains the prion-like features of neurodegeneration in Alzheimer’s disease, Parkinson’s disease, and amyotrophic lateral sclerosis. J. Mech. Phys. Solid. 124, 264e281. Wyatt, T., Harris, A., Lam, M., Cheng, Q., Bellis, J., Dimitracopoulos, A., Kabla, A., Charras, G., Baum, B., 2015. Emergence of homeostatic epithelial packing and stress dissipation through divisions oriented along the long cell axis. Proc. Natl. Acad. Sci. U S A 112 (18), 5726e5731. Xu, W., Mezencev, R., Kim, B., Wang, L., McDonald, J., Sulchek, T., 2012. Cell stiffness is a biomarker of the metastatic potential of ovarian cancer cells. PloS One 7 (10) e46609. Yamaguchi, Y., Shinotsuka, N., Nonomura, K., Takemoto, K., Kuida, K., Yosida, H., Miura, M., 2011. Live imaging of apoptosis in a novel transgenic mouse highlights its role in neural tube closure. J. Cell Biol. 195 (6), 1047e1060.

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C H A P T E R

13 The application of peridynamics for ice modeling Qing Wang, Lei Ju, Yanzhuo Xue College of Ship Building Engineering, Harbin Engineering University, China

O U T L I N E 1. Introduction 1.1 Structure and properties of ice 1.1.1 Structure of natural ice 1.1.2 Mechanical properties 1.2 Constitutive for ice 1.2.1 Elastic-brittle constitutive 1.2.2 Ductile constitutive 1.2.3 Ductile-brittle transition 1.3 Advantages and research status of using peridynamics to study ice 1.3.1 Advantages 1.3.2 Research status

276 276 276 277 280 281 281 282 283 283 283

2. Numerical study of mechanical properties of ice 2.1 Pre-crack propagation under tension of 2D flat ice 2.2 Wing crack propagation in 3D ice body 2.3 Three-point bending test of ice 2.4 Ice impacting on reinforced plate structure 2.5 The interaction between ice and cylindrical structure

284 284 284 287 289 291

3. Numerical simulation of interaction between level ice and sloping structure 294 3.1 Numerical model 294 3.1.1 Contact 294

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00002-0

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© 2021 Elsevier Inc. All rights reserved.

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3.1.2 Buoyancy 3.2 2D numerical analysis of interaction between ice and sloping structure 3.2.1 Numerical results 3.2.2 Influence factors of damage of ice 3.3 3D numerical analysis of interaction between ice and sloping structure

294

295 295 297 297

4. Research on numerical simulation of ice breaking by underwater explosion based on BBPD method 299 4.1 Explosive load 299 4.2 Numerical modeling and analysis 300 5. Numerical simulation of continuous icebreaking based on hybrid modeling method 303 5.1 Calculation model 303 5.2 Numerical results 303 References

305

Further reading

308

1. Introduction Ice has been studied for more than 100 years. The study of ice and the interaction between ice and structures have been done systematically in the countries of high-latitude and cold sea areas, for carrying out large-scale exploitation of offshore oil and other resources in the ice area, which promotes the development of research in the field of ice engineering.

1.1 Structure and properties of ice 1.1.1 Structure of natural ice Ice is a complex, natural, composite material, which includes pure ice, brine, and air. The properties of ice are complex and highly susceptible to environmental conditions. Due to the different temperatures, salinities, and densities at different places and at different times, the crystal structure of ice is also different; hence the ice shows different properties. Polarizing filters can be used to investigate the microstructure of ice (Langway, 1958). Granular ice (T1) is generally considered to be a homogeneous, isotropic material. It is small relatively and can form from flooded snow. Columnar ice (S2) can be very long and wide, and its c-axis

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is in the horizontal direction. Columnar ice (S1) has very large grains; it forms on the calm lakes, and its c-axis is in the vertical direction. Columnar ice is an inhomogeneous anisotropic material. The ice crystal structure and properties of granular ice and columnar ice are very different. The frazil ice (S4) usually forms from supercooled waters in flowing rivers, because of the accumulation of jagged grains. The microstructure (nm) of ice is more colorful. Nineteen kinds of ice phases have been discovered, such as the new ice XVIII (Millot et al., 2019) and ice XVII (del Rosso et al., 2016), along with amorphous ice and glassy water (Matsui et al., 2017; Huang et al., 2016, 2017). Hexagonal ice (Ih) (Ro¨ttger et al., 1994), the naturally occurring form of ice on Earth, can be seen in daily life, as it manifests as frost and sea ice. The microstructures of ice can be identify by molecular dynamics (MD) (Wang et al., 2020). 1.1.2 Mechanical properties • Compressive strength Compression failure is one of the main forms of sea ice failure from many sea ice failure experiments. Based on the 283 compression test data, Timco and Frederking proposed the sea ice ice strength calculation formula to calculate the strength of sea ice sheets (Timco and Frederking, 1990, 1991). For horizontally loaded columnar ice, the uni-axial strength is rffiffiffiffiffiffiffiffi 
vT 0:22 sc ¼ 37ð_εÞ 1 (13.1) 270 For vertically loaded columnar ice, the uni-axial strength is rffiffiffiffiffiffiffiffi 
vT 0:22 1 sc ¼ 160ð_εÞ 200 For granular ice, the uni-axial strength is rffiffiffiffiffiffiffiffi 
vT sc ¼ 49ð_εÞ0:22 1  280

(13.2)

(13.3)

The above formula is applicable to the case where the strain rate is between 107s1~2  104s1. At a higher strain rate, the sea ice will produce brittle failure. • Tensile strength Tensile strength is used to represent the maximum tensile force that can be sustained before sea ice fracture. Fig. 13.1 shows the variation curve of the tensile strength of ice under horizontal load with temperature in the current year. Shown in Fig. 13.2, in Richter-Mange and Jones’

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13. The application of peridynamics for ice modeling

FIGURE 13.1 Curve of tensile strength with temperature (Timco and Frederking, 2010). Modified from Kuehn, G.A., 1990. The structure and tensile behavior of first-year sea ice and laboratorygrown saline ice. J. Offshore Mech. Arctic Eng. 112 (4).

research, they found that under horizontal tensile loading, the tensile strength of sea ice decreases with the increase of porosity with the nonlinear overall trend. The relation between tensile strength and porosity can be obtained by ignoring the strain rate, because of its little influence on the tensile strength, st ¼ 4:278vT0:6455 MPa

(13.4)

Peyton and Dykins carried out multidirection loading tests based on the direction of ice growth for columnar ice. The test results showed that the tensile strength parallel to the direction of growth is three times perpendicular to the direction of ice growth (Peyton, 1996; Dykins, 1967, 1968). • Ductile-to-brittle transition Ice exhibits two mechanical properties under different strain rates: toughness and brittleness (Schulson, 2001), shown in Fig. 13.3. When the

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FIGURE 13.2 Curve of tensile strength with total porosity (Timco and Frederking, 2010). Modified from Richter-Menge, J.A., Jones, K.F., 1993. The tensile strength of first-year sea ice. J. Glaciol. 39 (133), 609e618.

FIGURE 13.3

Schematic sketch illustrating the ductile-to-brittle transition. The curves show hypothetical compressive stress-strain curves at progressively increasing strain rates reprinted (Schulson, 1990).

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FIGURE 13.4

13. The application of peridynamics for ice modeling

Compression strength and strain rate curves at different temperatures

(Schulson, 2001).

loading rate is low, the ice shows the toughness property. Even if the deformation is greater than 10%, and the material will not be damaged. When the loading rate is high, the ice exhibits brittleness. The failure will occur, even if the deformation is only about 0.1%. As shown in Fig. 13.4, the ultimate stress of ductile ice increases with increasing strain rate, but decreases with increasing temperature. The ductile, mechanical properties of ice occur and strain rate will affect its compression strength, when the compressive strain rate is less than 103s1. Moreover, with the increase of the stress ratio, the corresponding strain in the ductile and brittle transition zone increases firstly and decreases lastly.

1.2 Constitutive for ice Furthermore, since stress state during ice crushing is mostly compressive, ductile-brittle transition of ice compressive strength should be a complex factor (Jones, 1982). According to Yue et al. (2009), during ice crushing, ice forces represent three modes regarding loading rate.

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(1) When loading rate is higher than 40 mm/s, ice force becomes irregular. At this state, ice is an elastic, brittle material. (2) When loading rate is between 20 and 40 mm/s, ice force is defined as locked-in ice force. Ice exhibits ductile-brittle transition. (3) When loading rate is lower than 20 mm/s, ice force is quasi-static. Ice is a ductile material with plastic, mechanical behavior. 1.2.1 Elastic-brittle constitutive In the process of contact between marine structure and sea ice, the form of sea ice failure around the marine structure is basically brittle failure. At about 10 C, the sea ice shows an elastic, brittle quality, when the strain rate is bigger than 103s1. At this point, it is appropriate to use the typical PMB material peridynamics model for numerical simulation of elastic, brittle ice. The relation between bond force and bond stretch in PMB materials was shown in Fig. 13.5. For the PMB materials, the force function of peridynamics can be expressed as, f ðh; xÞ ¼ c  sðt; h; xÞ  mðt; h; xÞ  1 if sðt0 ; xÞ < s0 mðt; xÞ ¼ 0 otherwise

(13.5) (13.6)

1.2.2 Ductile constitutive Generally, the sea ice is regarded as a ductile material when the strain rate is less than 103s1. At this point, the material will not be damaged even if the deformation is greater than 10%. In order to satisfy the ductile material properties of sea ice, the force function was improved based on PMB materials. Considering that there are no physical quantities in classical mechanics such as stress/strain in peridynamics, a piecewise function was established, which was making an analogy with the ideal elastoplastic constitutive model in classical plastic mechanics, in Fig. 13.6. We define a parameter sy as the yield strength of bond stretch.

FIGURE 13.5

The relation between bond force and bond stretch in PMB materials

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FIGURE 13.6 The relation between bond force and bond stretch in ductile materials.

When the bond stretch does not reach the yield strength (sðt; xÞ < sy ),the linear elastic relationship between the force function f ðh; xÞ and bond stretch sðt; h; xÞ between material points remains. When the bond stretch reaches the yield strength (sy  sðt; xÞ < s0 ), the value of the force function f ðh; xÞ, namely the bond force, does not increase. In other words, the bond can still be extended without bond fracture, but until the bond stretch reaches the limit s0 and a permanent fracture occurs. For this material, the force function of peridynamics can be expressed as,  c  sðt; h; xÞ; if sðt; xÞ < sy f ðh; xÞ ¼ (13.7) c  sy ; if sy  sðt; xÞ < s0 1.2.3 Ductile-brittle transition Because the ductile-brittle transition affects the compression strength of ice, it also affects the ice load on the marine engineering structures. In order to simulate the mechanical properties of ice, a peridynamic model of ductile-brittle transformation controlled by strain rate is proposed. The derivative of the bond stretch with respect to time in the force function is used as the control quantity of the model transformation of the control function. Similar to strain in classical mechanics, bond stretch in peridynamics is also a parameter to measure the degree of material deformation, so the bond stretch is approximately equal to strain. The rate of change of bond stretch with time, namely the derivative of bond stretch with respect to time s,_ is called the rate of change of bond length, s_ ¼

ds Ds ¼ dt Dt

(13.8)

After determining the rate of change of bond length s,_ it can be used as a cutoff point for ductile-brittle transition. When s_ is less than this value, the force function is the ductile model, and when s_ reaches or exceeds this value, the force function is the elastic-brittle model.

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1.3 Advantages and research status of using peridynamics to study ice 1.3.1 Advantages Peridynamic theory has extensively been used in predicting the damage and fracture processes of brittle materials (Ha and Bobaru, 2011), composite laminate structures (Askari et al., 2006), and reinforced concrete materials (Kilic and Madenci, 2009) owing to its abilities of representing and capturing discontinuous deformations in solids and structures during their failure processes. So, the peridynamics is very suitable for the simulation of the sea ice breaking process (Ye, 2018). 1.3.2 Research status Scholars’ research mainly includes mechanical properties of ice, icee water interactions, and iceestructure interactions. The “iceestructure interactions” includes iceeship, iceepropeller, submarine surfacing through icedtypical structure (such as cylinder). For the research of mechanical properties of ice, Xue et al. (2018) used the peridynamics method to carry out the experimental simulation of the three-point bending of the ice, and determined the results simulated by the PD method were consistent with the test results. For the iceewater interactions, Liu et al. (2020) developed a coupling method between the bond-based peridynamics model for solids and the updated Lagrangian particle hydrodynamics (ULPH) model of fluids for simulating interaction between ice and seawater. In the research of iceeship interactions, Xue et al. (2019) researched the ice loads for a ship navigating in level ice applying a numerical method, which was developed based on peridynamics. Liu et al. (2018) calculated ice loads and simulated the shipeice interaction process in different operating conditions by peridynamics. For the propellereice contact process, Xiong (2020) examined the influence mechanism of the shadowing effect on the sea ice failure mode along with the loads acting on the propellers. Ye et al. (2017) investigated the propellereice contact process and the dynamic loads numerically. For the study of interaction between a submarine and an ice sheet, Ye et al. (2019) established a numerical model which simplified to be the interaction between a submarine and an ice sheet for simulating submarine surfacing through the ice. In order to determine some basic properties, scholars have conducted some studies on the interaction between ice and typical structures. Jia et al. (2020) simulated the extrusion and rupture of ice and the vibration of platform structure in the iceestructure interaction. Jia et al. (2019) also simulated the interaction between sea ice and wide vertical structures by the bond-based peridynamics method. Song et al. (2019a) proposed a state-based peridynamics with adaptive particle refinement for

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13. The application of peridynamics for ice modeling

simulating water ice crater formation due to impact loads. It was found that the peridynamic simulation results and the experiments matched well except for some minor differences discussed. Liu et al. (2016) simulated the interaction between level ice and a cylindrical, vertical, rigid structure at different velocities in state-based peridynamics. Wang and Wang (2018) simulated the progress of interaction between level ice and strengthened plate. Song et al. (2019b) simulated the behavior of the ice under impact loads applied by a rigid ball by the modified Druckere Prager plasticity model.

2. Numerical study of mechanical properties of ice This section mainly introduces the common applications of PD in the simulation of mechanical characteristics of ice, including the tensile simulation of flat ice with pre-cracked (Liu, 2016), the propagation process of wing cracks under compression of ice (Liu, 2016), three-point bending test ice (Lu, 2018), the contact between the reinforced structure and ice (Liu, 2016), and the contact between the rigid, vertical structure and ice (Jia et al., 2020). Since the Poisson’s ratio of ice can basically be considered as 0.3, which is similar to the fixed value of 1/3 (1/4) in the BBPD, therefore, both SBPD and BBPD can be used for simulation. The BBPD is used in this chapter.

2.1 Pre-crack propagation under tension of 2D flat ice As shown in Fig. 13.7, the two-dimensional ice plate is stretched, and there is a horizontal pre-crack on the left boundary of the ice plate. The lower boundary of the ice plate is fixedly supported, A velocity boundary condition is imposed on the upper boundary. From the calculation results in Fig. 13.8, it can be seen that as the upper boundary tensile loading, the high stress area first concentrates on the tip of the pre-crack, and then the stress increases and gradually expands to most of the plate surface. After the stress at the crack tip increases to a certain extent, the pre-crack begins to expand.

2.2 Wing crack propagation in 3D ice body As shown in Fig. 13.9, a velocity load is applied to the upper surface of a 3D ice block model, and there is an inclined pre-crack inside the ice block. The pre-crack direction is 60 degree counterclockwise from the horizontal, and the lower surface is fixedly supported.

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2. Numerical study of mechanical properties of ice

FIGURE 13.7

(a)

Boundary conditions.

(b)

(c)

FIGURE 13.8 Simulation results of BBPD. (A) t ¼ 8:334  105 s. (B) t ¼ 1:042  104 s. (C) t ¼ 1:250  104 s.

FIGURE 13.9 3D ice body with inclined crack. II. New applications in peridynamics

286

13. The application of peridynamics for ice modeling

(a)

FIGURE 13.10

6:75  103 s.

(b)

(c)

Simulation results of BBPD. (A) t ¼ 0:0 s. (B) t ¼ 1:8  103 s. (C) t ¼

Fig. 13.10 shows the propagation process of wing cracks under compression of elastic, brittle ice. The color indicates the degree of local destruction of the material point. Fig. 13.11 is a sea ice compression experiment image. The cracks in the experiment first nucleate in the ice body, the initial cracks expand, and then expand in the vertical direction. The wing crack propagation process simulated by the peridynamics fits the experimental results well.

FIGURE 13.11

Experimental results (Schulson, 2001).

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287

2.3 Three-point bending test of ice As shown in Fig. 13.12, the middle part of the 3D ice beam is subjected to external force, and the two ends are simply supported. This simulation is for comparison and verification with real test (Lu, 2017). The quasi-static method is used before the sample is damaged to speed up the calculation efficiency. And then the dynamic algorithm is used for simulation. Since the limit load is known, before reaching the limit load, a quasi-static algorithm can be used to apply different loads to obtain the corresponding displacement at time 1:21 s. When the limit load is about to be reached, a dynamic algorithm is used for calculation. The comparison of load and displacement curves between simulation and experiment is shown in Fig. 13.13. The simulation curve and the experimental curve have basically the same trend. The relative error of the deformation process is less than 5%.

FIGURE 13.12

Three-point bending test.

FIGURE 13.13

Correspondence between load and displacement.

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13. The application of peridynamics for ice modeling

(a)

(b)

(c)

(d)

(e)

(f )

FIGURE 13.14 Crack growth process. (A) t ¼ 46.2 ms. (B) t ¼ 1696.2 ms. (C) t ¼ 1861.2 ms. (D) t ¼ 2191.2 ms. (E) t ¼ 9946.2 ms. (F) t ¼ 13,200 ms.

Fig. 13.14 shows the change process of the model damage with time during the dynamic simulation. The crack is generated from the bottom of the middle of the model and penetrates the model section in a short time. The position of the crack and the fracture result are consistent with the experimental observation results. The simulation of three-point bending test can also be calculated using NSPD. The specific parameters and theoretical analysis can refer to the original literature (Lu, 2017). And the results are shown in Table 13.1 and Fig. 13.15. TABLE 13.1 Comparison of numerical simulation and experimental results. Fracture time(s)

Midpoint stress (Mpa)

0.47s midpoint deflection (mm)

NOPD

0.496

2.40

0.3495

Experiment

0.470

2.50

0.3500

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(a) Midpoint deflection as a function of time

289

(b) Midpoint stress as a function of time

FIGURE 13.15 Comparison of experimental and simulation results. (A) Midpoint deflection as a function of time. (B) Midpoint stress as a function of time.

2.4 Ice impacting on reinforced plate structure As shown in Fig. 13.16, the flat ice body moves toward the strengthening plate, and after the ice contacts the plate, the boundary of the ice continues to push the entire ice at the same speed and squeeze the plate. In the calculation, the ice material is simplified to elastic and brittle material, and the reinforced plate material is simplified to elastic material. The aforementioned models are all of the same material, and subsequent models need to simulate by two materials. And the T section is set as a rigid body without deformation. Ma¨a¨tta¨nen et al. (2011) conducted experiments of freshwater flat ice squeeze reinforcement board with a ratio of 1:3. Fig. 13.17 shows the visual result of the contact between the flat ice and reinforced plate structure, and the color represents the equivalent density. It can be seen from Fig.13.18A that after the flat ice body contacts with the

FIGURE 13.16

Top view of flat ice-reinforced plate model.

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FIGURE 13.17 Simulation results of ice impacting on reinforced plate structure.

(a)

(b) FIGURE 13.18 Ice force as a function of time. (A) PD. (B) Experiment.

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291

reinforcing plate, a high-stress area appears at the contact position first, then the high-stress area gradually spreads, and finally concentrates in the four areas divided by the T section and the ice body in the high-stress area in Fig. 13.18B. Fig. 13.18 shows the output ice force as time progresses. The trend of the ice force output by the PD is similar to the ice force in the experiment. It can be found that specific ice force analysis can refer to relevant literature (Ma¨a¨tta¨nen et al., 2011). In the design of polar sailing ships, it is generally assumed that the load on the hull shell is larger at the reinforced support, and smaller at the free span without reinforced support, as shown in Fig. 13.19A. The ice force distribution in the y direction is shown in Fig. 13.19B.

2.5 The interaction between ice and cylindrical structure As shown in Fig. 13.20, the vertical structure collides with the sea ice. Except for the side in contact with the structure, the other three sides are rigid support. The direction in which the structure moves is the positive direction of the x-axis of the coordinate system. The upright structure satisfies the vibration equation of single degree of freedom (Fig. 13.21): M $ u€ þ C$u_ þ K$u ¼ Fc

FIGURE 13.19

(13.9)

Relationship between ice force and position. (A) Theoretical distribution. (B) Simulation result.

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FIGURE 13.20

13. The application of peridynamics for ice modeling

Numerical model of the interaction between the sea ice and cylindrical

structure.

FIGURE 13.21

Vibration model of the structure.

Compare and analyze with the existing literature (Ji et al., 2013), in which the DEM method is used. The material parameters are consistent with this simulation. The simulation comparison results are as shown in Table 13.2. It can be found that the error between the two is within an acceptable range.

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TABLE 13.2 Comparison of simulation results of PD and DEM. Result of PD

Result of DEM

The maximum of horizontal ice load Fmax ðkNÞ

101.07

100.08

0.99%

The mean value of horizontal ice load Fmean ðkNÞ

48.05

44.42

8.17%

The maximum of vibration displacement umax ðmmÞ

0.25

0.22

13.63%

The mean value of vibration displacement umean ðmmÞ

0.06

0.09

33.33%

35.33

34.31

2.97%

The maximum of vibration acceleration amax ðgalÞ

Error

Similarly, Liu et al. (2016) also simulates the interaction between the vertical structure and ice, except that other structural parameters are used in the article, and the vibration response of the structure is considered (Fig. 13.21). Therefore, the numerical results are not compared here, only the simulation phenomenons are observed as shown in Figs. 13.22 and 13.23.

FIGURE 13.22

Virtual reports of the numerical simulation (step 21400).

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FIGURE 13.23

13. The application of peridynamics for ice modeling

Simulation results and local enlarged drawing.

3. Numerical simulation of interaction between level ice and sloping structure 3.1 Numerical model 3.1.1 Contact Contact is defined by repelling force, which takes effect when two bodies come close and prevents particles of different bodies from occupying the same spatial position. Contact force is regarded as the short-range force that could decompose into normal force perpendicular to the contact surface and friction force along the contact surface. 3.1.2 Buoyancy The impact of fluid on the damage of level ice should be considered in the simulation. As shown in Fig. 13.24, if the centroid of particle is above

FIGURE 13.24

Buoyancy model.

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295

FIGURE 13.25 Peridynamic model of ice and sloping structure.

the upperline, only gravity is added into the body force term; if below the lowerline, both gravity and buoyancy are added into the body force term; if between the upperline and lowerline, the buoyancy is calculated according to the coordinate of particle.

3.2 2D numerical analysis of interaction between ice and sloping structure Considering that ice shows brittle feature under higher moving velocities, it is idealized as elastic-brittle material, satisfying a linear-elastic constitutive model (Fig. 13.25). 3.2.1 Numerical results The damage of level ice is shown in Figs. 13.26 and 13.27. At the beginning of interaction, level ice moves toward sloping structure and makes contact with it. Local damage is found among particles at front end of level ice. Those shattered, fully damaged particles (shown in red) are regarded as crushed ice floes that peeled off the front end of level ice. Cracks appear inside the level ice when it approaches closer to the sloping structure. According to the analytical formula for horizontal ice load in 2D contact model between level ice and sloping structure proposed by Croasdale and Cammaert (1994), theoretical horizontal ice load in this case is 45.92 N.

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13. The application of peridynamics for ice modeling

(a)

(b)

(c)

(d)

Side view of interaction between ice and sloping structure. (A) t ¼ 0.066 s. (B) t ¼ 0.078 s. (C) t ¼ 0.099 s. (D) t ¼ 0.132 s.

FIGURE 13.26

FIGURE 13.27

Local damage at the front end of level ice (t ¼ 0.075 s).

II. New applications in peridynamics

3. Numerical simulation of interaction

5

x 10

297

4

Force (N)

4 3 2 1 0 0

FIGURE 13.28

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (s)

Horizontal ice load on the sloping structure.

The ice load on the sloping structure could be calculated by integrating the contact force with the volume of the particle. The horizontal ice load calculated is shown in Fig. 13.28. Considering the compression at the front end of level ice, the horizontal ice load is relatively higher in the initial stage. The flexural deformation of level ice is noted from t ¼ 0.0672 s. And the ice load reaches its peak at t ¼ 0.0726 s, when cracks begin to form at the bottom of level ice. The maximum horizontal ice load calculated is 4.47  104 N, with an error of 2.61% comparing with the analytical solution. 3.2.2 Influence factors of damage of ice The impact of ice thickness, ice speed, and slope angle on the interaction between level ice and sloping structure is studied using numerical method. Analyzing the numerical results, the following rules can be found. With the increase of ice thickness, the damage radius and the horizontal ice load also increase. The increase of ice speed leads to the increase of damage radius, and the horizontal ice load does not change. The damage radius increases with the increase of slope angle, and the horizontal ice load has the inverse trend. These rules have similar trends to the analytical formula.

3.3 3D numerical analysis of interaction between ice and sloping structure If level ice is wider than ice-facing surface of sloping structure, the contact between the two should be calculated with 3D modeling. The crack propagation of ice with limited width during the interaction between ice and sloping structure is analyzed in this section.

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The numerical results are shown in Fig. 13.29, indicating some typical phenomenon like crack initiation, propagation, and local fracture of level ice during the interaction.

FIGURE 13.29 the numerical results. (A) Crushing failure of level ice. (B) Failure at the boundary of level ice. (C) Branching of radial crack. (D) Initiation of circumferential crack. (E) Initiation of local fracture.

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4. Research on numerical simulation

4. Research on numerical simulation of ice breaking by underwater explosion based on BBPD method The implementation of underwater explosion is a fast and effective measure to prevent and control ice flood disasters. In this section, the BBPD approach was applied to investigate the fragmentation of ice cover by blast loads of an underwater explosion. The simulation results show a good agreement with the experimental data. Furthermore, the parameters that influence the fracture radius of the ice cover were also discussed.

4.1 Explosive load Underwater explosion involves many difficult problems, such as problems of fluidesolid coupling and additional water mass and liquid jets. In most cases, damage done to structures occurs early on and is due to the striking of the shock wave (Liang and Tai, 2006). The present investigation thus considers only the effects of the shock wave (Fig. 13.30). The empirical equations to calculate the pressure are proposed by Geers and Hunter (2003). The problem of fluidesolid coupling cannot be ignored for the dynamic response of a plate in the event of an underwater explosion. According to Taylor’s plate theory (Taylors, 1941), which is shown in Fig. 13.31, the pressure reflected on the fluidestructure interaction surface can be predicted reasonably accurately when the pressure from an underwater explosion impinges upon a flexible surface, such as an ice sheet.

6.0E+7

Pressure/Pa

5.0E+7 4.0E+7 3.0E+7 2.0E+7 1.0E+7 0 0.000

FIGURE 13.30

0.002

0.004 0.006 Time/s

0.008

Recession curve of shock wave pressure.

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0.010

300

13. The application of peridynamics for ice modeling

VS

αα

Plate

c Pr

Pi c FIGURE 13.31

Sketch of Taylor’s plate theory.

4.2 Numerical modeling and analysis As shown in Fig. 13.32, the sides of the three-dimensional ice plate are rigidly fixed, and there is a pre-hole in the center of the ice plate. The results of numerical simulation are shown in Fig. 13.33. The figure depicts the area of destruction in the central part of the ice sheet clearly. At t ¼ 21.2348 ms, the process of damage ends and the red area reaches a

FIGURE 13.32

Sketch of calculation model.

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4. Research on numerical simulation

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 13.33 Destruction process of ice for explosive 1.2 m under water. (A) t ¼ 2.0189 ms. (B) t ¼ 6.4357 ms. (C) t ¼ 11.4842 ms. (D) t ¼ 15.1677 ms. (E) t ¼ 18.1789 ms. (F) t ¼ 21.2348 ms.

maximum radius of 5.0 m. Other three locations of the explosive underwater are also simulated. The processes of damage are similar to Fig. 13.33 and the numerical results are given in Table 13.3. The forms of crack propagation are also simulated employing the present model, as shown in Fig. 13.34. Circular and radial cracks appear in the process of damage. A photograph of the test field (Liu et al., 2010) is shown in Fig. 13.35 and the diagram in Fig. 13.36 depicts simply the form of cracks in the test field (Meng et al., 2013). The results of numerical TABLE 13.3 Comparison of the results. The depth underwater of the explosive (m)

The radius of the damaged area in experiment (m)

The radius of the damaged area in numerical model (m)

Error (%)

1.2

5.22

5.0

4.21

1.5

5.54

5.3

4.33

1.8

5.26

5.1

3.04

2.1

4.70

4.5

4.25

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13. The application of peridynamics for ice modeling

(a)

FIGURE 13.34 (C) t ¼ 21.2348 ms.

(b)

(c)

Forms of crack propagation. (A) t ¼ 4.3761 ms. (B) t ¼ 11.5613 ms.

FIGURE 13.35

Damaged area of the test field (Liu et al., 2010).

FIGURE 13.36

Diagram of cracks.

simulation are consistent with experimental results. Therefore, it is reasonable that the peridynamic method can be used to simulate successfully ice fragmentation generated by a shock wave in the event of an underwater explosion.

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5. Numerical simulation of continuous icebreaking

303

This study (Wang, 2018) also investigates the factors affecting the radius of breaking ice, such as the mass of the explosive, the thickness of the ice, and the location of the explosive underwater, by employing the numerical model.

5. Numerical simulation of continuous icebreaking based on hybrid modeling method Icebreaker is one of main equipment for scientific investigations and operations at polar region. This section simulates the continuous icebreaking process by the method coupling of FEM and peridynamics.

5.1 Calculation model To enhance the computational efficiency, the ice sheet is modeled by coupling peridynamics and finite element method. The side of ice sheet which contacts with the icebreaker is set as a free boundary, while the other three sides are remote boundaries. The ship simulated in this section is a certain type of icebreaker. The numerical model of ship and ice sheet is shown in Fig. 13.37. Only the bow of the ship is modeled.

5.2 Numerical results Fig. 13.38 shows the results obtained from numerical simulation. After several repeated processes, a channel with a width slightly larger than the width of the ship is opened in the ice sheet.

FIGURE 13.37

Model of ice sheet and ship.

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13. The application of peridynamics for ice modeling

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

FIGURE 13.38 The numerical simulation of icebreaker sailing in level ice. (A) t ¼ 0.98 s. (B) t ¼ 1.95 s. (C) t ¼ 3.91 s. (D) t ¼ 8.79 s. (E) t ¼ 10.74 s. (F) t ¼ 15.63 s. (G) t ¼ 20.51 s. (H) t ¼ 29.30 s.

Fig. 13.39 shows the ice load during the interaction between the ship and ice sheet. Based on the numerical data, the maximum value of the total ice force in each cycle is between 4.0 and 5.0 MN. According to Lindqvist’s empirical formula (Lindqvist, 1989) for ice force, the resistance calculated is about 3.519 MN, which is in the same order of magnitude as maximum ice force obtained by the numerical simulation.

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References

305

FIGURE 13.39

Ice load during the icebreaking. (A) total ice load. (B) ice load in x-direction. (C) ice load in y-direction. (D) ice load in z-direction.

Scholars also studied the influence factors of ice load. According to the numerical results (Lu, 2018), the maximum ice load under different navigational speeds and ice thicknesses has a similar trend to that calculated by the Lindqvist’s empirical formula.

References Askari, E., Xu, J.F., Silling, S.A., 2006. Peridynamic analysis of damage and failure in composites. In: Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibition. No. 2006-88, Reno, Nevada. Croasdale, K.R., Cammaert, A.B., 1994. An improved method for the calculation of ice loads on sloping structures in first-year ice. Power Technol. Eng. 28 (3), 174e179. del Rosso, L., Celli, M., Ulivi, L., 2016. New porous water ice metastable at atmospheric pressure obtained by emptying a hydrogen-filled ice. Nat. Commun. 7, 13394. Dykins, J.E., 1967. Tensile properties of sea ice grown in a confined system. In: Proceedings of the International Conference on Low Temperature Science. Physics of Snow and Ice, vol. 1. Institute of Low Temperature Science, Sapporo, Japan, pp. 523e537.

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13. The application of peridynamics for ice modeling

Dykins, J.E., 1968. Tensile and flexural properties of saline ice. In: Riehl, N., Bullemer, B., Engelhardt, H. (Eds.), Physics of Ice. Plenum Press, New York, NY, pp. 251e270. Geers, T.L., Hunter, K.S., 2003. An integrated wave-effects model for an underwater explosion bubble. J. Acoust. Soc. Am. 111 (4), 1584e1601. Ha, Y.D., Bobaru, F., 2011. Dynamic brittle fracture captured with peridynamics. Proc. ASME Int. Mech. Eng. Congr. Expos. 78 (6), 1156e1168. Huang, Y., Zhu, C., Wang, L., Cao, X., et al., 2016. A new phase diagram of water under negative pressure: the rise of the lowest-density clathrate s-III. Sci. Adv. 2, e1501010. Huang, Y., Zhu, C., Wang, L., 2017. Prediction of a new ice clathrate with record low density: a potential candidate as ice XIX in guest-free form. Chem. Phys. Lett. 186e191. Ji, S., Di, S., Li, Z., Bi, X., 2013. Discrete element modelling of interaction between sea ice and vertical offshore structures. Eng. Mech. 30 (1), 463e469. Jia, B., Ju, L., Wang, Q., 2019. Numerical simulation of dynamic interaction between ice and wide vertical structure based on peridynamics. CMES-Comput. Model. Eng. Sci. 121 (2), 501e522. Jia, B., Wang, Q., Li, W.J., Wang, J.W., 2020. Peridynamic simulation of the interaction between sea ice and cylindrical structure. J. Harbin Eng. Univ. 41 (1), 52e59. Jones, S.J., 1982. The confined compressive strength of polycrystalline ice. J. Glaciol. 28, 171e177. Kilic, B., Madenci, E., 2009. Structure stability and failure analysis using peridynamic theory. Int. J. Non Lin. Mech. 44 (8), 845e854. Langway, C.C., 1958. Ice Fabrics and the Universal Stage. Technical Report, vol. 62. U.S. Army Snow and Permafrost Research Establishment, Corps of Engineers, Wilmette, Illinois. Li, Z.J., Jia, Q., Huang, W.F., et al., 2009. Characteristics of ice crystals air bubbles and densities of fresh ice in a reservoir. J. Hydraul. Eng. 40 (11), 1333e1338. Liang, C.C., Tai, Y.S., 2006. Shock responses of a surface ship subjected to noncontact underwater explosions. Ocean Eng. 33 (5e6), 748e772. Lindqvist, G., 1989. A straightforward method for calculation of ice resistance of ship. Proc. POAC 722e735. Liu, M.H., 2016. Numerical Simulation of Sea Ice Damage Based on Peridynamics. Harbin Engineering University, Heilongjiang. Liu, D.C., Meng, W.Y., Zhang, D.X., et al., 2010. Analysis of the dynamic response of the icecap structure under the action of explosion wave. J. North China Inst. Water Conserv. Hydro. Power 31 (4), 25e28. Liu, R.W., et al., 2018. Simulation of ship navigation in ice rubble based on peridynamics. Ocean Eng. 148, 286e298. Liu, M.H., Wang, Q., Lu, W., 2016. Peridynamic simulation of brittle-ice crushed by a vertical structure. Int. J. Naval Archit. Ocean Eng. 9 (2), 209e218. Liu, R., Yan, J.L., Li, S.F., 2020. Modeling and simulation of iceewater interactions by coupling peridynamics with updated Lagrangian particle hydrodynamics. Comput. Part. Mech. 241e255. Lu, W., 2017. The Study of the Numerical Simulation Method of Peridynamic Based on the Bending Fracture of Sea Ice. Harbin Engineering University, Heilongjiang. Lu, X.K., 2018. Calculation of Ice Load for Icebreaker Based on Coupling of Peridynamic and Finite Element Method. Harbin Engineering University, Heilongjiang. Ma¨a¨tta¨nen, M., Marjavaara, P., Saarinen, S., et al., 2011. Ice crushing tests with variable structural flexibility. Cold Reg. Sci. Technol. 67 (3), 120-12. Matsui, T., Hirata, M., Yagasaki, T., Matsumoto, M., et al., 2017. Communication: hypothetical ultralow-density ice polymorphs. J. Chem. Phys. 147 (9), 91101.

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References

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Meng, W.Y., Liu, X., Hu, J.Q., 2013. Numerical simulation and experiment research of ice blasting based on shaped charge technology. J. North China Univ. Water Res. Electr. Power 34 (3), 44e47. Michel, B., Ramseier, R.O., 1971. Classification of river and lake ice. Can. Geotech. J. 8 (1), 36e45. Millot, M., Coppari, F., Rygg, J.R., Correa Barrios, A., et al., 2019. Nanosecond X-ray diffraction of shock-compressed superionic water ice. Nature 569, 251e255. Peyton, H.R., 1966. Sea Ice Strength. University of Alaska Report UAG-182. Geophysical Institute, Fairbanks, AK, USA, p. 187. Ro¨ttger, K., Endriss, A., Ihringer, J., Doyle, S., Kuhs, W., 1994. Lattice constants and thermal expansion of H2O and D2O ice Ih between 10 and 265 K. Acta Crystallogr. B 50, 644e648. Schulson, E.M., 1990. The brittle compressive fracture of ice. Acta Metall. Mater. 38 (10), 1963e1976. Schulson, E.M., 2001. Brittle failure of ice. Rev. Mineral. Geochem. 51 (1), 1839e1887. Song, Y., Yan, J., Li, S., Kang, Z., 2019a. Peridynamic modeling and simulation of ice craters by impact. CMES-Comput. Model. Eng. Sci. 121 (2), 465e492. Song, Y., Yu, H., Kang, Z., 2019b. Numerical study on ice fragmentation by impact based on non-ordinary state-based peridynamics. J. Micromech. Mol. Phys. 4 (01), 1850006. Taylor, G.T., 1941. The Pressure and Impulse of Submarine Explosion Waves on Plates. Ministry of Home Security Report, FC235. Timco, G.W., Frederking, R.M.W., 1990. Compressive strength of sea ice sheets. Cold Reg. Sci. Technol. 17 (3), 227e240. Timco, G.W., Frederking, R.M.W., 1991. Seasonal Compressive Strength of Beaufort Sea Ice Sheets. Ice-Structure Interaction. Timco, G.W., Frederking, R.M.W., 2010. A review of the engineering properties of sea ice. Cold Reg. Sci. Technol. 60 (2), 107e129. Wang, Y., 2018. Research on Numerical Simulation of Ice Breaking by Underwater Explosion Based on Improved Peridynamics. Harbin Engineering University, Heilongjiang. Wang, Q., Wang, Y., et al., 2018. Simulation of brittle-ice contacting with stiffened plate with peridynamics. J. Ship Mech. 22 (03), 339e352. Wang, C., et al., 2020. Comparative study of the ReaxFF and potential models with density functional theory for simulating hexagonal ice. Comput. Mater. Sci. 177, 109546. Xiong, W.P., et al., 2020. Analysis of shadowing effect of propeller-ice milling conditions with peridynamics. Ocean Eng. 195. Xue, Y.Z., Liu, R.W., et al., 2019. Numerical simulations of the ice load of a ship navigating in level ice using peridynamics. CMES-Comput. Model. Eng. Sci. 121 (2), 523e550. Xue, Y.Z., Lu, X.K., Wang, Q., et al., 2018. Simulation of three-point bending test of ice based on peridynamic. J. Harbin Eng. Univ. 039 (004), 607e613. Ye, L.Y., 2018. Study on Prediction Method of Dynamic Behavior and Propeller Strength under Propellor-Ice Contact. Harbin Engineering University, Heilongjiang. Ye, L.Y., et al., 2017. Propeller-ice contact modeling with peridynamics. Ocean Eng. 139, 54e64. Ye, L.Y., et al., 2019. Peridynamic solution for submarine surfacing through ice. Ships Offshore Struct. 1e15. Yue, Q., Guo, F., Karna, T., 2009. Dynamic ice forces of slender vertical structures due to ice crushing. Cold Reg. Sci. Technol. 56 (2e3), 77e83.

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Further reading Frankenstein, G., Garner, R., 1967. Equations for determining the brine volume of sea ice from 0.5 C to 22.9 C. J. Glaciol. 6 (6), 943e944. Huang, Y., Shi, Q., Song, A., 2007. Model test study of the interaction between ice and a compliant vertical narrow structure. Cold Reg. Sci. Technol. 49 (2), 151e160. Iliescu, D., Baker, I., 2007. The structure and mechanical properties of river and lake ice. Cold Reg. Sci. Technol. 48 (3), 202e217. Kovacs, A., 1966. Sea Ice. Part 1. Bulk Salinity versus Ice Floe Thickness. Sea Ice. Part. Bulk Salinity versus Ice Floe Thickness. Schulson, E.M., Buck, S.E., 1995. The ductile-to-brittle transition and ductile failure envelopes of orthotropic ice under biaxial compression. Acta Metall. Mater. 43 (10), 3661e3668. Todd, F.H., 1961. Ship Hull Vibration. Princeton, E. Arnold.

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C H A P T E R

14 Fiber-reinforced composites modeling using peridynamics Masaaki Nishikawa, Naoki Matsuda, Masaki Hojo Department of Mechanical Engineering and Science, Kyoto University, Kyoto, Japan

O U T L I N E 1. Introduction

310

2. Peridynamics for composite materials 2.1 Theoretical background 2.2 Two different versions of PD model for composite ply Model (a): Oterkus-Madenci’s ply model including fiber and matrix bonds Model (b): Ghajari-Iannucci-Curtis’ ply model using continuous function of bond constants 2.3 Interlaminar bond and failure model

311 311 312

3. Numerical examples 3.1 Modeling of curvilinear fiber path 3.2 Multiple-site, multiple-type damage in laminated composites 3.3 Integrated framework for manufacturing and design of composites

316 316 320 321

4. Conclusions and future outlook

324

Acknowledgments

324

References

325

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00009-3

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312 314 315

© 2021 Elsevier Inc. All rights reserved.

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14. Fiber-reinforced composites modeling using peridynamics

1. Introduction Carbon fiber reinforced plastics (CFRPs) are increasingly being used in lightweight aircraft structures. CFRPs can be integrally molded into a monolithic structure by shaping an intermediate material, called prepreg, which is a fiber bed material impregnated with a semi-hardened resin. The strength properties of composite structures are not only determined by the properties of the constituent materials. It is possible to design and tailor the properties by layering intermediate materials during manufacturing. Thus, the barriers between material and structure design have been lowered. In conventional aircraft structural design, load transfer is controlled at the unit of structural members, such as the skin and stringer structure of the fuselage and the introduction of rib members into the wing structure, in order to optimize the load supported by the structure. In the case of composite materials, the characteristics of the anisotropic strength are utilized in the design, since the strength in the fiber direction and the orthogonal direction is different. Currently, the manufacturing of composite structures has greatly advanced through the automation of the layup process of prepreg materials, called automated tape laying (ATL) and automated fiber placement (AFP) techniques (Boisse, 2015). With the development of such manufacturing technology for composite materials, the fiber direction is allowed to be controlled and stacked for the placement of curved fibers, and the materials and processes are designed to prevent defects such as gaps/overlaps, fiber misalignments, and wrinkles during manufacturing. This is a new attempt to control the fiber orientation to form the structure. For example, it was recently shown by Lopes et al. (2010) that the initial buckling load under compressive loading can be made almost equivalent to that of a conventional structure without holes, by setting the fiber direction to optimize the load transfer. It has become possible to optimize the structural strength by controlling fiber orientation. It is important to establish a framework for understanding the effect of fiber orientation during the manufacturing process on structural strength, while investigating the failure mechanism based on the material microstructure during manufacturing. The stiffness and strength of carbon fiber differ by an order of magnitude from that of the resin material, so that loads in the aircraft structure transfer in the direction of the fibers, and the resin fails when accidental loads are applied in directions other than the fiber direction. The purpose of our study is to establish a method for understanding the structural strength with the resulting local fiber orientation determined during the manufacturing process. To this end, we developed a method to estimate the effect of local fiber orientation on the structural strength of composites from the viewpoint of the damage

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2. Peridynamics for composite materials

311

process. Combining these simulation techniques with advanced manufacturing technology opens up the possibility of designing novel composite structures with controlled arbitrary geometries and fiber orientations. This chapter introduces and reviews our numerical approaches based on peridynamics (PD) for the damage modeling with local fiber orientations, which has been investigated in our previous literature (Nishikawa, 2017; Nishikawa et al., 2017, 2019). Here we introduce three numerical examples. (1) Single-ply composite material with curvilinear fiber path (2) Multiple-site and multiple-type damage in laminated composites (3) An integrated framework of forming and mechanical analyses These numerical examples highlight the features of PD modeling that can accommodate arbitrary fiber orientation and multiple-type and multiple-site damage in the composites.

2. Peridynamics for composite materials The present study focuses on a new approach to model the damage in composites with arbitrary fiber orientation using PD models. Peridynamics is a physical approach to modeling the damage in solid materials proposed by Dr. Silling and his colleagues (Silling and Askari, 2005; Askari et al., 2006). The main advantage of the PD model is that it defines the constitutive law of the bonds and takes into account an energy-based criterion based on the fracture mechanics approach, similarly to cohesive zone modeling. By using the bond failure criterion, there is no need to predetermine the damage path. Therefore, the PD model is suitable for the simulation of the multiple-site and multiple-type damage (cracks, delamination, and fiber breakages) in composite laminates.

2.1 Theoretical background The bond-based PD model is expressed in the following integral form for equations of motion by Silling and Askari (2005). Z ru€ ¼ f ðx0  x; u0  uÞdH þ b (14.1) H

x0 and

where r is density, x are the positions of two material points, H is the neighborhood of x, u is the displacement, u€ is the acceleration, and b is the body force. f is the PD force defined by the following PD constitutive law. f ¼ cs

y0  y jy0  yj

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(14.2)

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where y is the position in the deformed configuration, and s is the mechanical stretch defined by s¼

jy0  yj  jx0  xj jy0  yj

(14.3)

The material constant c is modeled so that the elastic strain energy is equivalent to that in the conventional continuum mechanics. To deal with the bond failure, a critical stretch for s is defined and related to the fracture toughness values. In the bond-based PD theory, c is the only material parameter, and Poisson’s ratio is fixed to 1/4 or 1/3 in three-dimensional and plane-strain or plane-stress analyses, respectively, for isotropic materials. The statebased PD (Madenci and Oterkus, 2014) or PD differential operator (Madenci et al., 2016) can be used as alternative methods if we would like to employ a strict material model using elastic modulus and Poisson’s ratio.

2.2 Two different versions of PD model for composite ply For the fiber-reinforced composite materials, two types of models can be employed to determine the bond constant c and critical stretch s. Model (a): Fiber and matrix bond model: Oterkus and Madenci (2012) Model (b): Orthotropic continuum bond model: Ghajari et al. (2014) Model (a): Oterkus-Madenci’s ply model including fiber and matrix bonds For fiber-reinforced composites in a two-dimensional problem, Oterkus and Madenci (2012) obtained the material constants c for the fiber and matrix bonds in a composite ply by equating the strain energy density of the PD theory to that of classical continuum theory (Oterkus and Madenci, 2012; Kahraman et al., 2015).  cf þ cm f ¼ q c¼ (14.4) cm fsq Q11  Q22 24Q12 ptd2 ; cm ¼ ; V ¼ q 1PQ N ptd3 xqi Vq q¼1 2 E1 n12 E2 E2 E1 Q11 ¼ ; Q12 ¼ ; Q22 ¼ ; n12 ¼ n21 1  n12 n21 1  n12 n21 1  n12 n21 E2 cf ¼

(14.5)

(14.6)

where cf and cm are the bond constants for the fibers and matrix. Q11, Q12, Q22 are the orthotropic elastic parameters, and E1, E2, n12, n21 are the engineering material constants of unidirectional composite ply. xqi, Vq denote the initial length of the bond between material points q and i and II. New applications in peridynamics

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313

the volume of material point q. N is the number of material points within its horizon d, t is the thickness of ply, and 4 and q are the angles between the material points and the fiber orientation angle. For the intralaminar failure, the matrix failure was considered. In the isotropic PD model, the critical stretch s0 can be related to the critical energy release rate G0 of the material as follows (Silling and Askari, 2005; Oterkus and Madenci, 2012; Ghajari et al., 2014; Gerstle et al., 2005). sffiffiffiffiffiffiffiffiffiffiffi 5G0 Em s0 ¼ ; km ¼ 9km d 3ð1  2nm Þ Silling  Askari ð2005Þ; Oterkus  Madenci ð2012Þ (14.7)

sffiffiffiffiffiffiffi 2 G0 s0 ¼ 2 cm t d

(14.8)

Gerstle et al. ð2005Þ; Ghajari et al. ð2014Þ Two different equations were obtained by the difference in the definition of the PD strain energy density. The horizon d was set to 3.015 Dx, and in the present study, the distance Dx between the nearest neighboring points was set to 0.5 mm in the model shown in Fig. 14.1, as described later. Thus, the critical stretch was 0.01518 (Eq. 14.7) and 0.01752 (Eq. 14.8). The PD simulations used composite ply properties of CFRP T800S/ 3900-2B at room temperature: E1 ¼ 152 GPa, E2 ¼ 8.00 GPa, n12 ¼ 0.34, n21 ¼ 0.02, listed in JAXA Advanced Composites Database (Morimoto et al., 2015). Here, the restriction of the bond-based PD model, n12 was set to 1/3. The properties of epoxy matrix were set to the values as listed in Madenci and Oterkus (2014): Matrix elastic modulus Em ¼ 3.792 GPa, bulk modulus km ¼ Em ¼ 3.792 GPa, shear modulus Gm ¼ 3/8 Em ¼ 1.422 GPa in the bond-based PD, critical energy release rate G0 ¼ 2.37  103 MPa-m. 50 Velocity B.C.

10

50

Tensile direction

Hole (I = 10) Velocity B.C. FIGURE 14.1

10 (Unit: mm)

Model for an open-holed composite ply under tension.

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Model (b): Ghajari-Iannucci-Curtis’ ply model using continuous function of bond constants Ghajari et al. (2014) generalized the equations by Silling and Askari (2005) to include anisotropy in the bond constant, using associated Legendre functions. cð4Þ ¼

4 X

A2m;0 P02m ðcos 4Þ

(14.9)

m¼0

Constant coefficients A2m;0 are given as functions of c1 and c2 (Ghajari et al., 2014). The constant coefficients were selected so that the orientation dependence on the bond constant could be well reproduced. Using the equivalence of strain energy densities between PD theory and classical continuum theory, c1 and c2 were numerically determined by Ghajari et al. (2014) as 15:41Q11  7:41Q22 pd3 t 8:08Q22  0:08Q11 c2 ¼ pd3 t Q12 ¼ Q66 ¼ 0:059Q11 þ 0:274Q22 c1 ¼

(14.10) (14.11) (14.12)

For the intralaminar failure, the critical stretch s0 ð4Þ is orientationdependent and was determined as s20 ð4Þ ¼

4 X

B2m;0 P02m ðcos 4Þ

(14.13)

m¼0

s201 ¼

500½ð4GIc1  11GIc2 Þc1 þ ð112GIc1  72GIc2 Þc2    td4 71c21 þ 3168c1 c2 þ 944c22

(14.14)

s202 ¼

500½ð31:5GIc2  5GIc1 Þc1 þ ð11GIc2  4GIc1 Þc2    td4 71c21 þ 3168c1 c2 þ 944c22

(14.15)

Constant coefficients B2m;0 are given as functions of s201 and s202 (Ghajari et al., 2014). Here, the orientation dependence of Mode-I intralaminar fracture toughness GIc was considered. GIc1 and GIc2 are the fracture toughness in 0 and 90 directions. In the bond-based PD model, the fracture mode cannot be separated between Mode-I (crack opening mode) and Mode-II (shear fracture mode). Thus, it should be noted that the prediction based on the bond-based PD model underestimates the strain at damage initiation. GIc of 90 ply was set as GIc2 ¼ 140 J/m2 used in our previous literature for carbon/epoxy composites (Okabe et al., 2008). GIIc of 0 ply for splitting failure was used as GIc1: 633 J/m2 for T700S/ 2500 (Taketa et al., 2008), instead of GIc, because GIc of 0 ply was about

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50 kJ/m2, as reported using compact tension (CT) tests by Laffan et al. (2010). As reported in Ghajari et al. (2014), the condition 0.72 < GIc1/ GIc2 < 4.7 should be satisfied so that s201 and s202 are not less than zero. The restriction was introduced by the assumption of the bond-based PD model. Moreover, we assumed that the bond with less than 10 directions did not break in order to capture the fracture path clearly in the simulations.

2.3 Interlaminar bond and failure model We also included interlaminar bonds between the plies in the model (Killic et al., 2009; Yi-le et al., 2014), in order to deal with the elastic deformation and failure at the interlaminar region. We utilized interlaminar model given by Oterkus and Madenci (2012) as follows. cin ¼ cis ¼

Em ðnormal directionÞ tVq

2Gm 1  2  ðshear directionÞ  pt t 2 2 d þ t ln 2 d þ t2

(14.16) (14.17)

Here it is noted that the definition of the force by the interlaminar shear bond is defined as follows, instead of Eq. (14.2). f ¼ cis 4ðDxÞ2

y0  y jy0  yj

(14.18)

where f is the shear angle of the interlaminar shear bond (Oterkus and Madenci, 2012). It should be noted that the force is not the same as PD force density (force per unit volume). The area ðDxÞ2 where the force acts is multiplied in Eq. (14.18). The critical stretch sin and critical shear angle were set as follows in order to consider the failure of the bonds in normal and shear directions, respectively. sffiffiffiffiffiffiffiffiffi 2GIc sin ¼ (14.19) tEm sffiffiffiffiffiffiffiffiffi GIIc 4c ¼ (14.20) tGm Following the approach in Oterkus and Madenci (2012), we set GIc ¼ 3/4 GIIc ¼ G0, and as a consequence fc ¼ sin. In the present study sin ¼ fc ¼ 0.0833 was used.

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3. Numerical examples 3.1 Modeling of curvilinear fiber path As the first step, we attempted to simulate the damage in composite plate around an open hole under tensile loading, using a PD model in a two-dimensional problem, and investigated the effectiveness of applying the PD model to the damage in composites with curvilinear fiber reinforcement. Especially, two approaches for the PD analysis of fiberreinforced composite materials proposed by Oterkus and Madenci (2012) and Ghajari et al. (2014) were compared and the advantages and disadvantages of the PD models for the damage in composites with arbitrary fiber orientation were discussed. In the present study, the simulations were performed using in-house developed Fortran codes based on the reference codes published in Madenci and Oterkus (2014). Fig. 14.1 illustrates the open-hole tension model used in the simulations. The model size was 50 mm  70 mm. A circular hole with a radius of 5 mm was placed in the center of the model and analyzed under twodimensional planar stress conditions with a distance of 0.5 mm between adjacent material points in the x- and y-directions. As a boundary condition, a tensile rate of 1  101 mm/ms in the y direction was applied to the material points within 10 mm length at both ends. The time step was Dt ¼ 1  104 ms and the total number of steps was 105. The maximum applied tensile strain was 4%. In order to suppress numerical oscillation in the dynamic analysis, simulations were performed with artificial damping factor. (1) Straight fiber reinforcement model First, the results for a single composite ply with straight fiber reinforcement are shown. Two different PD theories (models (a) and (b)) were compared with varying fiber orientation angles of q ¼ 0 , 30 , and 45 . Figs. 14.2 and 14.3 present the numerical results for the straight fiber reinforcement, when the fiber orientation angles were varied. The Oterkus-Madenci’s ply model (model (a), Eq. 14.8) (Fig. 14.2) that represents fiber and matrix bonds could not capture the fracture path along the fiber direction in 30 and 45 directions, because the fiber bond between material points could not be clearly defined unless the bond orientation coincided with the fiber direction. In contrast, the results using GhajariIannucci-Curtis’ ply model (model (b)) (Fig. 14.3) showed clear fracture paths along the fiber direction. Therefore, model (b) including anisotropy of bond constants and bond failure criteria is favorable for simulating damage in composites with arbitrary fiber orientation. However, it should be noted that the simulation parameters (especially, artificial damping factor) in PD simulations affect the prediction of the

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3. Numerical examples

Tensile direction (y-direction)

(a) 0° ply

Damage did not occur along the fiber direction

(b) 30° ply

(c) 45° ply

FIGURE 14.2 Straight fiber reinforcement model with model (a): Oterkus and Madenci (2012). (“DISPY” in the figure denotes the displacement in y-direction.). Damage occurred along the fiber direction

(a) 0° ply

(b) 30° ply

(c) 45° ply

FIGURE 14.3 Straight fiber reinforcement model with model (b): Ghajari et al. (2014). (“‘DMG” in the figure denotes the damage variable whose value ranges from 0 to 1.).

fracture pattern in composites, since the material failure is determined based on the calculated displacement field. Slight changes of fiber orientation angle (less than 10 direction) were not sufficiently captured in the simulations. This problem can be solved by improving the discretization of the model. (2) Curvilinear fiber reinforcement model Automation of manufacturing techniques has been actively investigated to reduce the cost and improve the quality of aircraft and automobile structures made of carbon fiber composites. In recent years, attempts have been made to increase the strength of the manufactured structures while maintaining their stiffness, by orienting the fiber direction in a curvilinear manner using automatic stacking technology (Lopes et al., 2010; Gu¨rdal et al., 2008). In the present study, a damage analysis method based on PD theory was proposed as a basis to discuss the damage in composites with arbitrary fiber orientations. Here, the single-ply composites with a curvilinear fiber path were simulated, as shown in Fig. 14.4. PD simulations were performed on a

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T0 = 0°, T1 = 80°

50 Velocity B.C.

10

T0 = 0°, T1 = 40° 50

Tensile direction

Hole (I = 10) Velocity B.C.

10 (Unit: mm)

FIGURE 14.4

T0 = 0°, T1 = 0°

Curvilinear fiber reinforcement model.

single composite ply with local fiber orientation under an open-hole tension condition. The geometry and boundary conditions of the numerical model is the same as that shown in Fig. 14.1. Three simple fiber paths were simulated in order to evaluate the results for various simulation conditions. The curved fiber path was defined by the following equation, which was also analyzed in the literature of Lopes et al. (2010). qðxÞ ¼ T0 þ ðT1 þ T0 Þ

jxj d

(14.21)

where T0 ; T1 are the fiber orientation angles at the center of the plate (x ¼ 0) and the change in the fiber orientation angle at a distance d away. Here, three patterns were analyzed: T0 ¼ 0 (fixed), and T1 ¼ 0 , 40 , and 80 (varied). Here, it should be noted that the fiber orientation angle is not continuously varied in the model, because the angle of the bond takes only discrete values: q ¼ 0 , 26.6 , 45 , 63.4 , and 90 . Ideally, the restriction can be released to some extent if we reduce the value of Dx and increase the value of d/Dx, while the computational model becomes larger. Thus, the PD model (b) (Eq. 14.9) is advantageous for application to models dealing with curvilinear fiber reinforcement due to the discretization problem. This is the main difference between the models (a) and (b). In this section, we present the results for model (b). Fig. 14.5 shows the distribution of displacement and damage variables obtained from the PD simulations. The damage variable is calculated by taking the ratio of the actual bond stretch to the critical stretch, defined as a value between 0 and 1, and taking the volume average for all bonds connecting to the material point. The location of the damage is cyclic, as shown in Fig. 14.5. Although the calculations were sometimes unstable

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Displacement in y-direction [mm]

319

Damage variable 0 (intact), 1 (completely damaged)

(a) T0 = 0°, T1 = 0°

(b) T0 = 0°, T1 = 40°

(c) T0 = 0°, T1 = 80°

FIGURE 14.5

Results for curvilinear fiber-reinforced model.

under some analytical conditions, it was generally possible to predict cracks as clear discontinuities with a damping factor of c ¼ 1.0 without judging the damage of bonds with fiber orientation less than 10 in the analysis. It should be noted that the excessive vibrations generated by the dynamic analysis are likely to lead to bond failure, since the bond strain is used to determine the failure. As with the particle method, discontinuities are difficult to deal with explicitly and may lead to a decrease in accuracy. Therefore, tuning of the simulation parameters is necessary to obtain valid results in PD simulations. For the comparison of the three fiber paths, the numerical results reproduced the change in the damage path due to the relaxation of the stress concentration in the case of curvilinear fiber reinforcement. As described by Lopes et al. (2010), the stress concentration due to the circular hole was relaxed in the case of curvilinear fiber reinforcement, and

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the damage occurred farther away from the hole, which is also the case in the PD simulation results. Therefore, the PD model can effectively reproduce the damage in the composite with arbitrary fiber orientation.

3.2 Multiple-site, multiple-type damage in laminated composites In order to investigate the damage tolerance of CFRP structures, it has become important to understand the effect of damage initiation under loading at the laminate scale. As a next step, the model was extended to muti-ply laminated composites and the effectiveness of the PD simulation to reproduce the multiple-type, multiple-site damage in composite laminates with arbitrary fiber orientations was investigated. In this section, we describe the details of our previous simulations (Nishikawa et al., 2017) using multi-ply composite laminates. The openhole tension condition was modeled in the same procedure as in Fig. 14.1. In this simulation with multi-ply composite laminates, the inplane model size reduced by 1/10 (model length ¼ 7 mm, model width ¼ 5 mm, hole radius ¼ 0.5 mm) in order to accurately capture the transverse crack spacing in the 90 ply due to discretization issues. In addition, the time step was reduced by 1/10 while keeping the same tensile velocity applied to the model. The PD model (b) (Eq. 14.9) was used in this simulation. Fig. 14.6 presents the numerical results for laminated configuration of [45/90/45]. In the figure, the results are shown when a tensile strain of 1.2% is applied. As shown in the figure, splitting failure occurred along the fiber direction in the outer plies, and the damage in the 90 ply spread from the edge of the open hole to the region of that splitting failure. The damage pattern around the hole was similar to the experimental results described in Yi-le et al. (2014). The transverse cracking damage in the 90 ply appeared at multiple sites, and thus the present simulation sufficiently addressed the stress recovery behavior through the interlaminar region of the laminates. As seen in the example, the simulation well reproduced the multipletype, multiple-site damage in laminated composites with arbitrary fiber orientation. Furthermore, the PD simulation does not require a predetermined damage path. This would be a great advantage over other damage simulation techniques in laminated composites.

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3. Numerical examples

ply 1 = 45° ply

ply 2 = 90° ply

Splitting failure occurred along the fiber direction

Damage spread along the splitting failure

ply 3 = 45° ply

FIGURE 14.6 Numerical results of the damage in composite laminates.

3.3 Integrated framework for manufacturing and design of composites Finally, an attempt was made to integrate the analysis of the manufacturing process with the mechanical analysis considering the fiber orientation after manufacturing. A two-step analysis procedure of forming and mechanical analyses is desirable to evaluate the deformation and damage in the fabricated composite structure based on local fiber orientations. Here we present a numerical example of the composite with the APPLY (Advanced Placed Ply) configuration (pseudo-woven fiber placement pattern), originally proposed by Nagelsmit et al. (Prof. Gu¨rdal’s research group) (Nagelsmit et al., 2011; Nagelsmit, 2013). In order to increase production rate and efficiency, automated fiber placement techniques have been developed for the advanced manufacturing technologies of monolithic, large-scale CFRP composite structures. Automated placement techniques are expected to allow manufacturers to control fiber orientation, eliminate manufacturing defects, and reduce manufacturing variability for the structural integrity of composite structures. A new concept of the automated placement, called AP-PLY, is a special fiber architecture that combines through-thickness reinforcements

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with the fiber placement process. This concept is effective for improving the impact damage tolerance of composite structures due to the throughthickness reinforcements. Manufacturing preforms with AP-PLY configuration also improves the formability of the preform to the complicated structure as well as dry woven fabrics. This section aims to establish simulation techniques for composites with AP-PLY configuration. Since AP-PLY configuration introduces complicated ply architecture after composites forming process, leading to a complicated damage process in composites, it is attempted to combine the forming simulation and PD simulation techniques. First, the forming simulation is conducted using LS-DYNA (Nishi and Hirashima, 2013), a commercially available dynamic explicit finite element analysis solver, in order to obtain the nodal coordinates of AP-PLY configuration. The configuration is transferred to the modeling of impact simulation using PD theory. In order to integrate the analysis of the manufacturing process and mechanical analysis, the following efforts were made. The schematic of the simulation procedure is illustrated in Fig. 14.7. (1) A method was developed for evaluating the basic material properties, which is important in the forming process of intermediate prepreg materials. Especially, friction test and analysis of thermoplastic CFRP tapes in the forming process were developed (Abo et al., 2019). The details are described in our previous literature. (2) The deformation of AP-PLY laminated tapes due to the surface pressure load during the forming process was calculated by utilizing forming simulation with LS-DYNA. A method for evaluating fiber orientation was developed by simulating the molding and shaping at meso-scale. From this procedure, the initial configuration of AP-PLY composites was obtained. (3) Bond-based PD simulation was performed by assigning the fiber orientations obtained from the evaluation of the forming process.

Obtain the initial configuration of AP-PLY using forming simulation by LS-DYNA Bond-based peridynamics simulation using wavy ply and inter-ply matrix bonds (a) Flowchart of the integrated simulation.

FIGURE 14.7

(b) Schematic of the PD model.

Schematic of the integrated framework using forming and PD simulations.

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3. Numerical examples

The simulation allows the damage under mechanical loading to be analyzed by taking into account the evaluated fiber orientation in the material. In this process, the fiber orientation obtained in the procedure (2) was used to determine the orientation of the tape material in the AP-PLY composites. The model of Oterkus and Madenci (model (a)) was used to represent the bond within the tapes after manufacturing. The inter-ply properties were defined to adjust the stiffness and elongation at break of the inter-ply bond between the tapes, and thus, the post-manufacturing matrix properties were defined as the properties of the inter-ply bonds. Since AP-PLY configuration cannot define the inter-ply region as clearly as the conventional laminates, the PD model is useful for dealing with wavy plies and their connections. Fig. 14.8 presented a numerical example of the AP-PLY composite. Although the accuracy and experimental verification of the developed simulations is not sufficient, it is possible to perform the simulations taking into account the fiber orientation inside the material, which varies with the manufacturing process. It is important to try to evaluate the material and structural design of composite materials based on local fiber orientation in an integrated manner, and it is necessary to establish a comprehensive analysis system to evaluate the entire process from the manufacturing process to mechanical properties for future composites manufacturing and design. *Magnified deformation in z-direction

(a) Forming simulation using LS-DYNA.

(b) PD model for wavy plies.

Impact location

(c) Contour of z-displacement.

FIGURE 14.8

Results of the impact simulation for the AP-PLY composite.

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4. Conclusions and future outlook The present study is the first step to deal with the multiple-type, multiple-site damage in composite laminates with arbitrary fiber orientations using a PD theory. Overall, it is useful in analyzing the damage initiation in composites with local fiber orientation. On the other hand, there is a limitation in the physical properties that can be inputted in the bond-based PD theory, which makes it difficult to assign detailed properties and fracture characteristics to the theory. There is also the problem of discretization, where the accuracy of the PD simulation is not obtained for slight changes in fiber orientation. The model discretization problem can be improved by adopting a model with large degrees of freedom. In addition, the PD model is considered to be equivalent to a low-order element in finite element analysis, and its accuracy under bending load may be inferior. In particular, the accuracy of the simulations needs to be further verified when applied to impact loads and other applications. These improvements are the subject of the future study. PD simulation was also used to address the multiple-type, multiplesite damage in multi-ply laminated composites (splitting failure and transverse cracks). The simulated results reproduced characteristic features of damage phenomena in laminated composites. The PD simulation will offer a great advantage in that it does not require pre-defined fracture paths. For the strength evaluation, other types of damage (including fiber breakage and/or fiber kinking damage under compression and so on) must be considered in the simulations. The effects of manufacturing defects, such as initial crack (slit, missing tow), gap and overlaps (Marouene et al., 2017), fiber bridging, etc., are also left as challenging topics in the state-of-the-art composite manufacturing, and in principle, the PD simulation will be able to address these effects if the accuracy of predicting the fracture path for a slight change in fiber orientation angle is improved. Although the calculations with the models of curvilinear fiber path presented in this chapter are concerned with single-layered plates, Lopes et al. (2010) also predicted that ply staggering in the lamination process improves the damage initiation strain. We would like to study the application of this kind of problem in the future.

Acknowledgments The authors acknowledge the support of JSOL Engineering Business Division (Dr. Nishi et al.) for the forming simulations of LS-DYNA. We also thank Prof. Oterkus (University of Strathclyde) for the fruitful discussion on the PD model. The present chapter introduced the contents and the figures reproduced from our previous literature (Nishikawa, 2017; Nishikawa et al., 2017, 2019) and part of the simulated results were recalculated from the original publications using the updated PD codes.

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References

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References Abo, Y., Tanaka, Y., Nishikawa, M., Iwashita, M., Yamada, K., Kawabe, K., Nishi, M., Matsuda, N., Hojo, M., 2019. Simulation and experiments for mechanical properties dominating the press molding using CFRTP preforms. In: Proc. 22nd International Conference on Composite Materials (ICCM-22) (Paper No. 523). In: http://www.iccmcentral.org/Proceedings/ICCM22proceedings/. Askari, E., Xu, J., Silling, S., 2006. Peridynamic analysis of damage and failure in composites. In: Proc. 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2006-88. Boisse, P., 2015. Advances in Composites Manufacturing and Design. Elsevier, pp. 79e92. Gerstle, W., Sau, N., Silling, S., 2005. Peridynamic modeling of plain and reinforced concrete structures. In: Proc. 18th International Conference on Structural Mechanics in Reactor Technology (SMiRT 18), SMiRT18-B01-2, pp. 54e68. Ghajari, M., Iannucci, L., Curtis, P., 2014. A peridynamic material model for the analysis of dynamic crack propagation in orthtropic media. Comput. Methods Appl. Mech. Eng. 276, 431e452. Gu¨rdal, Z., Tatting, B.F., Wu, K.C., 2008. Variable stiffness composite panels: effects of stiffness variation on the in-plane and buckling response. Compos. A 39, 911e922. Kahraman, T., Yolum, U., Guler, M.A., 2015. Implementation of peridynamic theory to LSDYNA for prediction of crack propagation in a composite lamina. In: 10th European LS-DYNA Conference 2015, Wu¨rzburg, Gemany. Killic, B., Agwai, A., Madenci, E., 2009. Peridynamic theory for progressive damage prediction in center-cracked composite laminates. Compos. Struct. 90, 149e151. Laffan, M.J., Pinho, S.T., Robinson, P., Iannucci, L., 2010. Measurement of the in situ ply fracture toughness associated with mode I fibre tensile failure in FRP. Part I: data reduction. Compos. Sci. Technol. 70, 606e613. Lopes, C.S., Gu¨rdal, Z., Camanho, P.P., 2010. Tailoring for strength of composite steered-fibre panels with cutouts. Compos. A 41, 1760e1767. Madenci, E., Oterkus, E., 2014. Peridynamic Theory and its Applications. Springer, New York. Madenci, E., Barut, A., Futch, M., 2016. Peridynamic differential operator and its applications. Comput. Methods Appl. Mech. Eng. 304, 408e451. Marouene, A., Legay, P., Boukhili, R., 2017. Experimental and numerical investigation on the open-hole compressive strength of AFP composites containing gaps and overlaps. J. Compos. Mater. 51 (26), 3631e3646. Morimoto, T., Sugimoto, S., Katoh, H., Hara, E., Yasuoka, T., Iwahori, Y., Ogasawara, T., Ito, S., 2015. JAXA Advanced Composite Database. JAXA Research and Development Memorandum, JAXA-RM-14-004 (in Japanese). (Revised in 2018, JAXA-RM-17-004). https://jaxa.repo.nii.ac.jp/. Nagelsmit, M., Kassapoglou, C., Gu¨rdal, Z., 2011. AP-PLY: a new fibre placement architecture for fabric replacement. SAMPE J. 47 (2), 36e45. Nagelsmit, M.H., 2013. Fibre Placement Architectures for Improved Damage Tolerance. PhD thesis of Delft University of Technology. https://repository.tudelft.nl/. Nishi, M., Hirashima, T., 2013. Approach for dry textile composite forming simulation. In: Proc. 19th International Conference on Composite Materials (ICCM-19), pp. 7486e7493. In: http://www.iccm-central.org/Proceedings/ICCM19proceedings/. Nishikawa, M., Matsuda, N., Hojo, M., 2017. Modeling of multiple-type, multiple-site damage in composite laminates using peridynamics theory. In: Proc. 32nd Annual Technical Conference of American Society for Composites. ASC).

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Nishikawa, M., Matsuda, N., Hojo, M., 2019. Peridynamic modeling of the impact-induced damage in composite materials with AP-PLY configuration. In: Presentation in 4th International Symposium on Automated Composites Manufacturing (ACM4), Montreal, Canada. Nishikawa, M., 2017. Examination of peridynamics simulation for damage initiation of composite materials with local fiber orientation. In: Proc. Computational Mechanics Conference of the Japan Society of Mechanical Engineers (JSME CMD207). https://doi.org/ 10.1299/jsmecmd.2017.30.010 in Japanese. Okabe, T., Nishikawa, M., Takeda, N., 2008. Numerical modeling of progressive damage in fiber reinforced plastic cross-ply laminates. Compos. Sci. Technol. 68, 2282e2289. Oterkus, E., Madenci, E., 2012. Peridynamics analysis of fiber-reinforced composite materials. J. Mech. Mater. Struct. 7, 45e84. Silling, S.A., Askari, E., 2005. A meshfree method based on the peridynamics model of solid mechanics. Comput. Struct. 83, 1526e1535. Taketa, I., Okabe, T., Kitano, A., 2008. A new compression-molding approach using unidirectionally arrayed chopped strands. Composites A 39, 1884e1890. Yi-le, H., Yin, Y., Hai, W., 2014. Peridynamic analytical method for progressive damage in notched composite laminates. Compos. Struct. 108, 801e810.

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15 Phase fieldebased peridynamics damage model: Applications to delamination of composite structures and inelastic response of ceramics Pranesh Roy, Anil Pathrikar, Debasish Roy Centre of Excellence in Advanced Mechanics of Materials, Indian Institute of Science, Bangalore, Karnataka, India

O U T L I N E 1. Introduction

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3. Phase fieldebased PD damage model for composites delamination 333 3.1 Governing equations 333 3.2 Bulk and interface constitutive models 334 4. Numerical illustrations on composites delamination 4.1 Mode I delamination 4.2 Mode II delamination 4.2.1 End loaded split test 4.2.2 End notched flexure test

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4.3 Mixed (I/II) mode delamination 4.3.1 Fixed ratio mixed-mode test

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5. DE damage model using phase fieldebased PD 5.1 Phase fieldebased PD damage formulation using complementary energy density 5.2 Rate of internal energy density 5.3 Equations of motion for spherically symmetric geometry and loading 5.3.1 Constitutive correspondence

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6. Numerical illustrations

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7. Concluding remarks

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1. Introduction Interlaminar damage or interface delamination has attracted significant research interest, as it is a major cause of failure in laminated composite structures used in various industries (e.g., the aerospace industry). Delamination may depend on many factors including fiber orientation, placement of plies, nature of loading, etc. As delamination involves a gradual decrease in the stiffness of the interface which ultimately leads to failure, predictive models based on damage mechanics and fracture mechanics have been exploited. Fracture mechanicsebased approaches, e.g., the virtual crack closure technique (VCCT), J-integral based approach, etc., use Griffith’s criterion, which has a few limitations, e.g., the lack of prior knowledge of the crack propagation path, additional criteria for spontaneous emergence, propagation, and branching of a crack, interaction among cracks, mesh dependency, etc. Notable among the damage mechanicsebased approaches are the pioneering work of Dugdale (1960) on the plastic yielding in crack tip, Barenblatt’s (1962) theory of brittle cracks, and the work of Hillerborg et al. (1976), which allows the stress in the crack face to depend on the crack opening displacement. Models combining fracture and damage mechanics have also been proposed, the cohesive zone model (CZM) being an example. All these models face roadblocks due to the choice of appropriate crack tracking methods, computational complexity, and the extensive tuning of material parameters for different loading conditions. Another area where these damage and fracture mechanicsebased approaches find application is the ballistic resistance of layered composite

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329

ceramic-metal armours which, due to their lightweight and high impact resistance, have extensive industrial usefulness. When a projectile hits the brittle ceramic front layer, it gets eroded and the ductile metal backing layer absorbs the kinetic energy through plastic deformation. Damage mechanics approaches are available, e.g., Ashby and Sammis (1990) and Johnson and Holmquist (1999) who have used phenomenological damage mechanics on the basis of the strength of intact and fractured ceramic materials and pressure-volume relations. Deshpande and Evans (2008) (DE) have proposed a micromechanics-based damage model with the following features: effect of crack density on the stiffness, a crack growth law, change of mechanism depending on the confining pressure, effect of grain size on the response, dilatation due to microcracking, etc. However, all these models suffer from spurious mesh dependency when cracks develop. Considering these limitations, we apply a peridynamics (PD) damage model incorporating phase field for both the delamination and ceramics damage problems and study its consequences. In this chapter, we will present a phase fieldebased PD model to study delamination problems in composite laminates. Delamination and subsequent fracture are modeled exploiting the cohesive behavior that phase field naturally offers by its design, even as proper choices of the degradation function and the critical energy release rate are required. We treat the bulk and the interface as different materials characterized by distinct degradation functions and critical energy release rates. The advantage of the proposed approach may be summarized as follows. (a) Different modes of loading (modes I and II and the mixed mode) can be treated as problems that differ only in boundary conditions, thus eliminating the need for a special treatment; (b) the empirical interaction criterion for mixed-mode delamination is eliminated; (c) it is suitable for problems involving general boundary conditions and weak layer locations; (d) intralaminar damage can be modeled; and (e) spontaneous emergence and propagation of cracks can be obtained without additional crack tracking algorithms. In demonstrating the efficacy of this proposal, a few validation exercises involving mode I, mode II, and mixed-mode delamination cases are carried out. We also reformulate the DE damage model for ceramics within the phase fieldebased PD setup with a view to eliminating some of the limitations of the DE model. This replaces the damage variable in the DE model, which evolves according to a phenomenological crack growth law in the microscale, by a phase field like evolution equation. Here, a gradient damage term and an associated length scale parameter are incorporated, which prevent an unphysical localization of damage as the mesh is refined. A phase fieldebased macroscopic description is able to show branching of macrocracks at correct speeds. As the damage evolves, the strain energy degrades. In the DE model, expressions of

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complementary energy density in different regimes are given which include terms characterizing degradation. Therefore, to account for degradation, the classical phase field equations are derived in terms of the complementary energy density. For validation of the proposed phase fieldebased PD-DE damage model, we solve a cavity expansion problem and compare our results with those reported in Deshpande and Evans (2008). In order to achieve this, 3D PD phase field equations of motion are dimensionally reduced to write 1D equations while taking advantage of the spherical symmetry of the structure and boundary conditions. While material parameters from the complementary energy density in the DE constitutive model are used in the PD via constitutive correspondence, parameters in the damage evolution law proposed in DE are eliminated and new parameters (e.g., a length scale corresponding to the gradient of the phase field and a mobility constant) are introduced. Our simulations are in close agreement with the results furnished in Deshpande and Evans (2008). The rest of the chapter is organized as follows. In Section 2, brief recaps of the cohesive zone model (CZM) and the DE ceramic constitutive model are presented. A phase fieldebased PD damage model for delamination of composites is furnished in Section 3. Results of a few numerical simulations for various modes of delamination are reported in Section 4. DE damage model using phase fieldebased PD is formulated in Section 5. To demonstrate the working of the proposed model, a few numerical simulations are presented in Section 6 on a dynamic cavity expansion problem and the results are compared with the DE model predictions. Finally, a few concluding remarks are offered in Section 7.

2. Review of cohesive zone model (CZM) and Deshpande-Evans (DE) model First, a brief review of the CZM is furnished for completeness. Following this, the DE model is recapitulated based on Deshpande and Evans (2008).

2.1 Cohesive zone model (CZM) CZM uses concepts from damage and fracture mechanics to model progressive damage and failure of interfaces. It requires a predefined crack and crack propagation path to capture the delamination process. The constitutive model for the interface is typically given in terms of a relation between traction and separation (displacement jump) at the interface (see Alfano and Crisfield, 2001). Traction initially increases with

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2. Review of cohesive zone model (CZM)

separation followed by a gradual decrease. Although CZM can successfully characterize the main features of the delamination process, it faces a few difficulties; e.g., (a) it requires an ad hoc high penalty stiffness to obtain the pre-crack behavior; (b) the peak value of traction in the tractionseparation (TS) curve has to be appropriately tuned to avoid numerical problems; (c) the softening part of the TS curve is fraught with numerical convergence issues (Turon et al., 2007); and (d) an empirical interaction criterion is required to model mixed-mode delamination (see Benzeggagh and Kenane, 1996).

2.2 Deshpande-Evans (DE) constitutive model In the DE constitutive model, an array of microcracks each with radius a and two wings of length l each are considered to evolve in a linear elastic medium subjected to principal stresses s1
s3 , with two wings aligned along the s1 direction and radius with an angle j along the s1 direction. The constitutive behavior of ceramics is divided into three triaxiality regimes. While regime I is characterized by no relative sliding of the faces of the inclined flaw, regimes II and III correspond to frictional sliding and loss of contact along the faces of the inclined crack, respectively. In the DE constitutive model, the distinction among regimes is realized through a stress triaxiality factor l ¼ ssme . Here, sm ¼ trðsÞ=3 is the mean stress and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi se ¼ ð3=2Þsdev : sdev the von-Mises effective stress where sdev ¼ s ð1 =3ÞtrðsÞI is the deviatoric stress. Stress triaxiality l and mode I stress intensity factor KI for regimes I, II, and III are shown in Table 15.1. Expressions for the parameters A, B, C, and E are furnished in Table 15.2. TABLE 15.1 Stress triaxiality l and mode I stress intensity factor KI for regimes I, II, and III. Regime I

Regime II

l
 AB

  ðB =AÞ < l
AB C2 A2

KI ¼ 0

I pKffiffiffiffi pa

¼ Asm þ Bse

Regime III   l > AB C2 A2 I pKffiffiffiffi pa

 1=2 ¼ C2 s2m þ E2 s2e

TABLE 15.2 Expressions for the parameters A, B, C, and E. A ¼ c1 ðc2 A3 c2 A1 þc3 Þ (a)

C¼Aþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g aðD=D0 Þ1=3 (c)

B ¼ pc1ffiffi ðc2 A3 þc2 A1 þc3 Þ

BC E ¼ pffiffiffiffiffiffiffiffiffiffiffi (d) 2 2 C A

3

(b)

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15. Phase fieldebased peridynamics damage model

TABLE 15.3 Expressions for the parameters c1 , c2 , c3 , A1 , A2 , and b. c1 ¼

1 3=2 p2 a3=2 ½ðD=D0 Þ1=3 1þðb=aÞ

h i2 c2 ¼ 1 þ 2 ðD=D0 Þ1=3  1

qffiffih i 1=2 A1 ¼ p b3 1 þ m2 m (d)

(a) !

(

2=3

D0 1D2=3

A3 ¼ A1

(b)

h i2 c3 ¼ 2a2 p2 ðD=D0 Þ1=3  1 (c)

) 1=2

ð1þm2 Þ þm ð1þm2 Þ1=2 m

(e)

b ¼ 0:1 (f)

We now provide c1 , c2 , c3 , A1 , A2 , and b in Table 15.3. Note that m is the friction coefficient. Based on lattice trapping and stress corrosion mechanisms, a crack growth rate is proposed in the DE constitutive model which is presented next. h pffiffiffiffiffiffiffiffiffiffiffi i l_ ¼ min l_0 ðKI =KIC Þm ; G=r0 (15.1) Here, l is the length of the wing crack, G the shear modulus of the uncracked ceramic, r0 the mass density, m the rate sensitivity exponent ð10
m
20Þ, KIC the mode I fracture toughness of the ceramic, and l_0 the reference crack growth rate at KI ¼ KIC . The upper bound in l_ is applied assuming that the crack cannot grow faster than the shear wave speed in an uncracked material. Using l, initial damage (D0 ) and current damage (D) variables are defined as: 4 D0 ¼ pðaaÞ3 f 3 4 D ¼ pðl þ aaÞ3 f 3

(15.2) (15.3)

Here, f is the crack density (number of cracks per unit volume). Now, expressions for the complementary energy density for different regimes are presented in Table 15.4.

TABLE 15.4 Expressions for complementary energy density for different regimes. Regime I

 1 W ¼ W0 ¼ 4G

2s2 3 e

 2 þ3ð12yÞ 1þy sm (a)

Regime II

2 0 W ¼ W0 þ 4a3pD Gð1þyÞðAsm þ Bse Þ (b)

Regime III

 2 2  2 2 2 (c) 0 W ¼ W0 þ 4a3pD Gð1þyÞ C sm þ E se

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3. Phase fieldebased PD damage model

In Table 15.4, W0 is the complementary energy density for the uncracked ceramic. Under high confining pressure, plastic deformation is the dominant failure mechanism in ceramic materials instead of brittle pl

damage. The plastic strain rate ε_ ij is given by: 8 dev  ðn1Þ 3s se > > > > < 2s0 s0

ε_ ¼  ε_ 0 > > > 3 ε_ 0 ð1nÞ=n sdev > : 2 ε_ t s0 pl

pl

if ε_ e < ε_ t (15.4) otherwise

ε_ 0 is the reference strain rate, n the strain  rate sensitivity exponent, ε_ t the pl

critical plastic strain rate, and s0 εe the flow stress at an equivalent

 pl pl plastic strain εe . s0 εe is expressed as:  M 

sY pl s0 ¼ (15.5) 1 þ εe =εY 2

Here, sY is the uniaxial yield strength, εY the plastic strain at s0 ¼ sY , and M is the strain-hardening exponent. As ceramics typically undergo small elastic strains, the DE constitutive model is employed using a relation between the objective rate of Cauchy stress and the rate of deformation which makes the model hypoelastic. An additive decomposition rate of deformation is used. d ¼ de þ dp

(15.6)

3. Phase fieldebased PD damage model for composites delamination The phase fieldebased PD governing equations and constitutive relations are furnished in this section.

3.1 Governing equations For a detailed description of the governing equations, refer to the companion chapter on phase fieldebased PD theory. Considering quasistatic loading conditions, the phase fieldebased PD equations of motion can be expressed as: Z n o T  T0 dV 0 þ f ¼ 0 (15.7) HðxÞ

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15. Phase fieldebased peridynamics damage model

Z HðxÞ

 P  P0 dV 0  p ¼ 0

(15.8)

Here, T is the PD force vector state, f the body force vector, P the conjugate state corresponding to relative phase field scalar state S ½xhxi ¼ s0  s. The expression for p is given as: p ¼ 2sjþ nle 

Gc ð1  sÞ 2l

(15.9)

Using constitutive correspondence, T and P can be expressed as: 1

Thxi ¼ wðjxjÞsK x

 1 P hxi ¼ wðjxjÞv$ xK

(15.10) (15.11)

where,

     s ¼ s2 þ h lhtrðεnl ÞiþI þ 2mεnlþ þ lhtrðεnl ÞiI þ 2mεnl v ¼ 2Gc lGS

(15.12) (15.13)

3.2 Bulk and interface constitutive models The bulk behavior of laminates is represented through an orthotropic constitutive model. Considering the basis vectors along the orthotropic directions of the laminate and assuming plane-strain conditions, the nonzero components of stress are related to the nonzero components of strain in the following matrix-vector form (Voigt notation): e feεg eg ¼ ℂ (15.14) fs where

2

e g ¼ f s11 fs

s22

s12 gT

feεg ¼ f ε11

ε22

ε12 gT

1  n23 n32 6 E E L 2 3 6 6 e ¼ 6 n12 þ n13 n32 ℂ 6 6 E E L 3 1 4 0

n21 þ n31 n23 E2 E3 L 1  n31 n13 E3 E1 L 0

(15.15) 3 0 7 7 7 7 0 7 7 5 2G12

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(15.16)

(15.17)

335

4. Numerical illustrations on composites delamination

and L is given by L¼

1  n12 n21  n23 n32  n31 n13  2n12 n23 n31 E1 E2 E3

(15.18)

e is symmetric if the following conditions are satisfied: Note that ℂ Eij Eji ¼ ni nj

i; j˛f1; 2; 3g

no summation over i and j

(15.19)

For the present purpose, without loss of generality, we have not considered evolution of phase field in the bulk material. In order to implement Eq. (15.14), we need to use components of the nonlocal small strain εnl . The interface is chosen to be an isotropic material which undergoes damage as phase field evolves. The degradation function is considered bilinear with two independent parameters (see Roy et al., 2017a,b) as: for 0
s
a b gðsÞ ¼ s a for a
s
1 gðsÞ ¼ b þ

(15.20)

1b ðs  aÞ 1a

with a; b˛½0; 1.

4. Numerical illustrations on composites delamination The capability of the proposed model is demonstrated through a few numerical simulations on mode-I, mode-II, and mixed-mode delamination cases in this section. Displacement controlled quasi-static loading and plane-strain conditions are assumed. To solve the phase fieldebased PD governing equations, we used a staggered algorithm following Cazes and Moe¨s (2015) which offers significant computational simplicity. Riemann sum approximations are used to write various integral terms in the discretized form. For a detailed description of discretization and implementation of boundary conditions through boundary patches, we refer to Breitenfeld et al. (2014).

4.1 Mode I delamination A double cantilever beam (DCB) of carbon fiber reinforced epoxy laminate (T300/977-2) is considered (see Fig. 15.1). Following Camanho

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15. Phase fieldebased peridynamics damage model

FIGURE 15.1 Double cantilever beam (DCB) test.

and Da´vila (2002), we choose the elastic material properties of the bulk and interface; these are provided in Tables 15.6 and 15.7. The interface degradation function parameters are also given in Table 15.7. The geometrical properties of the specimen are reported in Table 15.5, where L is the length, B the thickness, H the total height, and a0 the initial crack length. A pair of opposing point loads are applied at top and bottom of the left face of the DCB (see Fig. 15.1). As displacements increase, the interface phase field parameter evolves resulting in delamination. Contour plot of the growth of the phase field parameter with increasing load steps is shown in Fig. 15.2. We also present the reaction force versus relative

TABLE 15.5 Geometrical properties. L (mm)

H (mm)

B (mm)

a0 (mm)

150

3.96

20

55

TABLE 15.6 Bulk material properties. E11 (GPa)

E22 ¼ E33 (GPa)

G12 ¼ G13 (GPa)

n12 ¼ n13

n23

150

11

6

0.25

0.45

TABLE 15.7 Interface material properties. Gc (N/m)

E (Pa)

n

a

b

352

5  107

0.25

0.5

0.03

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4. Numerical illustrations on composites delamination

337

FIGURE 15.2 Spatial variation of phase field: applied y-displacement ðuy Þ at left boundary (A) 1:4  103 m (B) 2:9  103 m and (C) 6:9  103 m.

FIGURE 15.3 Reaction force versus relative displacement curve.

displacement (twice the applied displacement) plot in Fig. 15.3 and compare it with the experimental evidence given in Camanho and Da´vila (2002). This shows a good agreement.

4.2 Mode II delamination Delamination under interlaminar in-plane sliding shear force is referred to as Mode II delamination. We consider an end loaded split (ELS) test and an end notched flexure (ENF) test, which are reflective of such delamination. II. New applications in peridynamics

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15. Phase fieldebased peridynamics damage model

FIGURE 15.4 End loaded split (ELS) test.

4.2.1 End loaded split test The model geometry of the ELS test setup is furnished in Fig. 15.4, where L is the beam length, B the thickness, H the height, and a0 the initial crack length. The displacement of the right boundary is specified as zero and the leftmost point of the bottom boundary is subjected to a displacement controlled point load P (see Fig. 15.4). The model geometry parameters, bulk properties, and interfacial properties are presented in Table 15.8 through 15.10. The contour plot for the progression of the phase field is shown in Fig. 15.5. We compare the reaction force versus applied displacement plot with experimental data given in Chen et al. (1999) which shows an acceptable match (see Fig. 15.6). 4.2.2 End notched flexure test The ENF test setup is shown in Fig. 15.7. L, B, H, and a0 , respectively, denote the length, width, total height, and initial crack length of the beam. The beam is simply supported with a displacement-controlled point load of magnitude P applied at the midpoint of the top face. The geometry parameters, bulk properties, and interfacial properties are indicated in Tables 15.11e15.13. The phase field evolution is shown in Fig. 15.8. We

TABLE 15.8 Geometrical properties. L (mm)

H (mm)

B (mm)

a0 (mm)

105

3.05

24

60

TABLE 15.9 Bulk material properties. E11 (GPa)

E22 ¼ E33 (GPa)

G12 ¼ G13 (GPa)

n12 ¼ n13

n23

100

8

6

0.27

0.45

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4. Numerical illustrations on composites delamination

TABLE 15.10

339

Interface material properties.

Gc (N/m)

E (Pa)

n

a

b

856

1  1011

0.25

0.85

0.05

FIGURE 15.5 Spatial variation of phase field: applied y-displacement ðuy Þ at left boundary (A) 2:9  103 m (B) 1:69  102 m and (C) 2:14  102 m.

FIGURE 15.6

Reaction force versus displacement curve.

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15. Phase fieldebased peridynamics damage model

FIGURE 15.7

End notched flexure (ENF) test.

TABLE 15.11 Geometrical properties. L (mm)

H (mm)

B (mm)

a0 (mm)

102

3.12

25.4

39.3

TABLE 15.12 Bulk material properties. E11 (GPa)

E22 ¼ E33 (GPa)

G12 ¼ G13 (GPa)

n12 ¼ n13

n23

122.7

10.1

5.5

0.25

0.45

TABLE 15.13 Interface material properties. Gc (N/m) 1719

E (Pa) 5

 1010

n

a

b

0.25

0.75

0.05

compare the reaction force versus applied displacement curve with the experimental data (see Chen et al., 1999), which once again demonstrates that our model is capable of simulating mode-II delamination (see Fig. 15.9).

4.3 Mixed (I/II) mode delamination If normal and tangential forces act together in a composite laminate, then the failure mode becomes quite complex. CZM treats this problem as a combination of mode I and mode II and uses an additional interaction criterion which needs to be respected. However, the present model dispenses with any such criterion and can simulate mixed mode naturally. We present a numerical simulation on fixed ratio mixed-mode (FRMM) test next.

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4. Numerical illustrations on composites delamination

341

FIGURE 15.8 Spatial variation of phase field: applied y-displacement ðuy Þ at mid-point (A) 1:2  103 m (B) 4:7  103 m and (C) 6:2  103 m.

FIGURE 15.9

Reaction force versus displacement curve.

4.3.1 Fixed ratio mixed-mode test The model geometry is furnished in Fig. 15.10. The geometrical, bulk material and interfacial properties are listed in Tables 15.14 through 15.16. The progressive failure of the interface is shown in Fig. 15.11. The comparison of the reaction force versus the applied displacement with the experimental data from Chen et al. (1999) is presented next which shows that almost all the experimental data points are in reasonable agreement with our simulation (see Fig. 15.12).

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15. Phase fieldebased peridynamics damage model

FIGURE 15.10 Fixed ratio mixed-mode test. TABLE 15.14 Geometrical properties. L (mm)

H (mm)

B (mm)

a0 (mm)

105

3.1

24

45

TABLE 15.15 Bulk material properties. E11 (GPa)

E22 ¼ E33 (GPa)

G12 ¼ G13 (GPa)

n12 ¼ n13

n23

130

8

6

0.27

0.45

TABLE 15.16 Interface material properties. Gc (N/m)

E (Pa)

n

a

b

434

1  109

0.25

0.62

0.1

FIGURE 15.11 Spatial variation of phase field: applied y-displacement ðuy Þ at left boundary (A) 4:9  103 m (B) 7:9  103 m and (C) 1:39  102 m.

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5. DE damage model using phase fieldebased PD

FIGURE 15.12

343

Reaction force versus displacement curve.

5. DE damage model using phase fieldebased PD We are now ready to formulate the DE damage model using our phase fieldebased PD. This eliminates a few limitations of the DE model. (a) The DE model does not have a length scale parameter and hence may suffer from spurious mesh dependency. (b) The crack growth law proposed in the micro-scale based on lattice trapping and stress corrosion mechanisms has the 1D shear wave speed as an upper bound on the crack growth rate. However, some experimental studies (Fineberg et al., 1992) have shown that the crack may branch at speeds lesser than the shear wave speed. Also, due to branching, microcracks will interact in a manner more complicated than what is proposed in the DE model. (c) The effect of change of orientation of wing cracks in the microscale as the stress state evolves is not considered. (d) The homogenization scheme employed to arrive at a scalar damage variable may not be appropriate. A new macroscopic damage evolution rule is proposed in this section through phase field which involves a gradient of the phase field term and a length scale parameter, thereby replacing the crack growth law in the DE constitutive model. The length scale parameter associated with the gradient term helps eliminate mesh dependency. Considerable computational simplicity may also accrue when the model is used in the phase fieldebased PD framework. In this section, first, we formulate the phase fieldebased PD damage model with complementary energy density assuming no dissipation. Then, we extend the formulation incorporating plasticity. Further on, considering a spherically symmetric geometry and loading conditions, modified equations of motion are derived. II. New applications in peridynamics

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15. Phase fieldebased peridynamics damage model

5.1 Phase fieldebased PD damage formulation using complementary energy density As the constitutive equations in the DE model are given in terms of the complementary energy density function jce ðs; sÞ, we need to use the Legendre transform of je ðε; sÞ: c

je ðs; sÞ ¼ supε ðs: εðs; sÞ  je ðε; sÞÞ

(15.21)

Here, s and ε are the nonlocal stress and strain measures, respectively. c With the understanding that ε ¼ D s jce , variation of je is given as: c

dje ¼ ds: ε þ s: dε  dje

(15.22)

We may now write the variation dL of the Lagrange density using the nonlocal measures as:  c dL ¼ djk  ds: ε þ s: dε  dje  djs (15.23) where, js is the nonlocal fracture energy density. Using Hamilton’s principle, one may readily derive the following equations of motion. Z (15.24) fT½xhxi  T½x0 hxigdV 0 þ f ¼ ru€ U

Z

U

ðP ½xhxi  P ½x0 hxiÞdV 0  p ¼ 0

(15.25)

Here, T and P are the PD force states conjugate to the relative displacement vector state and the relative phase field scalar state, respectively. p is given by: p ¼ D s js  D s jce

(15.26)

5.2 Rate of internal energy density PD power balance equation can be expressed as (see Silling and Lehoucq, 2010):       e ¼ W sup U e e þ W abs U (15.27) K_ U where K is the kinetic energy, W abs the absorbed power, and W sup the e of the material body U. Their explicit exsupplied power on a part U pressions are: Z   1 e _ ydV _ (15.28) K U ¼ e ry$ U2

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5. DE damage model using phase fieldebased PD

  e ¼ W abs U

Z

Z e U

U

345

Z Z

Z

  0 0 0 0 _ T $ y_  y_ dV dV þ e P s_  s_ dV dV þ e psdV U U

Z

U

Z

Z    0 0 0 e _ W sup U ¼ e T $ y_  T $y_ dV dV þ e f $ ydV U U=e U U Z Z 

þ e P s_0  P0 s_ dV 0 dV U U=e U

(15.29)

(15.30)

Following Silling and Lehoucq (2010), the first law of thermodynamics is written as:           e ¼ W sup U e þ K_ U e þW U e e þQ U E_ U (15.31)     e rate is the heat supplied and W U e the rate of plastic work. where, Q U     e and W U e are stated next: Explicit expressions of Q U Z Z Z   0 0 e Q U ¼ e qðx ; x; tÞdV dV þ e sðx; tÞdV (15.32) U U=e U U Z   e W U ¼ e cg_ p dV (15.33) U

Here, sðx; tÞ is the heat source, qðx0 ; x; tÞ the rate of heat transport from to x and c the scalar microscopic stress conjugate to the rate of equivalent plastic strain g_ p . Using Eqs. (15.27) and (15.31), we write:         e ¼ W abs U e þW U e e þQ U (15.34) E_ U

x0

5.3 Equations of motion for spherically symmetric geometry and loading Spherical cavity expansion under internal pressure is used as a benchmark problem, which can present some important features related to a penetration problem. Therefore, we intend to solve such a problem using the proposed PD phase field model. To do so, we need to recast the governing equations assuming spherical symmetry in the structure and loading conditions. First, we represent all the field variables in the component form along radial (r), inclination (4) and azimuth (f) directions using spherical coordinates. Due to the spherical symmetry, the components of the displacement vector along 4 and f directions are zero, i.e., u4 ¼ uf ¼ 0 and the partial derivatives of displacement u with respect to 4 and f are also zero. The non-zero component ur of the displacement vector depends only on the radial coordinate r. As T and P are two-point

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functions, they depend on both r and r0 . We denote the components of T as T r , T 4 , and T f , respectively. T may then be written as: T ¼ T r ðr; 4; f; r0 ; 40 ; f0 Þer þ T 4 ðr; 4; f; r0 ; 40 ; f0 Þe4 þ T f ðr; 4; f; r0 ; 40 ; f0 Þef (15.35) The reference configuration of the spherically symmetric body is given by: U : ¼ ½r1 ; r2   ½0; p  ½0; 2p. Power balance equation is utilized to derive the equations of motion under spherical symmetry. We denote the components of the deformation vector as yr ¼ xr þ ur , y4 ¼ x4 , and yf ¼ xf . xr , x4 , and xf represent the components of the position vector. Writing the absorbed power for the entire body W abs ðUÞ, in the spherical coordinate system (see Eq. 15.29), we get: Z r2 Z r2 h Z r2 i

  2 0 2 0 0 0 e e e _ 2 dr W abs ðUÞ ¼ psr T1 y_ r  T2 y_r þ P s_  s_ r dr r dr þ 4p r1

r1

r1

(15.36) where, e1 ¼ T e2 ¼ T

Z

2p 0

Z

Z

Z

p

0

2p

0

e¼ P

0

Z

2p 0

2p

p

0

Z

p

0

Z

0

Z Z

2p

Z

p

Z

0

p

0

T $ e0 r sin 40 sin 4d40 df0 d4df

(15.37)

T $ er sin 40 sin 4d40 df0 d4df

(15.38)

2p 0

Z

p

P sin 40 sin 4d40 df0 d4df

(15.39)

0

The expressions for the rate of kinetic energy and the supplied power for the entire body in spherical coordinates are furnished below: Z 2p Z p Z r2 r€ yr y_ r r2 sin 4drd4df (15.40) K_ ðUÞ ¼ 0

W

sup ðUÞ ¼

0

Z

2p 0

Z

r1

0

p

Z

r2

r1

fr y_r r2 sin4drd4df

(15.41)

Now, substituting Eqs. (15.36), (15.40), and (15.41) in the power balance Eq. (15.27), we get: Z r2 Z r2 Z r2 h i

  e 1 y_0  T e2 y_ þ P e s_0  s_ r0 2 dr0 r2 dr 4p r€ yr y_ r r2 dr þ T r r Z rr12 Z rr12 r1 _ 2 dr ¼ 4p þ4p psr fr y_r r2 dr (15.42) r1

r1

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5. DE damage model using phase fieldebased PD

347

Using a change of variables r4r0 and the fact that y_r and s_ are independent and arbitrary, we may write: Z r2   e2  T e0 1 r0 2 dr0 þ 4pfr ¼ 4pry€ (15.43) T r r1

Z

r2  r1

 eP e0 r0 2 dr0  4pp ¼ 0 P

(15.44)

These are the Euler-Lagrange equations which govern the dynamics of the dimensionally reduced system. 5.3.1 Constitutive correspondence In order to utilize the complementary energy density function proposed in Deshpande and Evans (2008), we adopt the constitutive correspondence approach (Silling et al., 2007). Using the equivalence between internal energy rates from classical and PD under homogeneous deformation, we relate the PD force states with the classical stress. We introduce the following definition of rr-component of the nonlocal deformation gradient (Frr ) as: Z   2 1 Frr ¼ w y0r  yr xr r0 dr0 Krr (15.45) H

Similarly, nonlocal gradient of the phase field (GS ) is written as: Z 1 2 GS ¼ wðs0  sÞxr r0 dr0 Krr (15.46) R

H

2 02 0 H wxr r dr

Here, Krr ¼ is the rr-component of the shape tensor. Apart from Frr , expressions for the other nonzero components of the nonlocal deformation gradient are presented below: F44 ¼ Fff ¼

yr xr

(15.47)

The classical rate of internal energy for the entire body may be written as (see also Silling and Lehoucq, 2010): Z 2p Z p Z r2   E_ c ðUÞ ¼ P: F_ þ Q$Vs_ þ Rs_  V$q þ s þ cg_ p r2 sin 4drd4df 0

0

r1

(15.48)

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15. Phase fieldebased peridynamics damage model

Expressing E_ c in terms of the non-zero components of the rate of _ we get: deformation gradient F, Z 2p Z p Z r2  _ _r E c ðUÞ ¼ Prr F_ rr þ P44 F_ 44 þ Pff F_ ff þ Qr ðVsÞ 0

Z ¼ 4p

r2 

r1

r1

0

 þRs_  V$q þ s þ cg_ p r2 sin 4drd4df

 _ r þ Rs_  V$q þ s þ cg_ p r2 dr Prr F_ rr þ 2P44 F_ 44 þ Qr ðVsÞ (15.49)

In Eq. (15.49), we have used the fact that no field variable is a function of 4 and f. The third step follows from the relation F_ 44 ¼ F_ ff . Now, substituting the nonlocal deformation gradient F in place of F, we may write: 2 Z r2 Z r2  2 y_ 1  4 E_ c ðUÞ ¼ 4p wPrr Krr xr y_0 r  y_r r0 dr0 þ 2P44 r x r r1 r1 Z þ

r2 r1

3

!

(15.50)

1 2 wQr Krr xr s_0  s_ r0 dr0 þ Rs_  V$q þ s þ cg_ p 5r2 dr

We now write the expression for Q as (see Eq. 15.32): Z 2p Z p Z r2 Z 2p Z p Z r2 2 Q ðUÞ ¼  qr0 sin 40 dr0 d40 df0 r2 sin 4drd4df 0 0 0 0 r r 1 1 Z 2p Z p Z r2 sr2 sin 4drd4df þ 0 0 r1 Z r2 Z r2 Z r2 2 e qr0 dr0 r2 dr þ 4p sr2 dr (15.51) ¼ r1

r1

r1

We have used the following abbreviation in the above equation: Z 2p Z p Z 2p Z p e q¼ qd40 df0 d4df (15.52) 0

0

0

0

W may be written as (see Eq. 15.33): Z 2p Z p Z r2 Z W ðUÞ ¼ cg_ p r2 sin 4drd4df ¼ 4p 0

0

r1

r2

r1

II. New applications in peridynamics

cg_ p r2 dr

(15.53)

6. Numerical illustrations

349

The rate of internal energy E_ may now be obtained using Eqs. (15.36), (15.51), and (15.53) as: Z ¼

r2

r1

Z

E_ ðUÞ ¼ W abs ðUÞ þ Q ðUÞ þ W ðUÞ Z r2 i

 2 0 2 0 0 0 e e e _ 2 dr psr T1 y_ r  T2 y_r þ P s_  s_ r dr r dr þ 4p r1 Z r2 Z r2 Z r2   2 e qr0 dr0 r2 dr þ 4p (15.54)  s þ cg_ p r2 dr

r2 h  r1

r1

r1

r1

Comparing the classical (E_ c ) and PD (E_ ) rates of internal energy from Eqs. (15.50) and (15.54), respectively, one may identify the following correspondence relations.

Here, a ¼

R

e 1 ¼ 4pwPrr K1 xr T rr

(15.55)

e2 ¼ 4pwPrr K1 xr  8pP44 T rr axr

(15.56)

e ¼ 4pwQr K1 xr P rr

(15.57)

R¼p

(15.58)

02 0 H r dr .

6. Numerical illustrations In order to assess the performance of the proposed phase fieldebased PD-DE damage model, we carry out a few numerical simulations. The dynamic cavity expansion problem is chosen, as it can capture some key features of impact scenarios in ceramics, for instance, the behavior of ceramics at low and high velocities, especially the propagation of elastic wave, damage and plasticity fronts. A constant velocity is applied at the inner surface of the sphere. The elastic wave with the compressive front generates and propagates to the outward direction and reflects back as a tensile wave front from the outer surface of the sphere. Under low impact velocities, due to moderate compression, the damage front initiates from the inner surface of the sphere in the wake of the elastic wave front and propagates outwards. When the elastic wave reflects back from the outer surface as a tensile wave, a damage front emerges and propagates inwards. However, under high impact velocities, the behavior is significantly different. Just after the impact, due to the presence of high compression, plasticity dominates brittle failure. Therefore, instead of a damage front, a plastic front emerges and propagates outwards in the wake of the elastic wave front. However, when the elastic wave reflects back from the outer surface as a tensile wave, the damage front initiates and starts propagating in the inward direction. II. New applications in peridynamics

350

15. Phase fieldebased peridynamics damage model

FIGURE 15.13

Geometry of spherical cavity and boundary conditions.

In the numerical simulations, taking advantage of the spherical geometry and loading conditions, only 1D equations are solved. Discretizing the 1D domain into a finite number of PD particles, each of which is associated with a finite volume, the integral terms are approximated through Riemann sums. For a detailed exposition on the numerical implementation of the PD equations, we refer to Breitenfeld et al. (2014). The inner and outer radii of the cavity (see Fig. 15.13) are 1 and 2 m, respectively. The effect of two velocities (100 and 1000 m/s) on the response is examined. Spatial variations of triaxiality, damage, equivalent plastic strain, and pressure time history are solved using the phase fieldebased PD and compared with the results reported in Deshpande and Evans (2008). The latter results are based on the FEM. The material parameters considered in the simulations are provided in Table 15.17. TABLE 15.17 Material properties for Al2 O3 (see Deshpande and Evans, 2008). dðmmÞ

GðGPaÞ

n

   r0 kg m3

pffiffiffiffi KIC ðMPa = m Þ

sY ðGPaÞ

εY

38

146

0.2

3700

3

4

0.002

M

n

0.1

10

a

  ε_ 0 s1

  ε_ t s1

l0 ðmm =sÞ

m

b

1

106

100

10

0.1

g pffiffiffi 1 2

2

  f m3

g1

m 0.75

1

II. New applications in peridynamics

1012

D0 0.01

7. Concluding remarks

351

While implementing the equations of motion, we apply a correction for dissipative relaxation in Eq. (15.44) as (see Roy et al., 2017a,b): Z r2

 eP e0 r0 2 dr0  4pp ¼ s_ (15.59) P e M r1 e is the mobility constant. In all the simulations presented here, Here, M e ¼ 0:003. we consider M From Fig. 15.14A, we observe that, for low velocity (v0 ¼ 100 m/s), the damage front initiates from the inner surface and propagates from the inner to the outer surface. After a reflection from the outer free surface, a tensile wave front develops because of which another damage front initiates at the outer surface and propagates toward inner surface. Both phase fieldebased PD and conventional DE damage model predict similar response. Fig. 15.14B shows how triaxiality is distributed in the domain at different time instants. As values of triaxiality tend to the positive side with progressing time, one may anticipate that compression damage at the inner surface happens in regime II. The time history of pressure at the inner surface is shown in Fig. 15.14B. We observe that a quick rise in pressure happens initially, followed by a slow increase. The sudden rise is due to the initial elastic response and the slow increase happens as the damage evolves. For the high-velocity case, i.e., v0 ¼ 1000 m/s, one may notice from Fig. 15.14D that there is no damage front initiation from the inner surface. This is due to the high compression at the inner surface; the state of the ceramic remains in regime I. After the reflection of the wave from the outer surface, a tensile damage front initiates from the outer surface and propagates toward inner surface. From Fig. 15.14E, one observes that, in the damage front, the value of triaxiality is mostly positive. Fig. 15.14E shows the pressure time history plot at the inner surface. Here, we see a sudden rise in pressure followed by a slow decrease with time.

7. Concluding remarks Toward exploiting the inherently cohesive character of the phase field, we have used a phase fieldeenhanced PD model to solve delamination problems and inelastic response of ceramic materials. Other than a relief from the smoothness requirements of the approximating fields, the present approach also eliminates certain aspects of empiricism in the conventional methods, for example, by rationalizing the bond-snapping criterion and treating delamination problems under various loading conditions (i.e., those inducing delamination under specific modes or a combination of them) as just problems with different boundary

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352

15. Phase fieldebased peridynamics damage model

FIGURE 15.14 (A) Spatial variation of damage D for v0 ¼ 100 m/s; (B) spatial variation of stress triaxiality l for v0 ¼ 100 m/s; (C) temporal variation of normalized pressure ðp =sf Þ for v0 ¼ 100 m/s; (D) spatial variation of damage D for v0 ¼ 1000 m/s; (E) spatial variation of stress triaxiality l for v0 ¼ 1000 m/s; (F) temporal variation of normalized pressure ðp =sY Þ pl for v0 ¼ 1000 m/s; (G) spatial variation of equivalent plastic strain εe for v0 ¼ 1000 m/s.

II. New applications in peridynamics

References

353

conditions. The advantages that accrue include the ability of the model to accommodate fragmentation of the material body. Problems that involve impact loading in metal-ceramic composites, intralaminar damage, and so on could be among possible future applications of this method. Extensions of the method to treat ductile damage and fracture or to include additional micromechanical features for higher predictive fidelity (e.g., micropolarity or other geometry-driven features related to defects) will also be of interest in future works.

References Alfano, G., Crisfield, M.A., 2001. Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues. Int. J. Numer. Methods Eng. 50 (7), 1701e1736. Ashby, M.F., Sammis, C.G., 1990. The damage mechanics of brittle solids in compression. Pure Appl. Geophys. 133 (3), 489e521. Barenblatt, G.I., 1962. The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 7, 55e129. Benzeggagh, M.L., Kenane, M., 1996. Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos. Sci. Technol. 56 (4), 439e449. Breitenfeld, M.S., Geubelle, P.H., Weckner, O., Silling, S.A., 2014. Non-ordinary state-based peridynamic analysis of stationary crack problems. Comput. Methods Appl. Mech. Eng. 272, 233e250. Camanho, P.P., Da´vila, C.G., 2002. Mixed-mode Decohesion Finite Elements for the Simulation of Delamination in Composite Materials, pp. 1e37. NASA/TM-2002-211737. Cazes, F., Moe¨s, N., 2015. Comparison of a phase-field model and of a thick level set model for brittle and quasi-brittle fracture. Int. J. Numer. Methods Eng. 103 (2), 114e143. Chen, J., Crisfield, M., Kinloch, A.J., Busso, E.P., Matthews, F.L., Qiu, Y., 1999. Predicting progressive delamination of composite material specimens via interface elements. Mech. Compos. Mater. Struct. 6 (4), 301e317. Deshpande, V.S., Evans, A.G., 2008. Inelastic deformation and energy dissipation in ceramics: a mechanism-based constitutive model. J. Mech. Phys. Solid. 56 (10), 3077e3100. Dugdale, D.S., 1960. Yielding of steel sheets containing slits. J. Mech. Phys. Solid. 8 (2), 100e104. Fineberg, J., Gross, S.P., Marder, M., Swinney, H.L., 1992. Instability in the propagation of fast cracks. Phys. Rev. B 45 (10), 5146. Hillerborg, A., Mode´er, M., Petersson, P.E., 1976. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concr. Res. 6 (6), 773e781. Johnson, G.R., Holmquist, T.J., 1999. Response of boron carbide subjected to large strains, high strain rates, and high pressures. J. Appl. Phys. 85 (12), 8060e8073. Roy, P., Pathrikar, A., Deepu, S.P., Roy, D., 2017. Peridynamics damage model through phase field theory. Int. J. Mech. Sci. 128e129, 181e193. Roy, P., Deepu, S.P., Pathrikar, A., Roy, D., Reddy, J.N., 2017. Phase field based peridynamics damage model for delamination of composite structures. Compos. Struct. 180, 972e993.

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Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E., 2007. Peridynamic states and constitutive modeling. J. Elasticity 88 (2), 151e184. Silling, S.A., Lehoucq, R.B., 2010. Peridynamic theory of solid mechanics. In: Advances in Applied Mechanics, vol. 44. Elsevier, pp. 73e168. Turon, A., Davila, C.G., Camanho, P.P., Costa, J., 2007. An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Eng. Fract. Mech. 74 (10), 1665e1682.

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C H A P T E R

16 Peridynamic modeling at nano-scale Xuefeng Liu1, Xiaoqiao He2, Chun Lu1, Erkan Oterkus3 1

Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, Guangdong, China; 2 Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong SAR, China; 3 Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom

O U T L I N E 1. Introduction

356

2. PD model for the failure of SLGS 2.1 Original PD formulation 2.2 PD model of SLGS 2.2.1 Establishment of the PD model 2.2.2 Determination of the PD parameters

357 357 358 358 360

3. PD simulation of the failure of SLGS 3.1 A CG idea for SLGS 3.2 Failure of SLGS 3.2.1 Validation of the PD model of SLGS 3.2.2 Failure modes of different SLGS 3.3 Discussion

361 361 362 362 364 367

4. Conclusion

368

References

368

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00012-3

355

© 2021 Elsevier Inc. All rights reserved.

356

16. Peridynamic modeling at nano-scale

1. Introduction Peridynamics (PD) is a new theory to describe the interaction between particles inside materials, which was introduced by Silling (2000) through reformulating the classical elasticity theory for discontinuities and longrange forces in the year of 2000. It has been proved that PD theory has great advantages over other methods in dealing with discontinuity related problems. In the past 20 years, PD theory has been widely applied in various discontinuity involved areas such as the failure investigations of different materials and structures like polymers (Lee and Hong, 2016), membranes (Silling and Bobaru, 2005; Taylor et al., 2016), fibers (Silling and Bobaru, 2005; Bobaru and Silling, 2004), concrete structures (Gerstle et al., 2007; Yang et al., 2018; Huang et al., 2015), polycrystalline materials (De Meo et al., 2016; Zhu et al., 2016), metals (Jafarzadeh et al., 2018; Sun and Sundararaghavan, 2014; Wu and Ren, 2015), geomaterials (Ha et al., 2015; Lai et al., 2015; Ouchi et al., 2015; Zhou et al., 2015), functionally graded materials (Cheng et al., 2015, 2018), and composite structures (Hu et al., 2012, 2015; Kilic et al., 2009; Oterkus and Madenci, 2012). The development and application of PD theory were reviewed and summarized in detail by Askari et al. (2008) and Javili et al. (2018). From above descriptions, it can be known that PD theory can be employed for discontinuous problems ranging from macro-scale to micro-scale even nano-scale, which mainly benefits from the length-scale parameter, horizon, in the PD theory. At nano-scale, classical continuum mechanics (CCM)ebased methods and fully atomistic simulations (such as molecular dynamic (MD) simulations) are traditionally employed. However, CCM-based methods are not applicable in the discontinuityrelated problems (i.e., crack growth) because the spatial derivatives in the CCM theory may lose their meaning at discontinuities while fully atomistic simulations are prohibitively expensive to investigate the large deformation and failure mechanisms at large nano-scale. Therefore, PD theory can be a good alternative in this case. It has been shown that PD methods are better than the above traditional approaches to study the mechanical characters of nano materials and structures. With PD theory, the nano-scale effect can be well captured (Ahadi and Melin, 2017). This chapter presents the application of PD modeling at nano-scale, especially, the study on graphene sheets by employing PD theory. To date, some investigations on graphene sheets have been conducted. Details can be found in the studies by Liu et al. (2018, 2019), Oterkus et al. (2015), Martowicz et al. (2015) and Diyaroglu et al. (2019). The remainder of this chapter is organized as follows. At first, the original PD theory is reviewed and the establishment of PD model of single layer graphene sheets (SLGS) is presented. Then, PD simulations of the failure of SLGS

II. New applications in peridynamics

2. PD model for the failure of SLGS

357

based on coarse-grain (CG) techniques are demonstrated. Finally, some conclusions are made. It should be mentioned here that all the PD simulations throughout this chapter are performed by running FORTRAN code in serial mode, and that the fully atomistic simulations employed in this chapter are performed by running the open-source classical MD simulation code (i.e., LAMMPS (Plimpton, 1995)).

2. PD model for the failure of SLGS 2.1 Original PD formulation The original PD theory is in the bond-based form, which is also called bond-based PD (BPD) theory. As shown in Fig. 16.1, in BPD theory, it assumes that a continuum body consists of many particles, and each particle interacts with others in the influence domain, H, called horizon with a radius of d. Each pair of particles is connected by a PD bond and interacts through the PD bond. The interaction is defined by PD force density vector ( f ), which contains all the constitutive information about the material body. f and f 0 are a pair of active and reactive forces. The governing equation of a certain particle k in BPD theory is written as. Z rk u€k ¼ f dV þ bk (16.1) Hk

where rk is the density, u€k is the acceleration, Hk is the corresponding horizon, bk is the body force density, and dV is the volume of each particle inside the horizon. f is a pairwise force density function in the PD bond, and is defined as f ¼ mcs 

FIGURE 16.1

yj  yk

 yj  yk 

BPD configurations before and after deformation.

II. New applications in peridynamics

(16.2)

358

16. Peridynamic modeling at nano-scale

in which c is a PD parameter called micro-modulus of the PD bond, yj and yk represent the deformed position vectors of particles j and k, j ,j denotes the length of the corresponding vector, and s is the stretch of the PD bond which is expressed as   y  y   jxj  xk j j k s¼ (16.3) jxj  xk j wherein xj and xk are the initial position vectors of particles j and k. m is a history-dependent scalar and is written as  1 s < sc (16.4) m¼ 0 s  sc When the bond stretch, s, is larger than the critical one, sc, m renders the pairwise force density (see Eq. (16.2)) to be zero, which means that the corresponding PD bond breaks and damage occur inside the material body.

2.2 PD model of SLGS 2.2.1 Establishment of the PD model According to the findings by Cadelano et al. (2009), the continuum constitutive nonlinear stress-strain relation for SLGS under uniaxial stretching can be written as s! ¼ D! ε 2 þ Eε! n n ! n n

(16.5a)

and ε! are the uniaxial stress and strain along the direction in which s! n n ! n , E is the 2D Young’s modulus, D! is an effective nonlinear (thirdn ! order) elastic modulus in the direction n and is given as 3 3 D! ¼ ð1  nÞ3 L3 þ ð1  nÞð1 þ nÞ2 L2 þ 3 cosð6aÞð1 þ nÞ3 L1 n 2 2

(16.5b)

where a is the chiral angle relative to the zigzag edge of SLGS, n is Poisson’s ratio, and L1 , L2 , and L3 are three nonlinear independent elastic coefficients which are related to the third-order elastic constants as shown in Cadelano et al. (2009). According to Eqs. (16.5a) and (16.5b), it can be observed that the Young’s modulus, E, of SLGS shows independence of the chiral angle or the chirality while the third-order elastic modulus, D! , presents periodic n dependence on the chiral angle or the chirality with a period of p= 3, which is also the period of the chiral structure of SLGS. Therefore, SLGS is isotropic under small deformation and its third-order elastic modulus is anisotropic and depends on the chiral angle.

II. New applications in peridynamics

2. PD model for the failure of SLGS

359

Based on the above discussions about the isotropy, anisotropy, and physical nonlinearity of SLGS, a BPD model (i.e., the PD force density function) can be constructed as f !0 ¼ c!0 s2 þ c0 s

(16.6a)

c!0 ¼ c2 cosð6bÞ þ c1

(16.6b)

n

n

with n

! wherein n0 stands for the orientation of the initial PD bond and is represented by the orientation angle of the PD bond, b (i.e., b ¼ a þ g), as shown in Fig. 16.2. c0 , c1 , and c2 are PD parameters to be determined in the BPD model. The mechanical characters of SLGS are included by considering the dependence of the PD force density function on the initial bond orientation and the current bond stretch. Under uniaxial stretching with strain of ε! along arbitrary direction ! n n (x direction in Fig. 16.2), the stretch of a single PD bond in Fig. 16.2 is written as 1 s ¼ ½sinðgÞcosðgÞ2 ε!2 þ ½cosðgÞ2 ε! n n 2

(16.7)

which is substituted into Eq. (16.6a), then the corresponding PD force density in the single PD bond can be obtained as.   1 f !0 ¼ fc2 cos½6ðg þ aÞ þ c1 g½cosðgÞ4 þ c0 ½sinðgÞcosðgÞ2 ε!2 n 2 n (16.8) þ c0 ½cosðgÞ2 ε! n

FIGURE 16.2 Schematics of SLGS PD material body under uniaxial stretching along x direction. xy coordinate system is a fixed and global system. x0 y0 coordinate system is a local one and can change with the chiral angle, a, in which x0 axis is along the SLGS zigzag edge and y0 axis is along the SLGS armchair edge.

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16. Peridynamic modeling at nano-scale

Through calculating the total PD force at particle k along the direction total total ! and then dividing F! by the surface dS (with normal vector of n , F! n n ! n ) at particle k in Fig. 16.2, the uniaxial stress at particle k along arbitrary direction ! n can finally be obtained as 

¼ s! n

 1 5 1 1 c2 cosð6aÞ þ c1 þ c0 phd3 ε!2 þ c0 phd3 ε! n n 192 48 32 8

(16.9)

in which h is the thickness of SLGS. 2.2.2 Determination of the PD parameters It should be noted that the stress-strain relation obtained from the PD calculation (i.e., Eq. 16.9) and the one in Cadelano et al. (2009). (i.e., Eq. 16.5a) under the same uniaxial stretching should be consistent with each other. Therefore, the PD parameters in the BPD model can be determined via uniaxial stress equivalence. Through performing MD simulations on the fully atomistic SLGS with different chirality under uniaxial stretching, the corresponding uniaxial stress-strain relations can be obtained, as presented in Fig. 16.3. From the figure, it can be known that the Young’s modulus of SLGS (calculated by taking the slope of linear part of the curves at small strain) is isotropic while the third-order elastic modulus, the fracture strength, and the fracture strain of SLGS are anisotropic and show dependence on the chiral angle or the chirality. Through curve fitting, the stress of each type of SLGS can be expressed with a quadratic formula in terms of the strain. It is proved that the results are consistent with the descriptions (see Eqs. (16.5a) and (16.5b)) by Cadelano et al. (2009). For each type of SLGS with a certain chiral angle, the stress-strain relation in Fig. 16.3 can be equated with the one calculated from Eq. (16.9). The PD parameters, i.e., c0 , c1 , and c2 , are finally derived as c0 ¼

7:4  1012 19:5  1012 19:2  1012 ; c ¼  ; c ¼ . 1 2 phd3 phd3 phd3

FIGURE 16.3

(16.10)

Stress-strain relations of SLGS with different chirality under uniaxial

stretching.

II. New applications in peridynamics

3. PD simulation of the failure of SLGS

361

In addition, another PD parameter of great importance (i.e., the critical stretch of the PD bond, sc) in the stretch-based damage criterion should be defined to determine the failure in the PD material body. When the stretch of the PD bond is larger than the critical value, the PD bond breaks in time and damage occurs inside the body. According to the dependence of the third-order elastic modulus of SLGS (i.e., D! in Eq. 16.5b) and the thirdn order micromodulus of the PD bond (i.e., c!0 in Eq. 16.6b) on their n

respective structural orientations and the dependence of the fracture strength and strain of SLGS on the chiral angle (see Fig. 16.3), it can reasonably be assumed that the critical stretch of the PD bond behaves similarly as well. Thus, the critical stretch, sc, can be written as sc!0 ¼ s1 cosð6bÞ þ s0 n

(16.11a)

in which scz  sca 2 scz þ sca s0 ¼ 2 s1 ¼

(16.11b) (16.11c)

wherein b is the orientation angle of the PD bond (See Fig. 16.2). If the critical stretches of the PD bond along the SLGS zigzag and armchair edges, scz and sca , are known, respectively, the critical stretch of the PD bond with any orientation angle can easily be determined. According to the critical stretches used for the zigzag and armchair SLGS in the studies by Liu et al. (2018) and Oterkus et al. (2015), the critical stretches, scz ¼ 0:13 and sca ¼ 0:1, are employed in the following PD simulations of the failure of SLGS.

3. PD simulation of the failure of SLGS 3.1 A CG idea for SLGS Although fully atomistic simulation methods (i.e., MD simulation, etc.) are powerful in the prediction of structural evolutions (i.e., dislocation motion, crack propagation, etc.) in nano materials, they are very computationally expensive for very large systems in which millions of atoms are contained. As a nonlocal continuum-based theory, it is shown that PD method can be an excellent alternative in such kind of studies by introducing a CG idea in the PD simulation. SLGS is a chirality-dependent structure which consists of hexagonal basic cells. Fig. 16.4 shows the CG idea employed in the PD simulation of the failure of SLGS. Four particles in the structure of lower-level in

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16. Peridynamic modeling at nano-scale

FIGURE 16.4 Schematics to show the CG idea used in PD simulation.

Fig. 16.4A can be coarse-grained as a particle in the CG structure (or the structure of high-level) in Fig. 16.4B. Specifically, four atoms in the fully atomistic SLGS structure are coarse-grained as a PD particle in the PD-CG SLGS structure and 4 PD particles can be coarse-grained as another one PD particle in the PD-CG SLGS structure of higher-level. In such a way, PD-CG SLGS structures of different level can be obtained and employed for future PD simulations. It should be strengthened that the chiral characters of atomistic SLGS structures can also be reflected in the PD-CG SLGS structures due to the usage of the CG technique as shown in Fig. 16.4.

3.2 Failure of SLGS 3.2.1 Validation of the PD model of SLGS Combining the established PD model in Section 2.2 and the CG technique in Section 3.1, a PD-CG model can be obtained. Then the PD-CG model is employed to simulate the fracture of SLGS with different chirality (i.e., armchair, chiral (a z 15 ), and zigzag). In the PD-CG simulations in this part, the size of each specimen is 20 nm  20 nm with particle spacing of 0.284 nm, the central crack in each specimen is about half of the size of the specimen and the horizon radius is three times of the particle spacing. To validate the current PD-CG model, the PD-CG simulation results are compared with the fully atomistic (i.e., MD) simulation results. In the MD simulations, fully atomistic SLGS with the same size and chirality as the PD-CG SLGS is employed and the atom spacing is 0.142 nm in each fully atomistic specimen. A PD particle in the PD-CG SLGS is composed of four atoms in the fully atomistic SLGS, which means four atoms are coarse-grained as a PD particle here. Under horizontal stretching condition, the obtained fracture forms of SLGS with different chirality from PD-CG and MD simulations are presented in Fig. 16.5. It can be observed from the figure that the PD-CG simulation results are highly consistent with the MD simulation results. It can be revealed in the simulation results that the crack propagation in SLGS shows dependence on the chiral structure of SLGS. The crack just

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3. PD simulation of the failure of SLGS

363

FIGURE 16.5 Fracture forms of SLGS with different chirality derived from PD and MD simulations. (A)e(C) The fracture forms of armchair, chiral (a z 15 ) and zigzag PD-CG SLGS, respectively, obtained from PD simulations. (D)e(F) The fracture forms of armchair, chiral (a z 15 ) and zigzag fully atomistic SLGS, respectively, obtained from MD simulations.

propagates straight ahead in armchair SLGS in Fig. 16.5A and D. In chiral (a z 15 ) SLGS in Fig. 16.5B and E, the crack propagates along the direction with an angle of 15 relative to the vertical direction. The angle between the crack path and the vertical direction is 30 in zigzag SLGS in Fig. 16.5C and F. Such fracture patterns in SLGS with different chirality were found as well by Hossain et al. (2018) and Rajasekaran and Parashar (2017). Comparing the crack paths in SLGS with different chirality shown in Fig. 16.5, it can be concluded that the crack in SLGS tends to propagate along the zigzag edge. The reason why the crack propagation shows dependence on the chirality of SLGS and is preferably along the zigzag edge in SLGS can be explained by the fracture parameter of great importance, the critical energy release rate (i.e. the fracture energy or the surface energy). In the findings by Zhang et al. (2014), it has been reported that the surface energy of the surfaces along the zigzag and  armchair edges in SLGS are 11.8 and 12.5 J m2 , respectively, from which it can be concluded that the crack surface along the zigzag edge can form more easily than the one along the armchair edge does because generation of the crack surface along the zigzag edge needs less energy. Therefore, it can be seen from the results that the crack in SLGS propagates preferably along the zigzag edge direction. Thus, the PD-CG model employed for the fracture simulation of SLGS is well validated as well.

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3.2.2 Failure modes of different SLGS Validation of the PD-CG model shows that the obtained PD-CG model can be used for the following failure simulations of SLGS, in which the influences of the chirality of SLGS, the size of SLGS, the degree of coarsegrain, etc. on the failure modes of SLGS are considered. In the PD-CG simulations in this part, the particle spacing is different in each SLGS specimen and can change with the employed CG technique of different degree. The horizon size in each case is chosen as 3 times of the particle spacing. In each case, each specimen is stretched by moving the left and right boundaries along the horizontal direction. The fracture patterns of different SLGS obtained from the PD-CG simulations are presented in Figs. 16.6e16.8 below. In Fig. 16.6, the size of each specimen is about 80 nm  80 nm. The central crack is about 25% of the specimen size. In this case, the CG technique, same as the one in Fig. 16.5AeC, is utilized. In each specimen, there exist about 61 thousand PD particles. In the corresponding fully atomistic SLGS, it contains 240 thousand atoms. For such large fully atomic systems, it might be a little difficult to conduct studies by using fully atomistic simulation methods with general computing resources. Different crack propagation forms can be seen in SLGS with different chirality in Fig. 16.6. In the armchair SLGS in Fig. 16.6A, the crack just

FIGURE 16.6 Fracture forms of armchair (A), chiral (a z 15 ) (B), and zigzag (C) PD-CG SLGS in which a PD particle is composed of four atoms in the fully atomistic SLGS.

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FIGURE 16.7 Fracture forms of armchair (A), chiral (a z 15 ) (B), and zigzag (C) PD-CG SLGS in which a PD particle contains 16 atoms in the corresponding fully atomistic system.

FIGURE 16.8 Fracture forms of armchair (A), chiral (a z 15 ) (B), and zigzag (C) PD-CG SLGS in which a PD particle consists of 64 atoms in the corresponding fully atomistic system.

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propagates forward straightly and the specimen breaks into two halves. The crack in the chiral (a z 15 ) SLGS in Fig. 16.6B propagates initially along the direction with an angle of 15 relative to the centrally vertical line and globally along the centrally vertical line. Similar to the fracture of the chiral (a z 15 ) SLGS, the crack initiates along the direction with an angle of 30 relative to the centrally vertical line and then propagates globally perpendicular to the loading direction in the zigzag SLGS as presented in Fig. 16.6C. In addition, many local zigzag crack patterns can be observed in the chiral and zigzag SLGS as shown in Fig. 16.6B and C, respectively. Fig. 16.7 presents the fracture forms of SLGS with global size of about 160 nm  160 nm and the central crack of about 25% of the specimen size. In this case, a PD particle is composed of four PD particles in the PD-CG SLGS in Fig. 16.6 (i.e., four PD particles in Fig. 16.6 are coarse-grained as a PD particle in Fig. 16.7 and 16 atoms in the fully atomistic SLGS are coarse-grained as a PD particle here) and the particle spacing in each specimen is 0.568 nm. About 61 thousand PD particles are contained in each specimen. Thus, about 980 thousand atoms exist in the corresponding fully atomistic SLGS. It is difficult to perform fully atomistic simulations on such kind of large atomic systems. Compared with the results in Fig. 16.6, similar failure modes are observed in Fig. 16.7 as well. It can be seen that the crack patterns including the crack initiation and the global crack propagation in Fig. 16.7 are almost consistent with the ones shown in Fig. 16.6. Therefore, detailed descriptions about the fracture forms of SLGS here are not presented repetitively. Additionally, a small number of local zigzag crack forms are presented in the chiral and zigzag SLGS in Fig. 16.7B and C, respectively. In the case shown in Fig. 16.8, the global size of each sample is about 700 nm  700 nm and the central crack in each sample is about 25% of the sample size. The size of the samples is basically at micro-scale. In this case, a PD particle is composed of four PD particles in the PD-CG SLGS in Fig. 16.7 (i.e., four PD particles in Fig. 16.7 are coarse-grained as a PD particle in Fig. 16.8 and 64 atoms in the fully atomistic SLGS are coarsegrained as a PD particle here) and the particle spacing in each sample is 1.136 nm. About 300 thousand PD particles are included in each sample. Thus, there are about 20 million atoms existing in the corresponding atomistic system, which makes it basically impossible to conduct fully atomistic simulations for the study on such a huge micro-scale atomistic system with the same computational resources as used in the PD simulations here. The observations from Fig. 16.8 are a little different from the ones in Figs. 16.6 and 16.7. The crack in each sample propagates globally along the centrally vertical line in the sample. In the armchair SLGS in Fig. 16.8A, the crack can propagate smoothly with occasional crack

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kinking. The crack initiation directions in the chiral (a z 15 ) and zigzag SLGS in Fig. 16.8B and C are not as obvious as the ones in Figs. 16.6B, 16.7B and 16.6C, 16.7C, respectively. In addition, the local zigzag crack forms can almost not be observed in the crack propagation process in the chiral (a z 15 ) and zigzag SLGS.

3.3 Discussion From the PD simulation results shown in Figs. 16.5e16.8, it can be seen that the fracture of SLGS can be different from each other due to the different size and chirality of SLGS and the usage of different CG techniques. Therefore, the effects of the size of SLGS, the chirality of SLGS, and the CG technique on the failure mode of SLGS are discussed in this part. From the simulation results throughout this whole chapter, two findings can be revealed. The first is that the crack propagation directions are totally different from each other in different SLGS as shown in Fig. 16.5AeC. The other one is that the fracture form of the armchair SLGS is very simple compared with the ones of other SLGS. The crack in armchair SLGS just propagates forward straightly and the sample just breaks simply into two halves as shown in Figs. 16.5A and 16.8A. The crack propagation forms in chiral (a z 15 ) and zigzag SLGS are a little more complicated, where crack kinking behaviors can occur in the crack propagation process as presented in Figs. 16.6Be16.8B and 16.6Ce16.8C. Such differences between the failure modes of different SLGS can be attributed to the effect of different chirality of SLGS. Comparing the results shown in Figs. 16.5B, 16.5C and 16.6B, 16.6C, the effect of the size on the failure modes of different SLGS can be revealed. The cracks in the SLGS of small size in Fig. 16.5B and C propagate straightly along the zigzag edge without depending on the stretching direction, respectively. However, the cracks in the SLGS of large size in Fig. 16.6B and C just initiate along the zigzag edge, and then propagate globally perpendicular to the stretching direction and locally in a zigzag way, respectively. Such crack forms in the large SLGS are consistent with the report by Liu et al. (2019) that the crack propagation pattern in zigzag SLGS should be the consequence of the competition between local and global level fractures. Comparing the results in Figs. 16.6B, 16.8B and 16.6C, 16.8C in which CG techniques of different level are utilized, the effect the CG technique on the failure modes of SLGS can be uncovered. Four atoms in fully atomistic SLGS are coarse-grained as a PD particle in Fig. 16.6, 16 atoms in fully atomistic SLGS are coarse-grained as a PD particle in Fig. 16.7, and 64 atoms in fully atomistic SLGS are coarse-grained as a PD particle in Fig. 16.8. With the degree of CG increasing, the fracture characters such as

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the crack initiation along the zigzag edge and the local zigzag crack pattern in the global crack propagation process become less and less obvious and disappear finally. It can also be concluded that employment of CG technique can be an efficient way to study the failure of SLGS when the information at atomic scale is not the main interest.

4. Conclusion The application of PD modeling at nano-scale is discussed and demonstrated in this chapter. Especially, through studying the fracture of SLGS with different chirality based on PD simulations, it can be concluded that PD method is of high efficiency and effectiveness to study the failure of large nano-scale materials or structures. Combining the PD model established based on stress equivalence and the CG technique with considering the chirality of SLGS, a PD-CG model is obtained. Based on PD-CG simulations, the fracture of different SLGS are investigated, in which the effect of the chirality of SLGS, the size of SLGS, the degree of CG, etc., on the fracture of SLGS are considered. Through comparing the PD-CG simulation results with other available results, the PD-CG model is well validated. It is found that the crack propagation in SLGS shows great dependence on the chirality. The crack in SLGS propagates more easily along the zigzag edges for the reason that the surface energy of the surface along the zigzag edge is the lowest. Due to the size effect, the crack propagation along the zigzag edge without being affected by the stretching direction is more obvious in SLGS of relatively small size. By adopting the CG technique of different level, the PD-CG model can be employed for the failure prediction of SLGS of very large size for which fully atomistic simulation is almost not possible. In addition, the PD model can probably be extended to the studies on other different chirality-related materials or structures of large nano-scale, if the information at atomic level does not matter. This work can also provide a significant insight on PD-MD coupling to solve cross-scale problems.

References Ahadi, A., Melin, S., 2017. Capturing nanoscale effects by peridynamics. Mech. Adv. Mater. Struct. 25, 1115e1120. Askari, E., Bobaru, F., Lehoucq, R.B., Parks, M.L., Silling, S.A., Weckner, O., 2008. Peridynamics for multiscale materials modeling. J. Phys. Conf. 125, 012078. Bobaru, F., Silling, S.A., 2004. Peridynamic 3D models of nanofiber networks and carbon nanotube-reinforced composites. In: AIP Conference Proceedings, pp. 1565e1570. Cadelano, E., Palla, P.L., Giordano, S., Colombo, L., 2009. Nonlinear elasticity of monolayer graphene. Phys. Rev. Lett. 102, 235502.

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Cheng, Z., Zhang, G., Wang, Y., Bobaru, F., 2015. A peridynamic model for dynamic fracture in functionally graded materials. Compos. Struct. 133, 529e546. Cheng, Z., Liu, Y., Zhao, J., Feng, H., Wu, Y., 2018. Numerical simulation of crack propagation and branching in functionally graded materials using peridynamic modeling. Eng. Fract. Mech. 191, 13e32. De Meo, D., Zhu, N., Oterkus, E., 2016. Peridynamic modeling of granular fracture in polycrystalline materials. J. Eng. Mater. Technol. 138, 041008. Diyaroglu, C., Behera, D., Madenci, E., Kaya, Y., Kedziora, G., Nepal, D., 2019. Peridynamic Modeling of Wrinkling in a Graphene Layer. AIAA Scitech 2019 Forum. Gerstle, W., Sau, N., Silling, S., 2007. Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250e1258. Ha, Y.D., Lee, J., Hong, J.W., 2015. Fracturing patterns of rock-like materials in compression captured with peridynamics. Eng. Fract. Mech. 144, 176e193. Hossain, M.Z., Ahmed, T., Silverman, B., Khawaja, M.S., Calderon, J., Rutten, A., Tse, S., 2018. Anisotropic toughness and strength in graphene and its atomistic origin. J. Mech. Phys. Solid. 110, 118e136. Hu, W., Ha, Y.D., Bobaru, F., 2012. Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput. Methods Appl. Mech. Eng. 217e220, 247e261. Hu, Y.L., De Carvalho, N.V., Madenci, E., 2015. Peridynamic modeling of delamination growth in composite laminates. Compos. Struct. 132, 610e620. Huang, D., Lu, G., Liu, Y., 2015. Nonlocal peridynamic modeling and simulation on crack propagation in concrete structures. Math. Probl Eng. 1e11. Jafarzadeh, S., Chen, Z., Bobaru, F., 2018. Peridynamic modeling of repassivation in pitting corrosion of stainless steel. Corrosion 74, 393e414. Javili, A., Morasata, R., Oterkus, E., Oterkus, S., 2018. Peridynamics review. Math. Mech. Solids 24, 3714e3739. Kilic, B., Agwai, A., Madenci, E., 2009. Peridynamic theory for progressive damage prediction in center-cracked composite laminates. Compos. Struct. 90, 141e151. Lai, X., Ren, B., Fan, H., Li, S., Wu, C.T., Regueiro, R.A., Liu, L., 2015. Peridynamics simulations of geomaterial fragmentation by impulse loads. Int. J. Numer. Anal. Methods GeoMech. 39, 1304e1330. Lee, J., Hong, J.-W., 2016. Dynamic crack branching and curving in brittle polymers. Int. J. Solid Struct. 100e101, 332e340. Liu, X., He, X., Wang, J., Sun, L., Oterkus, E., 2018. An ordinary state-based peridynamic model for the fracture of zigzag graphene sheets. Proc. Math. Phys. Eng. Sci. 474, 20180019. Liu, X., Bie, Z., Wang, J., Sun, L., Tian, M., Oterkus, E., He, X., 2019. Investigation on fracture of pre-cracked single-layer graphene sheets. Comput. Mater. Sci. 159, 365e375. Martowicz, A., Staszewski, W.J., Ruzzene, M., Uhl, T., 2015. Peridynamics as an analysis tool for wave propagation in graphene nanoribbons. Sens. Smart Struct. Technol Civil Mech. Aerosp. Syst. 2015, 94350I. Oterkus, E., Madenci, E., 2012. Peridynamic analysis of fiber-reinforced composite materials. J. Mech. Mater. Struct. 7, 45e84. Oterkus, E., Diyaroglu, C., Zhu, N., Oterkus, S., Madenci, E., 2015. Utilization of Peridynamic Theory for Modeling at the Nano-Scale, Nanopackaging: From Nanomaterials to the Atomic Scale, pp. 1e16. Ouchi, H., Katiyar, A., York, J., Foster, J.T., Sharma, M.M., 2015. A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamics approach. Comput. Mech. 55, 561e576. Plimpton, S., 1995. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1e19.

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Rajasekaran, G., Parashar, A., 2017. Enhancement of fracture toughness of graphene via crack bridging with stone-thrower-wales defects. Diam. Relat. Mater. 74, 90e99. Silling, S.A., Bobaru, F., 2005. Peridynamic modeling of membranes and fibers. Int. J. Non Lin. Mech. 40, 395e409. Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solid. 48, 175e209. Sun, S., Sundararaghavan, V., 2014. A peridynamic implementation of crystal plasticity. Int. J. Solid Struct. 51, 3350e3360. Taylor, M., Gozen, I., Patel, S., Jesorka, A., Bertoldi, K., 2016. Peridynamic modeling of ruptures in biomembranes. PLoS One 11 e0165947. Wu, C.T., Ren, B., 2015. A stabilized non-ordinary state-based peridynamics for the nonlocal ductile material failure analysis in metal machining process. Comput. Methods Appl. Mech. Eng. 291, 197e216. Yang, D., Dong, W., Liu, X., Yi, S., He, X., 2018. Investigation on mode-I crack propagation in concrete using bond-based peridynamics with a new damage model. Eng. Fract. Mech. 199, 567e581. Zhang, P., Ma, L., Fan, F., Zeng, Z., Peng, C., Loya, P.E., Liu, Z., Gong, Y., Zhang, J., Zhang, X., Ajayan, P.M., Zhu, T., Lou, J., 2014. Fracture toughness of graphene. Nat. Commun. 5, 3782. Zhou, X., Gu, X., Wang, Y., 2015. Numerical simulations of propagation, bifurcation and coalescence of cracks in rocks. Int. J. Rock Mech. Min. Sci. 80, 241e254. Zhu, N., De Meo, D., Oterkus, E., 2016. Modelling of granular fracture in polycrystalline materials using ordinary state-based peridynamics. Materials 9.

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C H A P T E R

17 Multiscale modeling with peridynamics Yakubu Kasimu Galadima, Wenxuan Xia, Erkan Oterkus, Selda Oterkus Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom

O U T L I N E 1. Introduction

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2. Coarsening approach 2.1 Coarsening of peridynamic model 2.2 Numerical implementation 2.3 Coarsening the micromodulus function 2.3.1 Coarsening of 1D micromodulus function 2.3.2 Coarsening of two-dimensional micromodulus functions

372 372 375 377 377 377

3. Model order reduction using static condensation 3.1 Reduced dynamic and static models 3.2 Reduced eigenvalue models

379 380 381

4. Homogenization approach

382

5. Conclusions

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© 2021 Elsevier Inc. All rights reserved.

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1. Introduction With the advancement of manufacturing technologies such as 3D printing, designing microstructured materials has become available. However, defects can still exist at the microscale in the form of microcracks and voids. With the current computational power, it is still challenging to fully model these materials at the microscale. Hence, multiscale methodologies appear to be the promising approach to model such structures. There are currently various different approaches available for multiscale modeling. Some of these methodologies have also been considered within peridynamic framework. Silling (2011) developed a coarsening approach so that the effect of microstructural behavior can be represented by using fewer degrees of freedom and presented this approach for one-dimensional (1D) problems. This approach was further extended for two-dimensional (2D) structures by Galadima et al. (2019). Although coarsening approach is promising, it is currently limited to static conditions. Another promising approach is model order reduction using static condensation. Galadima et al. (2020) recently proposed a model order reduction methodology for linear peridynamic systems using static condensation. Although the number of degrees of freedom is reduced in model order reduction methodology similar to coarsening approach, both static and dynamic problems can be analyzed. Finally, homogenization techniques have been developed within peridynamic framework. Madenci et al. (2018) introduced peridynamic unit cell homogenization approach to obtain thermoelastic properties of heterogeneous microstructures with defects. In this chapter, a brief summary of all these three approaches, i.e., coarsening, model order reduction using static condensation, and homogenization, is presented.

2. Coarsening approach 2.1 Coarsening of peridynamic model The coarsening method for linear peridynamics as proposed by Silling (2011) is a bottom-up multiscale method that aims at representing a complex system with a coarsened (or simplified) version of it. The coarsening procedure consists of a succession of model substitution with each subsequent model representing a coarsened or less complex version of the previous model. Information about material properties to be used in a coarsened model is derived from the preceding more detailed model.

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FIGURE 17.1 A peridynamic body showing (A) detailed level 0 body, (B) coarsened level 1 body, and (C) coarsened level 2 body.

In order to coarsen a detailed model, let B 0 be a linear peridynamic body as shown Fig. 17.1A. Let the motion of every point x ˛ B0 be governed by the linearized equilibrium peridynamic equation of motion, Z € tÞ ¼ Cðx; x0 Þðuðx0 ; tÞ  uðx; tÞÞdVx0 þ bðx; tÞ rðxÞuðx; (17.1) 0

Hx 0

0

where H x represents the family of x with x0 denoting particles inside H x while C0 : B 0  B 0 /[ is a tensor-valued micromodulus function that contains intrinsic material properties of B 0 . Let the set of all linearly admissible displacement fields on B 0 be A 0 . Let r0 be a positive number that delimits the neighborhood of x in B 0 such that jx0  xj > r0

0 C0 ðx; x0 Þ ¼ 0

cx; x0 ˛ B 0

(17.2)

Let B1 3B 0 so that B 0 and B1 are called level 0 body and level 1 body, respectively. Let A 1 be the set of all admissible displacement fields on B 1 . 1

0

Let x also be a point in B1 and let H x ¼ H x XB 1 be its family. Let r1 be a positive number that delimits the neighborhood of points that belong to the family of x in B1 . The goal of this coarsening technique is to be able to use a subset B 1 to describe the response of the detailed body B 0. To this end, suppose that u1 ˛ A 1 is given, and suppose that u0 ˛A 0 satisfies the compatibility condition 1

u0 ðpÞ ¼ u1 ðpÞ

cp ˛ H x

(17.3)

0

Neglecting interaction between H x and its exterior let’s assume that 1

the displacement field u0 for points outside of H x satisfy the equilibrium equation L0 ðzÞ þ bðzÞ ¼ 0

0

1

cz ˛ H x  H x

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(17.4)

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where

Z

  C0 ðz; pÞ u0 ðpÞ  u0 ðzÞ dVp

L0 ðzÞ ¼ H

0

cz ˛ H x

(17.5)

0 x

Let us make the simplifying assumption that no body force is applied 1

outside of H x . 0

1

bðzÞ ¼ 0 cz ˛ H x  H x

(17.6)

Let us further assume that given the displacement field u1 Eqs. (17.3) 0

and (17.4) have unique solution u0 on H x and let S0;1 x be the resolvent kernel that generates this solution Z  1  0 0 u ðpÞ ¼ S0;1 (17.7) x ðp; qÞ u ðqÞ dVq cp ˛ H x 1

Hx

It can be inferred from Eqs. (17.3) and (17.7) that 0

1 S0;1 x ðp; qÞ ¼ IDðp  qÞ cp ˛ R x ; cq ˛ H x

(17.8)

In Eq. (17.8), I is the isotropic tensor and D is the 3D Dirac delta 0

function. If all points in H x are assumed to undergo a linear rigid translation through an arbitrary displacement vector e u, then it can be deduced from Eq. (17.7) that 2 3 Z 0 6 7 e u ¼ 4 S0;1 u cp ˛ H x (17.9) x ðp; qÞdVq 5e R 1x

From Eq. (17.9) we infer that Z S0;1 x ðp; qÞdVq ¼ 1 H

0

cp ˛ H x

(17.10)

1 x

Subtracting u0 ðzÞ from both sides of Eq. (17.7) and taking into account the identity in Eq. (17.10) yields Z  1  1 1 u0 ðpÞ  u0 ðzÞ ¼ S0;1 (17.11) x ðp; qÞ u ðqÞ  u ðzÞ dVq cp; z ˛ H x 1

Hx

Substituting Eq. (17.11) into Eq. (17.5) gives 3 2 Z Z 7 6  1  0;1 1 7 L0 ðzÞ ¼ C0 ðz; pÞ6 4 Sx ðp; qÞ u ðqÞ  u ðzÞ dVq 5dVp 0

Hx

0

cz ˛ H x

1

Hx

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Reversing the order of integration and rearranging terms yield 2 3 Z Z 6 7 1  0 0 0;1 1 7 L0 ðzÞ ¼ 6 4 C ðz; pÞSx ðp; qÞdVp 5 u ðqÞ  u ðzÞ dVq cz ˛ H x 1

0

Hx Hx

(17.13) Since x is an arbitrary point in B 1 , if we denote the force density at any such choice of x by L0 ðxÞ ¼ L1 ðxÞ cx ˛ B 1

(17.14)

then following from Eq. (17.13) and Eq. (17.14), we obtain Z   L1 ðxÞ ¼ C1 ðx; qÞ u1 ðqÞ  u1 ðxÞ dVq cx ˛ B 1

(17.15)

1

Hx

where C1 : B 1  B1 is the coarsened level 1 micromodulus function defined by Z 1 C1 ðz; qÞ ¼ C0 ðx; pÞS0;1 (17.16) x ðp; qÞdVp cx; q ˛ B 1

Hx

The force density for any level m is similarly obtained as Z m L ðxÞ ¼ Cm ðx; qÞðum ðqÞ  um ðxÞÞdVq cx ˛ B m H

(17.17)

m x

m

where Cm : B  B m is defined by Z m C ðz; qÞ ¼ Cm1 ðx; pÞSm1;m ðp; qÞdVp x H

cx; q ˛ B m

(17.18)

m x

2.2 Numerical implementation The numerical implementation of the coarsening procedure proceeds with the discretization of B0 into nodes each with a volume in the undeformed configuration. For simplicity, all nodes will be assumed to have equal volume v. The discretized form of Eq. (17.1) as suggested in Silling and Askari (2005) is X   ru€0i ¼ v C0ik u0k  u0i þ bi (17.19) j

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It follows that the coarsened micromodulus for level 1 body from Eq. (17.18) may be written in discrete form as X C1ij ¼ v C0ik S0;1 (17.20) kj k

In order to evaluate Eq. (17.20), the resolvent kernel S0;1 kj will have to be computed. A method of computing S0;1 kj was suggested in Silling (2011). The method involves concatenating two systems of equations as follows:     define two vectors u0 and u1 to represent displacement fields in the   1 level 0 and level 1 bodies, respectively. In u0 , the nodes in H x are grouped at the top and the displacements are constrained to satisfy Eq. (17.3). The remaining degrees of freedom corresponding to nodes in 0

1

H x  H x are then determined by Eqs. (17.3)e(17.5). The procedure described above will yield a matrix equation of the form: 38 2 9 8 9 0 > > > > > 1 0 0 0 . u11 > > > u1 > > > > > 7> 6 > > > > > > > > 7> 6 « > > > > > > > « « 7> 6 > > > > > > > 7> 6 > > > > > > > > > > > 7 6. 0 0 1 > > > 1 0 0. > > > > u 7> R > > uR > 6 = < > = 7< 6 7 6 « ¼ (17.21) 7 6 « « > > > 7> 6 > > > > > > > > 7 6. 0 0 0 > 0 > >u > > > > Ci;i1 Pi Ci;iþ1 . 7> 6 > > > > i > > > 7> 6 > > > > > > > 7> 6 « > > > > > > > 7> 6 « « > > > > > > > 5> 4 > > > > > > > > 0 > : ; 0 : 0 > ; . CN;N1 PN uN 1

where R is the number of nodes in H x and N is the number of nodes in 0

H x . The diagonal elements Pi of the matrix are given by X Pi ¼ C0ij

(17.22)

jsi

In abbreviated form Eq. (17.21) can be written as   ½A u0 ¼ fbg

(17.23)

In one dimension ½A is N  N square matrix, while in two dimensions it is 2N  2N. Let ½A1 be the inverse of ½A. S0;1 kj is simply the leftmost R

column of ½A1 in one dimension or the leftmost 2R columns of ½A1 in two dimensions.

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FIGURE 17.2

377

Schematic of coarsening process of a one-dimensional bar.

2.3 Coarsening the micromodulus function This section will illustrate the typical form of coarsened micromodulus functions. One-dimensional as well as 2D cases of coarsened micromodulus function will be illustrated. 2.3.1 Coarsening of 1D micromodulus function To illustrate the coarsening of a 1D problem, consider a homogeneous bar of length 1 m and elastic modulus of 200 GPa. Let the micromodulus function describing the material of the bar be of the form; 8 > < 2E ; if jxj  d 2 CðxÞ ¼ Ad jxj (17.24) > : 0; if jxj > d The horizon in the level 0 body is given as d0 ¼ 25 mm. In order to numerically solve this problem, the bar is discretized into 400 nodes with the spacing between each node Dx ¼ 2:5 mm. To coarsen the model from level 0 body to level 1, every forth node in level 0 body is retained. Similarly, to coarsen level 1 to level 2, every second node in level 1 body is retained. Schematic representation of this coarsening process is shown in Fig. 17.2. A plot of the micromodulus function of level 0 as well as the coarsened level 1 and level 2 bodies is shown in Fig. 17.3. It can be seen from the plot that the coarsened micromodulus functions are characterized by sharp peaks consistent with the fact that the coarsened micromodulus function is defined only at the retained nodes. 2.3.2 Coarsening of two-dimensional micromodulus functions The procedure for coarsening 2D micromodulus function will be illustrated with the following problem. Consider a 500  500 mm square

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17. Multiscale modeling with peridynamics

FIGURE 17.3 Micromodulus functions for the detailed and coarsened models.

plate with a thickness of 50 mm. The elastic modulus of the material of the plate is 200GPa and a micromodulus function of the form 2 3 8  x2x xx xy  > > > jxj 6 7 > < 36E 1 4 5; if jq  xj  d 3 3 d (17.25) CðxÞ ¼ pd hjxj 2 x x x > x y y > > > : 0; if jq  xj > d Coarsening of the detailed model is carried out as shown in Fig. 17.4. Every second row and column of level 0 body are retained in the level 1 body, and similarly, every second row and column in the level 1 body are retained in level 2 body. Fig. 17.5 shows the micromodulus functions of level 0 and coarsened levels 1 and 2.

FIGURE 17.4 Level 0, 1, and 2 bodies of plate model.

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3. Model order reduction using static condensation

379

FIGURE 17.5 Micromodulus functions C0, C1, and C2 of the plate.

3. Model order reduction using static condensation Similar to most hierarchical multiscale protocols, the aim of this Model Order Reduction (MOR) technique is to represent a complex model with a reduced number of degrees of freedom. To achieve the objective of reducing the order of a given model, consider a linear PD body B. Let the motion of any arbitrary point x˛B 0 be governed by Eq. (17.1) and let the discretized form of Eq. (17.1) be ru€ni ¼

ri X

Cðxj  xi Þðuj  ui ÞVj þ bni

(17.26)

j

where ri denotes the number of nodes in the family of the ith node. Let B be discretized into n number nodes, the assembled PD equilibrium equation for all nodes in B may be written in matrix notation as € þ ½Cfug ¼ fbg ½Mfug

(17.27)

where fug ˛ Rn is a vector of displacement associated with all DoFs in the system, fbg ˛ Rn is a vector that collects all applied body forces, ½M ˛ Rnn is a diagonal matrix of mass density, and ½C ˛ Rnn is the micromodulus matrix, then ½M ˛ Rnn .

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17. Multiscale modeling with peridynamics

3.1 Reduced dynamic and static models The objective in this section is to derive the expression for reduced dynamic and static model. This will be achieved by replacing the original dynamic or static system with a reduced DoF system while maintaining the kinematic characteristics of the original system. Let fua g3fun g and fud g3fun g be the primary (active) and passive (deleted) DoFs such that:  fua g (17.28) fun g ¼ fug ¼ fud g Let the active DoFs be the DoFs to be retained in the reduced model and let the passive DoFs be the DoFs to be condensed out in the model order reduction process. Eq. (17.27) can be partitioned as follows: 9 8 9 38 38 2 9 2 ½Caa  ½Cad  > ½Maa  ½Mad  < fu€a g = = > < fba g > = < fua g > 7 7 6 6 ¼ (17.29) þ4 5 5 4 > > > : > ; ; : ; : € u f g ½Mda  ½Mdd  ½Cda  ½Cdd  fud g fbd g d The second of Eq. (17.29) when multiplied out yields: ½Mda fu€a g þ ½Mdd fu€d g þ ½Cda fua g þ ½Cdd fud g ¼ fbd g

(17.30)

If we assume that the external forces and inertia effect on the deleted DoFs are negligible, then Eq. (17.30) leads to fud g ¼ ½RG fua g

(17.31)

where RG ˛Rda is the Guyan condensation matrix which is given by:

 ½RG  ¼  C1 (17.32) dd Cda Introducing Eq. (17.32) into Eq. (17.28) yields: fug ¼ ½TG fua g

(17.33)

In Eq. (17.33), ½TG is a linear transformation matrix that maps quantities in the reduced model onto the full model and is defined as

I ½TG  ¼ (17.34) RG  ˛Rna

where I ˛Raa is an identity matrix. If we assume the micromodulus function C to be time invariant, then the second time derivative of Eq. (17.33) yields: € ¼ ½TG fu€a g fug

(17.35)

Substituting Eqs. (17.33) and (17.35) into Eq. (17.27) gives: ½MG fu€a g þ ½CG fua g ¼ fbG g

(17.36)

where the matrices ½MG  ˛ Raa and ½CG  ˛Raa and the vector fbG g ˛ Ra are the condensed mass matrix, condensed micromodulus function, and II. New applications in peridynamics

3. Model order reduction using static condensation

381

condensed body force vector, respectively, of the reduced model and are defined as: ½MG  ¼ ½TG T ½M½TG ;

½CG ½TG T ½C½TG ;

fbG g ¼ ½TG fbg

(17.37)

Eq. (17.36) is the dynamic equilibrium equation of the reduced system. If we neglect dynamic effect and assume B to be in static equilibrium, then Eq. (17.36) specializes to ½CG fua g ¼ fbG g

(17.38)

Eq. (17.38) is the static equilibrium equation of the reduced system. Since the majority of storage requirement and computational effort required to implement this condensation technique is used in the computation of ½Cdd 1 , a computationally more efficient way to achieve the condensation of the PD static model is to employ the standard GaussJordan elimination procedure (Paz, 1983).

3.2 Reduced eigenvalue models In this section, the reduction process of eigenvalue problems will be illustrated. Let the solution to Eq. (17.1) be given by the general plane wave equation: uðx; tÞ ¼ Ae iðkxutÞ

(17.39)

where u is the displacement of a point located at x, A is the amplitude of wave, k is the wave number, u is the wave frequency, and t is the time. Substituting Eq. (17.39) into Eq. (17.1) and assuming no external body load is applied gives: Z   2 u rðxÞuðx; tÞ ¼ Cðx; qÞ eikx  1 uðx; tÞdx (17.40) Hx

where x ¼ q  x. It will be noticed that uðx; tÞ in Eq. (17.40) is independent of x and as such is treated as a constant in the integral on the right hand side. Taking Euler’s transformation of Eq. (17.40) yields Z u2 rðxÞuðx; tÞ ¼ Cðx; qÞðcosðkxÞ þ i sinðkxÞ  1Þdx$uðx; tÞ (17.41) Hx

Since the micromodulus function is an even function and sinðkxÞ is an odd function, then Eq. (17.41) reduces to Z 2 u rðxÞuðx; tÞ ¼ Cðx; qÞð1  cosðkxÞÞdx$uðx; tÞ (17.42) Hx

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17. Multiscale modeling with peridynamics

From the definition of Cðx; qÞ given in Eq. (17.1), it is inferred that the integral in Eq. (17.42) is a matrix which will be denoted as Z (17.43) Dðk; xÞ ¼ Cðx; qÞð1  cosðkxÞÞdx Hx

So that Eq. (17.42) can be written as u2 rðxÞuðx; tÞ ¼ Dðk; xÞ$uðx; tÞ

(17.44)

Eq. (17.44) is an eigenvalue problem with the following characteristic equation   Dðk; xÞ  u2 rðxÞ ¼ 0 (17.45) Dðk; xÞ is called the dispersion matrix and first appeared in Silling (2000). The solution to Eq. (17.45) yields eigenvalues and hence eigenvectors which correspond to the natural frequencies and natural modes of the system. In matrix form, Eq. (17.45) may be written as   ½D  u2 ½M ¼ 0 (17.46) where ½M ¼ r½I is the mass density matrix. The reduced order eigenvalue problem may be stated as:   ½DG   u2 ½MG  ¼ 0 (17.47) where ½DG  is the statically condensed dispersion matrix defined as ½DG  ¼ ½TG T ½D½TG 

(17.48)

4. Homogenization approach If the microscopic detail of a heterogeneous material can be defined by either a “Representative Volume Element” (RVE) for a statistic homogeneous medium or a “Repeating Unit Cell” (UC) for a periodic microstructured material, microscopic analysis might be performed to obtain a homogenized description for this material. At least two distinct scales coexist in the process of homogenization: the macroscopic scale (x) and microscopic scale ε (y). For illustrative purposes, let’s consider only linear elastic deformation. The constitutive relations for the original heterogeneous material can be assumed as sε ðyÞ ¼ CðyÞ$εε ðyÞ εε ðyÞ ¼ SðyÞ$sε ðyÞ

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(17.49)

4. Homogenization approach

383

where C and S are functions of location called stiffness tensor and compliance tensor, respectively, which are the inverse of each other, C ¼ S1 . sε and εε are called microscopic stress field and microscopic strain field. Let’s assume the microstructure of a composite is periodic, thus the micromechanical analysis can be performed within a UC. Any individual UC can be effectively approximated as a material point in the macroscopic analysis. Homogenization replaces the original heterogeneous material with a fictitious homogeneous medium which has the constitutive relations of sðxÞ ¼ C ðxÞ$εðxÞ εðxÞ ¼ S ðxÞ$sðxÞ

(17.50)

in which C and S are called effective stiffness tensor (or effective material property matrix) and effective compliance tensor, which satisfy the relations of C¼ S1 . s and ε are called macroscopic stress and strain, respectively, which are constant values within a cell’s domain ε. It is assumed that the macroscopic displacement field u can be expressed by the microscopic displacement field uε as Z 1 uε ðy; xÞdV uðxÞ ¼ (17.51) V where V represents the volume of the UC, and microscopic displacement field uε can be expressed in terms of volume averaged displacement field and displacement fluctuation function e uε as uε ðy; xÞ ¼ uðxÞ þ εðxÞ$x þ e uε ðyÞ

(17.52)

in which εðxÞ is the average strain vector defined as εðxÞ ¼

vuðxÞ vx

(17.53)

In the absence of body force, the following equilibrium condition needs to be satisfied sij;j ¼ 0

(17.54a)

or vsxx vsxz vsxy þ þ ¼0 vx vz vy vsyy vsyz vsyx þ þ ¼0 vy vz vx vszz vszy vszx þ þ ¼0 vz vy vx

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(17.54b)

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17. Multiscale modeling with peridynamics

where x, y, and z are subjected to microscopic scale parameter ε. Then according to average stress theorem, the volume average of the stress field inside the body is equal to the constant stress tensor sij along the border. Z 1 sε dV ¼ s (17.55) V For UCs subjected to Periodic Boundary Condition (PBC), Mathematical Homogenization Theory (MHT) which focuses on the determination of stress concentration tensor HðyÞ can be carried out to obtain the effective material properties. Although, the application of MHT is limited to UCs with PBC setup and not compatible with other boundary condition types, MHT is highly efficient compared with other homogenization methods. A bond-based peridynamic homogenization scheme based on MHT has been developed by Madenci et al. (2018). Here, a more generalized representative volume element homogenization method using state-based peridynamics will be discussed. Unlike finite element RVE homogenization method, boundary conditions in peridynamic homogenization is enforced on a fictitious boundary area with certain layers of node. Take PBC, for example,  tþ i  ti ¼ 0    þ  ¼ u þ ε  x uþ x ij i j i j

(17.56)

in which t is the traction, u is the displacement. Superscript represents the corresponding surface pairs. The relationship of surface pairs is defined in peridynamics as shown in Fig. 17.6. From Eq. (17.52) and Eq. (17.56), the microscopic strain field εε can be further obtained as εε ðyÞ εε ðyÞ ¼ εðxÞ þ e

(17.57)

Substituting the microscopic strain Eq. (17.57) and constitutive relations Eq. (17.49) into Eq. (17.54), the displacement-based formulation for RVE homogenization analysis can be obtained as

  A0 CðxÞ εðxÞ þ e εε ðyÞ ¼ 0 (17.58) in which A0 is the derivative operator, 2 0 v=vx 0 0 6 6 v=vz A0 ¼ 6 0 v=vy 0 4 0 0 v=vz v=vy

3 v=vz

v=vy

7 7 v=vx 7 5 v=vx 0 0

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(17.59)

4. Homogenization approach

385

FIGURE 17.6 Application of boundary conditions.

Substituting Eq. (17.55) into the constitutive relations Eq. (17.50) it is further obtained that 1R ε s dV C ¼ V ε

(17.60)

in which C is the effective stiffness tensor and can be obtained from the resulting stress field fluctuation of the UC subjected to macroscopic strain condition εi ði ¼ 1; .; 6Þ εT1 ¼ ½cs ; 0; 0; 0; 0; 0 εT2 ¼ ½0; cs ; 0; 0; 0; 0 εT3 ¼ ½0; 0; cs ; 0; 0; 0 εT4 ¼ ½0; 0; 0; cs =2; 0; 0 εT5 ¼ ½0; 0; 0; 0; cs =2; 0 εT6 ¼ ½0; 0; 0; 0; 0; cs =2

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(17.61)

386

17. Multiscale modeling with peridynamics

in which cs is the scale factor and equals to a small number. Effective property matrix C can then be assembled from 8 9 8 < sxx > 9 C1;i > > > i > > > > > > > > > > > yy > > > > > > > > C < s > 2;i > > > > i > > > > > > > > > > > > , > > < < szz < C3;i > = = > > i (17.62) C4;i > ¼ > < syz > > cs ; ði ¼ 1; .; 6Þ > > > > > i > > > > > > > > C > > > > > > > zx 5;i > > > > > < s > > > > > i > > > > > > > > : > ; > : ; xy C6;i < si > where the angle brackets denote the volume average over the UC domain. This can be done in numerical peridynamics analysis as six separate loading steps.

5. Conclusions In this chapter three different multiscale methodologies used in peridynamic framework including coarsening, model order reduction using static condensation, and homogenization were covered and briefly explained. By using such strategies, significant benefits can be obtained especially in terms of computational time.

References Galadima, Y., Oterkus, E., Oterkus, S., 2019. Two-dimensional implementation of the coarsening method for linear peridynamics. AIMS Mater. Sci. 6 (2), 252e275. Galadima, Y., Oterkus, E., Oterkus, S., 2020. Model Order Reduction of Linear Peridynamic Systems Using Static Condensation. Mathematics and Mechanics of Solids. Madenci, E., Barut, A., Phan, N., 2018. Peridynamic unit cell homogenization for thermoelastic properties of heterogenous microstructures with defects. Compos. Struct. 188, 104e115. Paz, M., 1983. Practical reduction of structural eigenproblems. J. Struct. Eng. 109 (11), 2591e2599. Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. JMPSA 48 (1), 175e209. Silling, S.A., 2011. A coarsening method for linear peridynamics. Int. J. Multiscale Comput. Eng. 9 (6). Silling, S.A., Askari, E., 2005. A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83 (17e18), 1526e1535.

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C H A P T E R

18 Application of peridynamics for rock mechanics and porous media Selda Oterkus1, Erdogan Madenci2, Erkan Oterkus1 1

Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom; 2 Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States

O U T L I N E 1. Introduction

388

2. Fully coupled poroelastic peridynamic formulation

389

3. Numerical implementation

390

4. Numerical results 4.1 Consolidation problem (1D) 4.2 Consolidation problem (2D) 4.3 Square plate with a hydraulically pressurized crack problem (2D)

391 391 395 396

5. Conclusions

398

References

400

Further reading

401

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00010-X

387

© 2021 Elsevier Inc. All rights reserved.

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18. Application of peridynamics for rock mechanics and porous media

1. Introduction The poroelasticity theory has important applications in geomechanics including oil exploration, gas-hydrate detection, seismic monitoring of CO2 storage, hydrogeology, etc. (Carcione et al., 2010). A recent popular application of poroelasticity is the analysis of hydraulic fracturing. Hydraulic fracturing has been widely used especially in United States to extract oil and gas from shale, which is a fine-grained sedimentary rock, by pumping a special fluid into a well and creating cracks inside the shale layer. Many different numerical approaches have been utilized to model hydraulic fracturing process including finite element method (Hunsweck et al., 2013; Ouyang et al., 1997), cohesive zone elements (Chen, 2012), and extended finite element method (Lecampion, 2009). In this chapter, an alternative approach, peridynamics, is presented. Peridynamics was introduced by Silling (2000) and has been used to analyze many different challenging problems, especially fracture (Oterkus and Madenci, 2012; Gao and Oterkus, 2019; Wang et al., 2018; Oterkus et al., 2012; Alpay and Madenci, 2013; Yang et al., 2019; Imachi et al., 2019; Zhu et al., 2016; Vazic et al., 2017; Basoglu et al., 2019; Nguyen and Oterkus, 2019; Huang et al., 2019; Li et al., 2020; Lu et al., 2020). Peridynamics was utilized for the analysis of rock mechanics and porous media in various studies in the literature. Among these, fully coupled poroelastic peridynamic formulation for fluid-filled fractures was developed by Oterkus et al. (2017). Turner (2013) presented a new formulation for incorporating the effects of pore pressure in peridynamic framework. Nadimi et al. (2016) used peridynamics to simulate the initiation and propagation of hydraulic fracturing. Ouchi et al. (2015) utilized statebased peridynamic formulation to simulate fluid-driven fractures in heterogeneous poroelastic medium. Ha et al. (2015) investigated the complex fracturing responses of a single flaw embedded in rock-like materials under compression. Rabczuk and Ren (2017) utilized dualhorizon peridynamics to analyze fracture in granular and rock-like materials. Wang et al. (2016) used non-ordinary state-based peridynamics by incorporating the maximum tensile stress criterion and the MohrCoulomb criterion to simulate crack initiation and propagation in rocks subjected to compressive loading. In another study, Wang et al. (2017) utilized conjugated bond-based peridynamics formulation to analyze fracture behavior in rock specimens. After presenting the fully coupled peridynamic formulation, several problems are considered to demonstrate the capability of the presented approach.

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2. Fully coupled poroelastic peridynamic formulation

2. Fully coupled poroelastic peridynamic formulation Peridynamics is a new nonlocal continuum mechanics formulation. Material points inside the solution domain are interacting with each other in a nonlocal manner. The range of nonlocal interactions is defined with the horizon, H. In this chapter, a fully coupled poroelastic peridynamic formulation is presented. According to Wang (2000), it is possible to make an analogy between poroelastic and thermoelastic formulations. Therefore, based on the fully coupled thermoelastic peridynamic formulation given by Oterkus et al. (2014), the governing equations of the fully coupled poroelastic peridynamic formulation can be written as Z   y0  y € tÞ ¼ c s  aB gP 0 ruðx; dV þ bðx; tÞ (18.1a) jy  yj H

 Z  qf 1 _ Pðx0 ; tÞ  Pðx; tÞ c _ a  g e dV þ Pðx; tÞ ¼ kP B 0 QB 2 rf jx  xj

(18.1b)

H

In Eq. (18.1a), r is density, u€ is the acceleration of the material point x, y is the position of the material point x in the deformed configuration, t is time, and b is the body load vector. c is denoted as bond constant and can be expressed in terms of material parameters of classical continuum mechanics as 8 2E > > > ð1DÞ > > Ad2 > > > < 9E c¼ (18.2) ð2DÞ > phd3 > > > > > 12E > > ð3DÞ : pd4 where E is the elastic modulus, A is the cross-sectional area, h is the thickness of the plate, and d is the horizon size. The stretch of the bond (interaction) between two material points x and x0 , s, can be defined as s¼

jy0  yj  jx0  xj jx0  xj

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(18.3)

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18. Application of peridynamics for rock mechanics and porous media

aB represents Biot parameter and the coefficient of fluid pore pressure g is defined as 8 > 1 > > ð1DÞ > > E > > > < ð1  nÞ (18.4) g¼ ð2DÞ > E > > > > > ð1  2nÞ > > ð3DÞ : E P denotes the average fluid pore pressure of the two interacting material points x and x0 which can be expressed as P¼

Pðx0 ; tÞ þ Pðx; tÞ 2

(18.5)

In Eq. (18.1b), QB is the Biot modulus, qf and rf are mass of fluid produced per unit time and the density of the fluid, and kP is the peridynamic parameter which is defined as 8 > 2kP > > ð1DÞ > > 2 > > > mv Ad > > < 6k P ð2DÞ kP ¼ (18.6) > pmv hd3 > > > > > > 6kP > > > : pm d4 ð3DÞ v

where kP is the permeability of the bulk material and mv is the fluid viscosity. The time rate of change of extension can be expressed as  y0  y  0 e_ ¼ 0 $ u_  u_ (18.7) jy  yj in which u_ is the velocity of the material point x.

3. Numerical implementation Analytical solution of integro-differential equations given in Eq. (18.1a,b) is usually not possible. Therefore, numerical techniques are widely utilized including meshless method. Moreover, the fully coupled poroelastic peridynamic equations given in Eq. (18.1a,b) can be solved by using staggered strategy (Armero and Simo, 1992). Eq. (18.1a) is solved to determine the displacement field and Eq. (18.1b) is solved to determine

II. New applications in peridynamics

4. Numerical results

391

the fluid pore pressure field. Hence, the governing equations given in Eq. (18.1a,b) can be written in discrete form for the material point k as N   yn  yn X n ðjÞ ðkÞ VðjÞ þ bðkÞ c snðkÞðjÞ  aB gPðkÞðjÞ  n  yn  yðjÞ j¼1 ðkÞ  ! N PnðjÞ  PnðkÞ c X q 1 _n   aB ge_nðkÞðjÞ VðjÞ þ f ðkÞ P ¼ kP   2 QB ðkÞ j¼1 r f ðkÞ x  x 

ru€nðkÞ ¼

ðjÞ

(18.8a)

(18.8b)

ðkÞ

where n

PðkÞðjÞ ¼

PnðjÞ þ PnðkÞ

2 n n  yðjÞ  yðkÞ  $ u_ nðjÞ  u_ nðkÞ e_nðkÞðjÞ ¼   ynðjÞ  ynðkÞ       n    yðjÞ  ynðkÞ   xnðjÞ  xnðkÞ  n   sðkÞðjÞ ¼  n  xðjÞ  xnðkÞ 

(18.9a) (18.9b)

(18.9c)

and N is the number of material points inside the horizon of the material point k, n is the time step number, and VðjÞ is the volume of the material point j. Application of boundary conditions is different than classical continuum mechanics. Boundary conditions can be applied by creating a fictitious region outside the actual solution domain with a size equivalent to the horizon size.

4. Numerical results 4.1 Consolidation problem (1D) The length of the one-dimensional (1D) model is specified as L ¼ 15 m as shown in Fig. 18.1. The density, elastic modulus, and Poisson’s ratio are r ¼ 1900 kg/m3, E ¼ 1:0  108 N/m2, and n ¼ 1=3, respectively. The Biot modulus and Darcy conductivity of the material are specified as Q ¼ 1 1:65  1010 N/m2 and k ¼ kp =m ¼ 1:02  109 m4/Ns, respectively. A constant pressure load of Po ¼ 1  104 N/m2 is applied at the top surface and bottom surface. Fluid drainage is only permitted at the top surface. Therefore, the boundary conditions are specified as: szz ðz ¼ 0; tÞ ¼  Po

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(18.10a)

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FIGURE 18.1 Geometry, boundary conditions, and discretization of the one-dimensional consolidation problem.

uz ðz ¼ L; tÞ ¼ 0

(18.10b)

Pðz ¼ 0; tÞ ¼ 0

(18.10c)

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393

4. Numerical results

vP ðz ¼ L; tÞ ¼ 0: vz

(18.10d)

Initial conditions are specified as: Pðz; t ¼ 0Þ ¼ yPo

(18.11a)

uz ðz; t ¼ 0Þ ¼ ai Po ðh  zÞ

(18.11b)

a 1 ¼ 1:0  108 N/m2 (inverse final i where y ¼ aa aa with ai ¼ 1þa2 aQ, a compressibility), and the Biot parameter a ¼ 0:5. The model is created by using a discretization size of Dx ¼ 0:015 m and the horizon size of d ¼ 3Dx. The time step size is specified as Dt ¼ 1  106 s. For the PD model initial conditions are implemented as:

Mechanical field uz ðzf ; t ¼ 0Þ ¼ ai Po ðh  zÞ;

zf ˛ðz < 0Þ

uz ð0 < z < L; t ¼ 0Þ ¼ ai Po ðh  zÞ

(18.12a) (18.12b)

uz ðzf ; t ¼ 0Þ ¼ 0;

zf ˛ðz > LÞ

(18.12c)

Pðzf ; t ¼ 0Þ ¼ 0;

zf ˛ðz < 0Þ

(18.13a)

Flow field Pð0 < z < L; t ¼ 0Þ ¼ yPo Pðzf ; t ¼ 0Þ ¼ yPo ;

zf ˛ðz > LÞ

(18.13b) (18.13c)

For the PD model boundary conditions are implemented as: Mechanical field bðzf ; t ¼ 0Þ ¼ Po =ð3dxÞ;

zf ˛ðz < 0Þ

(18.14a)

uz ðzf ; t ¼ 0Þ ¼ 0;

zf ˛ðz > LÞ

(18.14b)

Pðzf ; t > 0Þ ¼ 0;

zf ˛ðz < 0Þ

(18.15a)

Flow field Pðzf ; t > 0Þ ¼ PðzÞ;

zf ˛ðz > LÞ

(18.15b)

The pore pressure and displacement variations along the 1D model at different times are shown in Figs. 18.2 and 18.3. Peridynamic solutions are compared against analytical results and a very good agreement is observed between the two solutions.

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FIGURE 18.2 Variation of the pore pressure along the one-dimensional model.

FIGURE 18.3 Variation of the vertical displacements along the one-dimensional model.

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4. Numerical results

395

4.2 Consolidation problem (2D) The consolidation problem considered in the previous section is now considered by using a two-dimensional (2D) model as shown in Fig. 18.4. The length and width of the model are specified as L ¼ 15 m and W ¼ 1:5 m, respectively. In addition to the boundary conditions described for the one-dimensional model, the following additional boundary conditions are applied: ux ðx ¼ 0; z; tÞ ¼ 0

(18.16a)

ux ðx ¼ W; z; tÞ ¼ 0

(18.16b)

vP ðx ¼ 0; z; tÞ ¼ 0 vx vP ðx ¼ W; z; tÞ ¼ 0: vx

FIGURE 18.4

(18.16c) (18.16d)

Geometry and boundary conditions of the 2D consolidation problem.

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FIGURE 18.5 Variation of the pore pressure along the central vertical axis of the 2D model at different times.

The model is created by using a discretization size of Dx ¼ 0:075 m and the horizon size of d ¼ 3Dx. The time step size is specified as Dt ¼ 1  105 s. The pore pressure and displacement variations in the 2D model at different times are shown in Figs. 18.5 and 18.6. Peridynamic solutions are compared against finite element analysis results obtained by using ANSYS, a commercial finite element software, and a very good agreement is observed between the two approaches.

4.3 Square plate with a hydraulically pressurized crack problem (2D) In the final numerical example, a square plate with a hydraulically pressurized crack problem is considered as shown in Fig. 18.7. The length and width of the solution domain are specified as L ¼ W ¼ 6 m. The crack has a length of 2a ¼ 0:3 m. The Biot parameter is specified as a ¼ 0:1. A hydraulic pressure of Po is applied on crack surfaces. Outer surfaces of the solution domain are fully constrained and no flow is permitted from these surfaces. The boundary conditions can be summarized as: ux ðx ¼ 0; z; tÞ ¼ 0

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(18.17a)

4. Numerical results

397

FIGURE 18.6 Variation of the vertical displacements along the central vertical axis of the 2D model at different times.

FIGURE 18.7 Geometry and boundary conditions of the square plate with a hydraulically pressurized crack problem.

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ux ðx ¼ W; z; tÞ ¼ 0

(18.17b)

uz ðx; z ¼ 0; tÞ ¼ 0

(18.17c)

uz ðx; z ¼ L; tÞ ¼ 0

(18.17d)

Pðx ¼ 0; z; tÞ ¼ 0

(18.17e)

Pðx ¼ W; z; tÞ ¼ 0

(18.17f)

Pðx; z ¼ 0; tÞ ¼ 0

(18.17g)

Pðx; z ¼ L; tÞ ¼ 0

(18.17h)

szz ðz ¼ L = 2; xÞ ¼  Po

t to

Pðz ¼ L = 2; xÞ ¼ Po

L=2  a < x < L=2 þ a t to

L=2  a < x < L=2 þ a

(18.17i) (18.17j)

The amount of hydraulic pressure is Po ¼ 2000 Pa with to ¼ 0:01 s. The initial conditions are given as ux ðx; z; t ¼ 0Þ ¼ 0

(18.18a)

uz ðx; z; t ¼ 0Þ ¼ 0

(18.18b)

u_ x ðx; z; t ¼ 0Þ ¼ 0

(18.18c)

u_ z ðx; z; t ¼ 0Þ ¼ 0

(18.18d)

Pðx; z; t ¼ 0Þ ¼ 0

(18.18e)

The model is created by using a discretization size of Dx ¼ 0:015 m and the horizon size of d ¼ 3Dx. The time step size is specified as Dt ¼ 1  106 s. The pore pressure and displacement variations in the 2D model at time t ¼ 6  103 s are shown in Fig. 18.8.

5. Conclusions In this chapter, fully coupled poroelastic formulation was presented. Governing equations were obtained by making analogy with thermoelasticity. To demonstrate the capability of the approach, two different example cases were considered including 1D and 2D consolidation problems and a square plate with a hydraulically pressurized crack problem. Displacement and pore pressure distributions were obtained by using the presented formulation.

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399

(a)

(b)

(c)

FIGURE 18.8

Variation of (A) horizontal displacements, (B) vertical displacements, and (C) pore pressure inside the solution domain at time t ¼ 6  103 s (displacements are magnified by 20000 for the deformed configuration). II. New applications in peridynamics

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18. Application of peridynamics for rock mechanics and porous media

References Alpay, S., Madenci, E., 2013. Crack growth prediction in fully-coupled thermal and deformation fields using peridynamic theory. In: 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 1477. Armero, F., Simo, J.C., 1992. A new unconditionally stable fractional step method for nonlinear coupled thermomechanical problems. Int. J. Num. Meth. Engg. 35 (4), 737e766. Basoglu, M.F., Zerin, Z., Kefal, A., Oterkus, E., 2019. A computational model of peridynamic theory for deflecting behavior of crack propagation with micro-cracks. Comput. Mater. Sci. 162, 33e46. Carcione, J.M., Morency, C., Santos, J.E., 2010. Computational poroelasticityda review. Geophysics 75 (5), 75A229e75A243. Chen, Z., 2012. Finite element modelling of viscosity-dominated hydraulic fractures. J. Petrol. Sci. Eng. 88, 136e144. Gao, Y., Oterkus, S., 2019. Ordinary state-based peridynamic modelling for fully coupled thermoelastic problems. Continuum Mech. Therm. 31 (4), 907e937. Ha, Y.D., Lee, J., Hong, J.W., 2015. Fracturing patterns of rock-like materials in compression captured with peridynamics. Eng. Fract. Mech. 144, 176e193. Huang, Y., Oterkus, S., Hou, H., Oterkus, E., Wei, Z., Zhang, S., 2019. Peridynamic model for visco-hyperelastic material deformation in different strain rates. Continuum Mech. Therm. 1e35. Hunsweck, M.J., Shen, Y., Lew, A.J., 2013. A finite element approach to the simulation of hydraulic fractures with lag. Int. J. Numer. Anal. Methods GeoMech. 37 (9), 993e1015. Imachi, M., Tanaka, S., Bui, T.Q., Oterkus, S., Oterkus, E., 2019. A computational approach based on ordinary state-based peridynamics with new transition bond for dynamic fracture analysis. Eng. Fract. Mech. 206, 359e374. Lecampion, B., 2009. An extended finite element method for hydraulic fracture problems. Commun. Numer. Methods Eng. 25 (2), 121e133. Li, M., Lu, W., Oterkus, E., Oterkus, S., 2020. Thermally-induced fracture analysis of polycrystalline materials by using peridynamics. Eng. Anal. Bound. Elem. 117, 167e187. Lu, W., Li, M., Vazic, B., Oterkus, S., Oterkus, E., Wang, Q., 2020. Peridynamic modelling of fracture in polycrystalline ice. J. Mech. 36 (2), 223e234. Nadimi, S., Miscovic, I., McLennan, J., 2016. A 3D peridynamic simulation of hydraulic fracture process in a heterogeneous medium. J. Petrol. Sci. Eng. 145, 444e452. Nguyen, C.T., Oterkus, S., 2019. Peridynamics for the thermomechanical behavior of shell structures. Eng. Fract. Mech. 219, 106623. Ouchi, H., Katiyar, A., York, J., Foster, J.T., Sharma, M.M., 2015. A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamics approach. Comput. Mech. 55 (3), 561e576. Ouyang, S., Carey, G.F., Yew, C.H., 1997. An adaptive finite element scheme for hydraulic fracturing with proppant transport. Int. J. Numer. Methods Fluid. 24 (7), 645e670. Oterkus, E., Madenci, E., 2012. Peridynamics for failure prediction in composites. In: 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference 20th AIAA/ASME/AHS Adaptive Structures Conference 14th AIAA, p. 1692. Oterkus, E., Guven, I., Madenci, E., 2012. Impact damage assessment by using peridynamic theory. Open Eng. 2 (4), 523e531. Oterkus, S., Madenci, E., Agwai, A., 2014. Fully coupled peridynamic thermomechanics. J. Mech. Phys. Sol. 64, 1e23. Oterkus, S., Madenci, E., Oterkus, E., 2017. Fully coupled poroelastic peridynamic formulation for fluid-filled fractures. Eng. Geol. 225, 19e28. Rabczuk, T., Ren, H., 2017. A peridynamics formulation for quasi-static fracture and contact in rock. Eng. Geol. 225, 42e48.

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Further reading

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Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solid. 48 (1), 175e209. Turner, D.Z., 2013. A non-local model for fluid-structure interaction with applications in hydraulic fracturing. Int. J. Comput. Methods Eng. Sci. Mech. 14 (5), 391e400. Vazic, B., Wang, H., Diyaroglu, C., Oterkus, S., Oterkus, E., 2017. Dynamic propagation of a macrocrack interacting with parallel small cracks. AIMS Mater. Sci. 4 (1), 118e136. Wang, H.F., 2000. Theory of linear poroelasticity with applications to geomechanics and hydrogeology, vol. 2. Princeton University Press. Wang, H., Oterkus, E., Oterkus, S., 2018. Predicting fracture evolution during lithiation process using peridynamics. Eng. Fract. Mech. 192, 176e191. Wang, Y., Zhou, X., Xu, X., 2016. Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics. Eng. Fract. Mech. 163, 248e273. Wang, Y., Zhou, X., Shou, Y., 2017. The modeling of crack propagation and coalescence in rocks under uniaxial compression using the novel conjugated bond-based peridynamics. Int. J. Mech. Sci. 128, 614e643. Yang, Z., Oterkus, E., Nguyen, C.T., Oterkus, S., 2019. Implementation of peridynamic beam and plate formulations in finite element framework. Continuum Mech. Therm. 31 (1), 301e315. Zhu, N., De Meo, D., Oterkus, E., 2016. Modelling of granular fracture in polycrystalline materials using ordinary state-based peridynamics. Materials 9 (12), 977.

Further reading Biot, M.A., 1941. General theory of three-dimensional consolidation. J. Appl. Phys. 12 (2), 155e164.

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C H A P T E R

19 Application of high-performance computing for peridynamics Cagan Diyaroglu, Bozo Vazic, Erkan Oterkus, Selda Oterkus Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom

O U T L I N E 1. Introduction

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404 405 407 408

3. Numerical results

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4. Conclusions

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References

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1. Introduction Peridynamic (PD) codes, which are written by using any programming language, e.g., C, Cþþ, FORTRAN, or Java, can be run in any available computing facility. This can be a desktop/laptop computer or a high performance computing (HPC) facility. Generally, researchers tend to

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00017-2

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solve large and complex problems in HPC facilities, which are mostly available to universities and research organizations. However, without making any modification to regular computer codes, the advantage of using a high performance computing facility may not be realized and even no improvement may be observed using such facilities compared to regular desktops. In this regard, small modifications to computer codes can make it possible to solve very large size problems. Moreover, in some cases, desktops/laptops can be sufficient to solve such large size problems rather than using HPC facilities. Nowadays, desktops/laptops include many central processing units (CPUs) and relatedly many cores in their architectures. Moreover, graphical processing units (GPUs) of computers have been improved above and beyond the expectations. Either one of them or both of them can be benefited in order to solve large and complicated problems in personal desktops/laptops. This chapter presents how it is possible to improve PD codes and make them efficient for large-scale problems with the help of parallel programming procedures. Moreover, how simple modifications with very basic knowledge of parallel programming skills can lead to significant increase in our calculation speeds is demonstrated. Specifically, CPU- and GPU-based approaches are explained for parallelizing PD codes.

2. Parallel programming of a PD code In order to solve large and complex problems in desktops or laptops within a reasonable time, codes must be modified to use many processors available in the architecture. By doing such modifications, the parts of the problem can be solved independently with each processor. These parts can either be the instructions of the code or the big data in the code, which are in the form of arrays or matrices. Such solution procedure is called parallel programming. However, in regular or serial solution procedure, the parts are executed or solved sequentially with only one processor. Fig. 19.1 shows the difference between serial and parallel programming procedures. Bearing in mind that the parts of the problem can be the instructions of the code and/or the data in the code, there are several ways of doing parallel programming. These are named as the single instruction multiple data (SIMD), the multiple instruction single data (MISD), and the multiple instruction multiple data (MIMD) procedures. In this section, SIMD parallel programming type is used to execute the time integration part of a PD code in which arrays with big data are solved concurrently with many processors as well as the same instruction, which is the time integration part, in the code is used by each processor.

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FIGURE 19.1 Types of programming procedures.

To be more precise, whenever a PD code does not solve any matrix equations, it is straightforward to parallelize the time integration part of the code. The solution methods, which do not include matrices, are Adaptive Dynamic Relaxation (ADR) and explicit methods. In these methods, we do not have to solve the equations simultaneously. More specifically, the PD force density functions, fðkÞðjÞ , are summed individually for each family member material point, j, in the time integration part and the unknown term, i.e., the acceleration or the displacement, is calculated. Due to independent calculation of the unknown term for each main material point k, SIMD parallel programming type is very convenient for time integration part of the code.

2.1 CPU-based approach The single instruction, which is the time integration part of the code, is solved with arrays, i.e., coordinate, displacement, and acceleration arrays. Especially, in three-dimensional complex problems these arrays become very large, and if they are dealt with many cores in parallel, significant time can be gained. In SIMD parallel programming type, the data in these arrays are separated into parts, which are called multiple data, and these multiple data or element groups of arrays are solved simultaneously with many cores available. If these cores are part of a single computer such that they share the same memory, the architecture is named as shared memory. In fact, shared memory architecture of the personal desktops or laptops is rather complicated but simply they share one common memory, as shown in Fig. 19.2. In some situations, many cores from different computers can

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FIGURE 19.2 Comparison of the shared and the distributed memory architectures.

also be linked together so that the memories are also connected together. This kind of architecture is named as distributed memory and mostly available in HPC facilities. Fig. 19.2 shows the general difference between the memory architectures. As depicted in Fig. 19.2, the shared memory parallelization can be done by commonly used software, OpenMP. It is an application program interface (API) as well as it supports C/Cþþ and FORTRAN programming languages. OpenMP manages the cores in CPUs by sharing the work between them, meaning that the cores can be run simultaneously by doing their own work using the shared memory. Apart from that, Fig. 19.2 also depicts the connection between different computer memories, which is done by a network. In this architecture type, parallelization is done by managing data between the CPUs so that the data is made available to any CPU within the network. This type of architecture is named as distributed memory and parallelization of many CPUs from different computers can be done by using the Message Passing Interface (MPI). The MPI is actually a library and is compatible with all programming languages. MPI provides connection between the different memories and also between the CPUs from different computers. The messages are passed among each CPU and the data transfers are managed between the shared memories. However, we only consider the shared memory parallelizing with OpenMP, here. In OpenMP, the cores are generally represented with the threads and each core can be considered as a thread. The master thread executes the parts of the code sequentially until the parallelized part is encountered. At the beginning of this part, slave threads reproduce from the master thread each representing the cores of the computer. Then, the master and the slave threads execute the specified part or parts of the code simultaneously until the end of the parallel section. At the end, slave threads join the master thread and it continues to execute the rest of the code in serial. Fig. 19.3 shows such an example of parallelized code with OpenMP. This code is written in FORTRAN but can easily be transformed into any programming language with slight modifications, please look at

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2. Parallel programming of a PD code

FIGURE 19.3

407

OpenMP implementation for an example PD code.

Barney, (2015) for more details. Besides, only the time integration part of the PD code is shown in here because solving other parts in serial may be sufficient. As can be seen from Fig. 19.3, parallelizing the code is very simple and only two new lines, which are denoted in green color, are inserted to the time integration part of the code. It is also important to mention that before the execution of the code, the total number of threads must be exported or defined with a following command in LINUX operating system and it is usually taken as equal to number of cores available in the computer. Export OMP_NUM_THREADS¼8 So, the number of threads that will be used by the code is 8. Until the first OpenMP command, which starts with “!$OMP PARALLEL DO,” the code is only executed with a master thread and at that point, master thread reproduces the slave threads. After that, the first following do loop, in which it calculates the total central force, i.e., force and then the unknown acceleration, acel, for each main point kk, is executed with these slave and master threads simultaneously. The meaning of the directives of a “!$OMP PARALLEL DO” command can be summarized as; -

-

The “DEFAULT(SHARED)” directive; it mentions that all variables available in the code will share the same memory. The “PRIVATE(kk, mnode, force.)” directive; it mentions that the variables inside the parenthesis are private to each thread. In other words, each thread creates its own copy of these variables and values of which can differ for each thread. For example, each thread calculates its own total force which belongs to the main point kk. The “SCHEDULE(DYNAMIC, CHUNK)” directive; it mentions that each thread executes some part of the arrays which is mentioned by CHUNK variable. The calculation of the chunk size can be seen in Fig. 19.3 as a CHUNK variable. Moreover, the “DYNAMIC” command forces free thread to be dynamically assigned to another residual chunk when it finishes its own chunk.

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At the end, after each thread calculates its own chunk and all the elements of the arrays are done, the threads join together with the “!$OMP END PARALLEL DO” command. After this point, only the master thread continues to execute the code. One of the important feature of the OpenMP is that whenever the thread encounters with “!$OMP END” command, if it is not assigned to any other job, it waits until all of the threads finish their own job. This feature of OpenMP makes parallel programming safe because it does not allow conflicts between the threads. Same steps repeat for each time step until the end of the desired time.

2.2 GPU-based approach Another way of parallelizing the time integration part of the code is to use the graphical processing units (GPUs) of the computers. Exploiting GPUs for parallel computing is very new and their history is not as old as CPU’s history, but their improvement on computational time is rather significant. From 1980s to the late 1990s graphics hardware was not programmable and it was only used for the purpose of graphics processing. Due to a high demand for realistic visualization of games, significant amount of money was invested into game industry and graphics hardware developers were forced for faster options. As a result, graphics hardware made a big impact in the game industry and scientists started to engage with the GPUs. The main idea was to exploit the high calculation capacity of the GPUs for scientific applications and its history goes back to the late 1990s. After several attempts, the graphics hardware or the GPU of the computers became increasingly programmable and the first GPU was released in 1999 by NVIDIA. However, there has been remarkable progress by means of scientific calculations since 2006. This was mainly because researchers must have had very good knowledge on the hardware of the GPUs in order to use them properly until that time. In 2006, the hardware and the software of GPU were unified and NVIDIA released the compute unified device architecture (CUDA). Since then, CUDA has been frequently used for the scientific computations. Now, there are many applications and studies, which used CUDA for engineering applications, available in the literature. In order to run CUDA in personal desktops/ laptops or in HPC facilities, CUDA-enabled hardware must be installed to the GPU architecture. Furthermore, if the programming language is C or Cþþ, CUDA can be run easily without installing its software. However, for FORTRAN, CUDA FORTRAN compiler must be installed. The way that the CPU and the GPU architectures in parallel programming work is different from each other. GPU tries to solve many simple equations simultaneously with small in size but excessive number of simple arithmetic logic units (ALUs). However, ALUs are much more

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powerful and complex in the modern CPUs because they try to solve more complicated problems in less time. Moreover, the control unit, which decodes the instructions from the input and which converts them into control signals, and the cache, which is a high speed static RAM, of the CPU occupy much more space than the control unit and the cache of the GPU. Please see Fig. 19.4 for a schematic comparison of CPU and GPU architectures. In this figure, DRAM indicates the dynamic random access memory. As mentioned earlier, the advantage of GPU comes from very high number of ALUs available on it and each ALU can execute very simple mathematical equations very quickly. Parallelizing a code by using CUDA is mainly an SIMD parallel programming type. As mentioned earlier, in this type of parallel programming, multiple data can be run simultaneously with single instruction of the code. In here, time integration part of the code is taken as a single instruction and the elements of the arrays, i.e., the coordinate, the displacement, and the acceleration arrays, can be solved in parallel. Before introducing the solution procedure in CUDA, it is essential to know some fundamental information on GPU architecture. Simply, GPU is composed of many number of processors and each of which is named as a thread. For example, NVDIA GTX680 graphics card includes 16,384 threads in its architecture. In CUDA, these threads are grouped by blocks and these blocks imitate the GPU architecture. GPU has streaming multiprocessors (SMs) on it and the number of SMs depend on the type of the graphics card used. Mainly, the SMs are general purpose processors, but they are designed very differently than the execution cores in CPUs. In this regard, blocks are distributed to each SM, which is available in the GPU, by the scheduler. The blocks are independent and they are executed by the streaming multiprocessors (SMs) sequentially or concurrently. In Fig. 19.5A, an example problem is shown and it is partitioned into eight parts. Each part of the problem is represented by a block and each block is

FIGURE 19.4

Schematic comparison of the CPU and the GPU architectures.

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a) GPU with four SMs

b) GPU with two SMs FIGURE 19.5 The solution of a problem with the GPU architecture. (A) GPU with four SMs. (B) GPU with two SMs.

scheduled to any available SM. In this GPU architecture, there are four SMs in total and each SM may execute in two blocks. Furthermore, depending on the availability of the SMs, the blocks are run sequentially or concurrently. The same problem can also be run with any other GPU, which has two SMs in total, as shown in Fig. 19.5B. In that case, each SM may execute four blocks and thus the solution time is higher than the former GPU architecture (CUDA C Programming Guide, 2015). Furthermore, it is clear that the same problem can be solved with two different architectures without changing the block numbers. This is actually a very

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2. Parallel programming of a PD code

411

important feature of CUDA because GPU architectures do not have any effect on the results but only on the solution time. GPU has also many kinds of memory on the device. These are the global memory, the shared memory, the constant memory, the texture memory, the per-thread local memory, and the registers. To be more precise, the most important ones are the -

Global memory; it is available to all threads on the device. Shared memory; it is only accessible by the threads in the same block. Per-thread local memory; it only represents one thread’s local memory.

For more information about the device memory architecture, please see (Diehl, 2012; Massimiliano and Gregory, 2014). Fig. 19.6 shows an example PD code parallelized with CUDA and the red color highlighted variables, attributes or signs are peculiar to CUDA in addition to usual FORTRAN commands. This code is written in PGI’s CUDA FORTRAN and it can be easily transformed into C/Cþþ programming language with slight modifications. Please see Kirk and Hwu (2012) for more details. Fig. 19.6A shows the main structure of the PD CUDA code. At the beginning of the program, two modules are introduced with the “USE” directives. One module is the “cudafor” which already exists in the CUDA library and it contains many CUDA definitions. The other is the user-defined module and here it is named as “time_integration” because it actually represents the time integration part of the code. In CUDA FORTRAN, the latter module is actually run on the GPU and it is shown in Fig. 19.6B separately. This part of the code is generally named as device code. On the other hand, the main program, shown in Fig. 19.6A, is run on the CPU and it is generally named as a host code. At the beginning of the host code, the null arrays, which will be used by the host and the device codes, are opened with its dimensions. The arrays, which are declared by the “REAL, DEVICE” and the “INTEGER, DEVICE” attributes, reside on the device memory. To be more precise, they are all located on the global memory of the device. These arrays are named with additional “_d” suffix different from the original arrays, which are located on the host memory. Here, only the coordinate, the displacement, the surface correction, the number of family members, and the family member arrays are shown and they are named as coord, disp, acorr, nfmem, and fmem, respectively. The element number of each array can either be equal to the total number of points, which is denoted by the tnode, or the total number of bonds, which is denoted by the bond, available in the PD domain. The rest of the PD code is written in the regular (serial) basis which already makes the serial parts very efficient for large-scale problems. The arrays, which are

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a) The main structure of the code

b) The parallelized module section of the code FIGURE 19.6

An example CUDA code. (A) The main structure of the code. (B) The parallelized module section of the code.

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2. Parallel programming of a PD code

413

created in the serial part of the code, reside on the host memory. The time integration part of the code is actually run on the device (GPU) and the built-up arrays must be transferred to the device memory. These transfer operations are actually achieved with the assignment statement, i.e., “¼,” as shown in Fig. 19.6A. In order to run the time integration part of the code on the device or on the GPU, the execution configuration, which is an intrinsic statement of the CUDA, is written as follows; CALL name of the subroutine > (the arrays and the variables that will be used by the device code). In the host code, the name of the subroutine is chosen as “solution” in Fig. 19.6A. There, the number of the blocks are calculated as number of blocks ¼ CEILINGðREALðtnodeÞ = threadsÞ

(19.1)

where tnode and threads are the integer variables in the code which are the total number of the material points and the number of the threads in each block, respectively. Each block has limited number of threads depending on the type of the GPU device and for the generally used Fermi-based architecture, it can be assumed as 1024 (Massimiliano and Gregory, 2014). So, the integer variable, threads, which defines the number of the threads in each block, can be chosen as equal to 1024. In Eq. (19.1), the ceiling function takes the closest integer greater than the value of the real number. Thus, the number of blocks is made to be sufficient for the specified number of threads and each block contains equal number of threads in the CUDA code. Closing the angle brackets in the CALL statement is followed by the indication of the arrays and the variables which will be used by the device code. The parallel part of the CUDA code is run on the device, which is also called as the device code or the kernel, given in Fig. 19.6B. It is a module, named as “time_integration,” and it contains subroutine, which is named as “solution.” The “solution” subroutine is called by the host code and it is run on the device. The SUBROUTINE statement in the module is preceded by the “ATTRIBUTES (GLOBAL)” statement and the GLOBAL attribute in here makes the subroutine visible to both the host and the device codes but it can only run on the device. As usual, the variables and the arrays are defined inside the brackets following the name of the subroutine. Also, they are declared as usual as in FORTRAN subroutine variables with the INTENT attribute. On the other hand, the variables, such as the reference length of the bond; rlen, the new length of the bond; nlen, the total force; force and the member node number; mnode, are declared with additional VALUE attribute in the CUDA code as shown in Fig. 19.6B. By doing this, these variables, which are already defined in the host code, are also made visible to the device memory. After that, the blocks are scheduled to the available number of SMs as mentioned before

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(see Fig. 19.6B). These blocks and related each thread inside the blocks execute the instructions of the code individually. The CUDA code numbers the blocks and the threads automatically, and the tricky part of making the code parallel is to use these numbers as the element number of the arrays. In Fig. 19.6B, it is assumed that four blocks are assigned to solve the device code, in which device has two SMs, and the number of the threads in each block is 1024. The numbering procedure of each block and each thread can also be seen in the same figure. Here, 1024  4 threads solve the same problem concurrently or sequentially with the SMs. Please also be aware that each block must include equal number of threads, which execute the same code; however, the total number of material points can be less than the total number of threads. In here, the outer loop of the serial code, which loops over the main material points, kk, is removed because each thread takes different kk value and solves the same code instruction in parallel. The proper numbering of kk can be achieved with the following equation, which can also be seen in Fig. 19.6B, as kk ¼ blockDim%x  ðblockIdx%x  1Þ þ threadIdx%x

(19.2)

where blockDim%x defines the total number of threads in each block, blockIdx%x defines the index number of a block, and threadIdx%x defines the index number of a thread in each block. All of these are CUDA functions and they represent actual CUDA threads or cores of the device by means of numbers. Please also keep in mind that the numbering is done in one dimension with %x suffix for one-dimensional arrays, and it is the simplest and the best way of parallelizing the code. However, multidimensional arrays can also be represented by CUDA threads such as using the %y suffix (Massimiliano and Gregory, 2014). Thereafter, each thread calculates its related material point’s total force, force, while summing up the contributions that come from each member material point, mnode. Thus, related acceleration, acel, of each main material point, kk, is calculated by each thread as well as the velocity and the displacement results are written to the certain places in the vel(kk) and in the disp(kk) arrays, respectively. Finally, all threads must be synchronized with the “CALL syncthreads()” function, before continuing to the following time steps. Because some threads may complete the job earlier than the others, while executing the same instructions of the code, synchronizing function does actually block the earlier threads until all finish the job. Since then, they are again released in order to execute the code simultaneously for the next time steps. When all time steps are completed, the device code returns to the host code just after the CALL statement and the desirable results can be transferred to the host memory with again the assignment statement, i.e.,

II. New applications in peridynamics

3. Numerical results

415

“¼.” In Fig. 19.6A only the displacement array, i.e., disp_d, is transferred to the host memory so that it can be post-processed in the host code.

3. Numerical results A one-dimensional clamped bar is shown in Fig. 19.7A. The length of the bar and the cross-sectional area are given as L ¼ 1 m and A ¼ 0:01  0:01 m2, respectively. It is made of an isotropic material and the Young’s modulus as well as the Poisson’s ratio are E ¼ 200  109 Pa and n ¼ 1= 4, respectively. The density of the material is r ¼ 7850 kg/m3. The bar is  subjected to an initial displacement gradient of duðxÞ dx ¼ 103 . Meshless discretization of the bar can be seen in Fig. 19.7B. The discretization size is specified as Dx ¼ 0:001 m and the horizon size is chosen as d ¼ 25:015Dx to demonstrate the capabilities of parallel programming procedures to reduce the solution time. The total number of time steps is specified as 700,000 with a time step size of Dt ¼ 1:4  108 s. The problem is solved with both serial and two different parallel programming procedures, CPU and GPU approaches.

((a))

(b) FIGURE 19.7

(A) Representative model, (B) Meshless discretization of vibration of a bar

problem.

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19. Application of high-performance computing for peridynamics

TABLE 19.1 Comparison of the solution times. Parallel programming procedures Serial programming

CPU-based; OpenMP

GPU-based; CUDA

Number of cores

1

8 cores

32 cores

448 CUDA cores

Solution times

544 s

122 s

77 s

77 s

Table 19.1 shows comparison of the solution times for two different parallel programming procedures as well as serial programming. Solving the same problem with different parallel programming procedures reduces the solution time depending on type of procedures used as well as the number of cores. CPU-based approach with getting the benefit of eight cores reduces the solution time nearly 4.5 times as compared to serial solution. Moreover, increasing the number of cores even more to 32 results in less solution time but the observed decrease is not in a linear manner. In GPU-based approach, TESLA S2050 graphics card is used, and it includes 14 SMs with 32 CUDA cores each. Moreover, the maximum thread processors per SM is 1536 so that it leads to 21,504 threads in total. Solving the same problem with such number of threads in GPU results in nearly seven times decrease in solution time as compared to serial solution. The same solution time can also be achieved by 32 CPU cores. However, the availability of such number of cores in any personal computer may not be possible unless HPC facility is used. On the other hand, only one Fermi-based architecture GPU card gives the same solution time with 32 CPU cores.

4. Conclusions This chapter presented how it is possible to improve PD codes and make them efficient for large-scale problems with the help of parallel programming procedures. Moreover, how simple modifications with very basic knowledge of parallel programming skills can lead to significant increase in our calculation speeds was demonstrated. Specifically, CPUand GPU-based approaches were explained for parallelizing PD codes. It was demonstrated that solving the same problem with different parallel programming procedures reduces the solution time depending on type of procedures used as well as the number of cores.

II. New applications in peridynamics

References

417

References Barney, B., 2015. OpenMP [WWW Document]. Lawrence Livermore Natl. Lab. URL. https:// computing.llnl.gov/tutorials/openMP/. (Accessed 11.2.15). Cuda, C., 2015. Programming Guide [WWW Document]. NVIDIA Corp. URL. https://docs. nvidia.com/cuda/cuda-c-programming-guide/. Accessed 11.3.15. Diehl, P., 2012. Implementierung eines Peridynamik-Verfahrens auf GPU. University of Stuttgart. Kirk, D.B., Hwu, W.W., 2012. Programming Massively Parallel Processors: A Hands-on Approach, Second. ed. Elsevier, USA. Massimiliano, F., Gregory, R., 2014. CUDA Fortran for Scientists and Engineers, First. ed. Elsevier, USA.

II. New applications in peridynamics

C H A P T E R

20 Application of artificial intelligence and machine learning in peridynamics Cong Tien Nguyen, Selda Oterkus, Erkan Oterkus Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, United Kingdom

O U T L I N E 1. Introduction

420

2. Linear regression

421

3. One-dimensional peridynamic machine learning formulation

422

4. Two-dimensional peridynamic machine learning formulation

424

5. Numerical results 5.1 One-dimensional bar subjected to axial loading 5.2 Vibration of a one-dimensional bar 5.3 Two-dimensional plate subjected to tension loading 5.4 Two-dimensional plate with a pre-existing crack subjected to tension loading

428 428 429 430

6. Conclusions

433

References

434

Peridynamic Modeling, Numerical Techniques, and Applications https://doi.org/10.1016/B978-0-12-820069-8.00015-9

419

432

© 2021 Elsevier Inc. All rights reserved.

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20. Application of artificial intelligence and machine learning in peridynamics

1. Introduction With the advancement of sensor technologies, data analysis, and computational resources, data-driven models have become important in many different scientific fields. However, for complex problems without having sufficient data, the accuracy of the data-driven approaches deteriorates. As an alternative, theory-guided data science which is a combination of physics-driven and data-driven models is a promising approach. In this chapter, such an approach is presented by coupling physics-driven peridynamic approach (Silling, 2000) and data-driven and linear-regression-based machine learning approach. The presented approach is different than the methodology developed by Kim et al. (2019), which utilized conventional neural networks to predict the damage pattern on a disk provided the location of a colliding object’s impact and to identify the collision location, angle, velocity, and size given the resulting damage pattern. There has been a rapid progress in peridynamics research. Among these Alpay and Madenci (2013) developed fully coupled peridynamic thermomechanical equations. Butt et al. (2017) derived the dispersion relationships for one-, two-, and three-dimensional cases within statebased peridynamic framework. Diyaroglu et al. (2017a) presented the implementation of peridynamic diffusion model within finite element framework. Diyaroglu et al. (2017b) developed peridynamic wetness formulation for time-dependent saturated concentration to compute moisture concentration in electronic packages. Diyaroglu et al. (2019) presented Euler-Bernoulli beam formulation within ordinary state-based peridynamic framework. Gao and Oterkus (2018) performed peridynamic analysis of marine composites under shock loads by considering thermomechanical coupling effects. Gao and Oterkus (2019) developed fully coupled thermomechanical analysis of laminated composites using ordinary state-based peridynamics. Imachi et al. (2019) introduced the new transition bond concept for dynamic fracture analysis within ordinary state-based peridynamics. Nguyen and Oterkus (2019a) proposed a peridynamic model to predict thermomechanical behavior of threedimensional (3D) shell structures with 6 degrees of freedom. Nguyen et al. (2019b) presented a peridynamic formulation for beam structures to predict damage in offshore structures. Nguyen et al. (2020a) introduced an ordinary state-based peridynamic model for geometrical nonlinear analysis. Nguyen et al. (2020b) investigated the effect of brittle crack propagation on the strength of ship structures by using peridynamics. Ren et al. (2016) developed dual-horizon peridynamics formulation which is especially suitable for nonuniform discretization. Roy et al. (2017) proposed a peridynamics damage model with the phase field as damage parameter. Vazic et al. (2017) investigated the dynamic propagation of a macrocrack interacting with parallel small cracks. Wang et al. (2018) utilized peridynamics to predict fracture evolution during II. New applications in peridynamics

421

2. Linear regression

lithiation process. Yang et al. (2019) presented implementation of peridynamic beam and plate formulations in finite element framework. Ye et al. (2017) developed a peridynamic model for propeller-ice contact analysis. To demonstrate the capability of the peridynamic machine learning approach presented in this chapter, four different numerical examples are considered including one-dimensional (1D) bar subjected to axial loading, vibration of a 1D bar, two-dimensional (2D) plate subjected to tension loading and 2D plate with a pre-existing crack subjected to tension loading.

2. Linear regression To obtain the machine learning model linear regression is utilized. In this section, linear regression process is briefly explained. For N number of data, aiðkÞ and M number of data sets, i.e., k ¼ 1; .; M, the numerical output, bðkÞ , can be expressed as bðkÞ ¼ a0 þ a1 x1ðkÞ þ . þ aN xNðkÞ þ εðkÞ

(20.1)

where εðkÞ represents the noise and ai is the unknown regression coefficient. To determine the unknown regression coefficient, least square minimization approach can be utilized by minimizing the sum of squared errors, F vF ¼ 0; vaj

j ¼ 0; 1; .; N

(20.2)

with F¼

M h  i2 X bðkÞ  a0 þ a1 x1ðkÞ þ . þ aN xNðkÞ

(20.3)

k¼1

which can be written in matrix form as aT aa ¼ aT b

(20.4)

where 2

1 61 6 a¼6 4«

a1ð1Þ a1ð2Þ

a2ð1Þ a2ð1Þ

. .

«

«

1

a1ðMÞ

a2ðMÞ

. .

3 aNð1Þ aNð2Þ 7 7 7 « 5 aNðMÞ

II. New applications in peridynamics

(20.5a)

422

20. Application of artificial intelligence and machine learning in peridynamics

8 9 a0 > > > > >

= 1 a¼ > « > > > > > : ; aN 8 9 bð1Þ > > > > >

= ð2Þ b¼ > « > > > > > : ; bðMÞ

(20.5b)

(20.5c)

3. One-dimensional peridynamic machine learning formulation One-dimensional peridynamic formulation can be written in discrete form for the material point i as, ru€ðiÞ ¼ c

NðiÞ X uðjÞ  uðiÞ

x

j¼1

VðjÞ þ bxðiÞ

(20.6)

where r is density, u and u€ denote displacement and acceleration, NðiÞ is the number of material points inside the horizon of the material point i, and V represents volume. The bond length is defined as     x ¼ xðjÞ  xðiÞ  (20.7) The bond constant c can be expressed in terms of elastic modulus, E, cross-sectional area, A, and horizon size, d, as c¼

2E Ad2

(20.8)

The body load vector is used to represent external force, Fx , which can be defined as bxðiÞ ¼

FxðiÞ ADx

(20.9)

where Dx is the discretization size. For a horizon size of d ¼ 3Dx, the following relationship can be established by using linear regression based on the equation of motion given in Eq. (20.6) for static condition as (see Fig. 20.1) uðiÞ ¼ a1 uði3Þ þ a2 uði2Þ þ a3 uði1Þ þ a4 uðiþ1Þ þ a5 uðiþ2Þ   FxðiÞ Dx þ a6 uðiþ3Þ þ a7 AE

II. New applications in peridynamics

(20.10)

423

3. One-dimensional peridynamic machine learning formulation

FIGURE 20.1

The material point i and its family members for d ¼ 3Dx.

where ak ðk ¼ 1; .; 7Þ represents regression coefficients. For static conditions, the body load vector can be written as N u  u  X FxðiÞ ðjÞ ðiÞ bxðiÞ ¼ ¼ c VðjÞ ADx x j¼1

(20.11)

By using Eq. (20.11), the last input parameter in Eq. (20.10) can be obtained as " #   N uðjÞ  uðjÞ P c VðjÞ ðDxÞ2 x FxðiÞ Dx j¼1 ¼ (20.12) AE E

FIGURE 20.2 Boundary conditions in modal analyses for one-dimensional model. II. New applications in peridynamics

424

20. Application of artificial intelligence and machine learning in peridynamics

Data sets can be obtained by considering all possible vibration modes for different boundary conditions as shown in Fig. 20.2. For this particular case, 50 data sets are obtained and among them 45 data sets are used for training purposes. The remaining data are used for testing purposes. By using Eq. (20.12) and data sets, the unknown regression constants can be obtained as a1 ¼ 0:0513, a2 ¼ 0:1496, a3 ¼ 0:2991, a4 ¼ 0:2991, a5 ¼ 0:1496, a6 ¼ 0:0513, and a7 ¼ 1.3594. The presented approach can be further extended to dynamic problems by using Eq. (20.10) as E 1  ru€ðiÞ ¼ 2 a1 uði3Þ þ a2 uði2Þ þ a3 uði1Þ þ a4 uðiþ1Þ Dx a7 (20.13)  þ a5 uðiþ2Þ þ a6 uðiþ3Þ  uðiÞ þ bxðiÞ

4. Two-dimensional peridynamic machine learning formulation Linearized form of two-dimensional peridynamic formulation can be written in discrete form for the material point i as rhu€ðiÞ ¼

N X

cmðiÞðjÞ sðiÞðjÞ cos fVðjÞ þ bxðiÞ

(20.14a)

cmðiÞðjÞ sðiÞðjÞ sin fVðjÞ þ byðiÞ

(20.14b)

j¼1

rh€ vðiÞ ¼

N X j¼1

€ v€ denote displacement and acceleration where r is density, u; v and u; components in x- and y-directions, respectively, and xðjÞ  xðiÞ ; x yðjÞ  yðiÞ sin f ¼ ; x

cos f ¼

and

 mðiÞðjÞ ¼

1 0

if interaction exists between i and j otherwise

The bond length is defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 x¼ xðjÞ  xðiÞ þ yðjÞ  yðiÞ

II. New applications in peridynamics

(20.15a) (20.15b)

(20.15c)

(20.16)

425

4. Two-dimensional peridynamic machine learning formulation

The bond constant c can be expressed in terms of elastic modulus E and horizon size d as c¼

9E pd3

(20.17)

The body load vectors are used to represent external forces, Fx and Fy , which can be defined as bxðiÞ ¼ byðiÞ ¼

FxðiÞ Dx2 FyðiÞ

(20.18a) (20.18b)

Dx2

For a horizon size of d ¼ 3Dx, the following relationship can be established by using linear regression based on the equation of motion given in Eq. (20.14a,b) for static condition as (see Fig. 20.3)     28 X

8 8 FxðiÞ þ b FyðiÞ aj uj þ bj vj þ a uðiÞ ¼ (20.19a) 3Eh 3Eh j¼1 vðiÞ ¼

28 X

gj uj þ hj vj þ g

j¼1



8 F 3Eh xðiÞ



 þh

8 F 3Eh yðiÞ

 (20.19b)

where aj ; bj ; gj ; hj ; a; b; g; h; ðj ¼ 1; .; 28Þ represent regression coefficients.

FIGURE 20.3

The material point i and its family members for d ¼ 3Dx.

II. New applications in peridynamics

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20. Application of artificial intelligence and machine learning in peridynamics

For static conditions, the body load vectors can be written as 1 0 N X (20.20a) bxðiÞ ¼  @ cmðiÞðjÞ sðiÞðjÞ cos fVðjÞ A 0 byðiÞ ¼  @

j¼1 N X

1 cmðiÞðjÞ sðiÞðjÞ sin fVðjÞ A

(20.20b)

j¼1

By using Eq. (20.20a,b), the last two input parameters in Eq. (20.19a,b) can be obtained as ! N P  mðiÞðjÞ sðiÞðjÞ cos fVðjÞ j¼1 8 8 FxðiÞ ¼ (20.21a) 3Eh 9p hDx ! N P  mðiÞðjÞ sðiÞðjÞ sin fVðjÞ j¼1 8 8 FyðiÞ ¼ (20.21b) 3Eh 9p hDx Data sets can be obtained by considering all possible vibration modes for different boundary conditions as shown in Fig. 20.4. For this particular case, 320 data sets are obtained and among them 310 data sets are used for training purposes. The remaining data are used for testing purposes. By using Eq. (20.21a,b) and data sets, the unknown regression constants can be obtained as aj ¼ 8 fa1 ; a2 ; .; a28 g 0:0244; 0:0173; 0:0509; 0:0711; > > > < 0:0503 0:1422; 0:0503; 0:0127; ¼ > 0; 0; 0:0127; > > 0; : 0:0127; 0:0173; 0:0509; 0:0711; bj ¼ 8 fb1 ; b2 ; .; b28 g 0; 0:0173; > > > < 0:0503; 0; ¼ > 0; 0; > > : 0:0254; 0:0173;

9 0:0509; 0:0173; 0:0127; > > > 0; 0; 0; = ; 0:0503; 0:1422; 0:0503; > > > ; 0:0509; 0:0173; 0:0244

0:0254; 0:0503;

0; 0:0254;

0:0254; 0;

0:0173; 0;

0;

0:0254;

0:0503;

0;

0:0254;

0;

0:0254;

0:0173;

II. New applications in peridynamics

9 0:0254 > > > = 0; ; 0:0503; > > > ; 0

427

4. Two-dimensional peridynamic machine learning formulation

FIGURE 20.4

Boundary conditions in modal analyses for 2D model.

gj ¼ fg1 ; g2 ; .; g28 g 8 0; 0:0173; > > > < 0:0503; 0; ¼ > 0; 0; > > : 0:0254; 0:0173; hj ¼ fh1 ; h2 ; .; h28 g 8 0; 0:0173; > > > < 0:0503; 0; ¼ > 0:1422; 0:0711; > > : 0:0509; 0:0173;

0:0254; 0:0503; 0; 0:0254;

9 0:0254 > > > = 0:0254; 0; 0; 0; ; 0:0254; 0:0503; 0; 0:0503; > > > ; 0; 0:0254; 0:0173; 0

0:0127;

0;

0:0503;

0:0509;

0:0244; 0:0127;

0:0509; 0;

0;

0:0254;

0:0173;

9 0:0127; 0:0173; 0:0509; > > > 0:0244; 0:0711; 0:1422; = ; > 0:0503; 0; 0:0503; > > ; 0:0127; 0:0173; 0

a ¼ 0:4421, b ¼ 0, g ¼ 0, and h ¼ 0:4421.

II. New applications in peridynamics

428

20. Application of artificial intelligence and machine learning in peridynamics

The presented approach can be further extended to dynamic problems by using Eq. (20.19a,b) as i 3Eh 1 h rhu€ðiÞ ¼ $ u þ b $ v  u (20.22a) a j j j j ðiÞ þ bxðiÞ 8Dx2 a i 3Eh 1 h rh€ vðiÞ ¼ (20.22b) gj $ uj þ hj $ vj  vðiÞ þ byðiÞ 2 8Dx g with uj ¼ fu1 ; u2 ; .; u28 g

(20.23a)

vj ¼ fv1 ; v2 ; .; v28 g

(20.23b)

5. Numerical results To demonstrate the capability of the coupled peridynamic machine learning approach, four different numerical examples are considered including 1D bar subjected to axial loading, vibration of a 1D bar, 2D plate subjected to tension loading and 2D plate with a pre-existing crack subjected to tension loading.

5.1 One-dimensional bar subjected to axial loading In the first example case, a 1D bar subjected to axial loading is considered as shown in Fig. 20.5A. The bar has a length of L ¼ 2 m and

FIGURE 20.5

One-dimensional bar subjected to axial loading; (A) geometry,

(B) discretization.

II. New applications in peridynamics

5. Numerical results

429

FIGURE 20.6 Displacement variation along the 1D bar subjected to an axial force of (A) Fx ¼ 1  108 N, (B) Fx ¼ 1  108 N.

cross-sectional area of A ¼ 0:1  0:1 m2. The bar has an elastic modulus of E ¼ 200  109 Pa. Two different loading conditions are considered; Fx ¼ 1  108 and Fx ¼ 1  108 N. The discretization size is specified as Dx ¼ 0:02 m. A fictitious region is introduced at the left edge to impose zero displacement boundary condition as shown in Fig. 20.5B. Axial displacements are obtained from both coupled peridynamic machine learning approach (ML-PD) and finite element analysis (FEA) for two different loading conditions. As shown in Fig. 20.6, a very good agreement is observed between the two solutions.

5.2 Vibration of a one-dimensional bar In the second example, vibration of a one-dimensional bar problem is considered. The bar has a length of L ¼ 2 m and cross-sectional area of A ¼ 0:1  0:1 m2. Elastic modulus and density of the bar are specified as E ¼ 70  109 N/m2 and r ¼ 2710 kg/m3, respectively. The bar is subjected to an initial strain of vu=vx ¼ 0:05. The discretization size is specified as Dx ¼ 0.002 m. A fictitious region is introduced at the left edge to impose zero displacement boundary condition. The problem is solved by three different approaches for comparison purposes including current peridynamic machine learning approach (ML-PD), regular peridynamics approach (PD), and finite element analysis (FEA). As shown in Fig. 20.7, all three approaches agree very well with each other.

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20. Application of artificial intelligence and machine learning in peridynamics

FIGURE 20.7 Variation of displacement uðmÞ of the material point located at x ¼ L= 2.

5.3 Two-dimensional plate subjected to tension loading In the third example case, a 2D plate subjected to tension loading problem is considered. The plate has a square shape with in-plane dimensions of L ¼ W ¼ 2 m and thickness of h ¼ 0:02 m as shown in Fig. 20.8A. The elastic modulus and Poisson’s ratio of the plate are specified as E ¼ 200  109 N/m2 and n ¼ 1=3. The plate is subjected to a

FIGURE 20.8 Two-dimensional plate subjected to axial loading; (A) geometry, (B) discretization (Note that PD regions are shown in blue, ML regions are shown in red).

II. New applications in peridynamics

5. Numerical results

431

tensional force per unit length of fx ¼ 4  108 N/m. The discretization size is specified as Dx ¼ 0.02 m. A fictitious region is introduced at the left edge to impose zero displacement boundary condition. The problem is solved by using a coupled regular peridynamic (PD) and peridynamic machine learning approach (ML-PD). As shown in Fig. 20.8B, PD approach is used around the boundary region where there is lack of interactions within the horizon. On the other hand, ML-PD approach is used for the remaining part of the solution domain. Figs. 20.9 and 20.10 show the variation of horizontal and vertical displacements obtained from FEA and coupled ML-PD approach. Fig. 20.11 also show the comparison between the two approaches along the central axes and a good agreement is observed between the two approaches.

FIGURE 20.9 Variation of horizontal displacements uðmÞ obtained by using (A) FEA, (B) ML-PD approach.

FIGURE 20.10

Variation of vertical displacements vðmÞ obtained by using (A) FEA, (B)

ML-PD approach.

II. New applications in peridynamics

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20. Application of artificial intelligence and machine learning in peridynamics

FIGURE 20.11

Variation of displacements (A) u along y ¼ W=2; (B) v along x ¼ L= 2.

5.4 Two-dimensional plate with a pre-existing crack subjected to tension loading For the final example, a 2D plate with a pre-existing crack subjected to tension loading is considered. The plate has a square shape with in-plane dimensions of L ¼ W ¼ 0:4 m and thickness of h ¼ 0:005 m as shown in Fig. 20.12. The length of the pre-existing crack is 2a ¼ 0:08 m. The elastic modulus, Poisson’s ratio, fracture toughness, and the critical energy release pffiffiffiffiffi rate are specified as E ¼ 200  109 N/m2, n ¼ 1=3, Kc ¼ 70  106 MPa m, and Gc ¼ 2:1778  104 J/m2, respectively. The loading is applied as jDvj ¼ 2  108 m. The discretization size is specified as Dx ¼ L=150. The evolution of the crack as the applied displacements increase is shown in Fig. 20.13. As the crack propagates, there is a constant update of

FIGURE 20.12

Plate with a pre-existing crack subjected to tension loading.

II. New applications in peridynamics

6. Conclusions

433

FIGURE 20.13 Damage evolution on the plate when the applied displacement equals to (A) 2  104 m, (B) 2:52  104 m, (C) 2:68  104 m, (D) 2:88  104 m (displacements are magnified by 100).

the regions being analyzed by the regular peridynamic approach (PD) and current peridynamic machine learning approach (ML-PD) as shown in Fig. 20.14.

6. Conclusions In this chapter, a peridynamic machine learning approach was presented based on linear regression. To demonstrate the capability of the coupled peridynamic machine learning approach, four different numerical examples were considered including ID bar subjected to axial loading, vibration of a 1D bar, 2D plate subjected to tension loading and 2D plate with a pre-existing crack subjected to tension loading. Current approach agrees well with finite element analysis results for the cases without having pre-existing crack. For the pre-existing case, a continuous update of the regions was performed for regular peridynamic and current peridynamic machine learning approach.

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20. Application of artificial intelligence and machine learning in peridynamics

FIGURE 20.14 Evolution of peridynamic machine learning and regular PD regions when the applied displacement equals to (A) 2  104 m, (B) 2:52  104 m, (C) 2:68  104 m, (D) 2:88  104 m (PD regions are shown in red and ML regions are shown in blue).

References Alpay, S., Madenci, E., 2013. Crack growth prediction in fully-coupled thermal and deformation fields using peridynamic theory. In: 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 1477. Butt, S.N., Timothy, J.J., Meschke, G., 2017. Wave dispersion and propagation in state-based peridynamics. Comput. Mech. 60 (5), 725e738. Diyaroglu, C., Oterkus, S., Oterkus, E., Madenci, E., 2017a. Peridynamic modeling of diffusion by using finite-element analysis. IEEE Trans. Compon. Packag. Manuf. Technol. 7 (11), 1823e1831. Diyaroglu, C., Oterkus, S., Oterkus, E., Madenci, E., Han, S., Hwang, Y., 2017b. Peridynamic wetness approach for moisture concentration analysis in electronic packages. Microelectron. Reliab. 70, 103e111. Diyaroglu, C., Oterkus, E., Oterkus, S., 2019. An EulereBernoulli beam formulation in an ordinary state-based peridynamic framework. Math. Mech. Solid 24 (2), 361e376. Gao, Y., Oterkus, S., 2018. Peridynamic analysis of marine composites under shock loads by considering thermomechanical coupling effects. J. Mar. Sci. Eng. 6 (2), 38.

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References

435

Gao, Y., Oterkus, S., 2019. Fully coupled thermomechanical analysis of laminated composites by using ordinary state based peridynamic theory. Compos. Struct. 207, 397e424. Imachi, M., Tanaka, S., Bui, T.Q., Oterkus, S., Oterkus, E., 2019. A computational approach based on ordinary state-based peridynamics with new transition bond for dynamic fracture analysis. Eng. Fract. Mech. 206, 359e374. Kim, M., Winovich, N., Lin, G., Jeong, W., 2019. Peri-net: analysis of crack patterns using deep neural networks. J. Peridynamics Nonlocal Model. 1 (2), 131e142. Nguyen, C.T., Oterkus, S., 2019a. Peridynamics for the thermomechanical behavior of shell structures. Eng. Fract. Mech. 219, 106623. Nguyen, C.T., Oterkus, S., 2019b. Peridynamics formulation for beam structures to predict damage in offshore structures. Ocean Eng. 173, 244e267. Nguyen, C.T., Oterkus, S., 2020c. Ordinary state-based peridynamic model for geometrically nonlinear analysis. Eng. Fract. Mech. 224, 106750. Nguyen, C.T., Oterkus, S., 2020d. Investigating the effect of brittle crack propagation on the strength of ship structures by using peridynamics. Ocean Eng. 209, 107472. Ren, H., Zhuang, X., Cai, Y., Rabczuk, T., 2016. Dual-horizon peridynamics. Int. J. Numer. Methods Eng. 108 (12), 1451e1476. Roy, P., Pathrikar, A., Deepu, S.P., Roy, D., 2017. Peridynamics damage model through phase field theory. Int. J. Mech. Sci. 128, 181e193. Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solid. 48 (1), 175e209. Vazic, B., Wang, H., Diyaroglu, C., Oterkus, S., Oterkus, E., 2017. Dynamic propagation of a macrocrack interacting with parallel small cracks. AIMS Mater. Sci. 4 (1), 118e136. Wang, H., Oterkus, E., Oterkus, S., 2018. Predicting fracture evolution during lithiation process using peridynamics. Eng. Fract. Mech. 192, 176e191. Yang, Z., Oterkus, E., Nguyen, C.T., Oterkus, S., 2019. Implementation of peridynamic beam and plate formulations in finite element framework. Continuum Mech. Therm. 31 (1), 301e315. Ye, L.Y., Wang, C., Chang, X., Zhang, H.Y., 2017. Propeller-ice contact modeling with peridynamics. Ocean Eng. 139, 54e64.

II. New applications in peridynamics

Index Note: ‘Page numbers followed by “f ” indicate figures and “t” indicate tables.’

A

C

Adaptive Dynamic Relaxation (ADR), 405 Advanced Placed Ply (APPLY), 321e322, 323f Arithmetic logic units (ALUs), 408e409 Atomic system, smoothing of, 3e5 Automated fiber placement (AFP) techniques, 310 Automated tape laying (ATL), 310 Axisymmetric analysis, peridynamics for, 58e59 classical axisymmetric equilibrium equations, 60e61 equation of motion, 64e67 failure criteria, 67e70 numerical results, 70e74 peridynamic theory, 61e63

Carbon fiber reinforced plastics (CFRPs), 310 Cauchy stress tensor, 119e120 Cell death, 265e266 Central processing units (CPUs), 403e408, 406f Classical axisymmetric equilibrium equations, 60e61 Classical continuum mechanics (CCM), 99, 114, 119, 135e136, 140, 185e186, 356 Classical Mindlin plate formulation, 102e104 Coarse-grain (CG) techniques, 356e357, 361e362 Coarsening approach, 372e378 micromodulus function, 377e378 numerical implementation, 375e376 peridynamic model, 372e375, 373f Cohesive zone model (CZM), 78, 328, 330e331 Cohesive Zone models, 114 Complementary energy density, 344 Composite ply, 312e315 Composites delamination, 333e335 Computational implementation, 237e238 Compute unified device architecture (CUDA), 408, 412f Consolidation problem (1D), 391e393, 392f flow field, 393 mechanical field, 393 Consolidation problem (2D), 395e396, 395f Constant strain triangular (CST) elements, 143 Constitutive correspondence, 84e85, 347e349 Contact, numerical model, 294 Continuum kinematicseinspired peridynamics (CPD) bond-based interactions, 224

B Beam and plate models, 98 beam subjected to transverse loading, 109e110 peridynamic Mindlin plate formulation, 104e109 peridynamic Timoshenko beam formulation, 99e102 classical Timoshenko beam formulation, 98e99 simply supported boundary conditions, 110e111 Biconjugate gradient stabilizationmethod (Bi-CGSTAB), 66 Boltzmann hereditary superposition integral, 214e215 Bond-based friction model, 184e185 Bond-based interactions, 224 Bond-based material models, 7, 22 Bond-based peridynamics (BB-PD), 38e39, 117, 185e187, 299e303, 357, 357f Bond breakage, 8 Bond breaking, criterion for, 86e87 Bond stretch criterion, 118 Buoyancy, 294e295, 295f

437

438

INDEX

Continuum kinematicseinspired peridynamics (CPD) (Continued) computational implementation, 237e238 definition, 224, 225f equilibrium, 236e237 examples, 240e241, 240fe241f external potential energy, 235 governing equations, 227e237 harmonic potentials, 238e240 horizon, 225e226 internal potential energy, 227e235 one-neighbor interactions, 228e229 three-neighbor interactions, 232e235 two-neighbor interactions, 230e232 kinematics, 225e227 local kinematics measures, 225e226 nonlocal kinematic measures, 226 Conventional FEM, 120 Conventional peridynamics formulation, 46 Coupled fluid diffusion, 24 Crack propagation heterogeneous materials, 54e55 solid bodies, 114 Curvilinear fiber reinforcement model, 317, 318fe319f Cylindrical coordinate system, 60, 60f

D 1D bimaterial bar under tension, 53e54 Deformation state, 12e13 Degrees of freedom (DoF), 98, 141 Delamination, 329 DeshpandeeEvans (DE) constitutive model, 331e333, 343e349 Dirac delta function, 20 Discrete element modeling (DEM), 184e185 Discretised version of coupling, 129e130 Discretization of peridynamic formulation, 164e165 1D micromodulus function, 377, 377fe378f 2D numerical analysis, 295e297, 296f 3D numerical analysis, 297e298 Dot product, 12 Double cantilever beam (DCB), 173, 335e336, 336f 2D spatial function, 142 Dual boundary element method (DBEM), 171

Dual-horizon peridynamics (DH-PD), 36e42, 252, 254 adaptivity and particles arrangement sensitivity, 47e51 dual-horizon concept, 39e41 equation of motion in, 44e46 forces in, 42e44 ghost force in traditional peridynamics, 38e39 test of spurious wave, 46e47 weak continuity along materials interfaces, 52e55 Ductile-brittle transition, 282 Ductile constitutive, 281e282, 285f Ductile material response, 23 Ductile-to-brittle transition, 278 Dynamic crack branching, 87e88 Dynamic crack propagation and arrest modeling crack arrest modeling with application phase, 174e176 numerical studies, 176e179 transition bond modeling, 173e174 Dynamic fracture modeling, 160e162 dynamic crack propagation and arrest modeling crack arrest modeling with application phase, 174e176 numerical studies, 176e179 transition bond modeling, 173e174 fracture modeling, 165e171 interaction integrals, 166e169 mixed-mode DSIFs for stationary cracks, 171e173 MLS approximation, 169e171 ordinary state-based peridynamics, 162e165

E Elastic-brittle constitutive, 281, 282f Elastic instability, 21e22 Elastic no-slip Mindlin law, 194 Elastic stateebased material model, 14e15 Element-free Galerkin method, 161 Emergent behavior, 258e260, 261fe263f Enrichment functions, 145, 150e151 Equations of motion, 345e349 Equilibrium, 236e237 Euler-Lagrange equations, 80, 99e101, 104e106, 111e112 Explosive load, 299

INDEX

eXtended Finite Element Method (XFEM), 140 in conjunction with peridynamics, 143e150 External potential energy, 235

F Fiber reinforced composites modeling Advanced Placed Ply (APPLY), 321e322, 323f automated fiber placement (AFP) techniques, 310 automated tape laying (ATL), 310 carbon fiber reinforced plastics (CFRPs), 310 composite materials, peridynamics for, 311e315 composite ply, 312e315 failure model, 315 Ghajari-Iannucci-Curtis ply model, 314e315 interlaminar bond, 315 Oterkus-Madenci’s ply model, 312e313, 313f theoretical background, 311e312 curvilinear fiber path modeling, 316e320 curvilinear fiber reinforcement model, 317, 318fe319f straight fiber reinforcement model, 316, 317f laminated composites, 320, 321f manufacturing/design of composites, integrated framework for, 321e323, 322fe323f multiple-site/multiple-type damage, 320 Finite element analysis (FEA), 429 Finite element discretization, 151 Finite element method (FEM), 160 Finite element nodes, 147, 149 Finite point method, 120e127 Fixed ratio mixed-mode (FRMM), 341, 342f Fracture modeling, 165e171 interaction integrals, 166e169 mixed-mode DSIFs for stationary cracks, 171e173 MLS approximation, 169e171 Frechet derivative, 14 Fully coupled poroelastic peridynamic formulation, 389e390

439

G Ghajari-Iannucci-Curtis ply model, 314e315 Ghost force in traditional peridynamics, 38e39 Ginzburg-Landau theory, 84 Graphical processing units (GPUs), 403e404, 408e415, 409fe410f Green’s function approach, 206

H Harmonic potentials, 238e240 Hertzian contact law, 193e194 Heterogeneous materials crack propagation in, 54e55 High-performance computing (HPC) Adaptive Dynamic Relaxation (ADR), 405 arithmetic logic units (ALUs), 408e409 central processing units (CPUs), 403e408, 406f compute unified device architecture (CUDA), 408, 412f graphical processing units (GPUs), 403e404, 408e415, 409fe410f Meshless discretization of vibration, 415f multiple instruction multiple data (MIMD), 404 multiple instruction single data (MISD), 404 numerical results, 415e416 parallel programming, 404e415 programming procedures, 405f representative model, 415f single instruction multiple data (SIMD), 404 Homogenization approach, 372, 382e386 Mathematical Homogenization Theory (MHT), 384 Periodic Boundary Condition (PBC), 384 Hooke’s Law, 99 Horizon, 225e226 Hydraulically pressurized crack problem (2D), square plate, 396e398, 397f Hydraulic fracturing, 51, 388

I Ice modeling BBPD method, 299e303 explosive load, 299 numerical modeling and analysis, 300e303, 302f

440

INDEX

Ice modeling (Continued) compressive strength, 277 constitutive for, 280e282 ductile-brittle transition, 282 ductile constitutive, 281e282, 285f elastic-brittle constitutive, 281, 282f cylindrical structure, 291e293, 292f damage of ice, influence factors, 297 2D numerical analysis, 295e297, 296f 3D numerical analysis, 297e298 ductile-to-brittle transition, 278 freshwater grain structures, 278f hybrid modeling method, icebreaking, 303e305 calculation model, 303 numerical results, 303e305, 305f mechanical properties, 277e280, 284e293 numerical model buoyancy, 294e295, 295f contact, 294 numerical results, 295e297, 296fe297f peridynamics advantages, 283 research status, 283e284 pre-crack propagation, 2D flat ice, 284 properties, 276e280 reinforced plate structure, ice impacting on, 289e291, 290f structure, 276e277 tensile strength, 277, 279f three-point bending test, 287e288, 287f wing crack propagation, 3D ice body, 284e286, 286f Inclined crack under tension, 153e156 Integro-differential equation, 115 Interaction integral, for propagating cracks, 168e169 Interaction integral for stationary cracks, 166e168 Interaction integrals, 166e169 Interlaminar bond, 315 Intermediate homogenization, 24e25 Internal energy density, 344e345 Internal potential energy, 227e235 one-neighbor interactions, 228e229 three-neighbor interactions, 232e235 two-neighbor interactions, 230e232 Internal ring crack under tension, 70e74

K Kalthoff-Winkler experiment, 90e92, 133e134

Kalthoff-Winkler test in 2D and 3D, 48e50 Kinematic correspondence, 84 Kinematics, 82 Korteweg-de Vries equation, 21

L Lagrange’s equation, 100e101 Laminated composites, 320, 321f Linear Elastic Fracture Mechanics (LEFM) fracture, 140 Linear microelastic model, 6e7 Linear peridynamic solid (LPS), 15 Linear viscoelastic isotropic material, 214e215 Local kinematics measures, 225e226

M Machine learning linear regression, 421e422 numerical results, 428e433 one-dimensional bar subjected to axial loading, 428e429 one-dimensional bar, vibration of, 429, 430f pre-existing crack subjected to tension loading, 432e433, 432f one-dimensional peridynamic machine learning formulation, 422e424 ordinary state-based peridynamic model, 420e421 two-dimensional peridynamic machine learning formulation, 424e428, 425f Material models, 5e18 linear microelastic model, 6e7 microelastic nucleation and growth model (MNG), 10e11 nonlinear and rate-dependent bondbased models, 11 non-ordinary state-based materials and the correspondence model, 15e18 ordinary state-based material models, 12e15 prototype microelastic brittle model, 8 Material stability, 21e22 Material variability, 24e25 Mesh-based methods, 161 Mesh-free scheme, 127 Meshless discretisation CCM, 120e127 PD, 127e128 vibration, 415f Meshless methods, 122

441

INDEX

Meshless numerical techniques, 22e23 Microelastic nucleation and growth model (MNG), 10e11 Micromodulus function 1D micromodulus function, 377, 377fe378f two-dimensional micromodulus functions, 377e378, 378fe379f Microplastic model, 11 Micropolar theories, 22 Mindlin plate formulation, 111e112 Mindlin plate theory, 102 “Mirror-mist-hackle” transition, 8 Mixed-mode DSIFs for stationary cracks, 171e173 Mode I delamination, 335e337 Mode II delamination, 337e340 end loaded split (ELS) test, 338, 338f end notched flexure test, 338e340, 340f Modeling inelasticity, 206 peridynamic plasticity formulation, 207e214 peridynamic viscoelasticity formulation, 214e217 plate under tensile loading, 217e218 pre-existing crack under tensile loading, 218e219 Model order reduction (MOR), 379e382 reduced dynamic models, 380e381 reduced eigenvalue models, 381e382 reduced static models, 380e381 Moving least squares (MLS) approximation, 169e171 Multiple instruction multiple data (MIMD), 404 Multiple instruction single data (MISD), 404 Multiple physical fields, 24 Multiscale modeling coarsening approach, 372e378 micromodulus function, 377e378 numerical implementation, 375e376 peridynamic model, 372e375, 373f homogenization approach, 372, 382e386 Mathematical Homogenization Theory (MHT), 384 Periodic Boundary Condition (PBC), 384 microscale, 372 model order reduction (MOR), 379e382 reduced dynamic models, 380e381 reduced eigenvalue models, 381e382

reduced static models, 380e381 one-dimensional (1D) problems, 372 static condensation, 372

N Neumann boundary conditions, 122 Newton’s laws, 2 Newton’s second law, 4 Nodal displacement vector, 147 Node release technique, 160 Node-to-surface contact algorithm, 184 Nonconvex elastic peridynamic materials, 19 Nondimensional weight function, 142 Nonlinear and rate-dependent bondbased models, 11 Nonlocal continuum mechanics formulation, 207e208 Nonlocal displacement gradient, 84 Nonlocal kinematic measures, 226 Non-ordinary state-based material models, 15e18 Non-ordinary state-based peridynamics, 388 Numerical illustrations dynamic crack branching, 87e88 Kalthoff-Winkler experiment, 90e92

O Ordinary state-based material models, 12e15 Ordinary State-Based Peridynamics (OSBPD), 116e117, 162e165 Oterkus-Madenci’s ply model, 312e313, 313f

P Pairwise bond force density, 4e5 Pairwise force function, 117 Parallel programming, 404e415 Partial stress method, 118 Partial stress tensor, 13, 36, 118 Particle-based contact model, 192, 192f, 201e202 PD differential operator (PDDO), 59 Peridynamic Differential Operator (PDDO), 23, 141e143 Peridynamic force density, 208e209, 214e215 Peridynamic machine learning approach (ML-PD), 431e433

442

INDEX

Peridynamic material points, displacements at, 144e147 Peridynamic Mindlin plate formulation, 104e109 Peridynamic plasticity formulation, 207e214 Peridynamics, 2e3 background, 252e256 biological materials, fracture in, 264 biological mechanism, 257f cell death, 265e266 dual horizon, 252, 254 emergent behavior, 258e260, 261fe263f across scales, 266e268 example, 264e268 growth implementations, 256e258 material models, 5e18 linear microelastic model, 6e7 microelastic nucleation and growth model (MNG), 10e11 nonlinear and rate-dependent bondbased models, 11 non-ordinary state-based materials and the correspondence model, 15e18 ordinary state-based material models, 12e15 prototype microelastic brittle model, 8 notation, 252e256, 253f obtained from smoothing of atomic system, 3e5 relation to local theory, 18e19 remodeling, 256e258 research trends in peridynamic theory, 20e25 better meshless numerical techniques, 22e23 ductile material response, 23 material stability, 21e22 material variability, 24e25 micropolar theories, 22 multiple physical fields, 24 special purpose material models, 21 wave dispersion, 21 shrinkage, 264e266 simple meshless discretization, 19e20 state-based peridynamics, 255e256 tissue growth, 264e266 cell division and, 265 tissue shrinkage, 265e266 Peridynamics damage model, 78e79 criterion for bond breaking, 86e87

numerical illustrations dynamic crack branching, 87e88 Kalthoff-Winkler experiment, 90e92, 91f phase field theory, 79e86 constitutive correspondence, 84e85 equations in explicit form, 85e86 governing equations, 82e84 kinematic correspondence, 84 kinematics, 82 Peridynamics formulation, 115e118 Peridynamic stress tensor, 13 Peridynamic theory, 39, 61e63, 115 Peridynamic Timoshenko beam formulation, 99e102 Peridynamic viscoelasticity formulation, 214e218 Petrov-Galerkin method, 22e23 Phase fieldebased peridynamics damage model cohesive zone model (CZM), 328, 330e331 complementary energy density, 344 composites delamination, 333e341 bulk and interface constitutive models, 334e335 governing equations, 333e334 DE damage model, 343e349 delamination, 329 DeshpandeeEvans (DE) constitutive model, 331e333 double cantilever beam (DCB), 335e336, 336f equations of motion, 345e349 fixed ratio mixed-mode (FRMM), 341, 342f internal energy density, 344e345 mixed (I/II) mode delamination, 340e341 mode I delamination, 335e337 mode II delamination, 337e340 end loaded split (ELS) test, 338, 338f end notched flexure test, 338e340, 340f numerical illustrations, 349e351 spherically symmetric geometry and loading, 345e349 constitutive correspondence, 347e349 virtual crack closure technique (VCCT), 328 Phase field theory, 78e86 constitutive correspondence, 84e85 equations in explicit form, 85e86

INDEX

governing equations, 82e84 kinematics, 82, 84 Plate under tensile loading, 217e218 Poroelasticity theory consolidation problem (1D), 391e393, 392f flow field, 393 mechanical field, 393 consolidation problem (2D), 395e396, 395f fully coupled poroelastic peridynamic formulation, 389e390 hydraulically pressurized crack problem (2D), square plate, 396e398, 397f hydraulic fracturing, 388 non-ordinary state-based peridynamics, 388 numerical implementation, 390e391 Pre-cracked plate subjected to traction, 131e132 Pre-crack propagation, 2D flat ice, 284 Pre-existing crack under tensile loading, 218e219 Prony coefficients, 219, 219t Propagating cracks, interaction integral for, 168e169 Prototype microelastic brittle model, 8

Q Quadratic polynomial basis functions, 130

R “Reaction force” horizons, 38e39 Reinforced plate structure, ice impacting on, 289e291, 290f Relative displacement vector, 116 Reproducing Kernel Particle Method (RKPM), 22e23 Research trends in peridynamic theory, 20e25 better meshless numerical techniques, 22e23 ductile material response, 23 material stability, 21e22 material variability, 24e25 micropolar theories, 22 multiple physical fields, 24 special purpose material models, 21 wave dispersion, 21 Rigid and deformable bodies, 184e192 bond-based peridynamic model, 185e187

443

contact model between impactor and target, 192e197 numerical results, 198e201 rigid impactor model, 187e192 Rigid impactor model, 187e192 Rigid sphere multiple sub-volume, 200e201 single sub-volume, 198e200

S Shrinkage, 264e266 Simple meshless discretization, 19e20 Single instruction multiple data (SIMD), 404 Single layer graphene sheets (SLGS) bond-based PD (BPD), 357, 357f classical continuum mechanics (CCM), 356 coarse-grain (CG) techniques, 356e357, 361e362 failure of, 362e367 modes, 364e367, 364fe365f validation, 362e363, 363f zigzag edge, 367 peridynamics (PD), 356, 358e361 establishment of, 358e360 parameters, 360e361 stress-strain relations of, 360f Solitary waves, 21 Special purpose material models, 21 Spherically symmetric geometry and loading, 345e349 Splice method, 115e120 Spurious wave test, 46e47 Stabilized correspondence model, 16, 17f Standard XFEM formulation, 144e145 State-based material models, 12e13 State-based peridynamics (SB-PD), 38e39, 255e256 State-based plasticity model, 206 State-valued functions, 116 Static condensation, 372 Stationary cracks interaction integral for, 166e168 mixed-mode DSIFs for, 171e173 Straight crack under tension, 151e153 Straight fiber reinforcement model, 316, 317f Strain energy density, 207e208, 210 Stress-strain relations, 360f Strong nonlocality, 2

444 T Tensile strength, 277, 279f Three-point bending test, 287e288, 287f Time-dependent peridynamic parameters, 215 Time integration approach, 128 Tissue growth, 265 cell division and, 265 Tissue shrinkage, 265e266 Two-dimensional micromodulus functions, 377e378, 378fe379f

INDEX

Virtual crack closure technique (VCCT), 328 Virtual work, principle of, 148e150

W Wave dispersion, 21 “Waves with zero velocity”, 16 Wing crack propagation, 3D ice body, 284e286, 286f

Y Young’s modulus, 121e122

V Variable scale homogeneous (VSH), 118 Velocity-Verlet time integration scheme, 128

Z “Zero energy mode” instabilities, 16 Zigzag edge, 367