PERIDYNAMIC THEORY FOR MODELING THREE-DIMENSIONAL

PERIDYNAMIC THEORY FOR MODELING THREE-DIMENSIONAL DAMAGE GROWTH IN METALLIC AND COMPOSITE STRUCTURES by Erkan Oterkus

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A Dissertation Submitted to the Faculty of the DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN AEROSPACE ENGINEERING In the Graduate College THE UNIVERSITY OF ARIZONA

2010

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THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Erkan Oterkus entitled Peridynamic Theory for Modeling Three-Dimensional Damage Growth in Metallic and Composite Structures and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy _______________________________________________________________________

Date: 12/01/2010

Erdogan Madenci _______________________________________________________________________

Date: 12/01/2010

Abe Askari _______________________________________________________________________

Date: 12/01/2010

Stewart Silling _______________________________________________________________________

Date: 12/01/2010

L. Rene Corrales _______________________________________________________________________

Date: 12/01/2010

Andrei Sanov Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. ________________________________________________ Date: 12/01/2010 Dissertation Director: Erdogan Madenci

3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: Erkan Oterkus

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ACKNOWLEDGEMENTS

I would like to thank to: - Dr.Erdogan Madenci for his help, guidance and support. - Dr. Atila Barut and Dr.Ibrahim Guven for their help and support. - Dr. Abe Askari, Dr. Rene Corrales, Dr. Andrei Sanov and Dr. Stewart Silling for serving on my committee. - All of my friends, especially Dr. Manabendra Das and Dr. Bahattin Kilic, for their friendship. - Finally, my family for their support.

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DEDICATION

Dedicated to my wife, Selda Alpay Oterkus

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TABLE OF CONTENTS

LIST OF FIGURES…………………………………………………………………….....9 ABSTRACT………………………………………………………………….…………..18 1. INRODUCTION…………………………………………………………………..…..19 2. PERIDYNAMIC ANALYSIS OF METALLIC STRUCTURES…….………….…...27 2.1. Introduction..……………………………………………………………………...27 2.2. State-based Peridynamic Theory...……………………………………………….27 2.3. Concept of State..……………………….……………………………………...…31 2.4. Governing Equations of State-based Peridynamic Theory.……..………………..35 2.5. Governing Equations of Bond-based Peridynamic Theory.………..…………….44 2.5.1. Pair-wise Response Function……………………………………….………..44 2.5.2. Boundary Conditions……………………………………………….………..48 2.5.3. Numerical Implementation…………………...…………………….………..50 2.5.4. Surface Correction Factors…………………...…………………….………..54 2.6. Numerical Results…..…………………………………………………………….55 2.6.1. Longitudinal Vibration of a Bar………...……………………….….………..57 2.6.2. Plate with a Pre-existing Crack under Velocity Boundary Conditions.… …..60 2.6.3. Plate with a Circular Cutout under Quasi-static Loading…………..………..65 3. PERIDYNAMIC ANALYSIS OF COMPOSITE STRUCTURES……..………….…69 3.1. Introduction………………………..………………………………………..…….69

7 TABLE OF CONTENTS - Continued

3.2. Peridynamic Analysis of a Lamina…...…………….…………………………….70 3.2.1. Critical Stretch Parameters for Fiber and Matrix Bonds...………………..…76 3.2.2. Surface Correction Factors for a Lamina.......................…………………..…78 3.3. Peridynamic Analysis of a Laminate…………...………………………………...79 3.4. Numerical Results..........………………………………………………………….82 3.4.1. A Lamina under Uniaxial Tension and Uniform Temperature Change......…82 3.4.2. Laminates under Uni-axial Tension………………………………...……..…86 3.4.3. A lamina with a Pre-existing Central Crack under Tension……..........…..…90 3.4.4. Laminates with a Pre-existing Central Crack under Tension……........…..…92 4. COUPLED FINITE ELEMENT METHOD AND PERIDYNAMIC THEORY……100 4.1. Introduction………..…………………………………………………………….100 4.2. Cylindrical Composite Shell under Internal Pressure…………...……….……...101 4.3. Failure Prediction in a Stiffened Composite Panel with a Central Slot………....105 4.3.1. Finite Element Analysis of the Entire Panel…….………………....……….105 4.3.2. Peridynamic Analysis of the Submodel ……………..……………....……..107 4.4. Numerical Results………………..………………………………………….......113 5. FINAL REMARKS AND FUTURE WORK……………………………………..…118 APPENDIX A. INFLUENCE FUNCTION……...……………………………….....…126 APPENDIX B. BALANCE LAWS…………..……………...…………………………129 B.1. Balance of Linear Momentum……….………………………………………….129 B.2. Balance of Angular Momentum……...…...…………………………………….130

8 TABLE OF CONTENTS - Continued

B.3. Conservation of Energy…………………...…………………………………….133 APPENDIX C. GOVERNING EQUATIONS OF CLASSICAL CONTINUUM MECHANICS……….………………………………………………………………….137 APPENDIX D. PERIDYNAMIC PARAMETERS……………….……………………150 APPENDIX E. PERIDYNAMIC SURFACE CORRACTION FACTOR…………......158 APPENDIX F. PERIDYNAMIC BOND CONSTANTS FOR ONE-DIMENSIONAL AND TWO-DIMENSIONAL STRUCTURES………….…………………...………...164 F.1. PD Material Constant for One-dimensional Structures……...………………….164 F.2. PD Material Constant for a Two-dimensional Plate………..…………..……….166 APPENDIX G. PERIDYNAMIC MATERIAL CONSTANTS OF A LAMINA...........171 APPENDIX H. SURFACE CORRECTION FACTORS FOR A COMPOSITE LAMINA………………….………………………………………………………….....182 APPENDIX I. PERIDYNAMIC INTERLAYER AND SHEAR BOND CONSTANTS OF A LAMINATE………………………………………...…………………………....190 APPENDIX J. CRITICAL STRETCH VALUES FOR BOND CONSTANTS.........…200 REFERENCES…….……………………………………………………………...……203

9 LIST OF FIGURES Figure 1.1. Relationship among length scales…….……………………………………..21 Figure 1.2. Deformation and force states in the peridynamic theory…………………....22 Figure 2.1. Kinematic description of a pair of PD material points……………………....28 Figure 2.2. (a) Deformation state, Y , and (b) force state, T …………………………....30 Figure 2.3. The “expansion” of the second-order tensor F ……………………………....32 Figure 2.4. Components of the position vector, ξ , between material points at x and x′ .34 Figure 2.5. Constitutive relation for elastic stretch between material points…………….46 Figure 2.6. Interactions among material points………………………………………….48 Figure 2.7. Boundary conditions: (a) domain of interest, (b) tractions in classical continuum mechanics, (c) interaction of a point in domain Ω + with domain Ω − , (d) force densities acting on domain Ω + due to domain Ω − …49 Figure 2.8. Discretization and material points in a one-dimensional region…………….51 Figure 2.9. Interaction of material points within the horizon………………...………….53 Figure 2.10. Surface effects in the domain of interest……….……………….………….55 Figure 2.11. Peridynamic model of a one-dimensional bar….……………….………….57 Figure 2.12. Variation of axial displacement at the center of the bar for a range of constrained-region lengths………………….….……………….………….59 Figure 2.13. Variation of axial displacement at the center of the bar for varying values of grid size.….……………….………………………………………………..59 Figure 2.14. Variation of the axial displacement at the center of the bar for varying values of horizon…………………………………...….……………….………….60

10 LIST OF FIGURES - Continued Figure 2.15. Square plate with a pre-existing central crack under velocity boundary conditions…………………………………….……………….……………61 Figure 2.16. Crack opening displacement near the crack tip at the end of 1250 time steps when no failure is allowed…………………..…………….………….62 Figure 2.17. Damage indicating self-similar crack growth at the end of 1250 time steps under the velocity boundary condition of V0 (t ) = 20 m/s ………………….63 Figure 2.18. Crack growth as a function of time…………….……………….………….63 Figure 2.19. Damage indicating crack branching at the end of 1000 time steps under a velocity boundary condition of V0 (t ) = 70 m/s .………………...………….64 Figure 2.20. Loading and geometry of the plate with a circular cutout.……...………….66 Figure 2.21. Variation of horizontal displacement along the central axis at the end of 1000 time steps when no failure is allowed……………..……...………….66 Figure 2.22. Variation of vertical displacement along the central axis at the end of 1000 time steps when no failure is allowed…………………...……...………….67 Figure 2.23. Damage plots for the plate with a circular cutout at the end of (a) 650 time steps, (b) 700 time steps, (c) 800 time steps, and (d) 1000 time steps…….68 Figure 3.1. PD horizon for a lamina with a fiber orientation of θ and PD bonds between material point i and other material points within its horizon..…………….70 Figure 3.2. Force-stretch relation for fiber and matrix bonds………....……...………….71 Figure 3.3. Components of the initial bond length between material points i and p…….74 Figure 3.4. Determination of critical stretch values s ft and smt ………………..….…….77

11 LIST OF FIGURES - Continued Figure 3.5. Surface effects in the domain of interest……….………....……...………….78 Figure 3.6. Four different bond constants for a fiber-reinforced composite material...….79 Figure 3.7. Loading and geometry of the unidirectional lamina under uniaxial tension and uniform temperature change…………………………………....………….83 Figure 3.8. Horizontal displacement along the central axis at the end of 8000 time steps……………………………………………………………………......84 Figure 3.9. Vertical displacement along the central axis at the end of 8000 time steps....84 Figure 3.10. Variation of horizontal displacement along the central axis at the end of 8000 time steps when no failure is allowed……………………...…….......85 Figure 3.11. Variation of vertical displacement along the central axis at the end of 8000 time steps when no failure is allowed……….……….….……...………….86 Figure 3.12. Loading and geometry of a composite laminate under uniaxial tension.......87 Figure 3.13. Horizontal displacement along the central axis in the 90 ply of the

[0 / 90 / 0 ] layup at the end of 8000 time steps..………………………...88 Figure 3.14. Vertical displacement along the central axis in the 90 ply of the [0 / 90 / 0 ] layup at the end of 8000 time steps....……………………….88 Figure 3.15. Horizontal displacement along the central axis in the 45 ply of the [0 / 45 / 0 ] layup at the end of 8000 time steps....……………………….89 Figure 3.16. Vertical displacement along the central axis in the 45 ply of the [0 / 45 / 0 ] layup at the end of 8000 time steps………………………….89 Figure 3.17. Loading and geometry of the unidirectional lamina with a crack under tension loading……………………………………………………………..90

12 LIST OF FIGURES - Continued Figure 3.18. Damage plots for a lamina having a central crack with a fiber orientation of (a) θ = 0 , (b) θ = 90 , and (c) θ = 45 …………………………………...91 Figure 3.19. Loading and geometry of a composite laminate with a crack under tension loading……………………………………………………………………..92 Figure 3.20. Matrix bond damage plots for a laminate of [0 / 90 / 0 ] with a pre-existing crack (a) Bottom ply, 0 , (b) Center ply, 90 , and (c) Top ply, 0 for smt ≠ sin = ϕ c …………………………………..………..………..94 Figure 3.21. Matrix bond damage plots for a laminate of [0 / 45 / 0 ] with a pre-existing crack (a) Bottom ply, 0 , (b) Center ply, 45 , and (c) Top ply, 0 for smt ≠ sin = ϕ c ..................................……………………………..95 Figure 3.22. Matrix bond damage plots for a laminate of [0 / 90 / 0 ] with a pre-existing crack (a) Bottom ply, 0 , (b) Center ply, 90 , and (c) Top ply, 0 for smt = sin = ϕ c .……..…………………………...………………..96 Figure 3.23. Shear bond damage plots for a laminate of [0 / 90 / 0 ] with a pre-existing crack (a) Bottom ply, 0 , (b) Center ply, 90 , and (c) Top ply, 0 for smt = sin = ϕ c .……..……………………………………………..97 Figure 3.24. Matrix bond damage plots for a laminate of [0 / 45 / 0 ] with a pre-existing crack (a) Bottom ply, 0 , (b) Center ply, 45 , and (c) Top ply, 0 for smt = sin = ϕ c .……………………………………………………98

13 LIST OF FIGURES - Continued Figure 3.25. Shear bond damage plots for a laminate of [0 / 45 / 0 ] with a pre-existing crack (a) Bottom ply, 0 , (b) Center ply, 45 , and (c) Top ply, 0 for smt = sin = ϕ c .………………………………..…………………..99 Figure 4.1. Cylindrical composite shell under internal pressure…………………….....101 Figure 4.2. Coupled FEM and PD analyses of a laminate: global FE model under internal pressure loading and PD submodel with displacement boundary conditions…………………………………………………………………102 Figure 4.3. FEM predictions of displacement contours: (a) u0 ( x, y ) , (b) v0 ( x, y ) , and (c) w0 ( x, y ) .…………………….....................................................................102

Figure 4.4. FEM predictions of rotation contours: (a) θ x ( x, y ) , (b) θ y ( x, y ) , and (c) θ z ( x, y ) …………………………………………...…………………….....103

Figure 4.5. Cut boundary displacements of a three-dimensional PD model: (a) u ( x, y, z ) , (b) v ( x, y, z ) , and (c) w ( x, y, z ) ………………………………………….....104 Figure 4.6. Comparison of out-of-plane deflections in the submodeling region: (a) FE analysis and (b) PD analysis………………………………………..….....104 Figure 4.7. Configured stiffened composite curved panel with a slot.………………....105 Figure 4.8. Finite element discretization of the configured panel and cut boundaries for submodeling..………………………………………………………..…....106 Figure 4.9. Displacement variation due to internal pressure loading (a) u, (b) v, and (c)

w………………………………………………………..………………....107

14 LIST OF FIGURES - Continued Figure 4.10. Displacement variation due to axial tension loading (a) u, (b) v, and (c)

w.……………………………………………………………………….....108 Figure 4.11. Peridynamic analysis: (a) submodel and (b) top view of the submodel......109 Figure 4.12. Boundary domain in the panel skin: (a) in a representative figure and (b) in the actual peridynamic model………………………………………….....109 Figure 4.13. Boundary domain in the stiffeners: (a) in a representative figure and (b) in the actual peridynamic model……………………………………….........110 Figure 4.14. Submodel: (a) no-fail zone representation and (b) top view of the submodel (orange: failure is not allowed; magenta: failure is allowed).………........111 Figure 4.15. Representation of a 34-ply panel skin by an equivalent 11-ply laminate....111 Figure 4.16. Application of displacement constraints due to internal pressure and axial tension loading…………………………………………………………....112 Figure 4.17. Displacement field in the submodel just before initial failure occurs: (a) u , (b) v , and (c) w .……………………………………………………….....114 Figure 4.18. Location of material points for monitoring displacement components.......115 Figure 4.19. Displacement variation of the monitored material points by time at the bottom ply: (a) lateral displacement, (b) axial displacement, and (c) out-of-plane displacement………………………………………..........115 Figure 4.20. Total damage plot of the panel skin: (a) sketch of experimental results; PD results corresponding to the axial tension loading of (b) 728.75 kips, (c) 825 kips, (d) 893.75 kips, (e) 962.5 kips, and (f) 1100 kips.…...….......117

15 LIST OF FIGURES - Continued Figure 5.1. Schematic of discrete crack growth steps (left) due to blocks of load cycles (right)…………………………………………….…………………….....120 Figure 5.2. Start, middle, and end of a load cycle.……………… ………………….....121 Figure 5.3. Degradation of the critical material stretch due to incremental load cycles..122 Figure A1. Isotropic expansion of homogeneous body……………………....………...127 Figure B1. Ordinary state-based PD interaction of two material points at x and x′ .…132 Figure B2. Bond-based PD interaction of two material points at x and x′ .…………...133 Figure C1. Material point interacting with others in its immediate vicinity……………137 Figure D1. Isotropic expansion loading……………....……………………....………...151 Figure D2. Simple shear loading……..……………....……………………....………...153 Figure E1. Material point x with a truncated horizon..………………………………...158 Figure E2. Material point x far away from external surface …………………..……...158 Figure E3. Discretization of the horizon of material point at x located far away from external surface………………………………..…………………..………..160 Figure E4. Construction of an ellipsoid for surface correction factors...………..……...161 Figure E5. (a) PD bond between material points at x( i ) and x( j ) and (b) the ellipsoid for the surface correction factor…………………………......………..……...162 Figure F1. PD modeling of a one-dimensional structure: (a) horizon and deformation of material points at x and x ′ , and (b) discretization……...………..……...165 Figure F2. (a) PD horizon for a two-dimensional plate and PD bonds between material point x and other material points within its horizon and (b) PD bond between material points x and x′ …………………….....………..……...167

16 LIST OF FIGURES - Continued Figure F3. A 2-dimensional plate subjected to isotropic expansion loading....………...168 Figure F4. A 2-dimensional plate subjected to pure shear loading……….......………...169 Figure G1. (a) PD horizon for a lamina with a fiber orientation of θ and PD bonds between material point i and other material points within its horizon and (b) PD bond between material points i and p with an orientation of φ …171 Figure G2. Deformed configuration of an angle lamina subjected to a combined mechanical and thermal loading……………………………………..…...172 Figure G3. Relative displacement between material points i and p……………..……...174 Figure G4. Relative displacement between material points i and p due to the strain field…………...…………………………………………………………….175 Figure H1. Material point x with a truncated horizon in a lamina.……………………182 Figure H2. Material point x far away from external surfaces of a lamina..………...…183 Figure H3. Construction of an ellipse for surface correction factors……....………...…187 Figure H4. (a) PD bond between material points at x( i ) and x( j ) and (b) the ellipse for the surface correction factor……………………………..……....………...…188 Figure I1. Interlayer and shear bonds between neighboring plies (only some of the interactions are depicted explicitly for clarity)…………………………...191 Figure I2. Shear bonds between material points b and a , and between material points d and c , in both undeformed and deformed configurations…...………...…192 Figure I3. (a) A composite laminate subjected to isotropic expansion loading (b) Deformation of interlayer and shear bonds between neighboring plies (only some of the interactions are depicted explicitly for clarity......…..…194

17 LIST OF FIGURES - Continued Figure I4. (a) A composite laminate subjected to simple shear loading. (b) Deformation of interlayer and shear bonds between neighboring plies (only some of the interactions are depicted explicitly for clarity)……....………………...…196 Figure I5. Shear bonds between material points b and a , and between material points d and c , in both undeformed and deformed configurations…...………...…197 Figure J1. Interlayer and shear bonds between material point, x and other material points located at the (k + 1)th ply………………………………..…...………...…201

18 ABSTRACT

A recently introduced nonlocal peridynamic theory removes the obstacles present in classical continuum mechanics that limit the prediction of crack initiation and growth in materials. It is also applicable at different length scales. This study presents an alternative approach for the derivation of peridynamic equations of motion based on the principle of virtual work. It also presents solutions for the longitudinal vibration of a bar subjected to an initial stretch, propagation of a pre-existing crack in a plate subjected to velocity boundary conditions, and crack initiation and growth in a plate with a circular cutout. Furthermore, damage growth in composites involves complex and progressive failure modes. Current computational tools are incapable of predicting failure in composite materials mainly due to their mathematical structure. However, the peridynamic theory removes these obstacles by taking into account non-local interactions between material points. Hence, an application of the peridynamic theory to predict how damage propagates in fiber reinforced composite materials subjected to mechanical and thermal loading conditions is presented. Finally, an analysis approach based on a merger of the finite element method and the peridynamic theory is proposed. Its validity is established through qualitative and quantitative comparisons against the test results for a stiffened composite curved panel with a central slot under combined internal pressure and axial tension. The predicted initial and final failure loads, as well as the final failure modes, are in close agreement with the experimental observations. This proposed approach demonstrates the capability of the PD approach to assess the durability of complex composite structures.

19 1. INTRODUCTION

Despite the development of many important concepts to predict material behavior and failure, the prediction of failure modes and residual strength of a material is a challenge within the framework of the finite element method (FEM). Governing equations of the FEM are based on the partial differential equations (PDEs) of classical continuum mechanics and on the requirement that the spatial derivatives of the PDEs do not, by definition, exist at crack tips or along crack surfaces. Therefore, the basic mathematical structure of the formulation breaks down whenever a crack appears in a body. Furthermore, these methods cannot address the nucleation of a crack in a continuous material. The field of fracture mechanics is primarily concerned with the evolution of pre-existing defects within a body, rather than the nucleation of new defects. Even when addressing crack growth, the FEM with traditional elements suffers from the inherent limitation that it requires remeshing after each incremental crack growth. In addition to the need to remesh, existing methods for fracture modeling also suffer from the need to supply a kinetic relation for crack growth, a mathematical statement that prescribes how a crack evolves a priori based on local conditions. It guides the analysis as to when a crack should initiate; how fast it should grow and in what direction; whether it should turn, branch, oscillate, arrest, etc. Considering the difficulty in obtaining and generalizing experimental fracture data, providing such a kinetic relation for crack growth clearly presents a major obstacle to fracture modeling using conventional methods. This prevents such methods from being applicable to problems in which multiple cracks grow and interact in complex patterns. Although the introduction of cohesive zone elements (Zhang

20 et al., 2002) and eXtended FEM (XFEM) (Li et al., 2001; Belytschko et al., 2000) eliminates the need to remesh, they still rely on external kinetic relations for injection of such elements while predicting crack growth paths. The difficulties encountered in the methods utilizing classical continuum mechanics can be overcome by performing molecular dynamics simulations or atomic lattice models. Atomistic methods (Abraham, 2002) although providing insight into the nature of fracture in certain materials, cannot be expected to provide a practical tool for the modeling of engineering structures. It is clear that the atomistic simulations are insufficient to model fracture processes in real-life structures. Moreover, the experiments of physicists have revealed that cohesive forces reach finite distances among atoms, yet the classical continuum theory lacks an internal length parameter that would permit modeling at different length scales. Hence, the classical continuum theory is valid only for very long wavelengths (Eringen, 1972). Therefore, the nonlocal continuum theory was introduced in an effort to account for the long-range effects (Eringen and Edelen, 1972; Kroner, 1967; Kunin, 1982). The relationship among the local and nonlocal continuum models and the molecular dynamics model is illustrated in Fig. 1.1. In the case of the local (classical) continuum model, the state of a material point is influenced by the material points located in its immediate vicinity. In the case of the nonlocal continuum model (PD theory), the state of a material point is influenced by material points located within a region of finite radius. As the radius becomes infinitely large, the nonlocal theory becomes the continuous version of the molecular dynamics model. Therefore, the nonlocal theory of continuous media establishes a connection between the classical (local) continuum mechanics and molecular dynamics models.

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Local model

Nonlocal model

Molecular dynamics

Figure 1.1. Relationship among length scales.

Because of the nonlocal nature of the theory, the stress field ahead of the crack tip is bounded as the crack tip is approached asymptotically (Eringen and Kim, 1974a; Eringen and Kim, 1974b), rather than unbounded as predicted by the classical continuum theory. As shown by Ari and Eringen (1983) and Elliott (1947), the nonlocal elasticity analysis of a Griffith crack is in agreement with the lattice model. Although this nonlocal continuum theory leads to finite stresses at the crack tips, the derivatives of the displacement field retained in the formulation have discontinuities arising from the presence of cracks. Another type of nonlocal theory circumvents this difficulty because it uses displacement fields rather than their derivatives (Kunin, 1982; Kunin, 1983; Rogula, 1982). Silling (2000) independently reintroduced a nonlocal theory that does not require spatial derivatives—the peridynamic (PD) theory. Compared to the previous nonlocal theory by Kunin (1982) and Rogula (1982), the PD theory is more general because it considers two- and three-dimensional in addition to one-dimensional media. Unlike the

22 nonlocal theory by Kunin (1983), the PD theory provides nonlinear material response with respect to displacements. Furthermore, the material response includes damage in the PD theory. Silling et al. (2007) generalized the PD theory by introducing two mathematical objects. As described in Fig. 2, the first one, called a deformation state, maps any distance between two material points onto its deformed image, and the second one, called a force state, provides the force density vector at a PD point influenced collectively by the deformation of all the other material points associated with that point. In this figure, the relative position vector ξ = x - x'

becomes

Y ξ = y - y'

in the deformed

configuration, and the corresponding force vector acting at material point x is T(x, t ) ξ . The relation between the deformation state and force state is the constitutive model for the material.

Undeformed state

Deformation state

Force state

Figure 1.2. Deformation and force states in the peridynamic theory.

The main difference between the PD theory and classical continuum mechanics is that the former is formulated using integral equations as opposed to derivatives of the

23 displacement components. This feature allows damage initiation and propagation at multiple sites, with arbitrary paths inside the material, without resorting to special crack growth criteria. In the PD theory, internal forces are expressed through nonlocal interactions between the material points within a continuous body, and damage is a part of the constitutive model. Interfaces between dissimilar materials have their own properties, and damage can propagate when and where it is energetically favorable for it to do so. In the peridynamic theory, material points interact with each other directly through the prescribed response function, which contains all of the constitutive information associated with the material. The response function includes a length parameter called internal length (horizon), δ . The locality of interactions depends on the horizon, and interactions become more local with a decreasing horizon. Hence, the classical theory of elasticity can be considered a limiting case of the peridynamic theory as the internal length approaches zero. For instance, it has been shown that the peridynamic theory reduces to the linear theory of elasticity with the proper choice of response function (Silling et al., 2003; Weckner and Abeyaratne, 2005). In another limiting case where the internal length approaches the inter-atomic distance, it was shown by Silling and Bobaru (2005) that van der Waals forces can be used as the response function to model nanoscale structures. Therefore, the peridynamic theory is capable of bridging the nano to macro length scales. With the PD theory, damage in the material is simulated in a much more realistic manner compared to the classical continuum-based methods. As the interactions between material points cease, cracks may initiate and align themselves along surfaces that form

24 cracks, yet the integral equations continue to remain valid. The PD theory has been utilized successfully for damage prediction in many problems. Silling (2003) considered a Kalthoff-Winkler experiment in which a plate having two parallel notches was hit by an impactor, and PD simulations successfully captured the angle of crack growth that was observed in the experiments. Later, impact damage was predicted by Silling and Askari (2004; 2005) using the PD theory. Silling and Askari (2005) also considered a plate with a center crack to show convergence of their numerical method. The PD theory was applied to damage analysis of plain and reinforced concrete structures by Gerstle et al. (2005). Furthermore, a new constitutive model was introduced for tearing and stretching of rubbery materials by Silling and Bobaru (2005), who predicted an oscillatory crack path when a blunt tool was forced through a membrane. The PD theory was also applied successfully by Askari et al. (2006) and Colavito et al. (2007a; 2007b) to predict damage in laminated composites subjected to low-velocity impact and damage in woven composites subjected to static indentation. In addition, Xu et al. (2007) considered notched laminated composites under biaxial loads. Warren et al. (2009) demonstrated the capability of the PD theory to capture the fracture behavior of a notched and unnotched isotropic bar under quasi-static loading, as well as other loading conditions while employing material models from the classical continuum mechanics. Foster et al. (2008) considered the elastic viscoplastic behavior of a bar under tension. The classical constitutive model was fit to high-strain rate stress-strain data taken from Kolsky bar experiments on 6061-T6 aluminum. They were able to reproduce the experimental results from the Taylor impact tests on the aluminum. Kilic et al. (2009) and Kilic and Madenci (2009a; 2009b; 2010a; 2010b) also considered the elastic response of many other

25 problems and compared PD predictions to experiments and other numerical/analytical solutions available in the literature. Also, Oterkus et al. (2010a) demonstrated that PD analysis is capable of capturing bearing and shear-out failure modes in bolted composite lap-joints. Based on the abovementioned discussion, Peridynamic theory is a new and promising technique to predict failure in materials. Hence, in this study, both the theory and applications of Peridynamic theory for metallic and composite structures are presented by following a systematic approach. Chapter 2 presents another perspective for rederiving the governing equations of the Peridynamic theory (Silling, 2000; Silling et al., 2007) based on the principle of virtual work. Also, it presents solutions to the longitudinal vibration of a bar, propagation of a pre-existing crack in a plate, and crack initiation and growth in a plate with a circular cutout. An application of the peridynamic theory in the analysis of fiber reinforced composite materials subjected to mechanical and thermal loading conditions is presented in Chapter 3. The PD approach to model a lamina is first validated against analytical solutions by considering uni-axial tension and uniform temperature change. Then, damage growth patterns from a pre-existing crack in a lamina for different fiber orientations are computed and compared against experimental observations. This approach is further extended to analyze composite laminates and to predict damage growth patterns from a pre-existing crack in two distinct laminate constructions under tension. In the absence of a crack, the PD displacement predictions are compared against those by classical

26 laminate theory. In the presence of a crack, damage patterns are qualitative compared against experimental observations. A coupling of the FEM with PD theory through a submodeling approach is introduced in Chapter 4 to investigate damage in complex composite structures by considering a stiffened composite curved panel with a central slot (NASA Panel 67) subjected to combined internal pressure and axial tension loading. The initial and final failure loads of the panel and failure modes due to the applied loading are evaluated by using PD theory and compared against test measurements and observations. Finally, Chapter 5 presents final remarks and some possible future work areas related with the Peridynamic theory.

27 2. PERIDYNAMIC ANALYSIS OF METALLIC STRUCTURES 2.1. Introduction Original formulation of the Peridynamic theory is first introduced by Silling (2000). Presently, this formulation is known as “Bond-based Peridynamic theory”. Afterwards, this original formulation is improved by using the state concept by Silling et al. (2007) and named as “State-based Peridynamic theory”. An alternative way of deriving the governing equations of State-based Peridynamic theory is demonstrated in Sec. 2.2. As a special case of State-based Peridynamic theory, some important concepts of Bond-based Peridynamic theory is highlighted in Sec. 2.3. Finally, solutions to three fundamental problems of solid mechanics, i.e. the longitudinal vibration of a bar, propagation of a preexisting crack in a plate, and crack initiation and growth in a plate with a circular cutout, by using Perdiynamic theory are shown in Sec. 2.4.

2.2. State-based Peridynamic Theory The PD theory is concerned with the physics of a material body at a point that interacts with all points within its range, as shown in Fig. 2.1. As in the classical (local) continuum theory, the material points of a body are continuous, as opposed to discrete in the case of molecular dynamics. However, the main difference between the PD- and classical continuum-based methods is that the former is formulated using integral equations, as opposed to partial differential equations that include spatial derivatives of the displacement components. These displacement derivatives do not appear in PD

28 equations, which allow the PD formulation to hold everywhere whether or not displacement discontinuities are present.

Figure 2.1. Kinematic description of a pair of PD material points.

According to the PD theory, the motion of the body is analyzed by considering the interaction of a PD material point, x , with the other, possibly infinitely many, material points in the body. Therefore, an infinite number of interactions may exist between the material point at location x and other material points. Hence, the PD state may contain particular information on an infinite number of interactions. However, the influence of the material points interacting with x is assumed to vanish beyond a local region (horizon), denoted by H shown in Fig. 2.1. Also, the horizon around material point x moves to a new position, denoted by t H .

29 The displacement vector field at a material point whose position is defined by a vector, x , is represented by uT (x, t ) = (u x , u y , u z ) . The prescribed body force vector is bT (x, t ) = (bx , by , bz ) and the mass density of the body is ρ . As shown in Fig. 2.1, the

distance from x to x′ is represented by the relative position vector ξ in the undeformed configuration. As the body deforms from its initial to the current configuration, tV , the material points, x and x′ , move to new positions by the displacement vectors, u and u′ , respectively. The governing equations of PD theory are nonlinear with respect to displacements, thus it is not limited to small deformations. The concept of representing functions in terms of infinite dimensional arrays (states) can also be applied to the PD theory. Therefore, a PD state can be described as an infinite dimensional of a particular quantity, such as the deformation of possibly infinitely many other material points associated with a material point at location x at some time t. As illustrated in Fig. 2.2, the PD theory concerns the deformation state, Y , and the force state, T . The material points at locations x and x ' with an initial relative position vector ξ = x '− x before deformation move to their new positions y and y' with a relative position vector ξ + η = y '− y after deformation. As described in Fig. 2.2a, their relative position vector (y '− y ) can be obtained by operating the deformation state, Y , on the relative position vector (x '− x) as (y '− y ) = Y ( x, t ) x '− x .

(2.1)

Similarly, the force that the material point at location x ' exerts on the material point at location x can be expressed as f = T ( x, t ) x '− x .

(2.2)

30 The difference between the force state and the deformation state is that the force state is dependent on the deformation state while the deformation state is independent. Therefore, the force state for the material point x depends on the relative displacements between this material point and the other material points within its horizon. Hence, the force state can also be written as T ( x, t ) = T ( Y ( x, t ) ) .

(a)

(2.3)

(b)

Figure 2.2. (a) Deformation state, Y , and (b) force state, T .

31 2.3. Concept of State A continuous function g ( x ) for −∞ < x < ∞ can be considered as a combination of an infinite number of discrete function values, g ( xi ) for i = 1,...., ∞ . These discrete function values can be stored in an infinite-dimensional array or a vector “state,” G as

 g ( x1 )      G =  g ( xi )  .      g ( x∞ ) 

(2.4)

The “state” concept is not restricted to continuous functions. It is also applicable to discontinuous functions. As explained by Silling et al. (2007), “states” can also be described as a general form of tensors. It is possible to convert states to tensors or vice versa. The process of converting a tensor to a state is referred to as “expansion” and the process of converting a state to a tensor as “reduction.” If a second-order tensor, F , operates on a vector ( x ( j ) − x( k ) ) , the corresponding vector ( y ( j ) − y ( k ) ) is obtained as

(y

( j)

− y ( k ) ) = F ( x( j ) − x( k ) ) ,

(2.5)

where j = 1,… , ∞ . All the ( y ( j ) − y ( k ) ) vectors can be stored in an infinite-dimensional array, or a vector state, Y :

32

 ( y (1) − y ( k ) )    Y=    ( y ( ∞ ) − y ( k ) ) 

or

 F ( x(1) − x( k ) )    Y= .   F ( x ( ∞ ) − x ( k ) ) 

(2.6)

In this equation, there is a direct relationship between the vector state Y and the second-order tensor F . This relationship can be expressed as the “expansion” of the second-order tensor F . The “expansion” process can be visualized as shown in Fig. 2.3. In this figure, the second-order tensor F operates on an infinite number of vectors, forming a circle, ( x ( j ) − x( k ) ) with j = 1,… , ∞ , and the resulting vectors, ( y ( j ) − y ( k ) ) , form an ellipse.

Figure 2.3. The “expansion” of the second-order tensor F .

The reverse transformation from a vector state to a second-order tensor, which is called the “reduction” process, can be obtained by the expression given by Silling et al. (2007). Hence, a vector state Y can be reduced to a second-order tensor F : F = ( Y ∗ X ) K −1 ,

(2.7)

where the vector state of reference position, X , and the shape tensor, K , are defined as

33

X x′ − x = x′ − x

(2.8)

K = X∗X.

(2.9)

and

As given by Silling et al. (2007), the product of two vector states ( Y ∗ X ) is defined as Y ∗ X = ∫ ω x′ − x Y x′ − x ⊗ X x′ − x dH ,

(2.10)

H

where the influence function, ω , is a scalar state, and ⊗ denotes the dyadic product of two vectors, i.e., C = a ⊗ b or Cij = ai b j . Therefore, the shape tensor, K , can be obtained as K = ∫ ω x′ − x X x′ − x ⊗ X x′ − x dH .

(2.11)

H

in which and ω ( x, t ) represents a scalar state influence (weight) function. The scalar state influence function provides a means to control the influence of PD points away from the current point. In this case, if the interaction between any two PD points ceases due to damage and/or a crack opening in the material, the weight function evaluated between those points becomes zero. The influence function, ω , derived in Appendix A, can be defined as

ω x′ − x =

δ x′ − x

,

(2.12)

with δ defining the radius of the horizon, H . The shape tensor, K , has a direct relationship with the volume of the horizon. Defining the position vector, ξ = x′ − x , in the form, the shape tensor can be rewritten as

34 K = ∫ ω ξ ξ ⊗ ξ dH

(2.13a)

H

or K ij = ∫ ω ξ ξi ξ j dH

with i, j = 1, 2,3 .

(2.13b)

H

The components (ξ x , ξ y , ξ z ) of the position vector ξ in reference to a Cartesian coordinate system ( x, y, z ) ,whose origin is located at x , between material points at x and x′ can be expressed as

ξ1 = ξ x = ξ sin (φ ) sin (θ ) ,

(2.14a)

ξ2 = ξ y = ξ cos (φ ) ,

(2.14b)

ξ3 = ξ z = ξ sin (φ ) cos (θ ) ,

(2.14c)

where ξ = ξ is the length of the position vector and definitions of angles φ and θ are shown in Fig. 2.4.

Figure 2.4. Components of the position vector, ξ , between material points at x and x′ .

35 The components of the shape tensor, K , become

δ ξ ξ dH , ξ i j H

K ij = ∫

i, j = 1, 2,3

(2.15a)

or δ 2π π

K ij = ∫

δ

∫∫ξ ξ ξ i

j

ξ 2 sin (φ ) dφ dθ dξ .

(2.15b)

0 0 0

After performing the integration in Eq. (2.15b), the components of the shape tensor, K , are obtained as K ij =

πδ5 3

δ ij ,

(2.16)

where δ ij is the Kronecker delta with i, j = 1, 2,3 . By defining the volume of the horizon, V = 4 / 3 π δ 3 , the shape tensor, K , can be expressed as

K=

Vδ2 I, 4

(2.17)

with I representing the identity matrix. Therefore, the shape tensor can be viewed as a quantity that serves as volume averaging of the product of vector states, ( Y ∗ X ) .

2.4. Governing Equations of State-based Peridynamic Theory The governing equations of the PD theory can be derived by applying the principle of virtual work, i.e., t1

δ ∫ (T − U )dt = 0 , t0

(2.18)

where T and U represent the total kinetic and potential energies in the body. This principle is satisfied by solving for the Lagrange’s equation

36 d  ∂L  ∂L = 0,  − dt  ∂u  ∂u

(2.19)

where the Lagrangian L is defined as L = T −U .

(2.20)

The total kinetic and potential energies in the body can be obtained by summing all kinetic and potential energies of the set of material points, respectively, 1 T = ∫ ρ u ( x, t ) ⋅ u ( x, t ) dV 2 V

(2.21a)

U = ∫ W ( x, t ) dV − ∫ b ( x, t ) ⋅ u ( x, t ) dV .

(2.21b)

and

V

V

Because the volume of each material point is infinitesimally small, for the limiting case of dV → 0 , the integration can be expressed in terms of infinite summation ∞

∫ (⋅) dV → ∑ (⋅)V(i ) .

V

(2.22)

i =1

Therefore, the kinetic and potential energies given in Eq. (2.21) can be rewritten as ∞ 1 T = ∑ ρ( i ) u (i ) ⋅ u ( i ) V( i ) i =1 2

(2.23a)

and ∞

∞

i =1

i =1

U = ∑ W(i ) V( i ) − ∑ ( b (i ) ⋅ u (i ) ) V(i ) ,

(2.23b)

where the strain energy density, W(i ) , of material point i can be expressed as a summation of micro potentials, w (i )( j ) , arising from the interaction of material point i and the others within its horizon in the form

37

W( i ) =

1 ∞ 1 ∑ w (i )( j ) ( u(1) , u(2) , 2 j =1 2

(

, u ( ∞ ) ) + w ( j )( i ) ( u (1) , u (2) ,

)

, u ( ∞ ) ) V( j ) .

(2.24)

By using Eq. (2.19), the Lagrange’s equation for material point k can be written as d ∂L ∂L − = 0. dt ∂u ( k ) ∂u ( k )

(2.25)

Invoking Eqs. (2.20) and (2.23), the Lagrange’s equation of the material point k can be obtained as ∞  ∞ 1 ∂w ( k )( j )  1 ∂w ( j )( k ) ρ ( k )u ( k ) V( k ) +  ∑ V( j ) V( k ) + ∑ V( k ) V( j ) − b ( k ) V( k )  = 0 ,  j =1 2 ∂u ( k )  j =1 2 ∂u ( k )  

(2.26)

in which it is assumed that the interactions not involving material point k do not have any effect on material point k ; i.e.,

∂w (α )( β ) ∂u ( k )

= 0,

if α ≠ k or β ≠ k .

(2.27)

This equation can also be written in a different form as ∞

1  ∂w ( k )( j ) ∂w ( j )( k )  +  V( j ) + b ( k ) , ∂u ( k )  2  ∂u ( k )

ρ ( k )u ( k ) = ∑ −   j =1

(2.28)

where, in general, w ( k )( j ) ≠ w ( j )( k ) . By considering the conservation of energy, it is shown in Appendix B that ∂w ( k )( j ) / ∂u ( k ) and ∂w ( j )( k ) / ∂u ( k ) represent the force density that the material point j exerts on the material point k and the force density that the material point k exerts on the material point j , respectively. With this interpretation, Eq. (2.28) can be rewritten as ∞

1 ( f( j )( k ) − f( k )( j ) )V( j ) + b( k ) , j =1 2

ρ ( k )u ( k ) = ∑ where

(2.29)

38

f( k )( j ) =

∂w ( k )( j )

and

∂u ( k )

f( j )( k ) = −

∂w ( j )( k )

(2.30a,b)

∂u ( k )

This definition of force density can also be related to the internal traction vector through the rederivation of the equilibrium equations for the local theory. The details are given in Appendix C. By utilizing the state concept, the force densities, f( k )( j ) and f( j )( k ) , can be stored in force state arrays that belong to material points k and j , respectively, as     T ( x ( k ) , t ) = −f( k )( j )     

and

    T ( x ( j ) , t ) = −f( j )( k )  .    

(2.31a,b)

The force densities f( k )( j ) and f( j )( k ) stored in vector states T ( x( k ) , t ) and T ( x( j ) , t ) can be

extracted again by operating the force states on the corresponding initial relative position vectors f( k )( j ) = −T ( x( k ) , t ) x( j ) − x( k )

and

f( j )( k ) = −T ( x( j ) , t ) x ( k ) − x( j ) . (2.32a,b)

By using Eq. (2.32), Lagrange’s equation of the material point k can be rewritten as ∞

(

)

ρ ( k )u ( k ) = ∑ T ( x( k ) , t ) x( j ) − x( k ) − T ( x( j ) , t ) x( k ) − x( j ) V( j ) + b ( k ) . j =1

(2.33)

By transforming the infinite summation back to integration and considering only the material points within the horizon, ∞

∑ ( ⋅) V j =1

( j)

→ ∫ ( ⋅) dV ' → ∫ ( ⋅) dH . V

(2.34)

H

Eq. (2.33) can be written in integral equation form as

ρ ( x ) u ( x, t ) = ∫ ( T ( x, t ) x '− x − T ( x ', t ) x − x ' ) dH + b ( x, t ) . H

(2.35)

39 This equation of motion is valid for both linear and nonlinear analyses. The nonlocal nature of the PD theory prohibits the application of tractions and point forces since their volume integrations result in a zero value. Therefore, the boundary conditions are applied over the volumes as body forces, displacements, and velocities in the PD theory. This governing equation should satisfy the balance of both linear and angular momentum conditions. As proven by Silling et al. (2007) and presented in Appendix B for the sake of completeness, Eq. (2.35) automatically satisfies the balance of linear momentum without any restriction on force state vectors. In order to ensure the balance of angular momentum, the required condition on the force state vectors is in the form

∫ ( Y ( x, t )

x '− x × T ( x, t ) x '− x ) dV ' = 0 .

(2.36)

V

The balance of angular momentum is automatically satisfied if the directions of the force vectors, T(x, t ) x '− x and T(x′, t ) x − x ' , are aligned with the relative position vector of the material points in the deformed configuration. In the case of force state vectors of equal magnitudes aligned in opposite directions, the PD theory limits the number of independent material constants to one for isotropic materials with a Poisson’s ratio of 0.25, and to two for orthotropic materials. It permits permanent deformation; however, it does not allow plastic incompressibility. In the case of force state vectors of different magnitudes, the PD theory decouples the deviatoric and volumetric deformations, thus enforcing plastic incompressibility, and is no longer limited to a Poisson’s ratio of 0.25. The peridynamic parameters for an isotropic material can be derived in terms engineering material constants as described in Appendix D.

40 A general form of a force state vector that satisfies the requirement of Eq. (2.36) necessary for balance of angular momentum can be derived by applying the principle of virtual displacements to Eq. (2.35) as

ρ ( x ) u ( x, t ) ⋅ ∆u = ∫ ( T ( x, t ) x′ − x − T ( x′, t ) x − x′ ) ⋅ ∆u dH + b ( x, t ) ⋅ ∆u ,

(2.37)

H

where ∆u represents the virtual displacement vector applied to the PD material point at x . This equation can also be written in matrix notation as

ρ ( x ) uT ( x, t ) ∆u = ∫ ( T ( x, t ) x′ − x − T ( x′, t ) x − x′

)

T

∆u dH + bT ( x, t ) ∆u .

(2.38)

H

Noting that T ( x, t ) x′ − x = T ( x′, t ) x − x′ = 0 for x′ ∉ H and integrating Eq. (2.38) throughout the body result in

∫ ( ρ ( x ) u ( x, t ) − b ( x, t ) ) ∆u dV = ∫ ∫ ( T ( x, t ) T

T

V

)

T

x′ − x

∆u dV ′dV

V V

− ∫ ∫ ( T ( x′, t ) x − x′

)

T

(2.39)

∆u dV ′dV .

VV

Exchanging the parameters x and x' in the second integral on the right-hand side of Eq. (2.39) leads to

∫ ∫ ( T ( x′, t )

x − x′

)

T

VV

∆u dV ′dV = ∫ ∫ ( T ( x, t ) x′ − x

)

T

∆u′ dVdV ′ .

(2.40)

VV

This relationship permits the right-hand side of Eq. (2.39) to be rewritten as

∫ ∫ ( T ( x, t )

VV

x′ − x

)

T

∆u dV ′dV − ∫ ∫ ( T ( x′, t ) x − x′ VV

= ∫ ∫ ( T ( x, t ) x′ − x

) ( ∆u − ∆u′) dV ′dV . T

)

T

∆u dV ′dV (2.41)

VV

The difference in virtual displacements of material points at locations x and x′ can be written in state form as

∆u′ − ∆u = ∆ Y ( x,t ) x′ − x .

(2.42)

41 Therefore, Eq. (2.41) can be rewritten as

∫ ∫ ( T ( x, t )

x′ − x

) ( ∆u − ∆u′) dV ′dV T

VV

= − ∫ ∫ ( T ( x, t ) x ′ − x

( ∆ Y ( x, t ) x′ − x ) dV ′dV .

)

T

(2.43)

VV

With this equation, Eq. (2.39) can be written in the form

∫ (ρ (x )u (x, t ) − b (x, t ))∆u dV = −∫ ∆W dV , T

T

I

V

(2.44)

V

where ∆WI corresponds to the virtual work of the internal forces at location x due to its interactions with all other material points: ∆WI = ∫ ( T ( x, t ) x′ − x

) ( ∆Y ( x, t ) T

x′ − x ) dV ′ .

(2.45)

V

Considering only the material points within the horizon, Eq. (2.45) can be rewritten as ∆WI = ∫ ( T ( x, t ) x′ − x

) ( ∆ Y ( x, t ) T

x′ − x ) dH .

(2.46)

H

The corresponding internal virtual work at location x in classical continuum mechanics can be expressed as ∆Wˆ I = tr(ST ∆E) = Sij ∆Eij ,

(2.47)

where S = ST is the second Piola-Kirchhoff (Kirchhoff) stress tensor, and the GreenLagrange strain tensor, E = ET , can be related to the deformation gradient tensor, F , E=

(

)

1 T F F−I . 2

(2.48)

The virtual form of the Green-Lagrange strain tensor can be written by using Eq. (2.48) as ∆E =

(

)

1 ∆F T F + FT ∆F = F T ∆F . 2

(2.49)

42 After substituting from Eq. (2.49) into Eq. (2.47), the internal virtual work expression in classical continuum mechanics takes the form ∆Wˆ I = tr(ST FT ∆F ) = tr(P ∆F ) ,

(2.50)

where P = ST FT is the First-Piola Kirchhoff (Lagrangian) stress tensor. By using the vector state reduction to a second-order tensor given in Eq. (2.7), the deformation gradient tensor, which corresponds to the deformation state in PD theory, can be obtained as F = ( Y ∗ X ) K −1 ,

(2.51)

whose virtual form can be obtained as ∆F = ( ∆ Y ∗ X ) K −1 ,

(2.52)

in which the explicit form of the shape tensor, K , is given by Eq. (2.17). The shape tensor, K , serves as a volume averaging quantity, and it is symmetric and diagonal. Substituting from Eq. (2.52) into the internal virtual work expression of classical continuum mechanics, Eq. (2.50) in conjunction with Eq. (2.10) results in     ∆Wˆ I = tr  P  ∫ ω x′ − x ∆ Y x′ − x ⊗ X x′ − x dH  K −1  .     H 

(2.53)

Using Eqs. (2.8) and (2.42), this equation can be expressed in indicial form as   ∆Wˆ I = Pij  ∫ ω x′ − x ( ∆ui′ − ∆ui )( xk′ − xk ) dH  K kj−1 . H 

(2.54)

Because the shape tensor is symmetric, this equation can be rearranged in the form ∆Wˆ I = ∫ ω x′ − x Pij K −jk1 ( xk′ − xk )( ∆ui′ − ∆ui ) dH , H

or, in matrix form,

(2.55a)

43 T ∆Wˆ I = ∫ (ω x′ − x P K −1 ( x′ − x ) ) ( ∆u′ − ∆u ) dH .

(2.55b)

H

After invoking Eq. (2.42) into Eq. (2.55b), equating the virtual work expressions from the PD theory and classical continuum mechanics, Eqs. (2.46) and (2.55b), respectively, results in

∫ ( T ( x, t )

x′ − x

) ( ∆ Y ( x, t ) T

x ′ − x ) dH

H

≡

−1 ∫ (ω x′ − x P K ( x′ − x ) ) ( ∆ Y ( x, t ) x′ − x ) dH . T

(2.56)

H

This requirement leads to the relation between the force state vector and the deformation gradient and stress tensors of classical continuum mechanics as

T ( x,t ) x′ − x ≡ ω x′ − x P K −1 ( x′ − x )

(2.57a)

T ( x,t ) x′ − x ≡ ω x′ − x F S K −1 ( x′ − x ) .

(2.57b)

or

Although the expression for the force vector state is identical to that derived by Silling et al. (2007) for hyperelastic materials, the PD formulation based on the principle of virtual displacements proves that the force vector state is valid for any material model provided that the Piola-Kirchhoff stress can be obtained directly or by using incremental procedures. Therefore, this equation also forms the basis for implementing any material behavior in the PD theory.

44 2.5. Governing Equations of Bond-based Peridynamic Theory 2.5.1. Pair-wise Response Function As described in Appendix B, the bond-based PD theory can be obtained if the force vectors T ( x,t ) x′ − x and T ( x′,t ) x − x′ are aligned in opposite directions and equal in magnitude as T ( x, t ) x′ − x =

1 (y′ − y ) f ( u ′ − u , x′ − x ) y′ − y 2

(2.58)

and T ( x′, t ) x − x′ =

1 (y − y ′) 1 (y′ − y ) . (2.59) f ( u ′ − u , x′ − x ) = − f ( u ′ − u , x′ − x ) y − y′ y′ − y 2 2

As shown by Silling et al. (2007), their substitution into Eq. (2.35) results in the equation of motion of the material point at , x

ρ ( x ) u ( x, t ) = ∫ f ( u′ − u, x′ − x ) dH + b ( x, t ) ,

(2.60)

H

in which f ( u′ − u, x′ − x ) is referred to as the pair-wise response function by Silling and Askari (2005), and can be defined as f ( u ′ − u , x′ − x ) =

(y ′ − y ) c ( s − αT ) , y′ − y

(2.61)

where the PD bond constant c can be related to the bulk modulus of the material and α is the coefficient of thermal expansion of the material. The derivation of these relations for c and α are described in Appendix D. As apparent from Eq. (2.61), the PD bond constant relates the stretch value to force density. Although referred to as the PD bond constant, it does not solely represent a material constant but also includes the total

45 volume of the horizon for volume averaging purposes and an influence function that determines the amount of contribution of each interaction to the associated material points. The mean value of the temperatures at material points x′ and x relative to the ambient temperature is denoted by T . The stretch s is defined as

s=

y′ − y − x′ − x ξ + η − ξ . = x′ − x ξ

(2.62)

It is the ratio of the change in distance to initial distance between points x′ and x. The response functions must obey the linear and angular admissibility conditions, which are examined in detail by Silling (2000). Any function satisfying the admissibility conditions is a valid response function. Therefore, the response functions are not restricted to be linear, which makes the peridynamic equation of motion given by Eq. (2.60) valid for both linear and nonlinear analyses. The response function relates the dependence of the interaction force on reference positions and displacements of any material point pairs. This interaction force can also be viewed as a bond force between the material points. The response function contains all the constitutive information associated with the material. In order to include failure in the material response, the response function can be modified through a history-dependent scalar-valued function µ (Silling and Bobaru, 2005) as f ( u′ − u, x′ − x ) = µ (ξ, t )c ( s − α T )

(y′ − y ) , y′ − y

where µ is a history-dependent scalar-valued function; it can be written as

(2.63)

46

1 if s (t ′, ξ ) − α T < s0 for all 0 < t ′ < t otherwise 0

µ (ξ, t ) = 

(2.64)

in which s0 is the critical stretch for failure to occur (Fig. 2.5). During the solution process, the displacements of each material point, as well as the stretches between pairs of material points, are computed and monitored. When the elastic stretch between two points exceeds the critical stretch, s0 , failure occurs, and these two points cease to interact with each other.

Figure 2.5. Constitutive relation for elastic stretch between material points. The inexplicit nature of local damage at a point, arising from the introduction of failure in the constitutive model, is removed by defining the local damage as

ϕ ( x, t ) = 1 −

∫ µ (ξ , t )dH

H

∫ dH

.

(2.65)

H

Thus, local damage is the weighted ratio of the number of the broken interactions to the total number of interactions. In the case of isotropic materials, the critical stretch value,

47 s0 , can be related to the equivalent energy release rate as derived by Silling and Askari (2005) s0 =

5 G0 , 9κ δ

(2.66)

where G0 and κ represent, respectively, the energy release rate and bulk modulus of the material. The energy release rate for a mode I opening is related to the stress intensity factor, K I , from G0 =

K I2 , E

(2.67)

where E represents the elastic modulus of the material. Because the peridynamic theory is nonlocal, material points interact across the interfaces. Hence, the response function needs to be specified for the interface, in addition to the response functions for individual materials. Therefore, the peridynamic theory is capable of modeling different interface strengths of the materials. If the domain consists of two dissimilar materials (Fig. 2.6), three different interactions need to be specified. Two of these interactions occur between material points having the same material, labeled 1 and 2 in Fig. 2.6, and the material properties used by the response function are trivially chosen to be those of the material point. In the case of interactions across the interface, labeled 3 in Fig. 2.6, the numerical experimentations revealed that displacement predictions are insensitive to properties of the interface material if the numbers of interactions across the interfaces are much smaller than those for material points having the same material. Also, when the numbers of interactions across the interfaces are comparable to those between the material points having the same material,

48 the use of the smaller of the bulk moduli of the two materials gives better results. Hence, the response function for interactions across the interface utilizes the material properties of the region having the smaller bulk modulus.

Figure 2.6. Interactions among material points. However, it is necessary to experimentally obtain the critical stretch value of a particular interface between dissimilar materials. The critical stretch value can be extracted from the double cantilever beam (DCB) tests and PD simulations in an inverse manner.

2.5.2. Boundary Conditions Since peridynamics is a nonlocal theory and its equations of motion utilize integrodifferential equations as opposed to partial differential equations in the case of the classical continuum theory, the application of boundary conditions is also very different from that of the classical continuum theory. The difference can be illustrated by considering a body that is subjected to mechanical loads. If this body is fictitiously

49 divided into two domains, Ω − and Ω + as shown in Fig. 2.7a, there must be a net force, F + , that is exerted to domain Ω + by domain Ω − so that force equilibrium is satisfied.

(a)

(b)

(c)

(d)

Figure 2.7. Boundary conditions: (a) domain of interest, (b) tractions in classical continuum mechanics, (c) interaction of a point in domain Ω + with domain Ω − , (d) force densities acting on domain Ω + due to domain Ω − . According to classical continuum mechanics, force F + can be determined by integrating surface tractions over the cross-sectional area, ∂Ω , of domains Ω − and Ω + as F+ =

∫ dA T ,

∂Ω

(2.68)

50 in which T is the surface traction (Fig. 2.7b). In the case of the peridynamic theory, the material points located in domain Ω + interact with the other material points in domain Ω − (Fig. 2.7c). Hence, the force densities, L , acting on points in domain Ω + must be determined by integrating the response function over domain Ω − as L(x) = ∫ − dV f (x, x′) . Ω

(2.69)

Finally, the force F + can be computed by volume integration of these force densities (Fig. 2.7d) over domain Ω + as F + = ∫ + dV L(x) . Ω

(2.70)

Hence, the tractions or point forces cannot be applied as boundary conditions since their volume integrations result in a zero value. Therefore, the boundary conditions are applied over the volumes as body forces, displacements, and velocities. As explained in Macek and Silling (2007), the thickness of the region over which the boundary conditions are applied should be comparable to the size of the horizon.

2.5.3. Numerical Implementation In order to solve Eq. (2.60), a collocation method is adopted and the numerical treatment involves the discretization of the domain of interest into subdomains. The domain can be discretized into subdomains by employing line, square, and cubic subdomains for one-, two-, and three dimensional regions, respectively. With this discretization, the volume integration in Eq. (2.60) is approximated, leading to

51 M

(

)

ρ ( x (i ) ) u ( x ( i ) , t ) = ∑ f u ( x ( j ) , t ) − u ( x (i ) , t ) , x ( j ) − x (i ) V( j ) + b ( x (i ) , t ) , j =1

(2.71)

where x( i ) is the position vector located at the ith collocation (material) point and M is the number of subdomains within the horizon of the ith material point. The position vector x( j ) represents the location of the jth collocation point. The volume of the jth cubic subdomain is V( j ) . A more advanced version of spatial integration of peridynamic equation is explained by Kilic (2008). As explained in Silling and Askari (2005), the

( )

truncation error in Eq. (2.71) is on the order of O(∆x ) + O ∆t 2 , where ∆x and ∆t correspond to incremental distance and incremental time, respectively.

In the case of a one-dimensional region, the discretization is achieved by N subdomains, with Gaussian integration (collocation) points representing the material points as shown in Fig. 2.8. Integration points are located at the center of each cubic subdomain with a weight of unity.

Figure 2.8. Discretization and material points in a one-dimensional region.

If the solution to Eq. (2.71) at the nth time step of ∆t (i.e., t = n∆t ) is represented as u (ni ) = u (i ) (t = n∆t ) , then it can be rewritten for this time step in the form

52 N

ρ( i )u (ni ) = ∑ f(ni )( j )V( j ) + b (ni ) ,

(2.72)

j =1

where f(ni )( j ) = f(ni )( j ) ( u (n j ) − u (ni ) , x( j ) − x (i ) ) represents the pair-wise interaction force between the ith and jth material points located at x( i ) and x( j ) . The pair-wise interaction force in the absence of thermal loading can be recast in a different form as f(ni )( j ) =

ξ (ni )( j ) + η(ni )( j ) ξ

n ( i )( j )

+η

n ( i )( j )

cs(ni )( j ) ,

(2.73)

in which the relative position and displacement vectors are defined as ξ (ni )( m ) = x ( m ) - x ( i ) and η(ni )( m ) = u (nm ) - u (ni ) . Thus, the stretch between the ith and jth material points at this time step, s(ni )( j ) , becomes n ( i )( j )

s

=

ξ (ni )( j ) + η(ni )( j ) − ξ (ni )( j ) ξ (ni )( j )

.

(2.74)

As shown in Fig. 2.9, if material point i interacts with other material points within a horizon of δ = 3∆x with ∆x = x(i ) − x(i +1) , the peridynamic equation becomes

ρ u (ni ) = f(ni )(i +1) ∆V(i +1) + f(ni )(i + 2) ∆V( i + 2) + f(ni )(i +3) ∆V(i +3) + f(ni )(i −1) ∆V(i −1) + f(ni )(i − 2) ∆V(i − 2) + f(ni )(i −3) ∆V( i −3) + b (ni ) .

(2.75)

53

Figure 2.9. Interaction of material points within the horizon.

The velocity and displacement at the next time step can be obtained by employing explicit forward and backward difference techniques in two steps, respectively. The first step determines the velocity at time n+1 using known acceleration and the known velocity at time n as u (ni+)1 = u (ni ) ∆t + u (ni ) .

(2.76)

The second step determines the displacement at time n+1 using the velocity at time n+1 from Eq. (2.76) and known displacement at time n as u (ni+)1 = u (ni+)1∆t + u (ni ) .

(2.77)

Similarly, the displacement and velocity of the (i+1)th material point can be obtained as u (ni++11) = u (ni++11) ∆t + u (ni +1)

(2.78)

u (ni++11) = u (ni +1) ∆t + u (ni +1) .

(2.79)

and

54

Although this explicit time integration scheme is straightforward, it is only conditionally stable. Therefore, a stability condition is necessary to obtain convergent results. A stability condition derived by Silling and Askari (2005) can be used to determine the time step size, ∆t . as ∆t