On the Equivalence between the Additive Hypo- Elasto-Plasticity and Multiplicative Hyper-Elasto- Plasticity Models and Adaptive Propagation of Discontinuities
Yang Jiao
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2018
© 2018
Yang Jiao
All Rights Reserved ABSTRACT
On the Equivalence between the Additive Hypo-Elasto-Plasticity and Multiplicative Hyper-
Elasto-Plasticity Models and Adaptive Propagation of Discontinuities
Yang Jiao
Ductile and brittle failure of solids are closely related to their plastic and fracture behavior, respectively. The two most common energy dissipation mechanisms in solids possess distinct kinematic characteristics, i.e. large strain and discontinuous displacement, both of which pose challenges to reliable, efficient numerical simulation of material failure in engineering structures.
This dissertation addresses the reliability and efficiency issues associated with the kinematic characteristics of plasticity and fracture.
At first, studies are conducted to understand the relation between two well recognized large strain plasticity models that enjoy widespread popularity in numerical simulation of plastic behavior of solids. These two models, termed the additive hypo-elasto-plasticity and multiplicative hyper-elasto-plasticity models, respectively, are regarded as two distinct strategies for extending the classical infinitesimal deformation plasticity theory into the large strain regime. One of the most recent variants of the additive models, which features the logarithmic stress rate, is shown to give rise to nonphysical energy dissipation during elastic unloading. A simple modification to the logarithmic stress rate is accordingly made to resolve such a physical inconsistency. This results in the additive hypo-elasto-plasticity models based on the kinetic logarithmic stress rate in which energy dissipation-free elastic response is produced whenever plastic flow is absent. It is then proved that for isotropic materials the multiplicative hyper-elasto-plasticity models coincide with the additive ones if a newly discovered objective stress rate is adopted. Such an objective stress rate, termed the modified kinetic logarithmic rate, reduces to the kinetic logarithmic rate in the absence of strain-induced anisotropy which is characterized as kinematic hardening in the present dissertation.
In the second part of the dissertation, the computational complexity of finite element analysis of the onset and propagation of interface cracks in layered materials is addressed. The study is conducted in the context of laminated composites in which interface fracture (delamination) is a dominant failure mode. In order to eliminate the complexities of remeshing for constant initiation and propagation of delamination, two hierarchical approaches, the extended finite element method
(XFEM) and the s-version of the finite element method (s-method) are studied in terms of their effectiveness in representing displacement discontinuity across delaminated interfaces. With one single layer of 20-node serendipity solid elements resolving delamination-free response of the layered materials, it is proved that the delamination representations based on the s-method and the
XFEM result in the same discretization space as the conventional non-hierarchical ply-by-ply approach which employs one layer of solid elements for each ply as well as double nodes on delaminated interfaces. Delamination indicators based on the s-method representation of delamination are then proposed to detect the onset and propagation of delamination. An adaptive methodology is accordingly developed in which the s-method displacement field enrichment for delamination is adaptively added to interface areas with high likelihood of delamination.
Numerical examples show that the computational cost of the adaptive s-method is significantly lower than that incurred by the conventional ply-by-ply approach despite the fact that the two approaches produce practically identical results.
Contents
List of Figures ...... vi
List of Tables ...... x
Acknowledgements ...... xi
Chapter 1 Introduction ...... 1
1.1 Overview ...... 1
1.2 Dissertation outline ...... 4
Chapter 2 Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic
Stress Rate ...... 7
2.1 Introduction ...... 7
2.2 Survey of existing additive hypo-elasto-plasticity models ...... 12
2.2.1 Kinematics ...... 12
2.2.2 Additive decomposition of the rate of deformation tensor...... 13
2.2.3 Classical formulations of finite strain additive hypo-elasto-plasticity ...... 14
2.2.4 Considerations of choice of objective stress rates...... 16
2.2.5 Integrability of the hypoelastic equation and the logarithmic rate ...... 17
2.2.6 The logarithmic spin ...... 20
2.2.7 Unloading stress ratchetting obstacle test ...... 21
2.3 Path dependency of the logarithmic rotation tensor ...... 30
2.3.1 Integration of the hypoelastic equation based on the logarithmic rate ...... 30 i
2.3.2 Illustration of the path dependency of the logarithmic rotation tensor ...... 33
2.4 Kinetic logarithmic spin ...... 36
2.4.1 Formulation of the kinetic logarithmic spin ...... 36
2.4.2 Verification of the kinetic logarithmic stress rate formulation ...... 44
2.5 Weak invariance of the elastoplasticity model based on the kinetic logarithmic rate ...... 48
2.6 Conclusions ...... 53
Chapter 3 A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-
Elasto-Plasticity Models: the Modified Kinetic Logarithmic Spin...... 54
3.1 Introduction ...... 54
3.2 Kinematics ...... 56
3.3 Typical Structure of the multiplicative hyper-elasto-plasticity models ...... 58
3.4 Comments regarding the relation between the multiplicative hyper-elasto-plasticity and additive hypo-
elasto-plasticity models ...... 65
3.5 Bridging the multiplicative hyper-elasto-plasticity and additive hypo-elasto-plasticity models ...... 66
3.5.1 General relations ...... 66
3.5.2 Consideration of specific cases ...... 84
3.6 Numerical Illustration ...... 86
3.6.1 Elastoplastic isotropy ...... 86
3.6.2 Strain-induced anisotropy in the form of kinematic hardening ...... 92
3.7 Conclusions ...... 96
Chapter 4 Analysis of Hierarchical and Non-Hierarchical Approaches to Representing
Delamination in Laminated Composites ...... 98
4.1 Introduction ...... 98
4.2 An Overview of representation of discontinuity via the s-method and the XFEM ...... 104
ii
4.2.1 The s-method ...... 104
4.2.2 The extended finite element method ...... 106
4.3 Incorporation of cohesive behavior at the interfaces ...... 107
4.4 Comparison of the s-method, the extended finite element method and the ply-by-ply discretization .... 108
4.4.1 Relations between the s-method, extended finite element method and the ply-by-ply discretization
approach in characterizing strong discontinuity...... 108
4.4.2 Representation of weak discontinuity using the s-method ...... 125
4.4.3 Advantages of the s-method ...... 127
4.5 Numerical validation ...... 128
4.5.1 Numerical experiment configuration ...... 129
4.5.2 Determination of cr ...... 131
4.5.3 Simulation results ...... 132
4.6 Conclusions ...... 135
Chapter 5 The Adaptive S-Method for Delamination Analysis in Laminated
Composites ...... 136
5.1 Introduction ...... 136
5.2 Adaptive delamination strategy ...... 139
5.2.1 Notion of superimposed node set...... 139
5.2.2 Combining weak and strong discontinuities ...... 141
5.2.3 Adaptive delamination algorithm ...... 142
5.2.4 Delamination Indicators ...... 144
5.3 Numerical examples ...... 152
5.3.1 A laminated plate problem ...... 153
5.3.2 A laminated shell problem ...... 163
5.4 Conclusions ...... 172 iii
Chapter 6 Conclusions ...... 174
6.1 Main contributions ...... 174
6.2 Directions for future research ...... 176
References ...... 178
Appendix A Numerical Integration of Additive Hypo-Elasto-Plasticity models ...... 198
A.1 Numerical integration of the additive hypo-elasto-plasticity model for J2 elastoplasticity based on the
logarithmic stress rate ...... 198
A.2 Numerical integration of the logarithmic spin tensor ...... 202
A.3 Numerical integration of the additive hypo-elasto-plasticity model for J2 elastoplasticity based on the
kinetic logarithmic stress rate ...... 204
Appendix B Eigenprojections of a Symmetric Second-Order Tensor ...... 212
Appendix C Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-
Elasto-Plasticity Models ...... 215
C.1 Multiplicative hyper-elasto-plasticity model of Dettmer and Reese [89] described in the current
configuration ...... 215
C.2 Equivalent expression of the spin tensor Ω ...... 218
C.3 Alternative expression of the plastic flow rule by Simo [124] ...... 219
C.4 Continuity of the spin tensor Ω ...... 220
C.5 Expressions of the scalar valued functions in Equation (3.97) ...... 222
C.6 Numerical approximation of the plastic work ...... 223
C.7 Numerical integration of the additive hypo-elasto-plasticity model based on the modified kinetic
logarithmic rate ...... 223
iv
Appendix D Hierarchical Representation of Delamination in Laminated Composites ....232
D.1 20-node isoparametric serendipity laminated element ...... 232
D.2 Proof of exact reproduction of shape functions of the 20-node isoparametric serendipity elements by local
elements of the same type ...... 233
D.3 Relations between the nodal coordinates of the global, local and superimposed elements ...... 240
Appendix E Cohesive Zone Model for Inter-Ply Interfacial Behavior ...... 244
v
List of Figures
Figure 2.1. Schematics of the three stages of the deformation in the unloading stress ratchetting obstacle test ...... 23
Figure 2.2. Stress responses and plastic work of the elastic-perfectly plastic material block 27
Figure 2.3. Stress responses of the pure hypoelastic material block ...... 28
Figure 2.4. Stress responses and plastic work of the elasto-plastic block with isotropic and kinematic hardening ...... 29
Figure 2.5. Logarithmic rotation angle in the material block shown in Section 2.2.7 ...... 35
Figure 2.6. Stress response and plastic work as obtained with the kinetic logarithmic stress rate formulation for the perfectly plastic material block ...... 46
Figure 2.7. Simple shear responses with elastic-perfectly plasticity: (a) normalized normal stress; (b) normalized shear stress ...... 47
Figure 2.8. Shear stresses in a simple shear problem assuming (a) linear kinematic hardening with and without plastic spin; (b) Armstrong–Frederick kinematic hardening (without plastic spin) ...... 47
Figure 3.1. Stress response and plastic work (normalized by the yield stress y ) of the simple shear problem with small elastic strain (see material parameters in Table 3.2): (a) perfect plasticity ( K 0); (b) elastoplasticity with isotropic hardening ...... 88
Figure 3.2. Stress response and plastic work (normalized by the yield stress y ) of the simple shear problem with large elastic strain (see material parameters in Table 3.): (a) perfect plasticity ( K 0); (b) elastoplasticity with isotropic hardening ...... 89
Figure 3.3. Cyclic homogeneous deformation of a unit square material block and the trajectory of the material point on the upper left corner ...... 90
Figure 3.4. Shear stress response and plastic work of the cyclic deformation problem with large elastic strain and perfect plasticity (adopting the material parameters in Table 3. with K 0) ...... 91
Figure 3.5. Shear stress response and plastic work of the cyclic deformation problem with large elastic strain and isotropic hardening (adopting the material parameters in Table 3.) ...... 92 vi
Figure 3.6. Shear stress response (normalized by the yield stress y ) of the simple shear problem with small elastic strain (see the material parameters in Table 3.4) ...... 95
Figure 3.7. Shear stress response (normalized by the yield stress y ) of the simple shear problem with moderately large elastic strain (see the material parameters in Table 3.5) ...... 96
Figure 4.1. Non-hierarchical finite element representation of delamination in laminated composites: (a) localized delamination; (b) Ply-by-ply discretization of delamination; (c) Laminate-by-laminate discretization of delamination...... 100
Figure 4.2. Displacement compatibility in non-hierarchical representation of delamination in laminated composites ...... 101
Figure 4.3. Superposition of displacement discontinuity at the delaminated interface (hidden nodes are not shown) ...... 104
Figure 4.4. 20-node hexahedral laminate element with a delaminated interface in its parametric space ...... 106
Figure 4.5. Three methods for representing strong discontinuity across delaminated interfaces: (a) problem description; (b) ply-by-ply discretization; (c) XFEM; (d) s- method...... 114
Figure 4.6. The s-method shape/basis functions along a vertical element edge passing through the th global node 1,2,3,4 ...... 120
Figure 4.7. The s-method shape/basis functions associated with nodes on a vertical straight line passing through the th global node 9,10,11,12 ...... 124
Figure 4.8. Representation of weak discontinuity through the s-method across an interface within the 20-node solid laminate element in its parametric space ...... 126
Figure 4.9. Configuration of the numerical experiment of the beam problem ...... 130
Figure 4.10. Final deformation for cr 0.001: (a) reference simulation based on ply-by-ply discretization; (b) the underlying mesh in the s-method simulation...... 130
Figure 4.11. Relations between reaction force and displacement at the upper loading edge for
different values of cr ...... 131
vii
Figure 4.12. Comparison of resultant reaction forces in the s-method and reference simulations: (a) reaction force at the upper loading edge; (b) reaction force at the lower loading edge...... 132
Figure 4.13. Damage state of Interface 1 in the ply-by-ply discretization and s-method simulations for various prescribed displacement d ...... 133
Figure 4.14. Damage state at Interface 5 in the ply-by-ply discretization and s-method simulations for various prescribed displacement d ...... 134
Figure 5.1. Two types of superimposed node sets representing strong discontinuity: (a) Edge superimposed node set; (b) Face superimposed node set...... 140
Figure 5.2. Combining weak and strong interply discontinuities for delamination modeling ...... 142
Figure 5.3. Flow chart of the adaptive delamination simulation ...... 144
Figure 5.4. Representation of a potential separation at a bonded interface in the surrogate edge problems for the (a) edge superimposed node set ijP, b and (b) face face superimposed node set klP, b ...... 145
Figure 5.5. Two special global element patches corresponding to the edge superimposed node edge sets m, n Pb : (a) a natural boundary patch; (b) a delamination-front patch ...... 147
Figure 5.6. Geometry, layup and loading of the plate problem ...... 154
Figure 5.7. Final deformed meshes: (a) reference simulation; (b) the global mesh in the adaptive simulation ...... 155
Figure 5.8. Comparison of the reaction force-displacement curves as obtained by the reference and adaptive simulations: (a) reaction force at the upper edge; (b) reaction force at the lower edge...... 156
Figure 5.9. Damage state in Interface 1 for different prescribed displacements in the reference and adaptive simulations: (a) d mm ; (b) d mm ; (c) d mm ; (d) d mm ...... 158
Figure 5.10. Damage evolution in Interface 9 for different prescribed displacements in the reference and adaptive simulations: (a) d mm ; (b) d mm ; (c) d mm ; (d) d mm ...... 161
viii
Figure 5.11. The final damage state in the reference and adaptive simulations: (a) Interface 2; (b) Interface 8...... 161
Figure 5.12. Numbers of degrees-of-freedom versus prescribed displacements in the reference and adaptive simulations ...... 162
Figure 5.13. Wall-clock time-displacement curves of the reference and adaptive simulations ...... 162
Figure 5.14. The final breakdown of the Wall time in the reference and adaptive simulations ...... 163
Figure 5.15. Geometry, layup and loading of the laminated shell problem ...... 164
Figure 5.16. One half of the deformed meshes at the end of the simulations: (a) reference simulation; (b) the global mesh in the adaptive simulation...... 165
Figure 5.17. Reaction forces as functions of the prescribed displacement: (a) vertical reaction force at the loading point; (b) horizontal reaction force at the right supporting edge...... 166
Figure 5.18. Damage state in Interface 1 as obtained by the reference and adaptive simulations for various prescribed displacements d : (a) d mm ; (b) d mm ; (c) d mm ; (d) d mm ...... 168
Figure 5.19. Damage state in Interface 2 as obtained by the reference and adaptive simulations for various prescribed displacements d : (a) d mm ; (b) d mm ; (c) d mm ; (d) d mm ...... 170
Figure 5.20. Contour plots of displacement separation in Interface 2 as obtained by the reference and adaptive simulations for two different prescribed displacements: (a) d mm ; (b) d mm ...... 170
Figure 5.21. Numbers of degrees-of-freedom in the reference and adaptive simulations versus prescribed displacement ...... 171
Figure 5.22. Time consumed by the reference and adaptive simulations as a function of the prescribed displacement ...... 171
Figure 5.23. The final wall time breakdown in the reference and adaptive simulations ...... 172
Figure D.1. A 20-node isoparametric serendipity laminate element in two different spaces: (a) the parametric space; (b) the physical space...... 233
ix
List of Tables
Table 2.1. Material parameters ...... 24
Table 2.2. Material Parameters for the Simple Shear problem ...... 47
Table 3.1. Comparison between the multiplicative hyper-elasto-plasticity and additive hypo- elasto-plasticity models a ...... 67
Table 3.2. Material parameters for small elastic strain ...... 87
Table 3.3. Material parameters for large elastic strain ...... 87
Table 3.4. Material parameters for small elastic strain [62] ...... 95
Table 3.5. Material parameters for moderately large elastic strain [89] ...... 95
Table 4.1. Elastic single ply properties ...... 128
Table 4.2. Cohesive law parameters ...... 129
Table 5.1. Elastic single ply properties ...... 153
Table 5.2. Cohesive law parameters ...... 153
x
Acknowledgements
First and foremost, I would like to express my eternal gratitude to my advisor Professor Jacob Fish for his continuous support, insightful guidance, kind patience and inspirational encouragement throughout my graduate study at Columbia University. As a tremendous mentor, Professor Fish has provided me with countless valuable insights into my research as well as precious freedom to explore the wonders of computational mechanics. It is definitely my fortune to have the opportunity to work with Professor Fish as his graduate student.
I am grateful to Professor Gerard A. Ateshian, Professor Haim Waisman, Professor Huiming Yin and Professor Steve WaiChing Sun for agreeing to serve on my dissertation committee. Their time and efforts in reading the dissertation and valuable comments are greatly appreciated. I would like to thank Dr. Stephen A. Langer and Dr. Andrew C.E. Reid from National Institute of Standards and Technology for their support of our collaborative work.
I owe a debt of gratitude to all my friends and colleagues at Columbia University and elsewhere:
Zheng Yuan, Zifeng Yuan, Nan Hu, Yongxiang Wang, Yang Liu, Hao Sun, Wentao Xu,
Dongming Feng, Muqing Liu, Fan Chen, Kun Wang, Ye Qian and many others. They made my life at Columbia eventful and memorable.
Finally, I would like to express my deepest gratitude to my parents, Jianguang Jiao and Ping Zhang.
It is because of their love and support that I have never felt lonely throughout this long journey.
xi
To my parents
xii
Chapter 1. Introduction
Chapter 1 Introduction
1.1 Overview
Responsible for ductile and brittle failure modes of solids, respectively, plasticity and fracture have been attracting intense interest among not only industrial practitioners but also academic researchers. In particular, significant research efforts have been undertaken to develop predictive plastic and fracture behavior simulation framework. The enormous interest in predictive simulation tools stems from the fact that the simulations are typically faster and lower cost compared to their experimental counterparts and, moreover, have in general much less limitations than analytical approaches on geometric shapes of structures, types of constitutive (material) models and setup of boundary conditions.
Despite their accessibility and generality as mentioned above, numerical approaches to studying ductile and brittle failures of solids face challenges posed by the physical complexity of plasticity and fracture. In fact, some of the challenges of realizing reliable, efficient numerical prediction of material failure in engineering structures result from the respective kinematic characteristics of plastic and fracture behavior of solids. It is well known that plasticity is one of the most common energy dissipation mechanisms featuring large strain while fracture is an energy dissipation mechanism characterized by initiation and propagation of cracks (i.e. discontinuities of the displacement field). In numerical simulation of plastic behavior of solids, the presence of large 1 Chapter 1. Introduction strain requires geometric nonlinearity to be involved in the constitutive models. Such constitutive models are often developed through extension of the corresponding classical constitutive models for infinitesimal deformation into the large strain regime. Distinct large-strain plasticity models, however, can be obtained as generalizations of the same classical infinitesimal-deformation model to the large strain regime. Understanding of the relation between these different models for the same purpose is therefore of crucial importance for ensuring reliability of the numerical approaches to studying plastic behavior of engineering structures. On the other hand, the kinematic characteristic of fracture requires accurate identification and representation of initiation and propagation of crack surfaces (across which displacement is discontinuous) in numerical simulation of fracture in solids. Since locations of the onset of cracks are usually not known a priori, excessive computational cost may be incurred to identify crack initiation throughout a structure. This may pose a serious computational efficiency problem in fracture analysis of practical structures which are usually on relatively large scales. As a result, it is of considerable significance to realize numerical methods capable of identifying and representing crack initiation and propagation in an efficient manner especially for large-scale structures.
The above two aspects of enhancing reliability or efficiency of numerical simulation of material failure are addressed in two parts of the dissertation for plastic and fracture behavior of solids, respectively.
In the first part, some well-recognized large-strain plasticity models, which are widely adopted in numerical simulation of plastic behavior of solids, are analyzed with relation between them revealed. Through proposing a novel objective stress rate (termed the kinetic logarithmic rate in the following development), improvements are made to a category of models based on additive
2 Chapter 1. Introduction decomposition of the rate of deformation tensor into elastic and plastic parts. In addition, it is revealed that another category of large-strain plasticity models, in which the deformation gradient is assumed to be multiplicatively decomposed into elastic and plastic parts, coincide with the former ones with a specific objective stress rate (termed the modified kinetic logarithmic rate in the following development) employed.
The second part addresses computational complexity of analyzing initiation and propagation of interface cracks in large structural components made of layered materials exemplified by laminated composites. To this end, attention is focused on hierarchical approaches to representing interface cracks (i.e. delamination) in finite element analysis of such structural components. A salient feature of these approaches is that finite element meshes do not need to conform to crack surfaces. This eliminates the complexities of adapting meshes to represent constantly changing geometric configurations of cracks caused by their propagation or initiation. Two hierarchical approaches (the extended finite element method and the s-version of the finite element method), are shown to be reliable in representing delamination in the sense that they are equivalent with the conventional approach which is based on remeshing. Furthermore, an adaptive methodology for finite element analysis of delamination propagation and initiation is developed based on the s- version of the finite element method. In addition to its striking accuracy, the methodology incurs significantly less computational expense than its conventional counterpart.
3 Chapter 1. Introduction
1.2 Dissertation outline
The aforementioned advances in numerical simulation of plastic behavior of solids are presented in Chapter 2 and Chapter 3 while those for fracture behavior elaborated in Chapter 4 and Chapter
5. The outline of the dissertation is summarized as follows.
In Chapter 2, the well-known classical large-strain plasticity models based on additive decomposition of the rate of deformation into elastic and plastic parts are critically examined. It is shown that even a relatively recent, self-consistency-oriented variant of such models exhibits physical inconsistency in an unloading process. A remedy is then proposed through a simple modification to the objective stress rate (referred to as the logarithmic rate) employed by the model.
The resulting objective stress rate is termed the kinetic logarithmic rate in the sense that it is dependent on stress.
Chapter 3 involves another category of large-strain plasticity models which assume multiplicative decomposition of the deformation gradient into elastic and plastic parts. Despite the fact that these models are usually considered to be distinct from their additive counterparts (which are studied in
Chapter 2), relation is revealed between these two categories of large-strain plasticity models. In fact, it is mathematically proved and subsequently demonstrated on several model problems that the multiplicative models coincide with the additive ones if a newly developed objective stress rate is employed in the latter. This objective stress rate, termed the modified kinetic logarithmic rate, reduces to the kinetic logarithmic rate in the absence of strain-induced anisotropy (which is characterized by kinematic hardening herein).
Chapter 4 studies reliability of two hierarchical approaches, the extended finite element method
(XFEM) and the s-version of the finite element method (s-method), in representing delamination
4 Chapter 1. Introduction in laminated composites. It is proved that the two approaches have the same discretization space
(i.e. approximation space) as the conventional non-hierarchical approach in which delamination must be represented through double nodes on material interfaces. The s-method and XFEM are therefore effective in modeling delamination in laminated composites in addition to the fact that the complexities of remeshing for constantly propagating delamination are eliminated due to the hierarchical nature of the two methods. Moreover, the s-method is shown to be able to bring about more straightforward transition from strong to weak discontinuity of displacement than the XFEM.
Since such transition is featured in interface crack problems, the s-method based representation of initiation and propagation of delamination may be more convenient to implement than that based on the XFEM.
Chapter 5 presents an adaptive methodology of analyzing initiation and propagation of delamination in large structural components made of laminated composites. The methodology is based on the s-version of the finite element method and a newly proposed notion of delamination indicators. While the former leads to hierarchical representation of delamination, the latter pinpoints locations with high risk of interface debonding (i.e. delamination) throughout the structural components. Once such locations have been identified, the s-method based representation of delamination is implemented therein to model the corresponding interface areas where the possible debonding behavior is described by a cohesive zone model (CZM). Numerical examples show that the adaptive methodology leads to accurate simulation of initiation and propagation of delamination in laminated composites. Furthermore, the computational cost of such simulation is significantly lower than that incurred by a conventional approach featuring ply-by- ply spatial discretization (i.e. meshing) of the structural components.
5 Chapter 1. Introduction
Finally, in Chapter 6 the conclusions of this dissertation along with some directions of future research are presented.
6 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
Chapter 2 Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
This chapter critically examines a half a century old idea of decomposing the rate of deformation into elastic and plastic parts. It is shown that even the most recent additive variant based on the logarithmic stress rate [1,2] is inconsistent with the notion of elasticity in a so-called unloading stress ratchetting obstacle test. An objective stress rate, termed the kinetic logarithmic stress rate due to its stress dependency, is accordingly proposed. It is proved that additive hypo-elasto- plasticity models based on the proposed kinetic logarithmic rate are able to produce elastic response during any unloading processes. These models can therefore pass the unloading stress ratchetting obstacle test. In addition, such models are shown to be weakly invariant under isochoric reference change for simple materials in the sense of Noll [3]. This chapter is reproduced from the paper [4] coauthored with Professor Jacob Fish whose inputs are gratefully acknowledged.
2.1 Introduction
Large strain plasticity, as observed in various materials, is one of the well-known phenomena requiring combined geometric and material nonlinear analysis of solids. A myriad of strategies have been developed to extend a well-established infinitesimal elastoplasticity theory to large
7 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate deformation regime [5]. Various approaches can be classified into two main categories. These include (i) the additive hypo-elasto-plasticity based on additive decomposition of the rate of deformation tensor, and (ii) the multiplicative hyper-elasto-plasticity. These approaches differ significantly in how the total deformation or motion is decomposed into elastic and plastic parts.
The former is a natural extension of the classical infinitesimal theory [6]. By this approach, the rate of deformation tensor, rather than the strain rate in the infinitesimal deformation theory, is additively decomposed into elastic and plastic parts. In addition, the time derivative of stress in the infinitesimal theory is replaced with an objective stress rate in order to ensure frame-invariance
(objectivity) of the constitutive model. This typically results in hypoelastic relation for representation of elastic response. These models are thus referred herein to as the additive hypo- elasto-plasticity models. The additive approaches include the classical theories employing conventional corotational objective stress rates, such as the Jaumann and Green-Naghdi rates [7–
14], and more recently developed logarithmic rate [2,15–18]. The multiplicative hyper-elasto- plasticity models are based on the assumption that the total deformation gradient is multiplicatively decomposed into the elastic and plastic deformation gradients [19–23]. Due to their adoption of hyperelastic relations in representing elastic response, these models are referred herein to as the multiplicative hyper-elasto-plasticity models. Noteworthy are other additive variants based on the additive decompositions of the Green-Lagrange strain [24–27], the Biot strain [27, 28] and the logarithmic strain [30–32] into elastic and plastic parts.
The additive hypo-elasto-plasticity models are widely if not exclusively adopted in commercial finite element codes due to simplicity of implementation and the fact that small deformation material models can be reused for large deformation problems within the widely adopted
8 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate corotational framework. Models based on the additive decomposition of the rate of deformation tensor, however, are known to suffer from a number of shortcomings. In the additive hypo-elasto- plasticity models, the elastic rate of deformation tensor is related to an objective stress rate whose definition is not unique [12,33–36]. To address the multiplicity issue, an appropriate choice of the objective stress rates is required to satisfy certain consistency requirements (other than the frame- invariance) imposed on the additive hypo-elasto-plastic split. One of the earliest and well-known consistency requirement is the Prager’s yield stationarity criterion [37,38]. This criterion brings about the preference of the corotational objective stress rate formulation [39]. Consequently, corotational objective stress rates, such as the Jaumann rate and the Green-Naghdi rate, have been widely adopted in the additive hypo-elasto-plasticity formulations. Some abnormal responses exhibited by these models, however, suggested for the need of additional consistency conditions to further narrow down the choice of objective stress rates. These abnormalities include the oscillatory stress response in the simple shear problem in the context of the most commonly used
Jaumann rate [11,40,41] and the non-physical energy dissipation during pure elastic deformation
[42,43].
As a remedy, Bruhns [2] proposed a self-consistency condition stating that an additive hypo-elasto- plasticity model should produce a dissipation-free hyperelastic response in the absence of plastic flow. The corotational logarithmic rate [1,44–46] was accordingly adopted since a hypoelastic model with constant material stiffness can be integrated to obtain a hyperelastic model provided that the logarithmic rate is employed [47,48] (see [49,50] for another remedy combining the
Jaumann rate and hyperelastic relation). As a result, the corresponding elastoplastic models are capable of representing energy dissipation-free elastic responses within the initial pure elastic
9 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate range prior to yield. In addition, the use of logarithmic stress rate rather than the classical Jaumann stress rate, eliminates the oscillatory stress response in the simple shear problem [1,2]. For this reason, many constitutive relations based on the logarithmic stress rate have been developed in the past two decades [2,15–17,51–59].
In the present chapter we first study the performance of the objective logarithmic stress rate based formulation on the elastic response in the post-failure regime. In fact, any unloading process after the initial yield should also be elastic and therefore free from energy dissipation according to the aforementioned self-consistency condition. A natural question may thus arise of whether the additive hypo-elasto-plasticity models based on the objective logarithmic stress rate are still elastic and dissipation-free throughout the unloading processes? We demonstrate that the logarithmic stress rate formulation [2,51–55,60] dissipates energy throughout the “elastic” process following unloading from the initial yield. The self-consistency condition is therefore not completely fulfilled with the logarithmic rate employed in the additive hypo-elasto-plasticity models. This has been first illustrated on a model problem in which loading, unloading and cyclic reloading are applied sequentially. The observed abnormal response, herein referred to as the unloading stress ratchetting, reveals a nonphysical energy dissipation during unloading processes after yield. The cause of the abnormal response is examined. It is shown that the spin tensor associated with the logarithmic stress rate is not integrable in the sense that the resulting rotation tensor is deformation- path-dependent. This fact is shown to be responsible for the unloading stress ratchetting.
To alleviate the unloading stress ratchetting and moreover achieve complete fulfillment of the self- consistency conditions, a new spin tensor, hereafter referred to as the stress-dependent or kinetic logarithmic spin is proposed. On one hand, the kinetic logarithmic spin, obtained through a simple
10 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate modification made to the logarithmic spin, coincides with the logarithmic spin prior to yield and therefore inherits its remarkable integrability properties in the initial elastic regime. On the other hand, the stress response in the post-yield unloading regime is no longer influenced by the deformation-path-dependent rotation tensor due to the consideration of the stress-free intermediate configuration. The response is shown to be independent of the deformation path and, moreover, free from energy dissipation.
We further show that the additive hypo-elasto-plasticity formulation based on the proposed kinetic logarithmic spin tensor possesses weak invariance of the elastoplastic constitutive relations under the isochoric change of the reference configuration. The notion of the weak invariance of a simple material in the sense of Noll [3] has been recently introduced by Shutov and Ihlemann [18] as an extension of Noll’s strong invariance [61]. It has been demonstrated in [18] that (i) except for those employing the Jaumann rate, none of the additive hypo-elasto-plasticity corotational formulations including the logarithmic stress rate based models are, unfortunately, weakly invariant under isochoric reference change, and (ii) only the framework of multiplicative hyper-elasto-plasticity can be formulated under certain conditions to be weakly invariant.
The chapter is organized as follows. In Section 2.2, a brief review of the existing additive hypo- elasto-plasticity models is presented. The nonphysical behavior of the objective logarithmic stress rate is illustrated in an unloading stress ratchetting obstacle test and the cause of it is discussed in
Section 2.3. The objective kinetic logarithmic stress rate is introduced in Section 2.4 to alleviate nonphysical behavior in the post-yield unloading regime. In Section 2.5, we show that such a modification renders the additive elastoplastic models based on the kinetic logarithmic stress rate
11 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate weakly invariant under isochoric reference change. Conclusions are drawn in Section 2.6.
Algorithmic details are given in Appendix A.
2.2 Survey of existing additive hypo-elasto-plasticity models
2.2.1 Kinematics
Consider a deformable body whose initial configuration is denoted by 0 in a three-dimensional
Euclidean space. 3 .. Each particle of the body can be identified with its initial position vector X .
Let x denote the position vector of a particle X in the current deformed configuration of the body. The motion or deformation of the body can thus be described as a mapping xxX ,t with t denoting the physical or pseudo time. The deformation gradient is then defined as
x F (2.1) X with its determinant restricted to positive values, i.e. J det0F . In light of the polar decomposition, the tensor F can be decomposed as
F=RU=VR (2.2) where R is a proper orthogonal tensor; U and V are symmetric second order tensors known as the right and left stretch tensors, respectively. The right and left Cauchy-Green deformation tensors, denoted as C and B , respectively, are defined as
CFFUBFT2T2 FV , (2.3)
12 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
The velocity v of a particle X is the material time derivative of the position vector x , i.e.
Dx v . The velocity gradient L can be expressed as Dt
v L F F 1 (2.4) x
The rate of deformation tensor D and the vorticity, or Jaumann spin, tensor W are the symmetric and skew-symmetric parts of the velocity gradient, respectively, denoted as
1 DLLLsym T (2.5) 2
1 WLLLskw T (2.6) 2
2.2.2 Additive decomposition of the rate of deformation tensor
The main hypothesis behind the finite strain additive hypo-elasto-plasticity is the additive decomposition of the rate of deformation tensor
D =DDˆˆep (2.7) where Dˆ e and Dˆ p denote the elastic and plastic parts of the rate of deformation tensor D , respectively. The stress power with respect to the initial configuration can be expressed as
p τD: (2.8) where τ is the Kirchhoff stress defined by the product of the Cauchy stress σ and the deformation gradient determinant J , i.e. τσ J . The assumption (2.7) can be therefore expressed in the form of the additive decomposition of the stress power p
p τ:: Dˆˆep τ D (2.9) ppep 13 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate where pe and p p denote the elastic and plastic parts of the stress power p , respectively. The additive split (2.9) of the stress power states the stress power can be decomposed into a conservative (recoverable) and dissipative (irrecoverable) parts [5]. For an elastoplastic deformation in the context of purely mechanical response, which is the focus of the present dissertation, the conservative part of the stress power contributes to the rate of change of the elastic energy while the dissipative part characterizes the rate of energy dissipation.
2.2.3 Classical formulations of finite strain additive hypo-elasto-plasticity
An elastic domain is defined in the space of the Kirchhoff stress τ as
τα,,0 (2.10) where α , termed the back stress, is an objective Eulerian stress-like variable that describes kinematic hardening; is a scalar variable for isotropic hardening. Both α and are regarded as internal variables. The domain in which equation (2.10) is satisfied is referred to as elastic domain in the sense that its interior points represent stress states which do not induce any plastic flow. The yield condition is thus expressed as the following boundary surface of the elastic domain:
τα,,0 (2.11)
The elastic part Dˆ e of the rate of deformation tensor is related to the Kirchhoff stress τ via the hypoelastic relation
τ = τ: Dˆ e (2.12)
14 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate where τ is a fourth-order stiffness tensor that may depend on stress τ ; τ is an objective rate of the Kirchhoff stress τ . Let τ denote the fourth-order compliance tensor, which is the inverse of the stiffness tensor τ satisfying
ττ : s (2.13)
s in (2.13) is the fourth-order symmetric identity tensor whose components in the Cartesian
s 1 coordinate system are ijklikjliljk . The hypoelastic relation (2.12) can then be written 2 in the following alternative form:
D=ˆ e ττ: (2.14)
The plastic part Dˆ p of the rate of deformation tensor is determined by the flow rule
pτα,, Dˆ p (2.15) τ where the scalar-valued function pτα,, represents the flow potential and the plastic multiplier [62] or consistency parameter [63].
The evolution equations of the back stress α and the isotropic hardening parameter can be expressed as
α A τ,, α (2.16)
K τα,, (2.17) where Aτα,, is a symmetric second-order tensor valued function and α an objective stress rate of the back stress α ; K τα,, is a scalar valued function.
15 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
The plastic multiplier is assumed to satisfy the Kuhn-Tucker complementarity conditions
,,,0,,,0 τατα (2.18) and the consistency requirement
τα, , 0 (2.19)
2.2.4 Considerations of choice of objective stress rates
In principle, there are infinite number of ways to define objective stress rates [64,65]. The elastoplastic model outlined in Section 2.2.3 is incomplete without defining an objective stress rate in (2.12) and (2.16). There are two main considerations in adopting a specific objective stress rate.
First, by the Prager’s consistency criterion [37,38], the objective stress rates in (2.12) and (2.16) should (i) be identical and (ii) corotational [39,66]. The corotational rate of an objective Eulerian second-order tensor T can be written as
T= T ΩT +TΩ (2.20) where Ω , termed the spin tensor, is a skew-symmetric second-order tensor satisfying ΩΩ T .
The most commonly used corotational objective stress rates are the Jaumann and Green-Naghdi rates. The spin tensors corresponding to the Jaumann and Green-Naghdi rates (respectively denoted by ΩJ and ΩGN herein) are defined as
ΩWJ (2.21)
ΩRRGNT (2.22)
Secondly, the physical soundness of the additive split of the stress power in (2.9) requires the energy dissipation to be absent whenever the plastic part of the rate of deformation vanishes, i.e.
16 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
Dˆ p 0. This is referred to as a self-consistency condition by Bruhns et at. [2]. It is evident that vanishing Dˆ p indicates vanishing plastic multiplier (i.e. ) as described in Section 2.2.3. This in turn reduces the elastoplastic relations to
τ = τ : D (2.23)
α 0 (2.24)
(2.25)
Consequently, the hypoelastic relation (2.23) should represent purely elastic response [2,39,48]. A proper objective stress rate is thus necessary to prevent energy dissipation resulting from (2.23).
2.2.5 Integrability of the hypoelastic equation and the logarithmic rate
The considerations above necessitate adoption of corotational stress rates that render the hypoelastic equation (2.23) capable of representing elastic response that is free from energy dissipation.
In fact, it is possible to integrate the rate form relation (2.23) to obtain an equivalent elastic form that explicitly represents elastic response prior to yield provided that certain conditions are satisfied. These conditions have been pointed out by Bernstein [67,68] in the context of the
Jaumann rate in (2.23) as the Bernstein’s integrability criterion. It is, however, far from being trivial to obtain the fourth-order tensor valued function τ that fulfills Bernstein’s integrability criterion.
Another noteworthy consideration is that the relation (2.23) is usually applied with constant stiffness tensor that is independent of the stress τ and thus renders the hypoelasticity model of grade zero. However, it has been demonstrated by Simo and Pister [43] that such a hypoelastic
17 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate model of grade zero is not able to represent an elastic response with commonly known objective stress rates. This can be readily observed from a nonphysical stress ratchetting response, i.e. accumulation of residual stresses that accompanies each cycle throughout a cyclic deformation path, which results from these hypoelastic models and exhibits obvious inconsistency with the notion of elasticity [69–73].
The above elastic integrability issue of the classical hypoelastic model has been successfully addressed by a relatively recent discovery of the logarithmic corotational rate [1,44–46,64]. The corresponding logarithmic spin tensor Ωl o g is defined so that the following relation holds:
D DVVlnlnlnln log ΩVVloglog Ω (2.26) Dt
D where lnV is the Hencky (or logarithmic) strain; and log denote the material derivative Dt and the logarithmic corotational rate, respectively.
With the logarithmic stress rate, the general hypoelastic relation (2.23) is written as:
D= ττ: log (2.27) where τ log denotes the logarithmic stress rate of the Kirchhoff stress and the relation (2.13) has been utilized.
It can be shown that (2.27) can be integrated to represent an elastic response prior to yield if the compliance tensor τ is written as [2,47]
2 τ τ c (2.28) ττ
18 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
where c τ denotes a scalar valued isotropic function of the Kirchhoff stress τ that represents the complementary energy potential of a hyperelastic material. The integrability criterion (2.28) appears to be more succinct than the general Bernstein’s integrability criterion. The salient feature of the logarithmic stress rate stems from the fact that the condition (2.28) permits the use of the hypoelastic model of grade zero featuring the most commonly used constant compliance and stiffness tensors if the Hencky’s hyperelasticity model [74] is considered. In this case the complementary energy potential can be written as
H 1 2 c ττττ :tr (2.29) 43 where and are the Lamé constants; tr denotes the trace of the corresponding tensor. In view of (2.28) and (2.13), the corresponding compliance and stiffness tensors are expressed as
1 HsII (2.30) 232
HsII2 (2.31) with I denoting the second-order identity tensor. Given (2.31), the hypoelastic relation (2.23) can be integrated to obtain the well-known Hencky’s elastic constitutive equation
τH: ln V tr ln V I 2 ln V (2.32) for the elastic regime prior to yield. It is noteworthy that the Hencky’s hyperelasticity model has demonstrated competence in representing hyperelastic responses of a wide class of materials undergoing moderately large deformation [75].
19 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
2.2.6 The logarithmic spin
Being one of the corotational rates, the logarithmic rate of an Eulerian second-order tensor T is defined by (2.20) with the spin tensor expressed as [1,2]
ΩWWBDlog , (2.33) where W B D, is a skew-symmetric second-order tensor, which is a function of the left Cauchy-
Green deformation tensor B and the rate of deformation tensor D , given by
m WBDPDP ,, cbb (2.34) 1
where m is the number of distinct eigenvalues of B ; b is the th distinct eigenvalue of B and
P the corresponding eigenprojection (see Appendix B or [76] for the definition and properties of
eigenprojections); the scalar valued function cbb , is given by
bb 2 cbb , (2.35) bbbb ln
In addition, considering the expression (B.9) of eigenprojections in Appendix B, the function
WBD , can be further expressed as
0,b1 b 2 b 3 WBDBD , 2c0 skw , b 1 b 2 b 3 (2.36) 2c skwBDBDBDB c skw22 c skw , b b b b 1 2 3 1 2 3 1
20 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
where skw denotes the skew-symmetric part of the corresponding second-order tensor; bi
i 1,2,3 denote the three eigenvalues of the left Cauchy-Green deformation tensor B ; the scalar
parameters ci i 0,1,2,3 are expressed as
121 bb c 12 (2.37) 0 bbbbbb1212121ln
i 3 1 3i 1 j 2 cbi ,1,2,3 (2.38) ij j1 1lnjj with
bbbbbb233112 (2.39) 123231312bbbbbb,,
2.2.7 Unloading stress ratchetting obstacle test
Due to its merits in constructing dissipation-free hypoelastic relations as described in Section 2.2.5, the logarithmic stress rate has been frequently applied in various constitutive models based on additive hypo-elasto-plasticity [2,15–17,51–60]. In these models, the relation (2.12) is usually employed with the logarithmic stress rate τ log of the Kirchhoff stress appearing on the left-hand side of the equation to describe the elastic response. It is evident that such a relation is capable of representing dissipation-free response within initial pure elastic range prior to yield as long as the condition (2.28) is satisfied [5]. The physical soundness of the stress power split (2.9) is therefore guaranteed at least in the initial elastic regime where the plastic flow Dˆ p is absent, indicating the total stress power . p . must be conservative. Hence a natural question arises as to whether the
21 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate hypoelastic model (2.12) remains energy dissipation-free after the initiation of plastic flow, which is often the case in the post-yield regime.
In the following model problem, herein referred to as the unloading stress ratchetting obstacle test, we will investigate the post-yield response of the hypoelastic model (2.12) in the context of the logarithmic stress rate formulation (2.33)-(2.39).
The self-consistent additive hypo-elasto-plasticity model [2] based on the J2 flow theory is employed in the model problem. The yield condition in (2.11) is thus of the Von Mises type:
τατα,, (2.40)
1 where TTT : for any second-order tensor T ; TTTI tr denotes the deviatoric 3 part of a second-order tensor T . Considering the stiffness tensor in (2.31), the hypoelastic relation
(2.12) is written as
τDIDlog tr2 ˆˆee (2.41)
The following associative flow rule is considered herein
τ,, α τ α Dˆ p (2.42) τ τα
Consider the Armstrong–Frederick model [77] for kinematic hardening of the back stress α
αDαlog cbˆ p (2.43)
22 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate where the scalars c and b are kinematic hardening parameters. When b 0 , the Armstrong–
Frederick model reduces to the classical linear kinematic model. The evolution of the state variable
for the isotropic hardening is assumed as
K (2.44) where the scalar K is an isotropic hardening modulus. The initial value of the state variable is
2 with denoting the yield stress. 3 y y
Figure 2.1. Schematics of the three stages of the deformation in the unloading stress ratchetting obstacle test
In the unloading stress ratchetting obstacle test, a 11 block of material is subjected to a homogeneous deformation history consisting of 3 stages as shown in Figure 2.1. At first, the block is stretched along the vertical direction. Once the block height reaches h 3 at time instant t 2s , the block starts to return to its initial configuration along the same deformation path as stage one.
After the block reaches its initial configuration at time instant t 4s , the third stage of deformation starts by subjecting the block to a cyclic deformation path. In this stage, the bottom edge of the
23 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate block is fixed as before while each material point on the upper edge moves along a circular path.
The motion path of a material point at the upper right corner of the block is shown by a grey dashed circle in Figure 2.1. The radius of the circular path is r 0 . 7 and the rate of change of the angle
is r a d s . Thus the period of the cyclic deformation is 2s.
The components of the deformation gradient in the Cartesian coordinate system shown in Figure
2.1 can be written as
10 F ,2t (2.45) 01 t
10 F ,4t (2.46) 05t
10.7sin4 t F ,4t (2.47) 010.7 1cos4 t
The material parameters are listed in Table 2.1 where E and denote the Young’s modulus and the Poisson’s ratio, respectively.
Table 2.1. Material parameters
E y c b K 1.9510 11 0.3 910 10 410 10 0 610 10
The stress response of the material block is computed through the numerical integration detailed in Appendix A.1.
24 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
We will also investigate the ability of the additive hypo-elasto-plasticity formulation based on the logarithmic stress rate to predict the total amount of plastic energy dissipation, or so-called plastic work W p , defined by
t Wpp t τD: ˆ ds (2.48) 0
Figure 2.2 depicts the stress response and plastic work of the elastic-perfectly plastic material block, which is modeled using vanishing hardening parameters, i.e. c b K 0 . It can be seen that stress ratchetting appears at t 4s (especially for the shear components) right after the material block starts undergoing the cyclic deformation path. The plot of the normalized plastic work
p (W Y ) in Figure 2.2, however, suggests that plastic dissipation does not occur until certain amount of time has elapsed since the beginning of the cyclic deformation at t 4s (marked by yellow background in Figure 2.2). This means that there is no plastic flow ( Dˆ p 0 ) and the material behavior is governed only by the relation (2.23) with the logarithmic stress rate adopted.
It is, nevertheless, within this period of time that the material block experiences the first two cycles of the deformation path 4s8st exhibiting stress ratchetting.
For comparison purposes, the stress responses for a purely hypoelastic material block subjected to the same deformation history (as illustrated in Figure 2.1) are shown in Figure 2.3. The material parameters, E and , for the logarithmic stress rate based hypoelastic relation are the same as those in Table 2.1. Such hypoelastic relation has been shown to be elastic and free from energy dissipation in Section 2.2.5. In fact, the stress responses exhibit perfect periodicity during cyclic loading t 4s as shown in Figure 2.3. The absence of stress ratchetting is consistent with the notion of elasticity.
25 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
In contrast, the response of elastoplastic block to cyclic loading (Figure 2.2) contradicts the notion of elasticity. For the formulation to be physically sound in the sense of (2.9), there should be no stress ratchetting which is inconsistent with the notion of elasticity in absence of plastic dissipation;
the stress ratchetting, as most obviously evidenced by the response of the shear component 12 , does take place without any plastic dissipation. The conclusion can thus be drawn that the hypoelastic relation (2.27) based on the logarithmic stress rate can no longer guarantee dissipation- free response without plastic flow once the initial yield takes place. Moreover, as observed from the evolution of plastic work in Figure 2.2, the nonphysical unloading stress ratchetting may even incur plastic flow and therefore plastic dissipation in the subsequent cycles once sufficient residual stresses have been accumulated to satisfy the yield condition.
Figure 2.4 shows the stress response and plastic work for an elastoplastic block with both isotropic and kinematic hardening (with the hardening parameters listed in Table 2.1). In comparison to elastic-perfectly plastic material block, there are more cycles without plastic flow, but with unphysical stress-ratchetting. This may be attributed to the hardening models that enlarge and shift the yield surface in the stress space after yield occurs in the initial loading stage 0s2st .
More loading cycles are therefore needed in the cyclic loading phase t 4s to accumulate sufficient residual stresses to expand and shift the yield surface. This is shown by a wider yellow area in Figure 2.4 compared to that in Figure 2.2.
26 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
0.4 0.2
y 0.0
-0.2
-0.4 -0.6 0 2 4 6 8 10 12 14 16
2.4
1.6
y
0.8
0.0
-0.8 0 2 4 6 8 10 12 14 16
1.5
y 1.0
0.5
0.0 0 2 4 6 8 10 12 14 16
0.4
y
/ p 0.2
W
0.0 0 2 4 6 8 10 12 14 16 t (s)
Figure 2.2. Stress responses and plastic work of the elastic-perfectly plastic material block
27 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
0.2 0.1
0.0
-0.1 -0.2
0 2 4 6 8 10 12 14 16
1.2
0.8
0.4
0.0 0 2 4 6 8 10 12 14 16
0.6
0.4
0.2
0.0
0 2 4 6 8 10 12 14 16 t (s)
Figure 2.3. Stress responses of the pure hypoelastic material block 0.4
y
/ p 0.2
W
0.0 0 2 4 6 8 10 12 14 16 t (s)
28 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
0.4 0.2
y 0.0
-0.2
-0.4 -0.6 0 2 4 6 8 10 12 14 16 2.8 2.4 2.0
y 1.6 1.2
0.8 0.4 0.0 -0.4 0 2 4 6 8 10 12 14 16 1.6 1.4 1.2
y 1.0 0.8
0.6 0.4 0.2 0.0 0 2 4 6 8 10 12 14 16 0.28 0.24 0.20 0.16
y
/ 0.12
p
W 0.08 0.04 0.00 0 2 4 6 8 10 12 14 16 t (s)
Figure 2.4. Stress responses and plastic work of the elasto-plastic block with isotropic and kinematic hardening
29 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
2.3 Path dependency of the logarithmic rotation tensor
In this section, the cause of the unloading stress ratchetting observed in the previous section is investigated. It is shown that such nonphysical responses are attributed to the deformation path dependency of the rotation tensor associated with the logarithmic spin.
2.3.1 Integration of the hypoelastic equation based on the logarithmic rate
Unloading process is characterized by vanishing plastic flow, i.e. Dˆ p 0 and therefore DD ˆ e .
For this reason, unloading responses are governed by the relation (2.27) along with (2.28) for a hypo-elasto-plasticity model based on the logarithmic stress rate. The unloading stress responses resulting from (2.27) can be better understood if the rate equation is written in an integrated form, which is the focus of this section.
Due to the isotropy of the scalar valued function c τ in (2.28), the following relation holds
T ccQ τQτ (2.49) for any orthogonal tensor Q which satisfies QQIT .
k Let h τ denote the derivative of c τ with respect to the Kirchhoff stress τ , i.e.
τ hk τ c (2.50) τ
According to the isotropic property of c τ described in (2.49), the second-order tensor valued function hk τ is also isotropic in the sense that
hQk τ Q T Q h k τ Q T (2.51)
30 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate for any orthogonal tensor Q . In addition, the fourth-order tensor valued function τ in (2.28) is also isotropic with respect to τ satisfying
TTT τQTQQQτQTQ:: (2.52) for any orthogonal tensor Q and symmetric second-order tensor T . As a skew-symmetric second- order tensor, the logarithmic spin Ωl o g can be expressed as
ΩQQlogloglog T (2.53) where Q l og is a proper orthogonal tensor and is referred to as the logarithmic rotation tensor.
Combining (2.53) and (2.26), the following relation holds
D D = QQVQQloglog Tloglog T ln (2.54) Dt
Similarly, the logarithmic rate τ log of the Kirchhoff stress can be expressed as
D τQQτlog QQloglog Tloglog T (2.55) Dt
Inserting (2.54) and (2.55) into (2.27) yields
DD QQVloglog Tloglog QQ= Tloglog Tloglogln: T τQQτ QQ (2.56) DtDt
Since τ satisfies (2.52), (2.56) can be rewritten as
DD QVlog Tloglog Qln: Tloglog = Tlog Q τ QQτ Q (2.57) DtDt
Due to (2.28) and (2.50), the right-hand side of (2.57) can be written as
31 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
2 ccττ DDD klogTklog : τhτQhτQ (2.58) τττ DtDtDt where τ = Q τ QlogTlog and the isotropic property (2.51) of hk τ is utilized.
Let us assume that the plastic multiplier at t t t 01, . Consequently, equation (2.57) (or
equivalently (2.27)) holds for t t t 01, . After integrating (2.57) in conjunction with (2.58) from
tt 0 to tt 1 , we obtain
log Tloglog Tkloglog Tloglog Tklog QVQQh111111000000lnln τQ= QVQQhτQ (2.59)
log where Q0 and V0 denote the logarithmic rotation tensor and the left stretch tensor at time t0 ;
l o g Q1 and V1 are the corresponding counterparts at time t1 . Denoting an incremental logarithmic
logloglog T rotation by QQQ 10, equation (2.59) can be written as
klogklog T lnlnVh1100 τQVhτQ (2.60)
Note that in the initial elastic range prior to yield, lnV=h k τ . This can be shown by simply
assuming the material is in the stress-free initial configuration at tt 0 and therefore remains in
k the initial elastic range at tt 1 because for t t01, t . In this case lnVh00τ 0 and
k (2.60) results in lnVh11 τ . However, once the yield have already occurred at a time tt 0 , we
k usually have ln0Vh00τ and the difference is not necessarily isotropic. This is the case in
the unloading stress ratchetting obstacle test at tst4 0 when the cyclic deformation starts.
Equation (2.60) relates two conservative (path-independent) second order tensors, i.e.
k logT lnVhnn τ computed with n 0,1, by orthogonal transformation Q . In Section 3.2, we
32 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate will show that the orthogonal transformation Ql o gT is path-dependent and thus equation (2.60)
can be only satisfied if and only if τ1 is path-dependent.
2.3.2 Illustration of the path dependency of the logarithmic rotation tensor
Two examples considered in this section illustrate the path dependency of the logarithmic rotation tensor Q l og as defined in (2.53).
In two-dimensional problems, the components of an orthogonal tensor Q l og in the Cartesian coordinate system can be written as
cossinloglog Qlog (2.61) loglog sincos where the rotation angle l og is a single independent variable determining the rotation tensor. Let the initial value of l og be 0 . From (2.53), we obtain the components of the logarithmic spin Ωl o g as
0 log log (2.62) Ω log 0
In the first model problem, two deformation paths between the same initial and final configurations are investigated. The first deformation path is expressed as
10 F 1 ,2 t (2.63) 01 t 2
The second deformation path is expressed as
33 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
10t ,1 t 01 F (2.64) 1 3cos1sin1tt ,12 t 2 sin13cos1tt
For Path 1, it is trivial to show that the rotation angle does not change, i.e.
log 0,2 t (2.65)
Consequently, we have
l o g 0 (2.66) which indicates no logarithmic rotation for this deformation path.
As to Path 2, the same conclusion can be readily obtained for t 1, i.e.
log 0,1 t (2.67)
For 12t , the rate of deformation and vorticity matrices can be written as
3 sin1cos1tt D (2.68) 8 cos1sin1tt
01 W (2.69) 8 10
The left Cauchy-Green deformation matrix B is expressed as
1 5 3costt 1 3sin 1 B (2.70) 2 3sintt 1 5 3cos 1
The eigenvalues of B are b1 4 and b2 1.
According to (2.33)-(2.39), the logarithmic spin matrix Ωlog is given by
34 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
log 3 01 Ω 4 (2.71) 8 ln 2 10
In light of (2.62), we have
log 3 4,2 t (2.72) 8ln 2
Combining (2.67) and (2.72), we obtain
log 3 log 40 (2.73) 8 ln 2
Thus, the Path 1 and Path 2 result in different logarithmic rotation tensors.
0.2
(rad)
log
0.0
-0.2
-0.4
-0.6
-0.8 Logarithmic Rotation Angle Rotation Logarithmic 4 6 8 10 12 14 16 t (s)
Figure 2.5. Logarithmic rotation angle in the material block shown in Section 2.2.7
In the second model problem, the logarithmic rotation angle log corresponding to the logarithmic rotation tensor Ωlog of the material block in Section 2.2.7 is investigated through the numerical integration introduced in Appendix A.2. The logarithmic rotation angle vs. time response is shown
35 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate in Figure 2.5. It can be seen that after each full deformation cycle a residual value of the rotation angle is accumulated.
The above two model problems clearly demonstrate the path dependency of the logarithmic rotation tensor Q l og , which gives rise to the path dependency and thus unloading ratchetting of the stress as observed in the previous section.
2.4 Kinetic logarithmic spin
A modification is proposed to the self-consistent additive hypo-elasto-plasticity model [2] in order to eliminate the unloading stress ratchetting. It will be shown that the modification renders the hypo-elasto-plasticity model capable of characterizing energy dissipation-free responses not only within the initial elastic range but at any instance of the deformation processes in absence of plastic flow, i.e. whenever Dˆ p 0.
2.4.1 Formulation of the kinetic logarithmic spin
A closer look at the definition (2.33) of the logarithmic spin Ωl o g , suggests that Ωl o g depends on a reference configuration since the left Cauchy-Green deformation tensor B in (2.33) necessitates
a reference definition. This reference configuration is evidently the initial configuration 0 according to the definitions of B and F in (2.3) and (2.2), respectively. Consequently, the hypo- elastic relation (2.12) and the back stress evolution (2.16), both containing the logarithmic stress
rate, depend on B and therefore the initial configuration 0 . In fact, the fluid-like characteristic of plasticity, the Schmid’s slip criterion and the hardening mechanism would all imply that it is the current stress along with its rate and the state variables that constitute the physical cause of the
36 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate plastic deformation and state change [5]. Thus, formulating finite deformation elasto-plasticity
models with respect to any fixed reference configuration (e.g. 0 ) is questionable.
Moreover, in conjunction with the deformation path dependency of the logarithmic rotation tensor
Ql o g (see Section 2.3), the reference configuration dependency of the logarithmic stress rate results in the unloading stress ratchetting illustrated in Section 2.2.7. In fact, the relation (2.60), which is obtained by integrating the hypoelastic relation (2.27), can represent elastic behavior if
k k ln0Vh00τ . In this case, (2.60) simply reduces to lnVh11 τ and the deformation path
l o g dependent Q has no effect on the relation. lnV0 and lnV1 , however, have to be defined with respect to the same reference configuration as Ωlog so that (2.60) holds (see the proof of (2.26) in
k [1]). Since this reference configuration is the stress-free initial configuration, lnVh00 τ is
l o g not necessarily zero once yield occurred at an earlier time tt 0 . The spin tensor Ω may
therefore be modified to eliminate its dependency on the initial configuration 0 .
As a remedy, the following stress-dependent so-called kinetic logarithmic spin is proposed:
ΩWWBτDklogk , (2.74) where the left Cauchy-Green deformation tensor B in the original definition (2.33) is replaced by a stress-dependent tensor Bk with the following definition:
kk2 B τVτ (2.75) and
kk V τ exp h τ (2.76)
37 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate where the symmetric second-order tensor valued function hk τ is already defined in (2.50) as
the derivative of an isotropic scalar valued function c τ with respect to τ . In (2.76), the exponential function expT of a symmetric second-order tensor T can be written as
m i expTE e i (2.77) i1
where m is the number of unique eigenvalues of T while i and Ei denote the ith eigenvalue and eigenprojection of T , respectively. Bk τ , V k τ and hk τ in (2.75) and (2.76) are referred to as the kinetic left Cauchy-Green deformation tensor, kinetic left stretch tensor and kinetic Hencky strain, respectively.
It will be shown that the additive hypo-elasto-plasticity model based on the kinetic logarithmic spin is physically consistent in the sense that there is no energy dissipation both in the initial elastic range and throughout any unloading processes.
Theorem 2.1. Given ττ 0 and FF 0 at instant tt 0 , then the hypo-elastic rate form
D= ττ: klog (2.78) is equivalent to the following hyper-elastic relation
τ hkeτV c ln (2.79) τ where
1 e k T 2 VFBF 0 (2.80)
1 k k k τ with FFF 0 , BB0τ 0 exp 2 h τ 0 , and defined in (2.28).
38 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
The above theorem is proved as follows. The relation (2.78) is assumed to hold throughout the
time interval tt01, . Let τ1 and F1 denote the values of the Kirchhoff stress τ and the
deformation gradient F at the time instant tt 1 , respectively.
A reference configuration is defined so that the corresponding position vector x satisfies
k 1 ddx V11 F X (2.81)
kkk with VV111τhτ exp .
The proof of the relation (2.26) in [1] suggests that this relation holds with respect to arbitrary reference configurations. The rate of deformation D can therefore be written as
D D =ln V ΩVVΩlog ln ln log (2.82) Dt with V and Ωl o g denoting the left stretch tensor and logarithmic spin with respect to the configuration , respectively, which can be written as
1 1 k T T 2 VFFBFF 1 1 1 (2.83)
ΩWWBDlog , (2.84)
kk2 2 where BV11 and BV .
Taking time derivative of both sides of (2.51) and utilizing (2.52), yields
ττΩ : Ωτh τ Ωkk Ωh τ (2.85)
D with ΩQQT for arbitrary orthogonal tensor Q . Note that the relation τ : τ hk τ is Dt used in obtaining (2.85). 39 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
As a result, according to (2.82) and (2.85), (2.78) can be written as
DD lnlnlnV ΩVVΩ=hτΩhτ+loglogkklogkkklog hτΩ (2.86) DtDt
At instant t1 , (2.86) can be written as
DDloglogkklogkkklog lnlnlnV ΩVVΩ=hτΩhτ+11111111 hτΩ (2.87) DtDt t tt11 t
l o g k l o g l o g k l o g where Ω1 , V1 and Ω1 denote the values of Ω , V and Ω at instant t1 , respectively.
Evaluation of (2.83) and (2.84) at instant t1 yields
kk VVh111exp τ (2.88)
log klog ΩΩ11 (2.89)
The relation (2.87) is therefore reduced to
DD lnV=h k τ (2.90) DtDt t tt11 t
In light of (2.76) and (2.75), (2.90) can be written as
DD lnB = ln Bk τ (2.91) Dt Dt t t11 t t
According to Equation (29) in [78] (see also [1], [76] or Equation (3.53) in Lemma 3.2 ), (2.91) can be further written as
m D k g b,0 b PB B τP (2.92) ,1 Dt tt 1
40 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
k where m is the number of unique eigenvalues of B1 while b and P are the th eigenvalue and eigenprojection, respectively (see Appendix B or [76] for the definition and properties of
eigenprojections); The function g b b, is written as
lnlnbb , bb gbb , (2.93) 1 , b
Pre-multiplying and post-multiplying both sides of (2.92) by P and P , 1 ,.. ., m yield
D k (2.94) PBBτP 0,,1,..., m Dt tt 1
Note that in obtaining the above equation it is kept in mind that gbb ,0 according to (2.93) and
0, PP (2.95) P ,
Computing the summation in (2.94) for all the values of and yields
mmmDD kk PB B τPPB B τP , 111DtDt t tt t 11 (2.96)
D k BBτ 0 Dt tt 1
m where the property PI of eigenprojections is utilized. Based on (2.83), (2.96) can be further 1 written as
41 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
D 1kTT1kTT (2.97) FFBFFFFB111 τFF 0 Dt tt 1
Consequently, the following expression is obtained
D FB1kT τF 0 (2.98) Dt tt 1
Since t1 is arbitrary, (2.98) holds at any instant whenever (2.78) holds. The following expression
can therefore be obtained by integrating (2.98) from t0 to the current instant t :
1kT1kT FBttτFFBτF 000 (2.99)
Rearranging of (2.99) yields the hyper-elastic relation (2.79), which completes the proof of
Theorem 2.1.
Remark 2.1. In the additive hypo-elasto-plasticity model based on the kinetic logarithmic stress rate, (2.78) and (2.79) hold whenever plastic flow is absent regardless of whether the yield has occurred.
Remark 2.2. The hypo-elasticity model (2.27) based on the logarithmic stress rate in [1,2,47] can
be regarded as a special case of that described in (2.78). In Theorem 2.1, let τ0 0 and FI0 .
τ Equation (2.80) thus reduces to VVe in which case (2.79) reduces to hk τV c ln . τ
Substituting (2.80) into (2.74) yields ΩΩkloglog suggesting that the kinetic logarithmic rate reduces to its original form within the initial elastic range.
Remark 2.3. An alternative to equation (2.74) is to define the spin with respect to the axes of plastic anisotropy. Deformation induced plastic anisotropy may arise, for instance, as a result of formation of crystallographic textures and residual stresses introduced by nonhomogeneous 42 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate deformation (or so-called Bauschinger effect). To deal with the deformation induced plastic anisotropy Besseling [79] introduced an orientation of an intermediate reference state by means of an orthogonal triad defining the directions of anisotropy or so-called substructure. For instance, in polycrystalline materials, directions of anisotropy are defined with respect to an average grain orientation [80]. Similar intermediate configurations in a slightly different framework, were proposed by Mandel [81], Dafalias [82], Kratochvil [83], Loret [84], Paulun and Pecherski [85],
Van Der Giessen [86] and others. To distinguish between the spin of the material (as defined by
W in the above references), and the spin of the triad defining the directions of anisotropy, or substructure, the concept of plastic spin tensor Wˆ p was introduced by Dafalias [82]. By this approach, the plastic spin is defined as the spin of the material relative to the direction of an anisotropy or substructure and thus W in (2.74) is replaced by WWWˆˆep , where assuming a simple flow rule (2.42), the plastic spin Wˆ p is given [82,84,85,87] by Wˆˆpp skw αD. In this case the kinetic logarithmic spin is modified as
ΩWWWBτDklogpk ˆ , (2.100)
Micromechanical aspects of the internal state variable α for a single crystal and polycrystal deformations among others were given in [80,86]. Dafalias [82] and Loret [84] considered to be constant. For other variants of see Paulun and Pecherski [85] and Aifantis [88].
Remark 2.4. In case of rigid plasticity with either isotropic or kinematic hardening,
WBD k ,0 , which coincides with a substructure spin considered in several previous studies
[82,85–87]. In absence of plastic spin, such a formulation with linear kinematic hardening (rather than Armstrong–Frederick kinematic hardening considered herein) gives rise to the classical stress 43 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate oscillation under (monotonic) simple shear. It is also instructive to point out that a multiplicative decomposition of the deformation gradient into elastic and plastic deformation gradients with linear kinematic hardening has been shown in [89] to exhibit similar stress oscillations.
Remark 2.5. The numerical integration algorithm based on the kinetic logarithmic spin formulation is given in Appendix A.3.
2.4.2 Verification of the kinetic logarithmic stress rate formulation
We first study the kinetic logarithmic rate formulation in the unloading stress ratchetting obstacle test outlined in Section 2.2.7. The stress response and the plastic work are shown in Figure 2.6 for a perfectly plastic material. It can be seen that all three stress components exhibit periodic response during the cyclic deformation stage t 4s . There is no unloading stress ratchetting and no spurious plastic flow is induced.
We now proceed to the classical simple shear problem for which the deformation gradient matrix
F can be written as
10 F 010 (2.101) 001
The material parameters are taken from Dettmer and Reese [89] and are summarized in Table 2.2.
The stress responses normalized by y are shown in Figure 2.7 and Figure 2.8 for elastic-perfectly plasticity ( c 0 and b 0 ) and elastoplasticity with kinematic hardening ( c 1 108 ), respectively. It can be seen in Figure 2.7 that the kinetic logarithmic rate formulation yields perfect plastic response for both the normal and shear stresses while the conventional logarithmic stress rate shows softening for the normal stress component. Figure 2.8a depicts shear stress responses
44 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate assuming linear kinematic hardening ( c 1 1 0 8 and b 0 ). Both the kinetic logarithmic rate formulation without plastic spin ( ) and the multiplicative hyper-elasto-plasticity formulation
(Model A in [89]) give rise to an oscillatory shear stress response. It can be seen that by
ˆ p incorporating plastic spin W with y (see Remark 2.3 in Section 2.4.1) the oscillatory shear response vanishes, but subsequent stiffening is observed.
The shear stress responses based on the Armstrong–Frederick kinematic hardening model are depicted in Figure 2.8b with the hardening parameters c 1 1 0 8 and b 2 . 7 . It can be seen that the kinetic logarithmic rate formulation and the multiplicative hyper-elasto-plasticity formulation give rise to very similar results with close stable values of the shear stress.
45 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
0.4
0.2
y 0.0
-0.2
-0.4 0 2 4 6 8 10 12 14 16
2.4
1.6
y
0.8
0.0
-0.8 0 2 4 6 8 10 12 14 16
1.5
y 1.0
0.5
0.0 0 2 4 6 8 10 12 14 16 0.4
0.3
y
/ 0.2
p
W 0.1
0.0 0 2 4 6 8 10 12 14 16 t (s)
Figure 2.6. Stress response and plastic work as obtained with the kinetic logarithmic stress
rate formulation for the perfectly plastic material block
46 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
Table 2.2. Material Parameters for the Simple Shear problem
E y c b K 2.2154 108 0 . 3 8 5 3.5 107 1 108 2 . 7 0
0.7
Y
Y
/
0.15 0.6
12
0.5 0.12 klog klog log 0.4 log 0.09 0.3 0.06 0.2 0.03 0.1
0.00 Normalized Shear Stress 0.0
Normalized Normal Stress 0 2 4 6 8 10 0 2 4 6 8 10 Shear Strain / 2 Shear Strain / 2
(a) (b) Figure 2.7. Simple shear responses with elastic-perfectly plasticity: (a) normalized normal stress; (b) normalized shear stress
1.2 14 klog, =0.0
Y Y
klog, =0.3/ 12 Y 1.0
Dettmer & Reese [78] 10 e p 0.8 (Based on F=F F ) 8 klog, =0.0 0.6 Dettmer & Reese [78] 6 e p (Based on F=F F ) 4 0.4 2 0.2 0
Normalized Shear Stress Normalized Shear Stress 0.0 0 2 4 6 8 10 0 2 4 6 8 10 Shear Strain Shear Strain
(a) (b) Figure 2.8. Shear stresses in a simple shear problem assuming (a) linear kinematic hardening with and without plastic spin; (b) Armstrong–Frederick kinematic hardening (without plastic spin)
47 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
2.5 Weak invariance of the elastoplasticity model based on the kinetic logarithmic rate
Herein we first introduce the notion of strong [61] and weak [18] invariance followed by a proof of weak invariance of the proposed additive elastoplasticity model based on the kinetic logarithmic rate with respect to an isochoric reference change. As a prelude, it is instructive to point out that with the exception of the Jaumann rate, which suffers from well-documented anomalies in a simple shear test, none of the common additive hypo-elasto-plasticity corotational models is weakly invariant under an arbitrary isochoric reference transformation as has been demonstrated in [18].
Consider a semi-infinite deformation history F sst,(,] and let P be an arbitrary transformation of the reference configuration, such that
F=new1 FPss (2.102)
F new s can be interpreted as a deformation gradient tensor with respect to a new reference configuration in which the position vector X new satisfies
ddXPXnew (2.103)
Noll [61] defined the (strong) invariance as
tt σ=FFtss FF new (2.104) ss
t where F F s denotes the response functional of the material. The strong invariance (2.104) s under arbitrary isochoric transformations, i.e. det1P , of the reference implies a fluid-like material behavior. Describing the state of an elastoplastic solid, unlike fluid, requires specification
48 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
of a set of state variables herein denoted by 0 that define the state of the material at time t0 . In this case, a general constitutive relation of a simple material in the sense of Noll [3] can be written as
t σ=Fts F 0 (2.105) st 0
Following [18], the constitutive relation (2.105) is considered to be weakly invariant under an arbitrary isochoric change of the reference configuration provided that for any constant unimodular
n e w second-order tensor P (i.e. d e t 1P ) there exist a new set of initial state variables 0 so that the following relation holds [18]:
tt newnew FFFFss00 (2.106) s ts00 t with F new s defined in (2.102).
It has been shown that the self-consistent additive hypo-elasto-plasticity model based on the logarithmic rate proposed in [2] can be written in the following compact form
τTDταlog ,,, (2.107)
αlog A D,,, τ α (2.108)
ˆ D,,,τα (2.109)
Note that in [2] the set of state variables at the initial instant t0 are defined as
Z0 τα 0,, 0 0 (2.110)
with τ 0 , α0 and 0 denoting the values of τ , α and at the initial time instant t0 , respectively.
49 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
For elastic-perfectly plastic materials, both AD , ,τα , and ˆ D, ,τα , vanish with α and equal to zero and constant, respectively. In this case, equations (2.107)-(2.110) reduce to
τ Tlog D τ , (2.111)
Z00 τ (2.112)
It has been demonstrated in [18] that the model described in (2.111) and (2.112) is not weakly invariant under isochoric change of the reference configuration.
We will now show that the additive hypo-elasto-plasticity model based on the kinetic logarithmic rate outlined in Section 2.4, however, possesses such weak invariance. Following the same procedure as that in obtaining the compact form (2.107)-(2.109) in [2], it is shown that the plastic multiplier in the model based on the kinetic logarithmic rate can also be expressed as
D,,,τα (2.113)
With the expression (2.113) inserted into the flow rule (2.15), the compact form of the additive hypo-elasto-plasticity model based on the kinetic logarithmic rate is readily obtained as follows:
τklog T D,,, τ α (2.114)
αADταklog ,,, (2.115)
ˆ D,,,τα (2.116)
The definition (2.74) leads to the following expression of the kinetic logarithmic spin:
ΩΩWklogklog D τ ,, (2.117)
Considering the definition (2.20) of the corotational rate along with (2.117), the kinetic logarithmic rates of the Kirchhoff stress τ and the back stress α are given by 50 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
τklog = τ + Z W,, D τ (2.118)
α=α+klog CWDτα ,,, (2.119) where Z W D,,τ and C W D, , , τα are second-order tensor valued functions.
Substituting (2.118) and (2.119) into (2.114) and (2.115), respectively, the following compact form is obtained for the additive hypo-elasto-plastic model based on the kinetic logarithmic stress rate:
τTWDτα ,,,, (2.120)
αAWDτα ,,,, (2.121)
ˆ D,,,τα (2.122)
Z0000 τα,, (2.123) where TWD ,,,,τα and AWD ,,,, τα are second-order tensor valued functions. The relations (2.120)-(2.122) constitute a system of ordinary differential equations with the initial conditions given by (2.123).
We will now show that the constitutive relation described by (2.120)-(2.123) is weakly invariant under isochoric reference change. Let us assume that the reference configuration is subjected to an isochoric transformation described by (2.103). The deformation gradient with respect to the new reference configuration is then expressed by (2.102). The new velocity gradient Lnew can thus be written as
1 LFFFFLnew new new 1 (2.124)
51 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
Consequently, in this case the new vorticity W n e w and rate of deformation Dnew tensors are given by
WLWnewnewskw (2.125)
DLDnewnewsym (2.126) where sy m and skw denote the symmetric and skew-symmetric parts of the corresponding second-order tensor, respectively. Equations (2.125) and (2.126) suggest that the ordinary differential equations (2.120)-(2.122) remain the same after the isochoric change of the reference configuration. The same solution is obtained if the corresponding initial conditions are identical to those given in (2.123), i.e.
new ZZ00 (2.127)
Consequently, the new Kirchhoff stress τ n e w satisfies
ττnew (2.128)
In addition, the unimodular tensor P leads to the following identity
JJnewnew=F=Fdetdet (2.129)
The relations (2.128) and (2.129) suggest that the Cauchy stress is not influenced by the reference change, i.e.
σσnew (2.130) with σnew denoting the Cauchy stress obtained by the model with the isochoric change of the reference configuration.
52 Chapter 2. Additive Hypo-Elasto-Plasticity Models Based on the Kinetic Logarithmic Stress Rate
We conclude that the additive hypo-elasto-plasticity model (2.120)-(2.123) based on the kinetic logarithmic rate is weakly invariant under the arbitrary isochoric change of the reference configuration.
2.6 Conclusions
An additive hypo-elasto-plasticity model based on the logarithmic stress rate has been shown to give rise to nonphysical energy dissipation in the form of unloading stress ratchetting in the post- yield unloading regime. The reason for such a physical inconsistency stems from the fact that the spin tensor associated with the logarithmic stress rate is not integrable. This fact renders the resulting logarithmic rotation tensor to be deformation-path-dependent, and in turn, causes the unloading stress ratchetting.
To alleviate the unloading stress ratchetting a stress-dependent kinetic logarithmic spin tensor has been introduced. The kinetic logarithmic spin tensor and the corresponding objective stress rate have been shown to possess the weak invariance of constitutive equations under the isochoric reference change.
53 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
Chapter 3 A Link between Multiplicative Hyper-Elasto- Plasticity and Additive Hypo-Elasto-Plasticity Models: the Modified Kinetic Logarithmic Spin
Through two theorems presented in this chapter, we prove and subsequently demonstrate in several numerical examples involving homogeneous deformation that for isotropic materials, hyper-elasto-plasticity models based on the multiplicative decomposition of deformation gradient coincide with additive hypo-elasto-plasticity models that employ the modified kinetic logarithmic stress rate. In the absence of strain-induced anisotropy (characterized by kinematic hardening herein), this objective stress rate coincides with the kinetic logarithmic rate developed in Chapter
2. We also show that other well-known additive decomposition models, such as those based on the
Jaumann and logarithmic rates, may considerably deviate from the multiplicative model. This chapter is reproduced from the paper under review [90] coauthored with Professor Jacob Fish whose inputs are gratefully acknowledged.
3.1 Introduction
The complexity of modelling large strain plasticity lies in the fact that large deformation of these materials usually involves not only irreversible plastic deformation, but also simultaneous 54 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin reversible elastic deformation. These two processes with contradictory characteristics are closely coupled in an inextricable manner with geometric nonlinearity. Quantifying elastic and plastic portions of the total finite deformation has been a subject of intensive research in both academia and practicing world. As introduced in Chapter 2, one of the common approaches is to split the rate of deformation into elastic and plastic parts in an additive manner [12] and another widely recognized approach is to consider the deformation gradient as the multiplicative product of elastic and plastic deformation gradients [91–94].
The multiplicative decomposition of the deformation gradient finds its roots in the slip theory of crystals [95–97]. It has been recently shown that such a kinematic assumption is a natural result of homogenization of crystalline slips [98,99]. Consequently, the elastic and plastic deformation gradients resulting from such a decomposition can be considered as internal state variables and the corresponding elastoplasticity models can be formulated in a thermodynamically consistent manner. Numerous elastoplasticity models have been developed based on the multiplicative decomposition of the deformation gradient (see [19,20,23,100–112] among others).
The additive split of the rate of deformation can be regarded as a direct generalization of the additive split of the strain rate in the classical infinitesimal elastoplasticity theory. It is therefore straightforward to extend the well-established infinitesimal theory into the large deformation regime based on this kinematic assumption (see Chapter 2). Due to the simplicity and clarity of the constitutive equations, the hypo-elasto-plasticity models have been enjoying popularity in both the academic research (see [2,13,14,113–120] among others) and commercial finite element codes.
The additive hypo-elasto-plasticity models and multiplicative hyper-elasto-plasticity models, which are based on the two aforementioned kinematic assumptions, respectively, are usually
55 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin known as two distinct categories of elastoplasticity models [5,62,63,121] and several comparative studies have been conducted to investigate the difference between the two [18,22,122,123]. In these studies, several specific objective stress rates are employed in the additive hypo-elasto- plasticity models.
In the present chapter, it is shown that a typical multiplicative hyper-elasto-plasticity model for isotropic materials coincides with an additive hypo-elasto-plasticity model if a newly developed objective stress rate, termed the modified kinetic logarithmic stress rate, is employed in the hypoelastic relation. In absence of strain-induced anisotropy characterized by kinematic hardening herein, the modified kinetic logarithmic stress rate reduces to the kinetic logarithmic stress rate proposed in Chapter 2.
The chapter is organized as follows. Some additional basic kinematic definitions and relations are introduced in Section 3.2. In Section 3.3, a summary of the typical structure of multiplicative hyper-elasto-plasticity models for isotropic materials is outlined. After making some comments regarding the relation between the multiplicative hyper-elasto-plasticity and additive hypo-elasto- plasticity models in Section 3.4, the link between the additive and multiplicative models is demonstrated in Section 3.5. In Section 3.6, numerical studies are conducted for an elasto-plastic solid subjected to large homogeneous deformation. Conclusions are drawn in Section 3.7.
3.2 Kinematics
The strain measure of interest in the present chapter is the Eulerian Hencky strain with the following definition:
hV ln (3.1) 56 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
For finite strain elastoplasticity, the deformation gradient can be multiplicatively decomposed into the elastic and plastic deformation gradients F e and F p [91–94]:
F F F ep (3.2)
The kinematic definitions based on F expressed by Equations (2.2)-(2.6) in Section 2.2.1 are assumed to apply to F e and F p as well which results in the elastic and plastic counterparts of those quantities defined in (2.2)-(2.6). The polar decomposition of F e and F p thus yields
F=RU=VReeeee (3.3)
F=ppppp RU=VR (3.4) where the proper orthogonal tensors, Re and Rp , are referred to as the elastic and plastic rotation tensors, respectively; the positive definite tensors U e and V e are the elastic right and left stretch tensors, respectively, while U p and V p denote their respective plastic counterparts. The elastic and plastic Cauchy-Green deformation tensors are then defined as
CFFUBFFVee Tee 2eee Te 2 , (3.5)
CFFUBFFVp pT p p2 , p p pT p2 (3.6)
Similarly to the Eulerian Hencky strain h defined in (3.1), the elastic Eulerian Hencky strain he is written as
hVee ln (3.7)
The elastic Eulerian Hencky strain he can be expressed in terms of the eigenvalues and eigenprojections of V e , i.e.
57 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
me eee hP ln v (3.8)
e e e e where m denotes the number of distinct eigenvalues of V ; v and P represent the th distinct eigenvalue and the corresponding eigenprojection of V e , respectively (see Appendix B or [76] for the definition and properties of eigenprojections). Note that the Einstein summation convention is not employed for subscripts denoted by Greek letters such as , and throughout the present chapter.
The elastic and plastic velocity gradient, rate of deformation and vorticity tensors are defined as
LFFDLWLeee 1eeee ,sym,skw (3.9)
LFFDLWLppp 1pppp ,sym,skw (3.10)
3.3 Typical Structure of the multiplicative hyper-elasto-plasticity models
Multiplicative hyper-elasto-plasticity models are based on the assumption that the deformation gradient F is multiplicatively decomposed according to the expression (3.2) [91–94]. The stress is determined by the following isotropic hyperelastic relation [124–126]:
Be τB2 e (3.11) Be where τ is the Kirchhoff stress defined as the Cauchy stress σ multiplied by the determinant J of the deformation gradient, i.e. τσ J ; BFFe e eT is the elastic left Cauchy-Green deformation tensor. The scalar valued function Be represents the elastic potential energy per
58 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin unit reference volume. The above equation implies that Be must be an isotropic function of
B e so that such a constitutive relation is frame-invariant. Consequently, (3.11) represents an isotropic hyperelastic response only.
No plastic flow is induced if the stress τ corresponds to an interior point of the elastic domain in the Kirchhoff stress space. The boundary of the elastic domain, termed the yield surface, can be expressed as
τα, , 0 (3.12) where α , known as the back stress, is an objective Eulerian stress-like internal variable that characterizes kinematic hardening. The scalar is an internal variable describing isotropic hardening; τα,, is a scalar valued function, which must be isotropic with respect to its
Eulerian tensor-valued independent variables, τ and α , due to frame invariance.
Once the plastic deformation develops, plastic flow is characterized as
pτα,, Dp (3.13) τ where the scalar valued function pτα,, , representing the flow potential, must also be isotropic with respect to τ and α , similarly to τα,, ; is a plastic multiplier; Dp is a kinematic description of the plastic flow which can be defined as [62]
DRDRpepeT (3.14)
There are various kinematic definitions of the plastic flow Dp (the reader may refer to the works of Simo [124]; Simo and Hughes [63]; Lion [127]; Dettmer and Reese [89]; Shutov and Kreißig
59 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
[128] and Vladimirov et al. [109] among others). Some of the definitions can be proved to be equivalent in isotropic cases [129]. In particular, the plastic flow rule employed by Dettmer and
Reese [89] (see also Vladimirov et al. [109] and Brepols et al. [123]) and that developed by Simo
[124] are shown in Appendices C.1 and C.3, respectively, to be equivalent to that described by
(3.13) in conjunction with (3.14).
Evolution of the internal variables, α and , are governed by
α A τ α ,, (3.15)
K τα,, (3.16) where Aτα,, is a symmetric second-order tensor valued function whereas K τα,, is scalar valued. Frame invariance requires that both Aτα,, and K τα,, should be isotropic with respect to τ and α ; α denotes an objective stress rate of the back stress α . The hardening laws employed by Eterovic and Bathe [102]; Lubarda [130]; Neto et al. [62] and Simo and Hughes
[63], fall into the category of evolution equations described by (3.15) and (3.16). In addition, it is shown in Appendix C.1 that a hardening model considered by Dettmer and Reese [89] can also be written in the form of (3.15).
The plastic multiplier is assumed to satisfy the following Kuhn-Tucker complementarity conditions [62,63]:
0,, ,0,,τ ατ ,0 α (3.17)
Remark 3.1. The hyperelastic relation (3.11) is obtained following a thermodynamical argument as shown by Simo [124] and Lin et al.[126] in the thermodynamics framework of Coleman and
60 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
Noll [131] and Coleman and Gurtin [132]. Following the same thermodynamical considerations, several alternative, but equivalent expressions of the hyperelastic relation (3.11) have been employed. Vladimirov et al. [133], for instance, employ the description of the isotropic hyperelastic relation in the reference configuration, i.e.
C e pp1T SFF2 e (3.18) C with S denoting the second Piola-Kirchhoff stress tensor. Simo and Hughes [63]; Lubarda [134]; and McAuliffe and Waisman [22] employ the corresponding Eulerian description, namely the push-forward of (3.18) into the current configuration with the deformation gradient F leading to
C e τ2 Fe F eT (3.19) C e
Another noteworthy equivalent expression of the hyperelastic relation (3.11) is based on the elastic
1 Eulerian Hencky strain hVBeeelnln (Neto et al., 2008, Lin et al., 2006): 2
he τ (3.20) he where he is an elastic potential energy function that satisfies
hBheee exp 2 (3.21)
Lemma 3.1. According to the kinematic definitions (3.2), (3.3), (3.4), (3.9) and (3.10), the definition (3.14) of the plastic flow Dp is equivalent to the following expression:
DVLVVVpsym e 1 e e 1 e (3.22)
61 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin where V e denotes the material derivative of the elastic left stretch tensor V e . Furthermore, the expression (3.22) is equivalent to
me 1 VPLPPLPPDPee 2eTee 2eeeeepe vvv v 2 ee (3.23) , vv
e e e e where m denotes the number of distinct eigenvalues of V ; v and P represent the th distinct eigenvalue and the corresponding eigenprojection of V e , respectively.
Proof. It is first shown that (3.14) is sufficient for (3.22). Considering F Fpe1 F (see (3.2)) in the definition of Dp in (3.10), the following expression is obtained:
DFLpe 1eesym FFF 1e (3.24)
D Note that the relation FFFFe 1ee 1e has been used in obtaining the equation above. Dt
Substituting (3.24) into the definition (3.14) of Dp and considering the polar decomposition
FVReee in (3.3) yield
DVLpe 1eesym VVV 1e (3.25) with symRRe T e 0 utilized.
Following a similar argument, it can be proved that (3.14) is also necessary for (3.22).
The equation (3.22) is next shown to be sufficient for (3.23). Pre-multiplying both sides of (3.22)
e e e e by P m and post-multiplying both sides of (3.22) by P m yield
eee1 e2eTee2e eeeepe e PVPPLPPLPPDP eev v 2, v v m (3.26) vv
62 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
Note that the following properties of the eigenprojections (see the expression (B.8) of eigenprojections in Appendix B) have been utilized in obtaining (3.26):
eeeeeee PVVPP vm (3.27)
ee1e1eee 1 PVVPP e m (3.28) v
e In addition, the eigenprojections P satisfies that (see (B.6) in Appendix B)
me e PI (3.29)
As a result, V e can be written as
mmmeee eeeeeee VPVPPVP (3.30) ,
Substituting (3.26) into (3.30) yields (3.23) in Lemma 3.1.
The necessity of (3.22) for (3.23) can be readily shown by substituting the expression of V e in
(3.23) into the right-hand side of (3.22). It is therefore proved that (3.14), (3.22) and (3.23) are equivalent to each other.
Remark 3.2. The expression (3.22) suggests that the plastic flow tensor Dp can be interpreted as the rate of deformation of the intermediate configuration which is determined by the following decomposition of the deformation gradient:
FVFep (3.31)
In fact, with Lp defined as LFFp p p 1 like Lp in (3.10), (3.22) indicates that DLpp sym .
63 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
Remark 3.3. In a kinematic hardening model investigated by Dettmer and Reese [89] (see
Appendix C.1 for description of the model in the current configuration), the objective stress rate
α of the back stress α in (3.15) is defined as
ααΩααΩ (3.32) with
ΩRRRWRee Tepe T (3.33)
Following a kinematic derivation similar to that for obtaining (3.22) from (3.14), (3.33) can be written in the following equivalent form (see Appendix C.2):
ΩVLVVVskw e 1 e e 1 e (3.34)
After substituting (3.23) in Lemma 3.1 into (3.34) and utilizing the relation BVee 2 , Ω can be expressed as a function of the elastic left Cauchy-Green deformation tensor B e , velocity gradient
L and the plastic flow Dp , i.e.
Ω BLepep,,, DWWBDD (3.35) where the skew-symmetric second order tensor valued function WBDD ep, is written as
me e p e e e p e WBDDPDDP ,, c b b (3.36)
e e ee 2 where m is the number of distinct eigenvalues of B ; bv is the th distinct eigenvalue of
e e e B and P is the corresponding eigenprojection. It is noteworthy that B share the same
e ee eigenprojections with V . In (3.36), the scalar valued function c b, b is expressed as 64 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
1 bbee cbb ee, (3.37) ee 1 bb
Remark 3.4. The kinematic hardening model adopted by Neto et al. [62], employs an objective stress rate α with the same form as that in (3.32). The spin tensor Ω therein is defined as
Ω R R eeT (3.38)
Since the plastic vorticity tensor W p is assumed to be zero by Neto et al. [62], the spin tensor Ω satisfies (3.33) as well. As a result, Ω can also be expressed by (3.34) and therefore (3.35).
3.4 Comments regarding the relation between the multiplicative hyper-elasto- plasticity and additive hypo-elasto-plasticity models
As introduced in Section 2.2.4, the objective stress rates may not be uniquely specified in the classical formulation of the additive hypo-elasto-plasticity models. Some known inconsistencies of these models, however, can be eliminated by defining appropriate objective stress rates in such a framework (see Sections 2.2.5 and 2.4). A number of efforts have been made to pursue objective stress rates that can best fit into the constitutive framework without inconsistencies (e.g. the work by Prager [37]; Lee [38]; Szabó and Balla [36]; Yang et al. [135]; Bruhns et al. [2]; Xiao et al. [39];
Jiao and Fish, [4]). Due to the Prager’s yielding stationarity criterion [37], most of the efforts focus on elastoplasticity models employing corotational objective stress rate in the hypoelastic relation
(2.12), i.e.
τ τ Ω τ τ Ω (3.39) in (2.12) with Ω denoting a skew-symmetric second order tensor.
65 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
On the other hand, the multiplicative hyper-elasto-plasticity models (see Section 3.3), rooted in a thermodynamically consistent framework, appear to exhibit fewer inconsistencies concerning energy dissipation. In particular, it is evident that the self-consistency condition proposed by
Bruhns et al. [2,5] can be automatically satisfied in the models if the plastic flow Dp is regarded as a natural counterpart of the plastic part Dˆ p of the rate of deformation D in the additive models
(see Section 2.2.4 for description of the self-consistency condition).
With the two seemingly distinct elastoplasticity frameworks, a series of questions may therefore arise including: Is it possible that the multiplicative hyper-elasto-plasticity models reduce to the additive hypo-elasto-plasticity models with some yet undiscovered objective stress rate? Does the kinetic logarithmic rate, which is shown in Chapter 2 to completely fulfill the self-consistency condition, provide the missing link between the two elastoplasticity frameworks? And if not, under what conditions does such an objective stress rate coincide with the kinetic logarithmic rate? Since there are numerous constitutive models formulated based on each of the two frameworks, such questions regarding their relations will be investigated in Section 3.5.
3.5 Bridging the multiplicative hyper-elasto-plasticity and additive hypo- elasto-plasticity models
3.5.1 General relations
We start by comparing the multiplicative hyper-elasto-plasticity and additive hypo-elasto- plasticity models (for brevity hereafter referred to as the multiplicative and additive models, respectively) which will be investigated in the section.
66 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
Table 3.1. Comparison between the multiplicative hyper-elasto-plasticity and additive hypo- elasto-plasticity models a
Multiplicative Hyper-Elasto- Additive Hypo-Elasto-Plasticity c Plasticity b
Eqn. Eqn. Expression Expression No. No. Kinematic (3.2) ep ˆˆep (2.7) Assumption F F F D D D
D=ˆ e ττ: (2.14) Elastic he (3.20) τ d Response e h ττΩττΩ (3.39)
Yield (3.12) τα,,0 (2.11) Condition pτα,, (3.13) Dp Plastic Flow τ pτα,, Dˆ p (2.15) Rule τ (3.14) DRDRp e p eT
Kinematic (3.15) αAτα ,, (2.16) Hardening Isotropic (3.16) K τα,, (2.17) Hardening Kuhn-Tucker (3.17) 0,,,0,,,0τ ατ α (2.18) Conditions a. See Section 3.2 for definitions of the kinematic quantities D , he , Re , Dp in this table. b. See Section 3.3 for more details. c. See Section 2.2.3 for more details. d. See Remark 3.1 in Section 3.3 for other equivalent forms of the hyperelastic relation.
It will be shown in the remaining of Section 3.5.1 that the additive model in Table 3.1 can produce the same material response as the multiplicative model if (i)
67 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
2 τ τ c (3.40) ττ in the hypoelastic relation (2.14) and (ii)
ΩΩΩWBτDKkkp , ˆ (3.41) in the definition (3.39) of the corotational objective stress rate τ .
In equation (3.40), c τ denotes the complimentary elastic energy potential corresponding to the hyperelastic material described by (3.20) in the multiplicative model. In equation (3.41), ΩK is termed the modified kinetic logarithmic spin; Dˆ p is the plastic part of the rate of deformation tensor;
Ωk , as the abbreviation for Ωklog in the present chapter, denotes the kinetic logarithmic spin which is defined as (see also (2.74) in Chapter 2)
ΩWWBτDkk , (3.42) with
Bkkτ exp 2 h τ (3.43)
τ hk τ c (3.44) τ
The skew-symmetric second-order tensor valued function WB kpτD, ˆ in (3.41), which is defined in a manner similar to WBD , employed by Xiao [1] for the logarithmic rate (see
Section 2.2.6), is written as
68 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
mk kpkkkpk ˆˆ WB τDPDP,, cbb (3.45)
k k k where m denotes the number of distinct eigenvalues of B ; b is the th distinct eigenvalue of
k k kk B while P is the corresponding eigenprojection; the scalar valued function c b, b is defined as
2 bbkk 2 cbbkk, (3.46) kk kk bb ln bb
Remark 3.5. As discussed in Section 2.2.5, the condition (3.40) for the compliance tensor τ permits use of constant compliance tensor. In the present context, this implies it is possible for the additive model employing constant compliance tensor H (see (2.30)) to deliver material response that is identical to the multiplicative model employing the Hencky’s hyperelasticity.
Remark 3.6. It will be shown in Section 3.5.2.1 that in the absence of kinematic hardening the term WB kpτD, ˆ in (3.41) vanishes rendering the modified kinetic logarithmic spin ΩK identical to the kinetic logarithmic spin Ωk . This indicates that the additive models, which employ the kinetic logarithmic spin Ωk and the corresponding integrability condition (2.28) as suggested by Jiao and Fish [4], are indeed equivalent to the corresponding multiplicative models in the absence of kinematic hardening.
Remark 3.7. It can be seen that there is no specific definition given in Table 3.1 for the objective stress rate α of the back stress α . The conclusion drawn herein applies to a general definition of the objective stress rate α which can be written as [39]
69 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
ααφτLFαφτLFαD ,,,,,,,,, ˆ p (3.47) where φ τ L F, , , α , and φτLFαD ,,,,, ˆ p denote two symmetric second-order tensor valued function; Dˆ p denotes Dp and Dˆ p in the multiplicative and additive models, respectively. The
ˆ p p dependency of φτLFαD ,,,,, on Dˆ is considered in the present dissertation so that the general expression (3.47) is able to describe the kinematic hardening law adopted by Dettmer and
Reese [89] and Neto et al. [62] (see Remark 3.3 and Remark 3.4 in Section 3.3). In addition, the function φτLFαD ,,,,, ˆ p is assumed to satisfy φτLFαDφτLFαD ,,,,,,,,,, aaˆˆpp for an arbitrary scalar a .
For subsequent derivation, we define the following second-order tensor
DDDep (3.48)
Comparing the flow rules (3.13) and (2.15) (see Table 3.1) as well as the kinematic relations (3.48) and (2.7), Dp and De in the multiplicative models are the counterparts of Dˆ p and Dˆ e in the additive models. With this in mind, the kinematic properties of De are further investigated in a manner inspired by Xiao et al. [1] and Hill [136].
Lemma 3.2. The tensor De defined in (3.48) satisfies
e e e e e P D h P 0, m (3.49)
70 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
e e e where m is the number of distinct eigenvalues of h ; P is the eigenprojection corresponding to the th distinct eigenvalue of he (see Appendix B or Carlson and Hoger [76] for the definition and properties of eigenprojections).
Proof. According to the expression (3.22) of Dp in Lemma 3.1 and the definition (3.48), De can be written as
DDVLVVVee1ee1esym (3.50)
Note that V e and V e1 share the same eigenvectors and eigenprojections with hVee ln . As a result, the expressions (3.27) and (3.28) hold (see the property (B.8) of eigenprojections in
Appendix B).
e e Pre- and post-multiplying both sides of (3.50) by P m yield
eeeeeeeeeee 1 P D PP DPP LPPVsym, P e m (3.51) v
e e Since LDW and P m is symmetric, (3.51) reduces to
eeeeeee 1 PDPPVPe , m (3.52) v
Furthermore, the identity given by Equation (29) in [78] (see also [76] and Equation (24) in [1])
me e e e suggests that the time derivative of a function e V fv P can be expressed as
me e e e e e e g v, v P V P (3.53) where g x, y is defined as
71 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
fxfy ,xy gxy , xy (3.54) fxxy ,
me eeee e e Since hVPlnln v, substituting for h in (3.53) and considering the property (B.7) of eigenprojections in Appendix B yields
eeeeeee 1 PhPPVPe , m (3.55) v
Combining (3.55) with (3.52) yields the relation (3.49) in Lemma 3.2.
Theorem 3.1. The tensor De defined in (3.48) can be written as the following corotational rate of the elastic Hencky strain he :
Dheeee ΩhhΩ (3.56) with the skew-symmetric second-order tensor Ω expressed as
ΩWWBDWBD e,, e p (3.57) where B e is the elastic left Cauchy-Green deformation tensor; WBD e , and WBD ep, are two skew-symmetric second-order tensor valued functions; WBD e , , which is expressed in
(2.34) with the independent variables B and D , is employed in the definition of the logarithmic spin [137]; WBD ep, is defined in a way similar to WBD e , as
me e p e e e p e WBDPDP ,, c b b (3.58)
72 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
e e e e where m is the number of distinct eigenvalues of B ; b and P denote the th distinct eigenvalue and its corresponding eigenprojection of B e respectively; the scalar coefficient
ee c b, b is written as
2 bbee 2 cbbee, (3.59) ee ee bb ln bb
Proof. The relation (3.56) in Theorem 3.1 is a tensor equation for the skew-symmetric second- order tensor Ω as the unknown. Following an argument similar to that given in Equations (14)-
(23) of [1], the equality (3.49) given in Lemma 3.2 ensures existence of Ω as the solution of the tensor equation (3.56) in Theorem 3.1 and moreover Ω can be written as
mmee1 ΩPW PPDhPeeeeee (3.60) ee ln vv
e ee 2 ee e e Note that V , BV and hV ln share the same eigenprojections P 1,...,m .
The solution given by (3.60) can be readily verified by substituting it into the corresponding equation (3.56) and by utilizing Lemma 3.2 and the properties of eigenprojections. It can be seen
e e e e that in case of triply coalescent eigenvalues of V and B (i.e. m 1, PI1 and therefore
ee hI ln v1 ), the expressions (3.57) and (3.60) both reduce to ΩW and the corresponding equation (3.56) is trivially satisfied in view of Lemma 3.2. Consequently, the rest of the proof only focuses on the cases of multiple distinct eigenvalues (i.e. me 1).
Applying (3.53) to the time derivative of hVee ln yields
73 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
ln vvee eeeeeee PhPPVPee ,., m (3.61) vv
ee where the property PP0 has been considered in obtaining (3.61).
p e e In addition, pre- and post-multiplying both sides of the expression (3.25) of D by P and P
,1,..., me , respectively, gives
e 2e 2 eeeeeeeee vv P VPPD PPWee P vv vv (3.62) ee 2vv epee ee PDP ,., m vv
e e where the properties (3.27) and (3.28) of the eigenprojections P m as well as the relation LDW have been utilized. Combining (3.62) and (3.61) yields
1 vve 2e 2 Peeeeeee h PPD PP W P ee e 2e 2 ln vv vv (3.63) ee 2vv epee e 2e 2 PD P ,., m vv
After substituting (3.63) and DDDep into (3.60), the following expression of the spin tensor
Ω is obtained:
me bbee 2 ΩWPDP ee ee ee bb ln bb (3.64) me 2 bbee 2 PDPe p e ee ee bb ln bb
74 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
e e 2 e e where bv 1,. . . ,m denote the distinct eigenvalues of B , which shares the same eigenprojections with V e and he . In obtaining the term W on the right-hand side of (3.64), the following relation has been utilized:
mmmmeeee eeeeee PWPPWPPWPW (3.65)
It is evident that the solution (3.64) of the equation (3.56) is identical to the expression (3.57) in
Theorem 3.1 and the proof is therefore completed.
Remark 3.8. With the spin tensor Ω as the unknown, the tensor equation (3.56) in Theorem 3.1 has an unique solution only if he (or equivalently V e or B e ) has three distinct eigenvalues (i.e. me 3) (see [78]). In addition, repeated eigenvalues of he (i.e. me 3) lead to infinite number of solutions of the equation (3.56). In these cases, the solution (3.57) provides values of the spin tensor Ω that ensure continuity of Ω when originally distinct eigenvalues coalesce (i.e. become identical). This is shown in Appendix C.4.
Theorem 3.2. Assume that the hyperelastic relation (3.20) is invertible. The hyperelastic relation
(3.20) and the kinematic definition (3.14) of the plastic flow in the multiplicative hyper-elasto- plasticity model are equivalent to the following equations:
DDDep (3.66)
D=e ττ : K (3.67)
τ hhekc (3.68) τ
75 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin where τ is a fourth-order tensor valued function of the Kirchhoff stress τ with the following definition:
2 τ τ c (3.69) ττ
where c τ denotes the complementary energy potential of the hyperelastic material described by (3.20). τK in (3.67) denotes the modified kinetic logarithmic stress rate defined as
ττΩττΩK KK (3.70) with the modified kinetic logarithmic spin tensor ΩK expressed as
ΩWWBDWBDKkkp ,, (3.71) where the skew-symmetric second-order tensor valued functions W , and W , are given in
(2.34) and (3.58), respectively. Bk is the kinetic left Cauchy-Green deformation tensor defined in
(3.43).
Proof. Let us first show that equations (3.66)-(3.68) in Theorem 3.2 are necessary for the hyperelastic relation (3.20) and kinematic definition (3.14) of the plastic flow. Since (3.20) is
invertible, there exists a complementary energy function c satisfying that
**** c τ τ: h h (3.72) where τ * and h* are symmetric second-order tensors satisfying the invertible relation
h* τ * . The inverse of the hyperelastic relation (3.20) can therefore be written as h*
76 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
τ he c (3.73) τ
Taking the time derivative of both sides of the hyperelastic relation (3.20) yields
2e h τh : e (3.74) hhee
Since the elastic potential energy function he is assumed to be isotropic with respect to he , the following expression holds for an arbitrary skew-symmetric second-order tensor Ω* (see the derivation of (2.85)):
2e h **e*ee* τΩττ ΩhΩhhΩ ee: (3.75) hh
Let Ω* in the above equation be equal to the spin tensor Ω expressed by (3.57) in Theorem 3.1.
The following relation is thus obtained:
2e h τΩ ττ ΩhΩ hhΩ : eee (3.76) hhee
According to the kinematic definition (3.14) and its equivalent form (3.22), Lemma 3.2 and
Theorem 3.1 can therefore be applied to obtain the following relation:
Dhe e Ω h e h e Ω (3.77)
DDDep (3.78)
In addition, due to the relation (3.72) and the invertibility of the hyperelastic relation (3.20), it is evident that the following holds:
77 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
2*2* c τh : s (3.79) ττhh****
h* with the invertible relation τ * satisfied. Applying (3.79) and (3.77) to the relation (3.76) h* yields
De τ: τ Ω τ τ Ω (3.80) with the notation in (3.69) utilized. In addition, the equation (3.73) indicates that hhek and therefore that BBek . As a result, the definition (3.71) of ΩK suggests that ΩΩ K in view of
(3.57). The equation (3.80) can therefore be written as
D=e ττ : K (3.81) with τ K expressed by the expression (3.70) in Theorem 3.2.
The obtained relations (3.73), (3.78) and (3.81) suggest that (3.66)-(3.68) are necessary for the hyperelastic relation (3.20) and the kinematic definition (3.14).
The sufficiency of the equations (3.66)-(3.68) will be investigated next. The following hyperelastic relation is obtained as the inverse of (3.68):
he τ (3.82) he where the function is expressed as
**** h τhτ: c (3.83)
78 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
* c τ for symmetric second-order tensors τ * and h* satisfying the invertible relation h* . τ *
Due to isotropy of the function c τ , the equation (3.67) can be rewritten as (see (2.85))
eKK D cccτττ D ΩΩ (3.84) Dt τττ
Substituting (3.66) and (3.68) into (3.84) yields
hDDepKeeK ΩhhΩ (3.85)
Note that the definitions (3.57) and (3.71) in conjunction with the relation (3.68) indicate that
ΩΩK . The equation (3.85) is therefore rewritten as
he D D p Ω h e h e Ω (3.86)
e e Pre-multiplying both sides of (3.86) by P m and post-multiplying both sides of (3.86)
e e by P m yields
eeee peeeee e P h P P D D P lnv v P ΩP , m (3.87)
ee According to the expressions of Ω in (3.57)-(3.59), (2.34) and (2.35), PΩP
me in (3.87) can be written as
e e eee e eeepe e ePWPPDPPDP c b ,,, b c b b e PΩP m (3.88) ee PWP ,
ee0, where the property PP e of eigenprojections has been utilized. P ,
79 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
Substituting (3.88) along with (3.55) and (3.61) into (3.87) yields
1 e2e e e2eT e eee pe eevPLPPLPPDP v 2, v v e e e vv PVP (3.89) vePDDP e p e , where the relations L D W and LT D W have been utilized.
me e e Noting that PI , V can be written as 1
mmmmeeee eeeeeeeeee VPVPP VPP VP (3.90) 1111
Substituting (3.89) into (3.90) yields
me e e e p e VPDDPv 1 e (3.91) m 1 ve2eTPLPPLPPDP e v e2e e 2 v eee v p e ee 1 vv
The expression (3.91) of V e is the same as (3.23) in Lemma 3.1 which indicates that the expressions (3.22) and (3.14) of the plastic flow must also hold. It is thus proved that the equations
(3.66)-(3.68) is sufficient for the hyperelastic relation (3.20) and the kinematic definition (3.14) of the plastic flow in the hyper-elasto-plasticity model.
Based on the above results, we proceed by showing that the multiplicative hyper-elasto-plasticity and the additive hypo-elasto-plasticity models in Table 3.1 can result in the same system of ordinary differential equations with respect to the state variables τ , α and provided certain conditions are satisfied.
80 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
In light of Theorem 3.2, the hyperelastic relation (3.20) and the kinematic definition (3.14) of the plastic flow in the hyper-elasto-plasticity models can be replaced with their equivalents shown in
(3.66)-(3.68). It is clear in Table 3.1 that such replacement of constitutive equations renders the multiplicative models in the same form as their additive counterparts. Furthermore, the following evolution equations can be written with respect to the state variables τ , α and :
τ=τDDΩττΩ : pKK (3.92)
αAταφτL ,,,,,,,,,,, FαφτL FαD p (3.93)
K τα,, (3.94)
The equation (3.92) is obtained by substituting (3.66) into (3.67). The fourth-order tensor τ is determined by
ττ : s (3.95)
The equation (3.93) results from combination of (3.15) and (3.47).
The plastic flow Dp in the ordinary differential equations above depends on the state variables τ ,
α and as well as the plastic multiplier through the flow rule (3.13). The spin tensor ΩK , as expressed in (3.71), is dependent on D , W , Bk and Dp while Bhkk exp 2 is dependent on the stress τ according to the definition (3.68).
It is evident that the right-hand sides of the ordinary differential equations (3.92), (3.93) and (3.94) all depend on the plastic multiplier in addition to τ , α , , F and L . In fact, the plastic multiplier itself is also dependent on the state variables τ , α and as well as on the kinematic
81 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin quantities F and L due to the Kuhn-Tucker conditions (3.17). The value of the plastic multiplier
is then determined according to the different loading cases as indicated by the Kuhn-Tucker conditions (3.17).
When the stress τ is located within the elastic domain, i.e. τα, , 0 , the Kuhn-Tucker conditions (3.17) require the plastic multiplier to vanish (i.e. 0 ). When the stress τ is on the yield surface, i.e. τα, , 0 , the plastic multiplier can be either a positive or zero value according to the Kuhn-Tucker conditions (3.17). In fact, the conditions (3.17) imply that the time derivative of the yield function τα,, must also be non-positive at this moment (i.e.
0if0 ) which serves as an additional consistency condition. The time derivative of the yield function can be written as
τ, ατ ,, ατ ,, α, ::τα (3.96) τα
Substituting the rate form expressions (3.92), (3.93) and (3.94) into (3.96) and utilizing the plastic flow rule (3.13) yield
ABτ,,,,,,,, L F ατ L F α (3.97) where Aτ,,,, L F α and Bτ,,,, L F α are scalar valued functions given in Appendix C.5.
The function Bτ,,,, L F α is assumed to be positive. If τα, , 0 and Aτ, L , F , α , 0 , the material undergoes unloading or neutral loading and therefore the plastic multiplier is zero (i.e.
82 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
0 ). If τα, , 0 and AτLFα,,,,0 , the aforementioned consistency condition (3.97)
AτLFα,,,, is active requiring that . BτLFα,,,,
Given the loading cases above, the plastic multiplier can be expressed in the following unified manner (see also [2]):
AτLFα,,,, H τα,, (3.98) BτLFα,,,, where H and , denoting the step function and the Macaulay bracket, respectively, are defined as
0,0x Hx (3.99) 1,0x
0,0x x (3.100) xx,0
With the expression of the plastic multiplier given in (3.98), equations (3.92), (3.93) and (3.94), along with some initial conditions of τ , α and , comprise the system of ordinary differential equations with respect to the state variables τ , α and .
Following the procedure similar to that in equations (3.92)-(3.100), it is evident that the additive hypo-elasto-plasticity model shown in Table 3.1 produces the same system of ordinary differential
2 τ equations as (3.92), (3.93) and (3.94) provided that: (i) τ c in the hypoelastic ττ relation (2.14) and (ii) ΩΩK WWBτDWBτD k ,, k ˆ p in the definition (3.39) of
83 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin the corotational objective stress rate τ . It is evident that the above two conditions are identical to those introduced in equations (3.40) and (3.41) suggesting that the statement corresponding to
(3.40) and (3.41) is true.
3.5.2 Consideration of specific cases
3.5.2.1 Absence of strain-induced anisotropy
In the elastoplastic constitutive models considered in the present dissertation, strain-induced anisotropy is represented by kinematic hardening, which is characterized by the back stress α . In the absence of strain-induced anisotropy constitutive functions τα,, , pτα,, and
K τα,, can be written as τ, , pτ, and K τ, , respectively. Note that the plastic flow rule is accordingly written as
pτ, Dˆ p (3.101) τ
The frame invariance requires the function pτ, to be isotropic with respect to its only Eulerian tensor valued independent variable τ . Consequently, the flow rule (3.101) suggests that Dˆ p must be coaxial with τ . In addition, the definitions (3.43) and (3.44) suggest that the Kirchhoff stress
τ is coaxial with the kinetic left Cauchy-Green deformation tensor Bk . The plastic flow Dˆ p can thus be written as
mk ˆ p p k DP d (3.102)
84 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
p ˆ p k k where d denotes the eigenvalue of D corresponding to the th eigenprojection P of B .
After substituting (3.102) into the expression (3.41) of the modified kinetic logarithmic spin ΩK and utilizing the properties of eigenprojections, the term WB kpτD, ˆ vanishes and (3.71) reduces to
ΩΩKk (3.103)
This indicates that the modified kinetic logarithmic spin ΩK coincides with the kinetic logarithmic spin Ωk [4] in the absence of kinematic hardening.
3.5.2.2 J2 Plasticity with associative plastic flow
In J2 plasticity the yield function τα,, is written as
τ,, α τ α (3.104)
1 where TTT : for any second-order tensor T and TTTI tr denotes the 3 deviatoric part of a second-order tensor T . The initial value of α and is 0 and 2 , 3 y
respectively, with y denoting the yield stress. Since the plastic flow is associative, the flow potential pτα,, in the plastic flow rule (2.15) is equal to the yield function τα,,. The plastic flow Dˆ p is therefore written as
τ,, α τ α Dˆ p (3.105) τ τα
85 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
Since τ is coaxial with Bk τ , following an argument similar to that in Section 3.5.2.1 yields
WB k ττ,0 (3.106)
Substituting (3.105) into the expression (3.41) and considering (3.106) yields
ΩΩWBταKkk , (3.107) τα
3.6 Numerical Illustration
The stress response predicted by the multiplicative model and its additive equivalent are studied herein for J2 plasticity with associative plastic flow rule. In addition, two other well-known additive models, based on the Jaumann rate and logarithmic rate [2], respectively, are also considered herein.
3.6.1 Elastoplastic isotropy
The case without strain-induced anisotropy (i.e. kinematic hardening) is considered first. Note that the condition (3.41) for the equivalence between the multiplicative and additive models reduces to
ΩΩΩKk in this case (see Section 3.5.2.1). The additive model with the kinetic logarithmic rate is therefore equivalent to the multiplicative model. The numerical integration methods presented in Appendix A are used to integrate the additive models. The multiplicative models are integrated through the numerical method in [62].
In addition to stress response, the plastic work done by the stress is investigated. For the additive models, the plastic power is defined as
86 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
W pp τD: ˆ (3.108)
As to the multiplicative models, the plastic power can be written as (see Equation (14.62) in [62]):
W pp τD: (3.109)
The numerical approximation to the plastic power is briefly discussed in Appendix C.6.
In the present section, the Hencky’s hyperelasticity model is adopted for the elastic response. The function K τα,, in the isotropic hardening law described by (3.16) or (2.17) is considered to be a constant, i.e.
K (3.110)
Two sets of material parameters, which differ only in the yield stress y and hardening modulus
K , are considered herein so that the aforementioned elastoplastic models can be compared for both small and large elastic strains. The material parameters are shown in Table 3.2 and Table 3. for small and large elastic strains, respectively. In the two tables, E and denote the Young’s modulus and the Poisson’s ratio, respectively.
Table 3.2. Material parameters for small elastic strain
E y K 1.9510 11 0 . 3 7.510 9 6 108
Table 3.3. Material parameters for large elastic strain
E y K 1.95 1011 0.3 9 1010 7.2 109
87 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
3.6.1.1 Simple shear
The classical simple shear problem is investigated with the multiplicative and additive models.
The deformation gradient matrix F for this problem can be written as
10 F 0 1 0 (3.111) 0 0 1
0.04 0.10
0.08
y 0.03
y 0.06 0.02 0.04
0.01
0.02 0.00 0.00 1.0 0.6
0.8
y
Multiplicative
y 0.6 Multiplicative 0.4 Additive, Klog Additive, Klog Additive, Log 0.4
0.2 Additive, Log Additive, Jaumann 0.2 Additive, Jaumann 0.0 0.0 6 8 5
6
y y 4 3 4
p p 2 2
W W 1 0 0 0 2 4 6 8 10 0 2 4 6 8 10
(a) (b)
Figure 3.1. Stress response and plastic work (normalized by the yield stress y ) of the simple shear problem with small elastic strain (see material parameters in Table 3.2): (a) perfect plasticity ( K 0); (b) elastoplasticity with isotropic hardening
88 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
0.4 0.8 0.3 0.6
y 0.2 y 0.4
0.2
0.1 0.0 0.0 0.6 0.8
0.4 0.6 y Multiplicative y Additive, Klog 0.4
0.2 Additive, Log
0.2
Additive, Jaumann 0.0 0.0 5 6 Multiplicative 4 Additive, Klog
Additive, Log y 3 y 4 Additive, Jaumann 2
2 p 1 p
W 0 W 0 0 2 4 6 8 10 0 2 4 6 8 10
(a) (b)
Figure 3.2. Stress response and plastic work (normalized by the yield stress y ) of the simple shear problem with large elastic strain (see material parameters in Table 3.): (a) perfect plasticity ( K 0); (b) elastoplasticity with isotropic hardening
It can be seen from Figure 3.1 that with rather small elastic strain all models deliver nearly identical stress response and plastic work except that the additive model with the logarithmic rate produces
softening response of the normal stress component 11 . This is true for both perfect plasticity (with
K 0 in Table 3.2) and elastoplasticity with isotropic hardening. If the elastic strain is large, however, only the additive model with the kinetic logarithmic rate is in good agreement with the multiplicative model as shown in Figure 3.2. Comparing Figure 3.2b with Figure 3.2a, it appears
89 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin that the difference between the results of the multiplicative model and those of the additive models with the Jaumann and logarithmic rates becomes more pronounced if isotropic hardening is considered.
3.6.1.2 Cyclic deformation featuring unloading and reloading
In the simple shear problem, the material is subjected to continuous loading with the plastic work increasing once yield occurs (see Figure 3.1 and Figure 3.2). Herein, a problem involving unloading and reloading is considered.
x2
r
1
x 1
Figure 3.3. Cyclic homogeneous deformation of a unit square material block and the
trajectory of the material point on the upper left corner
Figure 3.3 shows a unit square material block subjected to a cyclic homogeneous deformation in which the motion of each material point on the upper edge of the block forms a circular trajectory while those on the bottom edge are fixed. The radius of the circular trajectory is denoted by r and the position of the material point moving along this path is determined by the angle as shown by Figure 3.3. The deformation gradient matrix can be expressed as
90 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
1sin0 r F 011cos0r (3.112) 001
The value of the radius r is set to 1 . 2.
0.6 0.4 0.2
y 0.0
-0.2
-0.4 -0.6 2.0 Multiplicative 1.6 Additive, Klog
Additive, Log y 1.2 Additive, Jaumann
p 0.8
W 0.4 0.0
Figure 3.4. Shear stress response and plastic work of the cyclic deformation problem with large elastic strain and perfect plasticity (adopting the material parameters in Table 3. with K 0)
The shear stress response and the plastic work normalized by y are shown in Figure 3.4 and
Figure 3.5 for perfect plasticity and elastoplasticity with isotropic hardening, respectively. It can
91 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin be seen from the two figures that the multiplicative model and the additive model with kinetic logarithmic rate yield identical results while considerable deviation from the multiplicative model can be observed for the other two additive models, in particular for the plastic work.
0.6 0.4 0.2
y 0.0 -0.2
-0.4 -0.6
Multiplicative 1.6 Additive, Klog 1.2 Additive, Log Additive, Jaumann y 0.8
p 0.4
W 0.0
Figure 3.5. Shear stress response and plastic work of the cyclic deformation problem with large elastic strain and isotropic hardening (adopting the material parameters in Table 3.)
3.6.2 Strain-induced anisotropy in the form of kinematic hardening
In addition to the isotropic elastoplasticity discussed above, strain-induced anisotropy is investigated herein in the form of kinematic hardening (see [138] for an alternative representation of strain-induced anisotropy).
92 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
Following a finite strain extension of the Armstrong-Frederick hardening law, the kinematic hardening equations (3.15) and (2.16) can be written as
α D α cbp (3.113) where c and b are material parameters. The objective stress rate α in the present additive model
(based on the modified kinetic logarithmic spin ΩK ) is adopted as that in [89] and [62]. Its expression in the present context is given in (3.32) in conjunction with (3.35)-(3.37). Note that B e therein is dependent on the stress through the inverse of the hyperelastic relation (3.20). Such stress dependency of B e is utilized in the additive model based on the modified kinetic logarithmic spin.
In addition, the two well-known additive models based on the Jaumann and logarithmic objective stress rates are also investigated. In these models, the spin tensor employed to define the corotational objective stress rates α of the back stress in the kinematic hardening law (3.113) are the same as those for the Kirchhoff stress τ in the hypoelastic relations (2.12). These additive models are integrated through the numerical integration methods described in Appendix A.
In contrast to the previous isotropic case, the modified kinetic logarithmic spin ΩK , as given in the expression (3.41) for the equivalence between the multiplicative and additive models, is in general not equal to the kinetic logarithmic spin Ωk because the arguments in Section 3.5.2.1 are not applicable herein and thus the additional term W is none-zero. A corresponding numerical integration method (see Appendix C.7) is employed to obtain approximations to the material response produced by the additive model based on the modified kinetic logarithmic stress rate.
The shear stress responses obtained through the various elastoplasticity models are investigated for the classical simple shear problem. At first, with small elastic strain (see the material
93 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin parameters in Table 3.4) the shear stress responses are shown in Figure 3.6. It can be seen that the results from the additive models exhibit very good agreement with those from the multiplicative one [62] except that the additive model based on the logarithmic rate produces significantly different results under large shear strain. In fact, small elastic strain indicates that B e (or Bk ) should be quite close to the identity tensor I . In this case, the modified kinetic logarithmic spin tensor ΩWWBDWBDKkkp ,, in the additive model (or Ω defined in (3.57) for the equivalent multiplicative model) should be rather close to the Jaumann spin tensor W since both
WBD k , and WBD kp, are continuous functions and vanish with BIk . The spin tensor
ΩWWBDlog , (see the definition in (2.33)) for the logarithmic rate, however, is dependent on B rather than B e . Since the difference between B and I increases with the shear deformation, the spin tensor Ωlog may significantly deviate from W under large shear deformation. This may result in the discrepancy between the results from the additive model with the logarithmic rate and those from the others. Figure 3.7 shows the shear stress responses produced by these models with moderately large elastic strain (see the material parameters in Table
3.5). It can be seen that only the additive model based on the modified kinetic logarithmic stress rate provides results that are close to those from the multiplicative model [89]. The small difference between the results from such an additive model and those from the multiplicative model may be attributed to the fact that the Neo-Hookean hyperelasticity is assumed in the multiplicative model developed by Dettmer and Reese [89] while the Hencky’s hyperelasticity model is adopted in the additive model based on the modified kinetic logarithmic stress rate. Since the elastic deformation is not excessively large, this fact does not result in large difference between the two models.
94 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
Table 3.4. Material parameters for small elastic strain [62]
E y c b 2 210 0. 3 0 . 4 5 3 6
Table 3.5. Material parameters for moderately large elastic strain [89]
E y c b 2 . 2 1 5 4 1 0 8 0 . 3 8 5 3 . 5 1 0 7 1 1 0 8 2.7
1.0
0.8
0.6 y Multiplicative (Neto et al., 2008) Additive, Present
Additive, Log
0.4
Additive, Jaumann
0.2
0.0 0 2 4 6 8
Figure 3.6. Shear stress response (normalized by the yield stress y ) of the simple shear problem with small elastic strain (see the material parameters in Table 3.4)
95 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
1.4 1.2 1.0
0.8
y
Multiplicative (Dettmer and Reese, 2004) 0.6
Additive, Present 0.4 Additive, Log Additive, Jaumann 0.2 0.0 0 4 8 12 16 20
Figure 3.7. Shear stress response (normalized by the yield stress y ) of the simple shear problem with moderately large elastic strain (see the material parameters in Table 3.5)
3.7 Conclusions
It has been mathematically proved and demonstrated on several numerical examples that for isotropic materials, a hyper-elasto-plasticity model based on the multiplicative decomposition of deformation gradient coincides with an additive hypo-elasto-plasticity model that employs the spin tensor based on the modified kinetic logarithmic rate. In the absence of strain-induced anisotropy
(kinematic hardening), this objective stress rate coincides with the kinetic logarithmic rate recently proposed by Jiao and Fish [4]. It has been also shown that other well-known additive models based on Jaumann and logarithmic rates considerably deviate from the multiplicative model.
96 Chapter 3. A Link between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models: the Modified Kinetic Logarithmic Spin
The focus of the present chapter was on the theoretical aspects. The corotational models based on the additive split of the rate of deformation are attractive from a practical point of view as they offer a systematic framework of extending a library of infinitesimal inelastic materials models to large deformation regime. Yet, the additive models, much more than their multiplicative counterparts, are known to suffer from various shortcoming documented herein and elsewhere.
The theoretical link established herein between the multiplicative framework and one of the models based on the additive decomposition framework, may provide a firm footing for developing both efficient and accurate corotational algorithms for large deformation plasticity.
97 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
Chapter 4 Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
In this chapter, two hierarchical approaches, the s-version of the finite element method (s-method) and the extended finite element method (XFEM), are compared to the classical non-hierarchical ply-by-ply discretization approach in terms of their effectiveness in modeling delamination in laminated composites. The two hierarchical methods are shown to have approximation
(discretization) spaces which are identical to that of the classical ply-by-ply discretization approach. These approaches therefore produce the same result in delamination analysis of laminated composites. This chapter is reproduced from the paper [139] coauthored with Professor
Jacob Fish whose inputs are gratefully acknowledged.
4.1 Introduction
Since the first commercial application of composites in the NASA/Boeing 737 Graphite-Epoxy horizontal stabilizer in 1980s [140], composites have become increasingly popular in design of major components of large aircrafts. Today more than 50% of Boeing 787 and 53% of A350 XWB are made of composites. Laminated composites whose in-plane dimensions are several orders of 98 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites magnitude larger than their thickness, are most commonly used composites architecture for these structural components. Conventional shell or plate elements deployed in aircraft industry are not capable of reliably predicting the onset and propagation of delamination, which is a dominant failure mode in laminated composites [141–143]. On the other hand, an alternative ply-by-ply discretization of individual layers practiced for delamination studies in small coupons or elements is computationally not feasible for large scale aerospace components. Thus, the challenge is to develop a practical approach for delamination analysis of large-scale aerospace components.
Delamination usually initiates and evolves in localized areas along one or more interfaces in laminated composite structures. Schematics of a delamination in a laminated composite structure is illustrated in Figure 4.1a where a portion of the laminated structure is depicted in a cross-section.
The most common approach to modeling delamination (Figure 4.1a) using finite elements is illustrated in Figure 4.1b. By this so-called ply-by-ply discretization approach, the inter-ply interfaces are explicitly resolved by inter-element boundaries. The displacement (strong) discontinuity across the delaminated interface areas is modeled using double nodes coming from the elements above and below the interface.
It is evident that this classical ply-by-ply discretization approach is in principle adequate to model any delamination in any laminated composite as long as the finite element mesh is properly tailored.
In practice, however, this ply-by-ply discretization illustrated in Figure 4.1b suffers from a number of shortcomings. First, the ply-by-ply discretization approach is non-hierarchical in nature and may lead to excessive computational cost. Laminated composite structures are typically modeled using plate or shell elements whose thickness is considerably smaller than the in-plane dimensions.
Plate or shell elements are typically adequate to resolving the responses of composites if there is
99 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites no delamination. An alternative to plate or shell elements is a solid laminated element, which is typically a quadratic or higher order solid element, that contains multiple plies through its thickness direction. Clearly, neither of the two approaches possesses the necessary kinematics to resolve delamination.
(a) Delamination in a portion of a laminated composite plate Element boundary
(b) Ply-by-ply finite element discretization of delamination Element boundary
(c) Laminate-by-laminate discretization of delamination
Figure 4.1. Non-hierarchical finite element representation of delamination in laminated composites: (a) localized delamination; (b) Ply-by-ply discretization of delamination; (c) Laminate-by-laminate discretization of delamination.
One possibility to at least partially remedy computational complexity of the ply-by-ply discretization is illustrated in Figure 4.1c. The computational cost can be reduced if the ply-by-ply
100 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites discretization is replaced by a stack of laminated solid elements connected at the delamination interfaces as shown in Figure 4.1c. Caution must be exercised to ensure C0 continuity of displacements across adjacent laminated solid elements as illustrated in Figure 4.2. One possibility to ensure the C0 continuity is to employ multiple point constraints. This, however, introduces additional bookkeeping. Computational complexity is further amplified with initiation and evolution of delamination that requires continuous remeshing.
Figure 4.2. Displacement compatibility in non-hierarchical representation of delamination in laminated composites
The computational complexity can be addressed using hierarchical representation of delamination.
By this approach, a smooth approximation field based on the mesh of a single layer of through- the-thickness solid laminated elements is first introduced to resolve a delamination-free response of the composite structure. Delamination is then modeled by enriching the smooth approximation space. The hierarchical approach has the following advantages:
(i) It may substantially reduce computational cost if the enrichment is used only where it is
needed (see Chapter 5 or [144]).
101 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
(ii) The compatibility of the displacement field can be readily assured by enforcing the
enrichment to vanish at the interface between delaminated and delamination-free regions.
Consequently, multiple point constraints are not needed.
(iii) Remeshing is avoided.
The s-version of the finite element method (s-method) [145] and the extended finite element method (XFEM) [146] provide two possible hierarchical frameworks in modelling delamination in laminated composites since both the s-method and the XFEM feature hierarchical decomposition of the approximation space in the finite element method. The origin of such a hierarchical decomposition is often attributed to the global-local finite element method of Mote
[147]. Comprehensive review of the early works can be found in the papers of Noor [148] and
Dong [149]. Most of the early attempts employ global functions in enriching finite element approximations. Such global enrichment functions include crack tip asymptotic fields [150] and shear band weak discontinuities [151]. Due to the global nature of the enrichment functions, these methods give rise to dense stiffness matrices and are often referred to as global enrichment methods (GEM). Local enrichment methods (LEM) and sparse global enrichment methods (SGEM) were therefore developed in order to enhance the sparsity pattern of the stiffness matrices.
Enrichment functions in LEM are limited to individual elements so that they can be condensed out at element level. These enrichment functions have been employed to represent discontinuous strains [152,153], curvatures [154], and displacements [155,156]. By contrast, enrichment functions in SGEM are not limited to individual elements. Methods belonging to the category of sparse global enrichment methods include: (i) multigrid-like scale bridging methods [157–160];
(ii) the extended finite element method (XFEM) [146,161–163] or the generalized finite element
102 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites method (GFEM) [164–166], which are based on the partition of unity concept [167,168]; and (iii) the s-version of the finite element method (s-method). The s-method has been applied to hierarchical representation of strong [169,170] and weak [171–174] discontinuities.
In the present chapter, a global mesh of laminated solid elements, (consisting of a single layer of through-the-thickness 20-node serendipity elements), is initially employed to model delamination- free response of the composite structure. The discontinuity of displacements across the inter-ply interfaces of a laminate is embedded by either the s-method or the XFEM. The equivalence of the two hierarchical approaches and the standard non-hierarchical ply-by-ply approach are proved.
Although the two hierarchical approaches are equivalent and the XFEM has proved to be effective in modelling delamination in laminated composites [175–178], the s-method is still an interesting and promising approach in this context because of its sparser matrix structure and seamless transition from weak to strong discontinuity.
The outline of the chapter is as follows. A representation of discontinuity in displacements across the inter-ply interfaces using the s-method and the XFEM is outlined in Section 4.2. Incorporation of cohesive behavior into inter-ply interfaces is briefly introduced in Section 4.3. In Section 4.4, we prove that the enrichments introduced by the s-method and the XFEM are equivalent to the ply-by-ply discretization approach with double nodes across the delaminated inter-ply interfaces.
A numerical example in Section 4.5 demonstrates the equivalence between the s-method and the classical ply-by-ply (or laminate-by-laminate) discretization approach.
103 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
4.2 An Overview of representation of discontinuity via the s-method and the XFEM
4.2.1 The s-method
Figure 4.3(top) depicts a structured superposition of two 20-node hexahedral elements over an underlying (hereafter to be referred as the global) 20-node hexahedral element containing multiple plies. The two superimposed elements are positioned below and above the delaminated interface, respectively. The formulation of the laminated hexahedral elements is detailed in Appendix D.1.
Note that the 20-node laminated hexahedral element rather than a simpler 8-node laminated hexahedral element is employed due to its ability to resolve bending.
Figure 4.3. Superposition of displacement discontinuity at the delaminated interface (hidden nodes are not shown)
The overall discretized displacement field uh can therefore be written as
nnGS h G S G G S S e u u u x d x d in (4.1) 11
104 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites where e represents the domain of the eth global element and x denotes the spatial coordinates;
G x is the shape function of the th global node in the global (or underlying) element and
S G S x is the shape function of the th superimposed node; d and d denote the displacement vectors of the th global and th superimposed nodes, respectively; nG and nS represent the numbers of global and superimposed nodes, respectively.
Remark 4.1. It is necessary to constrain certain nodes in the superimposed elements to ensure that the subspaces reproduced by the global and superimposed meshes are linearly independent.
Figure 4.3 depicts which nodes in the superimposed elements are constrained to avoid linear dependency. For more details see Section 4.4.1.
Remark 4.2. The overall displacement field obtained via Equation (4.1) will be shown in Section
4.4.1 to be equivalent to that of the ply-by-ply (or laminate-by-laminate) discretization approach consisting of two 20-node elements (identical to the two superimposed elements) with all their nodes unconstrained as shown in Figure 4.3(bottom). The element superposition illustrated in
Figure 4.3 can be generalized to model multiple delaminated interfaces in a single global element.
Each delaminated interface would correspond to a pair of superimposed elements which are above and below the interface, respectively.
Remark 4.3. While Figure 4.3 focusses on superposition of displacement (or strong) discontinuity, a similar framework can be employed to superimpose displacement gradient (weak) discontinuity. The attractive feature of the s-method is that the weak discontinuity across the interface can be imposed by simply constraining the degrees-of-freedom corresponding to the displacement jump across the interface of interest. This issue is further discussed in Section 4.4.2.
105 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
4.2.2 The extended finite element method
The extended finite element method (XFEM) incorporates enrichment functions, such as the
Heaviside function or any other analytical solutions of choice, in addition to the global finite element shape functions [161,179,180]. If a displacement field is characterized by strong discontinuity across the delamination, the enrichment functions are constructed through the
Heaviside function H defined as
H (4.2) 1,
Such an enrichment function in a solid laminated element is depicted in Figure 4.4 where a 20- node hexahedral element is shown with a delaminated inter-ply interface in its parametric space.
Figure 4.4. 20-node hexahedral laminate element with a delaminated interface in its parametric space
The delaminated interface, expressed as c 0 , is a plane perpendicular to the axis in the parametric space (see Appendix D.1). The discretized displacement uh depicted in Figure
4.4 can be written as
106 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
2020 hGGc ee uda H on 12 (4.3) 11
G where is the element global shape function associated with the th node;
20 G d represents the standard finite element approximation on a single element; and 1
20 Gc H a denotes the displacement discontinuity across the delaminated 1 interface.
4.3 Incorporation of cohesive behavior at the interfaces
A cohesive zone model is used to simulate the inter-ply interfacial behavior. A cohesive traction- separation relation is employed to model the interface behavior between the upper and lower superimposed elements shown in Figure 4.3. Since incorporation of such a relation does not directly influence the kinematics of the solid elements, it is not shown in Figure 4.3. The three- point Newton-Cotes formula in each spatial direction is adopted to integrate the weak form at the interfaces [181]. The integration scheme results in nine integration points at a delaminated interface. Eight of these integration points coincide with the superimposed nodes at the interface and the ninth one resides at the interface centroid.
The cohesive zone model used in the present work is outlined in Appendix E.
107 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
4.4 Comparison of the s-method, the extended finite element method and the ply-by-ply discretization
It will be shown in this section that both the s-method and the XFEM are equivalent to the ply-by- ply discretization in representing strong discontinuity of displacement field across the interfaces in laminate elements. As for the weak discontinuity of the displacement field across the interfaces, the s-method provides the displacement discretization that is equivalent to the ply-by-ply discretization. With the XFEM, on the other hand, it is not trivial to reproduce weakly discontinuous displacement field that is equivalent to the ply-by-ply discretization (See Sections
4.4.2 and 4.4.3 for details).
4.4.1 Relations between the s-method, extended finite element method and the ply-by-ply
discretization approach in characterizing strong discontinuity
The s-method and the XFEM discretize displacement field in laminate elements with delaminated interfaces in two different but equivalent ways. In fact, both of these methods are equivalent to the ply-by-ply discretization approach in terms of discretization of displacement fields in delaminated elements.
The displacement field discretized through any of the aforementioned methods is a linear combination of basis functions of the corresponding method. The basis functions in the s-method are the shape functions associated with the global and unconstrained superimposed mesh nodes, while in XFEM, the basis functions are those associated with the standard (global) finite elements and the enrichment, which in the present dissertation, is the product of the shifted Heaviside function and the standard finite element shape functions (see Equation (4.3)). The three methods
108 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites are equivalent in that their basis functions span the same function space. In other words, any discretized displacement field in one of the methods can be exactly reproduced by the other two.
In order to prove the equivalence of the three methods, a general case with multiple delaminated interfaces is studied. Figure 4.5 shows the representation of multiple delaminations through the ply-by-ply discretization approach, the XFEM, and the s-method. The three methods are illustrated with n interface cracks taking place along n interfaces, respectively, within the global delamination-free 20-node isoparametric serendipity element. Note that the three methods are considered in the parametric space of the global solid laminate element where the interfaces are planes perpendicular to the axis. The delaminated interfaces are numbered from bottom up as
c shown in Figure 4.5 where i denotes the position of the ith delaminated interface.
In the ply-by-ply discretization approach (see Figure 4.5b), the domain e is discretized by
n 1 elements. The discretized displacement field uh on e can thus be expressed as
n120 h iie ud in (4.4) i11
i i where n 1 is the number of through-the-thickness local elements; and d represent the shape function and nodal displacement vector of the th node in the ith local element,
h i respectively. Herein both u and d are 13 row matrices. It should be noted that the value of
i the shape function is zero in any local element other than the ith local element. This leads to a potential discontinuity of the displacement field across the delaminated interfaces in e . The concise matrix form of Equation (4.4) can be written as
109 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
u Nh d in e (4.5) where N is the row matrix consisting of the 2 0 1n basis functions and d is the corresponding matrix of nodal values having the size of 2 0 1n 3 . N can be written as the following block row matrix:
121 n N (4.6) and d is given in the block form as
T 1 T2 T1 T n dddd (4.7)
In Equations (4.6) and (4.7), i and di in1,2,...,1 are defined as
i i i i in 1,2,..., 1 (4.8) 1 2 20
ddddiiiiTTTT in1,2,...,1 (4.9) 1220
The relation between the discretized displacement field and the basis functions in the ply-by-ply discretization approach can therefore be clearly expressed via Equation (4.5).
In the XFEM (see Figure 4.5c), the displacement field in the domain e is discretized as follows:
20n 20 X G X G c ie u d H i a in (4.10) 1i 1 1
G where the superscript X denotes XFEM; is the global finite element shape function associated
e X with the th node in the global element on ; d denotes the vector of coefficients of the
G c shape function ; n is the number of delaminated interfaces and i the coordinate of the ith
110 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
c delaminated interface; H i , which is the shifted Heaviside function, serves as the
i enrichment function for the ith delaminated interface; a is the vector of coefficients
Gc X X i corresponding to the enrichment H i . Note that u , d , and a are 13 row matrices.
X G It is evident from Equation (4.10) that the basis functions of the function space of u are
Gc 1,2,...,20 and H i in1,2,..., ; 1,2,...,20 . Similarly to the previous standard non-hierarchical approach, Equation (4.10) can be written in matrix form as
uNdXXX in e (4.11) where NX is the row vector of the 201n basis functions and dX is the corresponding coefficient matrix having the size of 2013n . The two matrices can be expressed in the following block form:
NX GHHH c G c G c G (4.12) 12 n and
TT dX d GX a1 T a 2 T an T (4.13) where G , defined in a manner similar to i in1,2,..., 1 in Equation (4.8), is the row matrix
T of the 20 standard shape functions of the global element; dGX and aiT in1,2,..., , which are similar to diT in1,2,...,1 in definition (see Equation (4.9)), are the 3 20 matrices
111 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
G cG containing the coefficients corresponding to the basis functions in and H i , respectively.
Similarly to the XFEM, the basis functions in the s-method (see Figure 4.5d) comprise of the shape functions of the global element in e , and the shape functions associated with the superimposed patches of elements:
2020 n SGGS SSiie udd in (4.14) 111 i
G GS where the superscript S denotes the s-method; and d are the shape function and nodal
e Si Si values corresponding to the th node of the global element in , respectively; and d denote the shape function and nodal values corresponding to the th superimposed node in the ith superimposed patch of two elements which characterize the ith delaminated interface. uS ,
GS Si d and d are 13 row matrices. The local numbering of nodes in a superimposed patch is shown in Figure 4.5d (on the second patch from the right). The basis functions comprising the
S G Si function space of u are 1,2,...,20 and in1,2,..., ; 1,2,...,20 . Equation
(4.14) can be expressed in matrix form as
uSSS N d in e (4.15) where NS is the row matrix consisting of the 201n basis functions and dS is the corresponding coefficient matrix having the size of 20n 1 3 . NS and dS can be written as the following block row matrices:
112 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
SGS 1S 2S n N (4.16) and
TT dddddSGS S 1 TS 2 TST n (4.17) where G and Si in1 ,2 ,.. ., are the 1 2 0 matrices containing the global shape functions of the global element and superimposed shape functions of the ith superimposed patch, respectively;
T dGS and dSTi in1 ,2 ,.. ., , which are similar to diT in1 ,2 ,.. ., 1 in definition (see
Equation (4.9)) are the 3 2 0 matrices of coefficients corresponding to G and Si , respectively.
With the concise discrete forms of the three methods (see Equations (4.5), (4.11), and (4.15)), the equivalence of the three methods will be subsequently proved. In order to prove that the function spaces of the discretized displacement fields are the same in the three methods, one only needs to prove the existence of two invertible matrices, TX and TS , defined as
NNXX T (4.18) and
NNSS T (4.19) where the superscripts X and S indicate the corresponding transformation matrices transform the basis functions of the standard non-hierarchical approach into the XFEM and the s-method, respectively.
113 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
(c) Approach 2: XFEM
Sol. 2
Sol. 1
Prob.
(b) Approach 1: Ply-by-ply discretization (a) The problem Sol. 3
(d) Approach 3: s-method
Figure 4.5. Three methods for representing strong discontinuity across delaminated interfaces: (a) problem description; (b) ply-by-ply discretization; (c) XFEM; (d) s-method.
114 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
Let us first assume the two invertible matrices exist. Equation (4.19) can therefore be inserted into
Equation (4.15) yielding
uNdNSSSSSSS TdNT d in e (4.20)
The equation above suggests that any discretized displacement field uS with an arbitrary coefficient matrix dS in the s-method can be reproduced by the ply-by-ply discretization approach with the corresponding coefficient matrix being TdSS. In addition, since TS is assumed to be invertible, Equation (4.19) can be rewritten as
1 NNT SS (4.21)
Substituting Equation (4.21) into Equation (4.5) yields
11 uh Nd N S T S d N S T S d in e (4.22)
The above equation suggests that any discretized displacement field uh with an arbitrary coefficient matrix d in the ply-by-ply discretization approach can be reproduced by the s-method
1 with the corresponding coefficient matrix beingTdS . Consequently, the function spaces in the two methods are identical since any function from either of the two spaces is always a member of the other. It is evident that the same conclusion can be made regarding the equivalence between the XFEM and the ply-by-ply discretization approach provided that the invertibility assumption about TX holds.
Having validated the sufficiency conditions regarding the equivalence of the three methods, we now proceed with the proof that such conditions can be indeed satisfied. The proof is presented in
115 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites two steps. First, it will be shown that there exist two matrices, TX and TS , which satisfy Equations
(4.18) and (4.19), respectively. Secondly, we will prove that TX and TS are invertible.
For the first step of the proof, it will be shown that any shape function of the global 20-node isoparametric serendipity element can be exactly reproduced by local elements of the same type if the nodal coordinates of the local element in the parametric space of the global element satisfy
LG iiaa10 LG iibbi10 1,2,...,20 (4.23) LG iicc10
GGG LLL where iii ,, and iii ,, are the coordinates of the ith nodes of the global and local
elements, respectively, in the parametric space of the global element; aabbcc010101,,,,, are
constants and a111 b,, c are not zero. The proof of this statement is given in Appendix D.2.
It can be readily verified that nodal coordinates of any local element in the ply-by-ply discretization approach and the global element have the relation expressed by Equation (4.23) in the parametric space of the global element. Furthermore, the same relation exists between the nodal coordinates of any local element in the ply-by-ply discretization approach and any superimposed element in the s-method in the parametric space of the superimposed element. These relations are shown in
Appendix D.3.
It is evident that the global shape functions adopted by both the XFEM and the s-method can be written as
n1 20 GGi i i i , , 1,2,...,20 (4.24) i11
116 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
iii where ,, denote the coordinates of the th node of the ith local element in the parametric space of the global element in e . Note that in Equation (4.24) the basis function
G is reproduced by all the n 1 local elements in their respective domains.
In addition, based on the definition of the Heaviside function, the value of the basis function
cG H i in XFEM equals zero below the ith delaminated interface and
G above it. The following expression can therefore be written:
n1 20 cGG jjjj Hin i ,,1,2,..., ;1,2,...,20 (4.25) ji 11 Furthermore, based on the proofs in Appendices D.2 and D.3, the shape functions corresponding to the superimposed nodes can be written as
i 20 Sijjjj ,,1,2,...,12 S i j11 in1,2,..., (4.26) n1 20 Sijjjj ,,13,14,...,20 ji 11
Note the numbering of the superimposed nodes in the superimposed patch (see Figure 4.5d) where the first twelve nodes are below the corresponding delaminated interface.
Equations (4.24)-(4.26) suggest that each basis function in both the XFEM and the s-method can be written as a linear combination of the basis functions in the ply-by-ply discretization approach.
This validates the existence of the two matrices, TX and TS , that satisfy Equations (4.18) and
(4.19), respectively.
With the existence proved, TX and TS will be now shown to be invertible. Instead of investigating the two matrices directly, the invertibility of the matrices will be proved by validating the following
117 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites sufficiency conditions: all the basis functions in either of the two methods (XFEM and s-method) are linearly independent.
At first reduction to absurdity is used to readily prove the sufficiency condition. Let us assume the basis functions in, for example, the s-method are linearly independent but the corresponding transformation matrix TS is not invertible. Accordingly, there must be a non-zero column matrix vS satisfying TSS v 0 . After post-multiplying both sides of Equation (4.19) by vS , one obtains
NvNTvSSSS 0. It is evident that the expression NvSS 0 contradicts the definition of linear independence of the s-method basis functions NS since vS is non-zero column matrix. TS , therefore, must be invertible if the s-method basis functions are linearly independent. Similarly linear independence of the XFEM basis functions assure invertibility of TX .
Next, the linear independence of the basis functions can be verified in both the s-method and the
XFEM. Consider two column matrices, cX and cS , of constant coefficients satisfying NcXX 0 and NcSS 0 in e . Based on the definition of linear independence, if cX and cS can only be zero column matrices, NX and NS are row vectors containing linearly independent basis functions.
In the following we will show that cS must indeed be zero to satisfy NcSS 0 proving that the basis functions are linearly independent in the s-method. Similar proof can be made for the XFEM.
Consider the s-method. It is evident that the entries of cS are coefficients associated with all the global and local shape functions. The element superposition scheme in Figure 4.5d is depicted in
Figure 4.6 along a vertical edge (in direction) of the global element. Note that the numbering of the global and superimposed nodes in Figure 4.6 is consistent with Figure 4.5d. The index in Figure 4.6 can be either 1, 2, 3, or 4 corresponding to a vertical element edge in Figure 4.5d. Let
118 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
e d g e us denote the vertical element edge passing through the th 1,2 ,3 ,4 global node by .
Since NcSS equals zero on the whole domain e of the global element, NcSS must vanish on the four vertical element edges. Considering only the shape functions associated with nodes on the vertical edge (see Figure 4.6), NcSS can be written as
S SGSGGSGGSG Nc ccc 441616 n (4.27) SSSSSSiiiiii edge ccc 881616 in1,2,3,4 i1
Si GS S where c and c represent the coefficients in c corresponding to the th superimposed node in the ith superimposed patch and the th global node, respectively. The shape functions appearing in Equation (4.27) are shown in Figure 4.6. According to the proofs given in Appendices
D.2 and D.3, each shape function in the s-method has the form described in Equation (D.2) within their corresponding elements in the parametric space of the global element. It is thus evident that the expressions of all the shape functions in Equation (4.27) are quadratic polynomials in on
e d g e their respective supports in 1,2,3,4 where the values of and are fixed. We will
SS e d g e denote Nc in by f .
G It is evident that among all the shape functions appearing in Equation (4.27), only
edge are not zero in at as shown in Figure 4.6. By evaluating the expression
GS in Equation (4.27) for , it can be readily seen that c 0 1,2,3,4 . Similarly,
GS evaluating f 0 for yields c 4 0 1,2,3,4 . Furthermore, f
1,2,3,4 is continuous everywhere in its domain resulting in limff lim for 00
119 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
Si any 0 in 1,1 . It can be clearly seen from Figure 4.6 that only the basis functions
e d g e in1,2,...,;1,2,3,4 are discontinuous along the corresponding element edges . They exhibit jump discontinuity across the corresponding delaminated interfaces, namely, l i m 0Si c i but l i m 1Si on e d g e with and in1,2 ,. . . , . As a result, by applying the c- i conditions limlimff jn1,2,..., to the expression in Equation cc jj
S j (4.27), one obtains c 0 .
Figure 4.6. The s-method shape/basis functions along a vertical element edge passing through the th global node 1,2,3,4
Based on the discussion above, Equation (4.27) can be simplified as
n S S GS Gedge SSSSi ii i Nc ccc16 168 816 16 in1,2,3,4 (4.28) i1
Si Si It is evident that in Equation (4.28) only 8 and 16 1,2,3,4;1,2,..., in have
c edge discontinuous first and second order derivatives with respect to at i on as shown in
120 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
Si Si e d g e Figure 4.6. In fact, the expressions of 8 and 16 1,2,3,4;1,2,...,in in can be expressed by Lagrangian polynomials as
c 1 i ,c cc i Si ii11c edge 8 1 i in 22 (4.29) c 0,1i 1,2,3,4;1,2,...,in
c 0,i c Si 1 edge i c in 16 ,1i cc11 (4.30) ii c 1 i 22 1,2,3,4;1,2,..., in
Despite the discontinuous derivatives of the functions in Equations (4.29) and (4.30), NcSS has continuous derivatives to any orders. It should therefore follow that limlimff and cc jj limlimff jn1,2,..., . Applying the above conditions to the cc jj expression in Equation (4.28) and considering Equations (4.29) and (4.30), the following systems of linear equations can be obtained:
SS jj cc8 16 cc0 jj11 SS jj 1,2,3,4; jn 1,2,..., (4.31) cc8 16 220 cc11 jj
121 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
The determinants of the coefficient matrices of the above systems of linear equations are
2 2 21 c jn1 ,2 ,.. ., . Since 11 c , the determinants of the coefficient matrices are j j
S j S j non-zero. As a result, both c 8 and c 16 1,2,3,4;1,2,...,jn can only be zero. Equation
S S GS GS edge (4.28) can thus be simplified as Nc c16 16 0 on with 1,2 ,3 ,4 . Evaluating the
edge GS expressions at in yields c 16 0 1,2 ,3 ,4 . It is now proved that all the coefficients of the basis functions associated with the global and superimposed nodes on the four
e d g e vertical edges 1,2,3,4 must vanish.
All the basis functions not shown in Figure 4.6 are associated with nodes located on the four vertical straight lines (along the direction) passing through the 9th, 10th, 11th, and 12th global nodes, respectively, as shown in Figure 4.5d. Let us denote the vertical line passing through the
l ine th 9,10,11,12 global node by . Similarly to Figure 4.6, Figure 4.7 depicts the element
l ine superimposition scheme in Figure 4.5d along one of the lines 9,10,11,12 . It should be
line noted that only the shape functions associated with the nodes in 9,10,11,12 are plotted in Figure 4.7. According to the expressions of the shape functions of the 20-node serendipity element, some nodes on the aforementioned vertical element edges shown in Figure 4.6 can
SS line contribute to the value of Nc in . The discussion above, however, proves that the coefficients associated with these nodes must be zero. This indicates that the value of NcSS on
line line 9,10,11,12 is determined only by the nodes on as shown in Figure 4.7.
SS line Consequently, the value of Nc on 9,10,11,12 can be expressed as
122 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
SSGSGGSG Nc cc 44 n (4.32) SSSSiiii line cc 4444 in9,10,11,12 i1
Si GS S where c 4 and c are the coefficients in c associated with the th node in the ith superimposed patch and the th node of the global element, respectively. Note that all the shape
line functions in Equation (4.32) are linear functions of on their respective supports in
9 ,1 0 ,1 1 ,1 2 . One can readily verify the statement through the expressions of the shape functions of the global element in its parametric space and the linear coordinate transformations between the parametric spaces of the global element and superimposed elements shown in
SS l ine Appendices D.2 and D.3. It is thus appropriate to denote the value of Nc on
9,10,11,12 by g which, based on the discussion above, are piecewise linear functions
G l ine of . It is evident that only 9,10,11,12 are not zero in for as shown in
GS Figure 4.7. Evaluating the expression in Equation (4.32) for yields c 0
GS 9,10,11,12 . Similarly, g 10 assures that c 4 0 9,10,11,12 . Note that the
Si Si shape functions, 4 and 4 9,10,11,12;1,2,..., in are discontinuous across the corresponding delamination cracks as shown in Figure 4.7. The linear combination of the shape
SS line functions Nc , however, is zero everywhere on 9,10,11,12 with continuous derivatives to any orders. Consequently, g should satisfy limlimgg and cc jj
123 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites limlimgg jn1 ,2 ,.. ., . Combining the above conditions with Equation (4.32) cc jj yields the following systems of linear equations:
SS jj cc440 9,10,11,12;1,2,...,jn (4.33) 110ccccSS jj jj 44
SS jj It is evident that the unique solutions to the above systems of linear equations are cc440
9,10,11,12;1,2,...,jn . All the entries in cS have now been shown to be zero.
Figure 4.7. The s-method shape/basis functions associated with nodes on a vertical straight line passing through the th global node 9,10,11,12
With cS shown to be a zero vector, the s-method basis functions in NS must be linearly independent. This validates the selection of unconstrained superimposed nodes in Figure 4.3 (see
Remark 4.1 in Section 4.2.1).
As for the XFEM, the same procedures can be used to readily prove cX 0.
124 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
Following the proof of linear independence of the basis functions in the two methods, the transformation matrices, TX and TS , as stated above, must be invertible. This concludes the proof that the three methods have the same approximation space.
4.4.2 Representation of weak discontinuity using the s-method
The subdomain of a weak discontinuity as a wrapper around the subdomain of a strong discontinuity is necessary to improve solution accuracy in the vicinity of strong discontinuities.
As stated in Remark 4.3 (Section 4.2.1), the weak discontinuity can be effectively represented by the s-method. Figure 4.8 shows the representation of the weak discontinuity across an interface of the 20-node isoparametric serendipity element in its parametric space. By comparing Figure 4.8 with Figure 4.3, it can be seen that all the degrees-of-freedom responsible for strong discontinuity across the interface in Figure 4.3 are constrained in Figure 4.8. This include the degrees-of- freedom corresponding to the relative displacement between each pair of double superimposed nodes at the mid-side of horizontal interface edges and those corresponding to the displacement of superimposed corner nodes at the delaminated interface. In the situation shown by Figure 4.5d, such constraints across the ith interface can be expressed as
Si d0 1,2,3,4 (4.34)
SSii dd8 (4.35)
125 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
Figure 4.8. Representation of weak discontinuity through the s-method across an interface within the 20-node solid laminate element in its parametric space
Constraining the degrees-of-freedom does not only leave the total degrees-of-freedom the same as in the ply-by-ply discretization approach as shown in Figure 4.8 but also makes the function space identical to that of the ply-by-ply discretization approach. This can be further clarified using Figure
4.5d. By constraining the degrees-of-freedom corresponding to the strong discontinuity shown in
Figure 4.5d, one changes the basis functions of the approximation space. The original basis
Si Si Si SSii functions, , 4 , and 12 1,2,3,4;1,2,...,in are replaced with 412
1,2,3,4;1,2,...,in while the remaining basis functions are unchanged. The resulting basis
G Si Si SSii functions are: , 8 , 16 , and 4 12 1,2,...,20; 1,2,3,4; in 1,2,..., . It is evident that the new basis functions remain linearly independent. This can be readily proven through reduction to absurdity. Furthermore, the new set of basis functions are continuous on e and based on Appendices D.2 and D.3, each of the basis function can be written as a linear combination of the basis functions in the corresponding non-hierarchical approach with single 126 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites nodes at the interfaces (as illustrated in Figure 4.8). Based on the proof in the previous section and the discussion above, we conclude that the function spaces spanned by the basis functions in the s-method and the ply-by-ply discretization approach are the same when weak discontinuity is present.
4.4.3 Advantages of the s-method
It is evident that both the s-method and XFEM, in which remeshing is not needed, have advantages over the ply-by-ply discretization approach. Herein we enumerate advantages of the s-method over
XFEM in modeling strong and weak discontinuities.
The s-method gives rise to sparser matrix structure in case of strong discontinuities. For example, in Figure 4.5d the strong discontinuity at a delaminated interface is governed by 12 superimposed nodes while in XFEM the strong discontinuity is determined by all 20 nodes. Furthermore, the displacement jumps at the superimposed nodes in the s-method at delaminated interfaces are directly described by the interface nodes. In XFEM, however, one has to evaluate the corresponding enrichment terms shown in Equation (4.10) to obtain the displacement jumps at these positions. Consequently, when cohesive zone models are employed, the cohesive forces only influence the residuals corresponding to the superimposed nodes at the delaminated interfaces in the s-method.
The s-method provides an easy pathway to switch from a weak to a strong discontinuity. As stated in Section 4.4.2, the weak discontinuity can be readily characterized in the s-method by constraining the degrees-of-freedom corresponding to strong discontinuity. The resulting function space is equivalent to the ply-by-ply discretization approach. In XFEM, however, it is not trivial
127 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites to constrain the relative displacement at an interface (see [182]). A weak discontinuity can be enforced implicitly using Lagrange multipliers [183] or by penalty method. An alternative method of representing weak discontinuity in XFEM is to introduce enrichment functions with weak discontinuity [184,185]. By this approach if a weak discontinuity is needed across material interfaces in a global element, all element nodes should have additional degrees-of-freedom corresponding to the enrichment functions. Compared to the s-method illustrated in Figure 4.8, more degrees-of-freedom are indeed needed by the XFEM based on such an enrichment approach.
Furthermore, the XFEM does not provide an explicit framework to convert a weak discontinuity into a strong discontinuity by simply changing the enrichment functions. This transition is crucial in propagating delamination fronts (see Chapter 5).
4.5 Numerical validation
In order to study the accuracy of the s-method, a reference solution based on the ply-by-ply discretization of individual layers with double nodes and cohesive elements placed along all the interfaces is constructed. The mesh for both methods is chosen appropriately to represent the same discretization space as discussed in Section 4.4.
The composite plies are assumed to be transversely isotropic. The elastic properties of the unidirectional fiber-reinforced composites are listed in Table 4.1. Cohesive law parameters (see
Appendix D.1) considered in this study are given in Table 4.2.
Table 4.1. Elastic single ply properties
E1(MPa) EE23 (MPa) 12 13 23 GG12 13 (MPa) 55000 9500 0.33 0.45 5500
128 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
Table 4.2. Cohesive law parameters
cn (MPa) cs (MPa) ct (MPa) cr 3.2 3.2 3.2 0.001 1 1 1 GI(N m m ) GII (N m m ) GIII (N m m ) s t 0.64 0.64 0.64 0.1 0.1
The onset of interface damage is controlled by cr (see Appendix D.1), which is a critical value of the non-dimensional effective interface separation indicating damage initiation. Note that a very
small value of cr may affect conditioning, while a large value may not be physical. Analysis of a
beam problem is conducted to first obtain a nearly optimal value of cr (shown in Table 4.2). The
same value of cr is then adopted for subsequent studies.
Visualizations in this section are realized through Gmsh [186].
4.5.1 Numerical experiment configuration
A laminated beam subjected to mode I loading as shown in Figure 4.9 is considered. The beam consists of six layers of unidirectional fiber-reinforced composites with initially five intact interfaces, i.e., the model has no initial delamination. The beam is fixed at the two horizontal edges at the left end. The right end of the beam is subjected to mode I prescribed displacement. The dimensions of the beam are 50 6 3mm3 . A total separation of 2mm is applied at the right end of the beam. Each layer has the same thickness. The layup configuration of the laminate is
45 / 45 / 0 / 90 / 45 / 45 . Fibers in the 0 ply are aligned along the x1 direction.
129 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
Figure 4.9. Configuration of the numerical experiment of the beam problem
(a) Reference simulation: ply-by-ply discretization
(b) s-method simulation: underlying mesh
Figure 4.10. Final deformation for cr 0.001: (a) reference simulation based on ply-by-ply discretization; (b) the underlying mesh in the s-method simulation.
The mesh of the ply-by-ply simulation and the global mesh of the s-method are shown in Figure
4.10, which depicts their deformed shapes at the end of the simulations with cr 0.001. The ply- by-ply beam model consists of 17 2 6 20-node hexahedral elements (see Figure 4.10a) and
17 2 5 16-node quadrilateral cohesive elements having zero thickness. The global mesh in the
130 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites s-method simulation consists of 1 7 2 1 20-node hexahedral elements as shown in Figure 4.10b.
Each of the 34 global elements has five superimposed patches representing strong discontinuities across all the five interfaces as exemplified by Figure 4.5d.
4.5.2 Determination of cr
The influence of the value of cr is investigated in this section. Figure 4.11 depicts the relations between the resultant reaction forces and the displacements at the upper loading edge for different
values of cr .
It can be seen that for values of cr smaller than 0.008 the solution practically does not change.
For subsequent studies cr has been selected as 0 . 0 0 1.
Figure 4.11. Relations between reaction force and displacement at the upper loading edge for
different values of cr
131 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
4.5.3 Simulation results
Figure 4.12 depicts the comparison of the resultant reaction force-displacement curves in the s- method and the ply-by-ply simulations. At both the upper and lower loading edges, the reaction forces in the s-method simulation are practically identical to those in the ply-by-ply simulation.
(a) Upper loading edge (b) Lower loading edge
Figure 4.12. Comparison of resultant reaction forces in the s-method and reference simulations: (a) reaction force at the upper loading edge; (b) reaction force at the lower loading edge.
Besides reaction forces at the loading end, further comparison is made regarding the damage state of the interfaces of the laminated beam. The damage state at interfaces is characterized by the damage variable in the present cohesive zone model (see Appendix E). A damage variable value of one indicates totally delaminated interface while a value of zero represents intact state of the interface.
132 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
Reference simulation s-method simulation
(a)
(b)
(c)
(d)
Figure 4.13. Damage state of Interface 1 in the ply-by-ply discretization and s-method simulations for various prescribed displacement d
The interfaces in the beam are numbered from the bottom up. Note that all the five interfaces are intact without any damage at the beginning of the simulations. Both the s-method and reference simulations suggest that interface damage develops primarily in Interfaces 1 and 5 while the other three interfaces are only slightly damaged throughout the simulations. Evolution of damage at the first and fifth (i.e. lowermost and uppermost) interfaces of the beam in the s-method simulation are compared to that of the reference simulation. Figure 4.13 and Figure 4.14 show the contour plots of damage state at Interfaces 1 and 5, respectively, in both the reference and s-method
133 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites simulations as functions of prescribed displacement. It can be seen that the results of damage evolution obtained by the s-method are practically identical to that of the reference solution. In the two simulations, damage first initiates at the loading end of the beam at both Interfaces 1 and 5.
With further increase of prescribed displacement, it develops a nonsymmetric pattern, which shows faster growth in Interface 5.
Reference simulation s-method simulation
(a)
(b)
(c)
(d)
Figure 4.14. Damage state at Interface 5 in the ply-by-ply discretization and s-method simulations for various prescribed displacement d
134 Chapter 4. Analysis of Hierarchical and Non-Hierarchical Approaches to Representing Delamination in Laminated Composites
4.6 Conclusions
The s-method and XFEM are investigated in resolving delamination in laminated composites. It has proved that both of these hierarchical methods are equivalent to the ply-by-ply discretization approach in modelling strong discontinuities across delaminated interfaces even though the s- method gives rise to sparser matrices. The s-method, however, is advantageous in modeling weak discontinuities and in straightforward transition from weak to strong discontinuity across interfaces.
The equivalence between the s-method and the ply-by-ply approach has been also validated through a numerical example.
Since the strong discontinuity is introduced at all the five interfaces in Section 4.5, i.e., no weak discontinuity is present, the advantage of the s-method over XFEM is only due to sparsity or roughly 10% speed-up in the example considered. Weak discontinuities are only required for problems involving propagation of strong discontinuities that are wrapped around by weak discontinuities region. This aspect is studied in the next chapter.
135 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
Chapter 5 The Adaptive S-Method for Delamination Analysis in Laminated Composites
In this chapter, a methodology aimed at addressing computational complexity of analyzing delamination in large structural components made of laminated composites is proposed. The chapter features delamination indicators that pinpoint the onset and propagation of delamination fronts with striking accuracy. Once the location of delamination has been identified, the discrete solution space of the classical laminated solid element is hierarchically enriched by a combination of weak and strong discontinuities through the s-method (see Chapter 4) to adaptively track the evolution of delamination fronts. Numerical examples suggest that despite an overhead that comes with adaptivity, the adaptive s-method is computationally advantageous over the classical ply-by- ply discretization especially as the problem size increases. This chapter is reproduced from the paper [144] coauthored with Professor Jacob Fish whose inputs are gratefully acknowledged.
5.1 Introduction
Cohesive zone models (CZMs), originated by Dugdale [187] and Barenblatt [188], proved to be effective and convenient in modelling initiation and propagation of delamination in laminated composites especially in conjunction with finite elements [176,181,189–196]. In finite element
136 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites codes, cohesive zone models are most often employed through either interface (cohesive) elements or mixed boundary conditions at interfaces that are regarded as internal boundaries [197].
Cohesive zone models fall into two categories: (i) intrinsic models [198,199] and (ii) extrinsic models [197,200,201]. The two categories differ in whether the damage initiation is inherently contained in the model (with intrinsic initiation criteria) or a separate (extrinsic) criterion for damage initiation is required. Extrinsic cohesive zone models equate damage initiation to the interface separation, and therefore, provide traction-separation relation only after the damage has already initiated. By contrast, intrinsic models include initial elastic (or hardening) traction- separation response prior to damage initiation. Consequently, intrinsic cohesive models have to be typically in place prior to onset of damage at an interface, while extrinsic cohesive zone models have to “burst into” the position at the interface with damage initiation. Accordingly, extrinsic cohesive zone models have to be implemented in an adaptive manner so that only damaged interface areas have cohesive elements or the aforementioned mixed boundary conditions. Intrinsic cohesive zone models, on the other hand, can be placed anywhere at the interfaces regardless of their damage state. The plausible convenience of intrinsic cohesive zone models lies in the fact that they require no adaptive features in finite element codes.
For delamination analysis of large-scale laminated composite structures consisting of multiple plies, adaptive simulations could be very appealing, no matter whether intrinsic or extrinsic cohesive zone models are used. If the interply interfaces are intact, there is usually little incentive to explicitly represent these interfaces in the finite element models and therefore, plate or shell elements are ideal choices for such structures. Existence of interface delamination, however, necessitates explicit representation of damaged or delaminated interfaces. In conventional CZM
137 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites based approaches, all the interfaces are explicitly represented together with layerwise (or ply-by- ply) discretization by solid elements (See Section 4.1). Moreover, if intrinsic cohesive zone models are used, all interfaces have to be equipped with degrees-of-freedom representing interface separation even if most of them may be intact [193]. Typically, finite element meshes have to be sufficiently refined to discretize each ply with solid elements. Considering the fact that delamination usually initiates and evolves in localized areas along very few interfaces, the numbers of degrees-of-freedom in conventional CZM based approaches are often excessively large, especially for laminated composites with numerous plies. For example, in the present chapter we show that more than 60% of the degrees-of-freedom might be redundant in a non-adaptive intrinsic
CZM based simulation. It is also noteworthy to point out that most of the cohesive zone based finite element analyses of delamination in the literature are aimed at validation of cohesive zone models rather than at studying computational viability of these models in the context of large-scale structural systems. Consequently, these numerical studies typically focus on benchmark layup configurations that are much simpler than those in practical applications. In summary, there is an obvious need for adaptive CZM based finite element analyses that would enable viable delamination analyses of large-scale laminated composite structures.
Various methods were developed to address the issue of adaptive mesh refinement or adaptive displacement field enrichment in the vicinity of moving cohesive zones that track evolving crack fronts. These methods include, but are not limited to, adaptive hierarchical enrichment of polynomial functions [202], use of standard finite elements [203], adaptive mesh refinement and coarsening [204], and adaptive mesh refinement in conjunction with the extended finite element method (XFEM) [205].
138 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
In the present chapter, the discontinuity of displacements across the delaminated interply interfaces in laminated composites is resolved using the adaptive s-method. Initially, a global mesh of typical plate or shell elements should be first employed without consideration of imperfect interply interfaces to resolve a delamination-free response of the corresponding composite structure. Each global element may contain multiple through-the-thickness plies. The 20-node laminated serendipity elements (see Appendix D.1) serve as an alternative to the laminated plate or shell elements in the present chapter. Once the likelihood of delamination has been detected by a so- called delamination indicator at some positon in some interface, a patch of discontinuous local mesh involving a cohesive zone model in this interface (see Chapter 4) is superimposed onto the underlying global mesh to resolve the potential discontinuity.
The outline of the chapter is as follows. In Section 5.2, we formulate the delamination indicators and adaptive strategy based on the s-method. Numerical examples in Section 5.3 study the accuracy and computational efficiency of the adaptive s-method in comparison to the classical ply- by-ply discretization approach. Conclusions are outlined in Section 5.4.
5.2 Adaptive delamination strategy
5.2.1 Notion of superimposed node set
For adaptive simulations considered herein, it is convenient to group the superimposed nodes into node sets each of which influences discontinuity of displacement field in a specific interface location. Figure 5.1 depicts the two types of superimposed node sets, positions of which are shown within their respective patches of adjacent global 20-node elements. The first type of superimposed node sets are located on global element edges parallel to the stack direction of plies as shown in
139 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
Figure 5.1a. The second type of superimposed node sets consist of double nodes positioned on the global element faces parallel to the stack direction as shown in Figure 5.1b. Herein the first and the second types of superimposed node sets are referred to as the edge and face superimposed node sets, respectively. Examples of the two types of superimposed node sets are depicted in the second superimposed patch in Figure 4.5d where superimposed nodes , 8 and 16
1,2 ,3 ,4 comprise four edge superimposed node sets and the four pairs of double nodes comprise four face superimposed node sets.
(a) Edge superimposed node set (b) Face superimposed node set
Figure 5.1. Two types of superimposed node sets representing strong discontinuity: (a) Edge superimposed node set; (b) Face superimposed node set.
A superimposed node set can be added to each global element edge or face that is parallel to the stack direction. It is thus necessary to number the element edges and faces in the global mesh. Note that any edge superimposed node set may correspond to a unique ordered pair of integers in the global mesh. The first integer corresponds to the element edge carrying the superimposed node set, whereas the second integer points to the interface across which the discontinuity may take place
140 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites because of the current edge superimposed node set. Similarly, any face superimposed node set is identified by a pair of identifiers with the first denoting the element face carrying the superimposed node set and the second pointing to the corresponding interface. The two integer pair sets including all the positions of edge and face superimposed node sets in a model, respectively, are denoted by
Piijjedge ||ZZ and Piijjface ||ZZ .
5.2.2 Combining weak and strong discontinuities
The superimposed node sets shown in Figure 5.1 represent strong discontinuities in the corresponding interface areas. The weak discontinuity can be also introduced by constraining displacements of the middle node in the edge superimposed node set and the relative displacement between the doubles nodes in the face superimposed node set (see Equations (4.34) and (4.35)).
The weak discontinuity is introduced to provide a gradual transition from a strong discontinuity region to the discontinuity-free global elements as shown in Figure 5.2. The two-dimensional mesh in Figure 5.2 depicts the cross-section of the three-dimensional global hexahedral elements at an interply interface. The dots shown in the mesh denote the positions of the superimposed node sets representing either strong or weak discontinuity at the interface. The green dots correspond to the superimposed node sets representing strong discontinuity while the blue dots represent weak discontinuity.
We will distinguish between three types of interface solutions: (a) perfectly intact interply solution and (b) weakly intact interply solution, and (c) discontinuous interply solution. The perfectly intact interply solution is a higher order through-the-thickness continuity solution provided by global elements. The weakly intact interface solution is the C0 through-the-thickness continuous solution provided by the ply-by-ply discretization approach. It corresponds to continuous displacement 141 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites field but discontinuous strain field. Finally, the discontinuous interply solution corresponds to the
C-1 through-the-thickness continuous solution representing delamination.
Figure 5.2. Combining weak and strong interply discontinuities for delamination modeling
5.2.3 Adaptive delamination algorithm
Consider first a classical intrinsic cohesive zone model for delamination analysis where cohesive elements are placed at all interply interfaces even though interface damage may be limited to local areas [193]. The displacement jump is typically very small at the intact interfaces since the elastic response is very stiff. Intact interfaces thus undergo stiff elastic traction-separation response until the separation becomes sufficiently large to reach an ultimate interface strength.
In the beginning of an adaptive simulation, there are initial superimposed node sets that represent initial delamination. Throughout the simulation, the proposed delamination indicators to be formulated in Section 5.2.4 attempt to identify the interface locations where the separation may reach critical value corresponding to the interface strength. Such interface locations are then
142 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites equipped with corresponding superimposed node sets representing strong discontinuity that enables incorporation of the cohesive zone model.
The adaptive cohesive interface insertion procedure is outlined in Figure 5.3. Assume that a converged discrete solution on a global mesh with a number of superimposed node sets have just been obtained for a certain load increment. All applicable positions at the intact interfaces are then probed by the delamination indicators for potential new delamination positions. Probes are conducted over all non-existing superimposed node sets and all existing superimposed node sets corresponding to weak discontinuities (see Section 5.2.2). Once all the applicable positions have been probed, the analysis proceeds to the next load increment if probe results show that no additional strong discontinuities are needed; otherwise, the corresponding new superimposed node sets are added and the new converged discrete solution is accordingly obtained at the same load increment since the finite element discretization has been changed by the addition of the new superimposed node sets. After the new converged discrete solution is obtained, delamination indicators are again invoked in the manner described above.
Figure 5.3 summarizes the flowchart of the adaptive delamination analysis. There are three unique elements (highlighted in yellow) exclusive to the adaptive strategy while the remaining steps are standard procedures in a nonlinear finite element code. The three unique steps are:
1. Invoke delamination indicators to test for potential delamination at every applicable position; 2. Identify positions in need of superimposed node sets where delamination may take place; and 3. Dynamically allocate storage for newly generated superimposed node sets consisting of: a. Superimposed node sets representing strong discontinuities at the positions predicted by delamination indicators (see Section 5.2.4); and
143 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
b. Superimposed node sets representing weak discontinuities at the newly developed delamination fronts (see Section 5.2.2).
Start
iinc=1
No End Yes Allocate storage spaces for the Form and solve the discrete newly generated superimposed nonlinear system of equations nodes in a dynamic data structure.
Test for delamination using delamination indicators
No Yes New superimposed node sets are needed?
Figure 5.3. Flow chart of the adaptive delamination simulation
5.2.4 Delamination Indicators
edgeedge faceface Let PPb and PPb be the two sets (Section 5.2.1) corresponding to edge and face superimposed node sets at the intact interfaces, respectively. The two node sets represent all the perfectly intact (depicted in cyan in Figure 5.2), weakly intact (yellow in Figure 5.2) areas. The superimposed node sets depicted by the blue dots in Figure 5.2 represent delamination front. Note
edge face that the two integer-pair sets, Pb and Pb , correspond to the collection of all not yet existent superimposed node sets and those representing weak discontinuity at the interfaces in the model.
144 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
Consider an edge superimposed node set depicted in Figure 5.4a located on the ith element edge in the global mesh capable of representing discontinuity at the jth interface intersecting the
edge element edge so that i j, P b . Likewise, consider a face superimposed node set in Figure 5.4b
face identified by an integer pair k l, P b . The numbering of the nodes within the superimposed node sets is illustrated in Figure 5.4.
(a) Edge superimposed node set (b) Face superimposed node set
Figure 5.4. Representation of a potential separation at a bonded interface in the surrogate
edge problems for the (a) edge superimposed node set ijP, b and (b) face superimposed node set
face klP, b .
The basic idea of the proposed delamination indicators is as follows. First, estimate the traction t b at an intact interface position corresponding to a superimposed node set whose integer-pair
edge face identification (ID) is included in either Pb or Pb . The likelihood of delamination can then be
assessed by inserting the estimated traction t b at that position into a traction-based damage
initiation criterion denoted as f tb 0 . If f tb 0 delamination is likely to occur and a superimposed node set is deemed to be necessary in the corresponding location. Note that in this 145 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites case of the present chapter, the intrinsic cohesive zone model will be incorporated at the originally
intact interface areas surrounding the superimposed node set. The traction t b is obtained through the traction-separation law which is defined in Appendix E.
The formulation of the delamination indicators is postulated on a sequence of surrogate problems.
e d g e face A surrogate problem is constructed for each superimposed node set in Pb and Pb . Consider
s e t an edge or face superimposed node set whose integer-pair ID mn, belongs to Pb . Note that the superscript “set” can be either “edge” or “face” to indicate the two different types of superimposed
set node sets illustrated in Figure 5.4. Let m denote the domain of the patch around the mth element edge or face. Such a patch is composed of all adjacent global elements corresponding to either a
set global element edge or face as shown in Figure 5.4. The boundary of the domain m is denoted
se t intset as m and the potentially discontinuous interface of interest is denoted by nm .
se t intset set We further introduce various partitions of m and nm based on the position of the patch m .
e x t If one of the patch boundaries intersects with the natural boundary t of the global problem
set t s e t u set domain as shown in Figure 5.5a, m is partitioned into natural m and essential m boundaries,
tsetsetext usetsetext intset such that mm t and mm \ t . Furthermore, if nm intersects with the delamination-front boundary (see the patches around the blue dots in Figure 5.2), then a portion of intset intset int set b set e set d set nm is not intact, i.e. delaminated. In this case, nm is defined as n m n m n m n m
b set e set e set d set d set b set b set with n m n m , n m n m and n m n m (see Figure 5.5b). Herein, nm
e set d set denotes the perfectly bounded intact interface area around the superimposed node set; n m n m
e set d set denote the portion of the interface with strong discontinuity where nm is elastic and nm has 146 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites irreversible damage. The surrogate problems for different superimposed node sets will be described for the most general case assuming all types of the patch boundaries and the interface partitions exist.
(a) Natural boundary patch (b) Delamination-front patch
Figure 5.5. Two special global element patches corresponding to the edge superimposed node
edge sets m, n Pb : (a) a natural boundary patch; (b) a delamination-front patch
set A surrogate problem corresponding to either an edge or face superimposed node set mnP, b is stated as follows. Given the converged discrete solution u over the entire problem domain, find
set set set the local perturbation u nmS in m such that for w nmS the following discrete weak form holds:
w :0σd w ts+ d w t d w t d w b d (5.1) sets t set int set+ int set set m m n m n m m
set where the function space nmS is defined as
set 0 set+ set- u set int set nS= mv x | v xC n m , n m , v x 0 on m , v x discontinous on n m (5.2)
147 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
0set+set- 0 s e t+ where C nmnm, denotes C continuous functions in the patch above nm and below
s e t - i n t s e t s nm the discontinuous interface nm ; σ is stress; t the prescribed surface traction on the
t s e t i n t s e t + i n t s e t natural boundary m ; nm and nm are the positive and negative sides of the interface i n t s e t nm , respectively. A side of an interface is considered positive if its outward normal vector points
i n t s e t + i n t s e t in the stacking direction of plies; t and t are the tractions acting on nm and nm ,
respectively; b is the prescribed body force; the symmetric gradient operator in sw is defined as
1 T www (5.3) s 2 where is the gradient operator.
Dependence of stress on the prior converged solution u and the unknown perturbation u is denoted as
set σσuus on m (5.4)
+ intset Tractions t and t at the interface nm are functions of the separation Δ , namely
b b b set t Δ t u u on nm e e e set t t t Δ t u u u u n nm (5.5) tdΔ t d u u u u on d set nm
+ intset where u and u are the values of u and u on nm , respectively; u and u represent the
int set b set corresponding values on nm . On the intact interface nm , 0 Δ u u resulting in
+ b b set Δ Δ u u ; t Δ represents the traction-separation relation at the intact interfaces nm ,
148 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites such that t 0b 0 ; te Δ and td Δare elastic and inelastic (with preexisting damage) traction- separation relations, respectively.
The trial and test functions, u and w , of the surrogate problem are defined as
iiedgeedge ud jjN3 onfor, edgeedge ijP (5.6) iiedgeedge i b ww jjN3 and
1 kface k face k face ud lNN12 l l 2 face face on k for k , l Pb (5.7) 1 kface k face k face ww lNN12 l l 2
edge where ijP, b denotes the integer-pair ID (see Section 5.2.1) of the edge superimposed node
i edge set at the intact interface; j N3 is the shape function of the third node in the edge superimposed
i edge i edge node set and j d is the corresponding nodal displacement (see Figure 5.4a); j w denotes the
face nodal values of the test function; k, l Pb is the integer-pair ID of the face superimposed node
k face k face set representing the intact interface areas; l N1 and l N2 are the shape functions of the first and
k face second superimposed nodes, respectively, in the face superimposed node set; l d is the
k face corresponding nodal displacement (see Figure 5.4b) and l w denotes nodal values of the test function.
Inserting (5.6) and (5.7) into the weak form of the surrogate problem (5.1) and taking advantage of the arbitrariness of the nodal values of the test functions yields:
mset m set m set m set m set m set m set m set m set set nrb n d n r e n d n r d n d n r int n d n r ext 0 for m , n P b (5.8)
149 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
edge For the edge superimposed node set i j, P b , the residuals in (5.8) are given as
iiiiirdtdedgeedgeedgebedgeedge NNd (5.9) jjjjjb33 b edge+ ji
iredge i d edge iN edge t e u u i N edge i d edge d (5.10) je j e edge+ j 3 j 3 j ji
iiiiirdtuudedgeedgeedge dedgeedge NNd (5.11) jjjjjd33 d edge+ ji
T iiiiirdBedgeedgeedgeedgeedge σudˆ Nd (5.12) jjjjjint3s3 edge i
iiirtbedgeedgesedge NdNd (5.13) jjj ext33 t edgeedge ii
i edge i edge where j B3 is the B matrix corresponding to the shape function j N3 . σˆ in (5.12) is the 61 stress matrix written in the Voigt notation.
face For the face superimposed node set klP, b , the residuals in (5.8) are given as
kkkkkrdtdfacefaceface bfaceface NNd (5.14) lllllb11 b face+ lk
kkkkrtuudfaceface efaceface NNd (5.15) lllle11 e face+ lk
krface kN face t d u u k N face k d face d (5.16) ldd face+ l 1 l 1 l lk
T kkkkkkrBBfacefacefacefaceface0.50.5 face σˆ ud NNd (5.17) llllllint12s12 face k
kkkkkrtbfacefaceface0.50.5 sfaceface NNdNNd (5.18) lllllext1212 t faceface kk
k face k face k face k face where l B1 and l B2 are the B matrices corresponding to the shape functions l N1 and l N2 , respectively. σˆ in (5.17) is the 61 stress matrix written in the Voigt notation.
150 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
m s e t Equation (5.8) represents a nonlinear system of three equations with three unknowns n d for
set each edge and face superimposed node set m, n P b . Further simplification follows from the perturbation analysis assuming that (a) u O where 0< ≪1, and (b) the stiffness of
tteb undamaged interface satisfies that Ke1O due to infinitesimality of the
d interface thickness. Consider the Taylor series expansion of s uu , t , te , tb around u 0 and 0 , respectively.
ssssu uuuu ... O u s O() O(1) td tttddd ... O O() O 1 O(1) (5.19) te tttKeeee O() O(1) O() 1 tb tt0Kbbe O() O(1) O() 1
m set Inserting (5.19) into (5.8) yields a linear system of equation for three unknowns n d for each
set edge and face superimposed node set m, nP b .
mset m set m set m set m set m set set nK n d n rext n r int 0 n r e 0 n r d 0 for m , n P b (5.20) where
2 iiKKedgeN edge d e for i , j P edge (5.21) jjbeedge+ edge+ 3b j i j i
151 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
2 kkKKfaceN face d e for k , l P face (5.22) llb face+ e face+ 1b l k l k
It is possible that not all quadrature points in a cohesive element are uniformly elastic or uniformly
e s e t d s e t damaged, and thus classification into either nm or nm need further definition. Hereafter, a
d s e t cohesive element is considered to be belonging to nm if the damage variable at the centroid is
e s e t nonzero. Otherwise, a cohesive element is considered to be elastic and belonging to nm . The likelihood of delamination at the position of a superimposed node set is typically related to a certain damage initiation criterion that can be expressed as f tb 0. The delamination is assumed to
bm set take place when f td n 0 . In the present chapter, the damage initiation criterion is assumed to be the same as that employed by the intrinsic cohesive zone model detailed in Appendix
E.
5.3 Numerical examples
In this section two numerical examples with increasing complexity are presented to study the accuracy and computational efficiency of the adaptive s-method for delamination analysis. All the laminated structures considered herein have no initial delamination. The initiation and subsequent propagation of delamination are guided by the adaptive strategy outlined in the previous section.
To assess the accuracy and computational efficiency of the proposed method, each numerical example is accompanied with a reference solution in which ply-by-ply discretization is adopted with double nodes and cohesive elements placed along all interfaces. Since the discretization space reproduced by the s-method is identical to the ply-by-ply discretization (see Chapter 4), the adaptive s-method should be able in principle to reproduce the reference solution provided that the 152 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites adaptive strategy is capable of identifying the onset and propagation of delamination fronts. The computational efficiency of the adaptive s-method in comparison to the brute force approach of placing cohesive elements at all interfaces currently exercised in practice depends on the adaptive s-method’s ability to pinpoint the critical node sets on one side, and the computational overhead that comes with adaptivity on the other. In principle, if the adaptive s-method is capable of significantly reducing the problem size with little overhead and without sacrificing accuracy, it would provide a viable alternative to conventional practices.
The composite plies are assumed to be transversely isotropic. The elastic properties of the unidirectional fiber-reinforced composites are listed in Table 5.1. Cohesive law parameters (see
Appendix E) considered in this study are given in Table 5.2.
Table 5.1. Elastic single ply properties
E1(MPa) EE23 (MPa) 12 13 23 GG12 13 (MPa) 55000 9500 0.33 0.45 5500
Table 5.2. Cohesive law parameters
cn (MPa) cs (MPa) ct (MPa) cr 3.2 3.2 3.2 0.001 1 1 1 GI(Nmm) GII (Nmm) GIII (Nmm) s t 0.64 0.64 0.64 0.1 0.1
All the mesh-related visualizations in this section are realized through Gmsh [186].
5.3.1 A laminated plate problem
An originally intact laminated plate with 10 layers of unidirectional fiber-reinforced composites is subjected to Mode I loading as shown in Figure 5.6. The size of the plate is 30 24 2mm3 . All
153 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites the plies in the plate have the same thickness. The stacking sequence from the bottom up is
90/45/0/ 45/ 90/45/0/ 45/ 90/45 . Fibers in 0 plies are aligned along the x1 axis.
The plate is fixed at the two horizontal edges on the left. The vertical displacement of 1mm and
1m m are prescribed at the upper and lower edges on the right, respectively. In the reference (ply- by-ply) simulation the plate model consists of 2 0 1 6 1 0 20-node hexahedral laminated elements each of which contains only one ply. In addition, 20 16 9 16-node quadrilateral cohesive elements of zero thickness are inserted at all the nine interply interfaces. The global mesh in the adaptive simulation contains one layer of 2 0 1 6 1 hexahedral laminated elements each of which contains all ten plies.
Figure 5.6. Geometry, layup and loading of the plate problem
The final deformed shapes of the foregoing meshes corresponding to the reference and adaptive simulations are shown in Figure 5.7. The contour plots denote vertical displacement. It is evident
154 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites that the total number of integration points associated with the bulk material (see Appendix D.1) in the reference simulation is identical to that in the adaptive simulation at all time. Thus, the CPU time used for element integration is roughly the same in both simulations.
(a) Mesh in the reference simulation
(b) Global mesh in the adaptive simulation
Figure 5.7. Final deformed meshes: (a) reference simulation; (b) the global mesh in the adaptive simulation
The resultant reaction forces in the adaptive simulation are compared to those of the reference simulation in Figure 5.8. It can be seen that the reaction force-displacement responses as obtained with the adaptive s-method and the reference solution are very close. The maximum relative error in the reaction forces is 3.9% while mostly the relative error is below1% .
155 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
(a) Upper edge (b) Lower edge
Figure 5.8. Comparison of the reaction force-displacement curves as obtained by the reference and adaptive simulations: (a) reaction force at the upper edge; (b) reaction force at the lower edge.
The damage state at the interfaces in the two simulations is subsequently compared. Interfaces are numbered from bottom up. Interfaces 1 and 9 are selected for detailed comparison since the two interfaces are most damaged. Figure 5.9 depicts the contour plots of damage in Interface 1 at different loading stages. Figure 5.9 also shows the positions of adaptively generated edge superimposed node sets that characterize either strong or weak discontinuity. It can be seen that the results obtained from the adaptive simulations agree very well with the reference simulation in terms of the delamination initiation and propagation at Interface 1. Similar observation can be made for Interface 9 as can be seen from Figure 5.10. Smaller areas of damaged zones are found in Interfaces 2 and 8 and good agreement is again observed between the reference and adaptive simulations as shown in Figure 5.11.
156 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
Superimposed node set with strong discontinuity Superimposed node set with weak discontinuity
Reference simulation Adaptive simulation
(a)
(b)
(c)
157 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
(d)
Figure 5.9. Damage state in Interface 1 for different prescribed displacements in the reference and adaptive simulations: (a) d mm ; (b) d mm ; (c) d mm ; (d) d mm .
In the remaining of this example, we study computational efficiency of the adaptive method.
Clearly adaptive models contain considerably fewer degrees-of-freedom than the reference ply- by-ply discretization. Figure 5.12 compares the numbers of degrees-of-freedom in the two models.
Note that while the reference simulation has a fixed number of degrees-of-freedom (reflected in a horizontal line in Figure 5.12), in the adaptive simulation, the number of degrees-of-freedom increases throughout the simulation. The number of degrees-of-freedom in the adaptive simulation reaches 2 1,5 2 6 by the end of the simulation. This is less than one third of the number of degrees- of-freedom in the reference simulation. Figure 5.13 shows the increasing wall-clock time of the two simulations with increasing prescribed displacements. The same running environment and solution control are used for both simulations. The simulations were conducted on an idle Intel®
Xeon® E5506 CPU (Intel Corporation, CA, USA) and 6GB RAM desktop. It can be seen that the amount of time for the adaptive simulation is considerably smaller than that of the reference simulation. With the completion of the loading process, the adaptive simulation provided the wall time saving of around 2.82 hours, which is 36.5% of the total wall time consumed by the reference
158 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites simulation. For prescribed displacements equal to 0 . 1 0 5 m m, the relative time saving was 7 1 . 1 % ; more than half of the time is saved until the prescribed displacement reaches 0 . 4 4 5 m m. Further speedup gains can be obtained if reduced integration is employed. The breakdown of the wall time is detailed in Figure 5.14. It can be seen that solving the linear systems of equations is the most time-consuming operation ( 6 . 3 7 hours or 8 2 . 5 % of the total running time) for the reference simulation. In the adaptive simulation, the linear system of equations is solved in less than one fourth of the time required by the reference simulation due to the much smaller systems of equations. The overhead in the adaptive simulation is primarily attributed to the increased number of Newton-Raphson iterations: 1,6 2 8 in the adaptive method versus 801 in the reference simulation. The reason lies in the iterative nature of the adaptive algorithm as shown in Figure 5.3.
Furthermore, while the number of degrees-of-freedom is considerably reduced by the adaptive model, the number of integration points associated with the bulk material remains the same. Due to increased number of iterations, reducing the CPU time of the integration process, such as by employing reduced integration instead of full integration considered in this study, would further increase the advantage of the adaptive model.
159 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
Superimposed node set with strong discontinuity Superimposed node set with weak discontinuity
Reference simulation Adaptive simulation
(a)
(b)
(c)
160 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
(d) Figure 5.10. Damage evolution in Interface 9 for different prescribed displacements in the reference and adaptive simulations: (a) d mm ; (b) d mm ; (c) d mm ; (d) d mm .
Superimposed node set with strong discontinuity Superimposed node set with weak discontinuity Reference simulation Adaptive simulation
(a) Interface 2
(b) Interface 8
Figure 5.11. The final damage state in the reference and adaptive simulations: (a) Interface 2; (b) Interface 8. 161 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
Figure 5.12. Numbers of degrees-of-freedom versus prescribed displacements in the reference and adaptive simulations
Figure 5.13. Wall-clock time-displacement curves of the reference and adaptive simulations
162 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
Figure 5.14. The final breakdown of the Wall time in the reference and adaptive simulations
5.3.2 A laminated shell problem
A laminated shell representing one sixth of a laminated cylindrical tube depicted in Figure 5.15 is analyzed. The cylinder is 4 0 m m in diameter, 3 0 m m length and 2 m m thickness. The two longitudinal edges of the inner shell surface are constrained and a downward prescribed displacement in the amount of 2mm is prescribed at the center point of the inner surface as shown in Figure 5.15. The shell consists of a stack of 10 layers of unidirectional fiber reinforced composites of equal thickness forming a layup configuration of
90/45/0/ 45/ 90/45/0/ 45/ 90/45. Fibers in the 0 plies are aligned along the x1 axis.
In other plies the orientation of fibers is determined by rotating the x1 direction around the outward normal to the cylindrical surface. The model in the reference simulation consists of 282010
20-node hexahedral laminated elements each of which contains only one ply. All the nine interply interfaces in the model are represented by 28 20 9 16-node interface cohesive elements of zero
163 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites thickness. The global mesh in the adaptive simulation contains one layer of 2 8 2 0 1 20-node hexahedral laminated elements each of which contains all ten plies.
The deformed meshes considered in this study are shown in Figure 5.16. To visualize the delamination, the two meshes are cut along the cross section that passes the loading point and the front half of the shell is removed. The contour plots on the deformed meshes depict the distribution of the vertical component of displacements.
Figure 5.15. Geometry, layup and loading of the laminated shell problem
We first study the accuracy of the adaptive method. The vertical reaction force at the loading point and the horizontal reaction force at the right supporting edge of the model as functions of the prescribed displacement are depicted in Figure 5.17. It can be seen that the reaction force- displacement responses as obtained by the adaptive simulation are almost undistinguishable from the reference simulation.
164 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
(a) Mesh in the reference simulation
(b) Global mesh in the adaptive simulation
Figure 5.16. One half of the deformed meshes at the end of the simulations: (a) reference simulation; (b) the global mesh in the adaptive simulation.
165 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
(a) Vertical reaction force (b) Horizontal reaction force at the loading point at the right supporting edge
Figure 5.17. Reaction forces as functions of the prescribed displacement: (a) vertical reaction force at the loading point; (b) horizontal reaction force at the right supporting edge.
Figure 5.18 and Figure 5.19 depict the damage state at interfaces as obtained with the reference and adaptive simulations. Following the same convention as in the previous test problem, the interfaces in the laminated shell are numbered from bottom up.
It can be seen (Figure 5.18) that the area above the loading point at the lowermost interface
(Interface 1) in the laminated shell is most damaged. Nearly identical results are obtained by the adaptive and reference simulations regarding the damage state in Interface 1. For visualization, the global mesh in the adaptive simulation is cut along interfaces of interest.
Considerably less damage (see Figure 5.19) has been observed in Interface 2 that is one ply above
Interface 1. It can be seen that the damage state in Interface 2 predicted by the adaptive simulation agrees well with the reference simulation. It is noteworthy to point out (see Figure 5.19c and Figure
5.19d) that the damage at Interface 2 ceases to develop when the magnitude of the prescribed displacement is above mm . Most of the newly generated superimposed node sets are
166 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites introduced in interface areas with increasing likelihood of delamination after the prescribed displacement reaches mm . This can be observed through Figure 5.20 where the contour plots of the magnitude of the separation at the interface are shown at the two instances.
Superimposed node set with strong discontinuity Superimposed node set with weak discontinuity
Reference simulation Adaptive simulation
(a)
(b)
(c)
167 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
(d)
Figure 5.18. Damage state in Interface 1 as obtained by the reference and adaptive simulations for various prescribed displacements d : (a) d mm ; (b) d mm ; (c) d mm ; (d) d mm .
We proceed to study computational efficiency of the adaptive method. Figure 5.21 depicts the increase in the number of degrees-of-freedom in the adaptive simulation as a function of the prescribed displacement. The fixed number of degrees-of-freedom in the reference simulation is illustrated by a horizontal line. By the end of the simulation, the number of degrees-of-freedom in the adaptive simulation reaches 32,078 in comparison to 124,643 in the reference simulation.
Figure 5.22 shows that the adaptive simulation is significantly faster than the reference simulation.
The two simulations were conducted on an idle Intel® Xeon® E3-1281 CPU (Intel Corporation,
CA, USA) and 16GB RAM machine. Identical load increments were used for both simulations.
Figure 5.22 is therefore an appropriate reflection of the computational efficiency of the two simulations throughout the loading history. It took 11.08 hours to complete the reference simulation in comparison to 3.93 hours for the adaptive simulation, which corresponds to 64.5% improvement. In the earlier stages when the delamination region is relatively small, the time saving are in the range of 80% to 90%. The breakdown of the wall time is given in Figure 5.23. It can be
168 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites seen that approximately 87% of the total time spent on solving linear equations in the reference simulation is saved by the adaptive simulation. Note that the advantage of adaptive simulations will further increase if reduced integration is employed.
Superimposed node set with strong discontinuity Superimposed node set with weak discontinuity
Reference simulation Adaptive simulation
(a)
(b)
(c)
169 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
(d)
Figure 5.19. Damage state in Interface 2 as obtained by the reference and adaptive simulations for various prescribed displacements d : (a) d mm ; (b) d mm ; (c) d mm ; (d) d mm .
Superimposed node set with strong discontinuity Superimposed node set with weak discontinuity Reference simulation Adaptive simulation
Magnitude of Separation at Interface 2 (μm) Magnitude of Separation at Interface 2 (μm) 0 0.092 0.18 0.28 0.37 0.46 0.55 0.65 0.74 0.83 0.92 0 0.096 0.19 0.29 0.38 0.48 0.57 0.67 0.76 0.86 0.96 3 3 (a)
Magnitude of Separation at Interface 2 (μm) Magnitude of Separation at Interface 2 (μm) 0 0.092 0.18 0.28 0.37 0.46 0.55 0.65 0.74 0.83 0.92 0 0.096 0.19 0.29 0.38 0.48 0.57 0.67 0.76 0.86 0.96 3 3 (b) Figure 5.20. Contour plots of displacement separation in Interface 2 as obtained by the reference and adaptive simulations for two different prescribed displacements: (a) d mm ; (b) d mm . 170 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
Figure 5.21. Numbers of degrees-of-freedom in the reference and adaptive simulations versus prescribed displacement
Figure 5.22. Time consumed by the reference and adaptive simulations as a function of the prescribed displacement
171 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites
1
Figure 5.23. The final wall time breakdown in the reference and adaptive simulations
5.4 Conclusions
The classical ply-by-ply discretization of individual layers may increase the size of the problem by an order of magnitude in comparison to laminated shell or plate elements. As an alternative, a hierarchical adaptive approach based on the s-version of the finite element method is proposed to address the computational efficiency issue of delamination analyses in large-scale laminated composite structures. The adaptive strategy features delamination indicators, which introduce cohesive interfaces only if and where they are needed.
Two problems have been analyzed: a smaller plate problem and a larger cylindrical shell problem both having 10 plies. The time savings offered by the adaptive model are more pronounced as the problem size increases. Furthermore, in practical applications when the delamination is localized to relatively small regions the advantage of adaptivity is considerably larger. This can be clearly
172 Chapter 5. The Adaptive S-Method for Delamination Analysis in Laminated Composites seen from Figure 5.22. When the prescribed displacement is equal to 0.7 mm, the adaptive approach is about five times faster than the classical ply-by-ply discretization.
The overhead in the adaptive simulation is primarily attributed to the increased number of iterations due to the iterative nature of the adaptive algorithm as shown in Figure 5.3. While the number of degrees-of-freedom is considerably reduced, the total number of integration points associated with bulk materials remains the same. Due to increased number of iterations, reducing the CPU time of the integration process, for instance by either employing reduced integration instead of full integration or by employing 14-point quadrature rule (instead of 27), would further increase the advantage of the adaptive model.
173 Chapter 6. Conclusions
Chapter 6 Conclusions
In the concluding chapter, the main contributions of this dissertation are outlined with directions for future research discussed.
6.1 Main contributions
The dissertation contributes to two aspects of enhancing reliability or efficiency of numerical simulation of failure in solids. For ductile failure which is most often characterized by large strain plasticity, a relation is revealed between two seemingly distinct large strain plasticity constitutive frameworks both of which enjoy widespread popularity in numerical approaches to studying plastic behavior of solids. For brittle failure which features discontinuous displacement field, an adaptive methodology is developed to realize reliable, efficient finite element analysis of initiation and propagation of delamination (displacement continuity) in layered materials. The contributions are detailed as follows:
The kinetic logarithmic stress rate: Through an unloading stress ratchetting obstacle test,
it is shown that additive hypo-elasto-plasticity models based on the logarithmic stress rate
[53,57,59] give rise to nonphysical energy dissipation in elastic unloading processes. The
kinetic logarithmic stress rate is accordingly proposed to eliminate such physical
inconsistency. It is proved that the kinetic logarithmic stress rate, which results from a
174 Chapter 6. Conclusions
simple modification to the definition of the logarithmic rate, renders the corresponding
additive hypo-elasto-plasticity models capable of representing energy dissipation-free
elastic responses whenever plastic flow is absent.
A link between the multiplicative hyper-elasto-plasticity and additive hypo-elasto-
plasticity models: It is proved that for isotropic materials the multiplicative models
coincide with the additive ones if a newly discovered objective stress rate is employed.
This objective stress rate, which reduces to the kinetic logarithmic rate in the absence of
strain-induced anisotropy, is termed the modified kinetic logarithmic stress rate. Such
equivalence between the two categories of elastoplasticity models is illustrated through
numerical model problems in which homogeneous deformation along with J2 plasticity
with associative flow rule is considered.
Hierarchical representation of delamination in laminated composites: One single layer
of 20-node serendipity solid laminated elements are employed to resolve delamination-free
response of shell structures made of laminated composites. It is proved in this case that
delamination representations based on the s-version of the finite element method (s-method)
and that based on the extended finite element method (XFEM) are equivalent to the
conventional ply-by-ply approach in the sense that they all have the same discretization
(approximation) space. In addition, it is shown that the s-method provides a more
straightforward pathway than the XFEM to switch from weak to strong discontinuities.
Adaptive delamination analysis: The adaptive s-method is developed to simulate the
onset and propagation of delamination throughout a large-scale structural component made
of laminated composites without incurring excessive computational cost. Based on the s-
175 Chapter 6. Conclusions
method representation of delaminated interfaces, such an adaptive approach features
delamination indicators which are proposed to detect the onset and propagation of
delamination. Numerical examples show that the computational cost of the adaptive s-
method is significantly lower than that incurred by the conventional ply-by-ply approach
despite the fact that the two approaches produce practically identical results. This
efficiency advantage of the adaptive s-method is more pronounced as the problem size
increases.
6.2 Directions for future research
Several directions of future research are discussed as follows:
In additive hypo-elasto-plasticity models, the hypoelastic relation and kinematic hardening
law contain objective stress rates of the Kirchhoff stress and back stress, respectively. A
theoretical question that has not been addressed is whether it is necessary to use the same
definition of corotational objective stress rate for the Kirchhoff stress and back stress in
additive hypo-elasto-plasticity models as suggested in [39]. The question arises from the
fact that most multiplicative hyper-elasto-plasticity models can be shown to indeed employ
different definitions of corotational objection stress rate for the Kirchhoff stress and back
stress in their equivalent additive hypo-elasto-plasticity forms based on the modified
kinetic logarithmic spin (see Section 3.5.1). There seems to be a conflict between many
existing multiplicative models and the requirement of identical corotational objective stress
rate definition for the Kirchhoff stress and back stress. This issue needs to be addressed in
future work.
176 Chapter 6. Conclusions
The analysis of large strain elastoplasticity models presented in the dissertation is limited
to isotropic materials in the sense that both the elastic response and yield condition are
isotropic. In addition, all the elastoplasticity models considered herein are rate independent.
The proposed kinetic logarithmic rate based formulation as well as the revealed relation
between the multiplicative and additive models may be generalized to materials with
inherent anisotropy or rate dependency, such as polymer matrix composites (PMC). Elastic
strains in a polymeric phase of PMC, while constrained by stiff inclusions, might be still
sizeable due to stretching of polymer chains. The aforementioned generalization would be
of interest to both academic community and practitioners.
The thickness of structural components made of laminated composites are usually several
orders of magnitude smaller than the in-plane dimensions. Shell and plate elements are
therefore especially suitable for modeling of such structural components. Computational
viability and efficiency of delamination analysis in these structural components may be
further increased if the solid element based adaptive s-method presented in this dissertation
is adapted to laminated shell or plate elements.
177 References
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197 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
Appendix A Numerical Integration of Additive Hypo-Elasto- Plasticity models
A.1 Numerical integration of the additive hypo-elasto-plasticity model for J2 elastoplasticity based on the logarithmic stress rate
Consider a hypo-elasto-plasticity model described by (2.7), (2.40)-(2.44), (2.18) and (2.19).
Assume that the Kirchhoff stress τ , the back stress α and the internal variable for isotropic
hardening are given at an instant tn and are denoted as τ n , αn and n , respectively. The values
τ n1 , αn1 and n1 of these variables at time tn1 ttnn1 will be numerically integrated as described herein.
Among the above governing equations, (2.41), (2.43) and (2.44), which describe either the responses of stresses or evolution of the internal variables in the rate form, will be first expressed in the integrated form.
Considering (2.7) and (2.31), the relation (2.41) can be rewritten as
τlog Hp: D Dˆ (A.1)
Inserting (2.55) and (2.42) into (A.1) yields
198 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
D QQloglog Tloglog TH τQQDn : (A.2) Dt where
τα n (A.3) τα
Due to isotropy of the fourth-order tensor H , (A.2) can be rewritten as
D Qlog TlogHlogτQQDQQnQ Tloglog Tlog : (A.4) Dt
Inserting (2.54) into (A.4) yields
DDlog TlogHlog Tloglog Tlog Q τ QQV QQn:ln Q (A.5) DtDt
H Following the integration of (A.5) from tn to tn1 and taking advantage of the isotropy of yields
loglog T τQτQnnnn1 t (A.6) Hloglog : lnln TloglogVQVQQQn Tloglog T QQ n1 ds nnnnnn111 t n with
log Tloglog T QQQnnn 1 (A.7)
log log where lnVn1 and Qn1 denote the values of lnV and Q at instant tn1 , respectively, while
log lnVn and Qn denote the corresponding values at instant tn .
Similarly to (A.6), equation (2.43) can be integrated to obtain
199 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
t α Qlog α Q log T c Q log n1 Q log T n Q log ds Q log T (A.8) n1 n n n n 1t n 1 n
Note that in the present section (Appendix A.1) the nonlinear kinematic hardening parameter b is assumed to be zero. The numerical integration of the nonlinear kinematic hardening law is considered in Appendix A.3.
Integrating (2.44) yields
t Kdsn1 (A.9) nn1 t n
The radial return mapping algorithm [63] allows for the following approximation with regard to
(A.6) and (A.8):
t n1 QnlogTloglogTlog QQnQ ds (A.10) t nnnn 111 n with
t tds n1 (A.11) nnn 1 t n and
τα n nn11 (A.12) n1 ταnn11
where n1 denotes the plastic multiplier at instant tn1 and tttnnn 1 is the length of the
time interval ttnn, 1 .
Combining equations (A.6), (A.8) and (A.9) the following relations can be obtained:
trial H τn1 τ n 1 n: n n 1 (A.13)
200 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
trial ααnnnnn111 c (A.14)
trial nnn11 K (A.15) with
trialloglog THloglog T τQτQVQVQnnnnnnnn11 :lnln (A.16)
trialloglog T αQαQnnnn1 (A.17)
t r i a l nn1 (A.18)
t r i a l t r i a l t r i a l l o g It is evident that τn1 , αn1 and n1 can be evaluated as long as Qn is known. The rotation
log Qn is approximated by the numerical integration algorithm detailed in Appendix A.2. The
values of τ n1 , αn1 and n1 are in turn obtained given n . The scalar n is obtained through the Kuhn-Tucker complementarity conditions as described below.
trialtrialtrial If ταnnn111,,0 , the process is considered to be elastic and n vanishes leading to
trial ττnn11 (A.19)
trial ααnn11 (A.20)
trial nn11 (A.21)
trial trial trial If ταn1, n 1 , n 1 0 , the process is considered to be plastic and n should be positive and
satisfy ταn1, n 1 , n 1 0 . In case of the plastic process, equations (A.12), (A.13), (A.14) yield
trial trial τn1 α n 1 τ n 1 α n 1 2 c n (A.22)
The Kuhn-Tucker complementarity conditions can therefore be written as
201 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
τn1,, α n 1 n 1 τ n 1 α n 1 n 1 (A.23) trial trial trial ταn1, n 1 , n 1 2 cK n 0
from which the value of n is given as
trialtrialtrial ταnnn111,, (A.24) n 2 cK
The values τ n1 , αn1 and n1 follow from (A.13)-(A.15).
p Based on (2.48), the plastic work Wn in the time interval ttnn, 1 is given as
t Wdsppn1 τD: ˆ (A.25) n t n and in view of (2.42), the plastic work can be approximated as
t Wdsp n1 τnτn:: (A.26) nnnn t 11 n
A.2 Numerical integration of the logarithmic spin tensor
The rotation tensor Q l og satisfies the ordinary differential equation (2.53) with initial value of
loglog QQ n (A.27) tt n
log The following algorithm aims at approximating the rotation tensor Q at tn1 ttnn1 .
log log Let Qn1 denote the value of Q at tn1 . Based on [63] it can be written as
log log log Qn1 exp t nΩQ n n (A.28)
202 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
l o g where Ωn is the logarithmic spin tensor at instant tttnnn 1 1 with 0 ,1 ; e x p denotes the exponential map of the corresponding skew-symmetric tensor. Consider a skew- symmetric tensor Ω written in the matrix form with its components expressed in the Cartesian coordinate system:
0 1231 Ω 0 (A.29) 1223 3123 0
The skew-symmetric matrix (A.29) can be expressed in the vector form as
T ω 233112 (A.30)
The tensor e xpΩ can then be written as
2 sin ω 1 sin2 ω expΩIΩΩ 2 (A.31) ωω22 where I is the second-order identity tensor and ω is the L2-norm of ω .
log The skew-symmetric tensor tnnΩ in (A.28) can be approximated as follows. According to
log (2.33), Ωn is written as
log ΩWWnnnn BD , (A.32)
with Wn , Bn and Dn denoting the values of the vorticity tensor W , the left Cauchy-Green
deformation tensor B and the rate of deformation tensor D at instant tn , respectively. Using
the generalized midpoint rule, Wn and Dn can be approximated as [63]
203 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
1 1 WFFFnnnn skw 1 (A.33) tn
1 T1 DFCCFnnnnn 1 (A.34) 2tn
where Fn , Fn and Fn1 denote the values of the deformation gradient F at instances tn , tn
and tn1 , respectively; C n and Cn1 are the values of the right Cauchy-Green deformation tensor
T CFF at tn and tn1 , respectively. The expression (2.36) indicates that the function
W B D, is linear with respect to D in the sense that the relation WBDWBD ,,aa holds for an arbitrary scalar a . Consequently, inserting (A.33) and (A.34) into (A.32) yields the
log following approximation of tnnΩ :
1 log1T1 tnΩFFFW nnnnnnnnnskw, BFCCF 11 (A.35) 2
A.3 Numerical integration of the additive hypo-elasto-plasticity model for J2 elastoplasticity based on the kinetic logarithmic stress rate
With the logarithmic stress rate log replaced by its kinetic counterpart kl og in (2.41) and
(2.43), the additive hypo-elasto-plasticity model based on the kinetic logarithmic stress rate can be completely described by (2.7), (2.40)-(2.44), (2.18) and (2.19). Suppose that the values of the
Kirchhoff stress τ , the back stress α and the internal variable for isotropic hardening, τ n , αn
and n , respectively, are given at instant tn . Herein the focus is on the approximation to the values
of the above variables at instant tn1 ttnn1 , i.e. τ n1 , αn1 and n1 .
204 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
Following an integration procedure similar to that in (A.1)-(A.9) of Appendix A.1, τ n1 , αn1 and
n1 can be expressed as
klogklog THklogp klog τQτQεεnnnnnn1 : (A.36)
klog klog T p klog αn1 Q n α n Q n cb ε n d n (A.37)
t Kn1 ds (A.38) nn1 t n with
t εQQDklogklogklog QQ Tklogklogn T1 ds (A.39) nnn 11t n
t εQQDQQp klogklogklog Tpklogklogn1 T ˆ ds (A.40) nnn 11t n
t d Qklog n1 Q klog T α Q klogds Q klog T (A.41) n n11t n n where Qklog is a proper orthogonal tensor satisfying the following ordinary differential equation:
QklogklogklogΩQ (A.42)
klog k l o g k l o g The values of Q at instants tn and tn1 are denoted as Qn and Qn1 , respectively. The tensor
klog Qn in (A.36) is the relative rotation tensor from tn to tn1 which is defined as
klog klog klog T QQQn n1 n (A.43)
Following approximation of the integrals in (A.39), (A.40) and (A.41), the following expressions are obtained
205 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
klogklogklog Tklogklog T εQQDQQnnnnnnn 11 21 21 21 t klogklog T QDQnnnn1 21 21 2 t (A.44) klogklog T Qnnn1 21 2 εQ
p klogklogklog Tpklogklog T ˆ εQQDQQnnnnnnn 11111 t (A.45) ˆ p tnnD 1
klogklog Tklogklog T dQQnnnnnnnn111111 αQQ t (A.46) tnnn 11α
kl og klog where Qn12 and Dn12 denote the values of the rotation tensor Q and the rate of deformation
1 klogklogklog T tensor D , respectively, at instant ttt ; Q=nnn1 QQ 211 2 is the relative nnn1 21 2
ˆ p rotation tensor from tn12 to tn1 and εDn t n n12 ; Dn1 is the value of the plastic rate of
ˆ p deformation tensor D at tn1 ; n1 and αn1 are the values of the plastic multiplier and back stress
at instant tn1 , respectively.
According to the approximation (A.34) in Appendix A.2, εn can be written as
1 εFCCF T1 (A.47) nnnnn2 1 211 2
where Fn12 is the value of the deformation gradient F at tn12; C n and Cn1 are the values of
T the right Cauchy-Green deformation tensor CFF at tn and tn1 , respectively.
p klog In addition, due to (2.42) and (A.3), εn can be rewritten as
p klog εn t n n1 n n 1 n n n 1 (A.48)
206 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
where nnn t 1 can be regarded as an approximation to the integral of the plastic multiplier
over the time interval ttnn, 1 . Consequently, (A.38) can be rewritten as
nnn 1 K (A.49)
k l o g kl og The above relative rotation tensors Qn and Qn12 are then approximated via numerical integration of the ordinary differential equation (A.42). According to (A.28) in Appendix A.2, the
k l o g approximation to Qn is written as
klogklog Qnnn exp t Ω (A.50)
klog where the parameter in (A.28) is set as 0 and thus Ωn denotes the value of the kinetic
klog logarithmic spin tensor Ω at instant tn .
klog Making the usual approximation to Qn12 (see [63] among others)
klogklog2 QQnn 12 (A.51) and therefore
klogklog 1 Qnnn12 expt Ω (A.52) 2
klog In view of (2.74), Ωn can be written as
klogk ΩWWnnnn BD , (A.53)
k k with Wn , Bn and Dn denoting the values of the tensors W , B and D at instant tn , respectively.
In light of the approximation (A.33) and (A.34) in Appendix A.2, Wn and Dn can be written as
207 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
1 1 WFFInnnskw 1 (A.54) tn
1 T1 DFCFInnnn 1 (A.55) 2tn
k The relations (2.75) and (2.76) suggest that Bn is a function of τ n , i.e.
kk BBnnn τ (A.56)
Inserting (A.54)-(A.56) into (A.53), the following approximation is obtained:
klog1kT1 1 tnΩF nnnnnnnn FIWskw, B11 τFCFI (A.57) 2
In summary, the above approximations of (A.36)-(A.38) yield
klogklog THklogklog T τQτQQεQnnnnnnnnn11 21 21 n : (A.58)
1 klog klog T αn11 Q n α n Q n c n n n (A.59) 1b n
nnn1 K (A.60)
τα n nn11 (A.61) n1 ταnn11
klog T klog T where the rotation tensors Qn and Qn12 are defined by (A.50), (A.52) and (A.57).
At first, the trial state is obtained by assuming n 0 in which case the trial values of τ n1 , αn1
and n1 can be expressed as
208 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
trialklogklog THklog τQτQεnnnnn1 : trialklogklog T αQαQnnnn1 (A.62) trial nn1
trialtrialtrial If ταnnn111,,0 , the trial values and the parameter n 0 satisfy the Kuhn-Tucker complementarity conditions and therefore the following relations hold:
trial ττnn11 (A.63)
trial ααnn11 (A.64)
trial nn11 (A.65)
trialtrialtrial If ταnnn111,,0 , the value of the parameter n would make ταnnn111,,0 to hold, i.e.
ταταnnnnnn111111,,0 (A.66)
Equation (A.66) along with (A.58)-(A.61) constitute a system of nonlinear equations. The system of nonlinear equations will be first rewritten in an alternative form to facilitate efficient numerical solution.
Note that since the back stress αn at tn is assumed to be traceless, i.e. tr0αn . Equations (A.59)
and (A.61) suggest that nn1 and αn1 should be traceless as well.
Considering (2.31), the equation (A.58) can then be written as
klog klog T klog τn11 Q n τ n Q n 2 ε n n n n (A.67)
trτn1 tr τ n 3 tr ε n (A.68)
209 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
Subtracting (A.59) from (A.67) and considering (A.61) yield that
c n 1 12nnη 1 1b nnη 1 (A.69) ητb klogklog Tklognnn QQnnn 2 ε 1b n where
ηταnnn111 (A.70)
ηn τ n α n (A.71)
Taking the norm on both sides of (A.69), the following holds
ητb c klogklog Tklognn n n ηQQεnnnnn1 22 (A.72) 11bb nn
It can be seen that n is the only unknown on the right-hand side of (A.72).
Substituting (A.70) and (A.60) into the consistency condition (A.66) yields
ηnnn1 K 0 (A.73)
Considering the expression (A.72), (A.73) indeed represents a nonlinear equation for n whose numerical solution can be obtained through the Newton-Raphson method.
With n obtained, ξn1 can then be computed by substituting n into (A.69). Consequently,
the values τ n1 , αn1 and n1 are obtained by evaluating (A.58), (A.59) and (A.60), respectively.
Remark A.1 If the plastic spin Wˆ p is considered herein as described in Remark 2.3 of Section
2.4.1, the backward Euler method is employed for the approximation of the rotation tensor
210 Appendix A. Numerical integration of additive hypo-elasto-plasticity models
k l o g corresponding to the spin expressed by (2.100). This renders the rotation tensors Qn and
kl og Qn12 dependent on τ n1 , αn1 and n through the exponential mapping (A.31). It is therefore
no longer trivial to obtain an explicit representation of the nonlinear equation for n similar to
(A.73). Instead, a system of nonlinear equations (A.58)-(A.61), (A.66) has to be solved simultaneously.
211 Appendix B. Eigenprojections of a symmetric second-order tensor
Appendix B Eigenprojections of a Symmetric Second-Order Tensor
Eigenprojections are often introduced to circumvent complexity induced by various possibilities of repeated eigenvalues (i.e. multiplicities of eigenvalues) of a matrix or tensor [76].
Consider a symmetric second-order tensor T in 3-dimentional space. The spectral decomposition of T results in the following expression:
3 Tvv (B.1) 1
where 123,, is the spectrum of T and vvv123,, is an orthonormal basis of eigenvectors of
T . It is evident that the orthonormal basis v1,, v 2 v 3 is not unique in case of repeated eigenvalues of T . In order to eliminate the nonuniqueness of (B.1), the following alternative expression is introduced:
m TE (B.2) 1
where m is the number of distinct eigenvalues of T ; denotes the th distinct eigenvalue of
T while E denotes the eigenprojection corresponding to . The eigenprojection E is defined
212 Appendix B. Eigenprojections of a symmetric second-order tensor
as the orthogonal projection operator on the null space of TI [76]. In fact, comparing (B.2)
with (B.1) yields the following expressions of E :
If the tensor T has no repeated eigenvalues (i.e. m 3 and ), the orthonormal basis
v123 v,, v is unique and E is written as
E v v , 1,2,3 (B.3)
If the tensor T has two distinct eigenvalues (i.e. m 2 ), one of them must be a simple eigenvalue
(herein denoted by 11 without loss of generality) while the other (i.e. 223) a repeated
one. The corresponding eigenprojections E1 and E 2 in this case are written as
EvvEvvvvIvv1 1 1, 2 2 2 3 3 1 1 (B.4)
If the tensor T has only one distinct eigenvalue (i.e. m 1 and 1123 ), the only
eigenprojection E1 is trivially the identity tensor, namely
EI1 (B.5)
It is evident that the eigenprojections E 1,...,m possess the following properties regardless of multiplicities of the eigenvalues:
m EI (B.6)
0, EE , m (B.7) E ,
ETTEE , m (B.8)
213 Appendix B. Eigenprojections of a symmetric second-order tensor
In addition, the eigenprojections E of the tensor T can be expressed in terms of T and its eigenvalues as follows [206]:
I,1m m E TI (B.9) ,1m
214 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
Appendix C Relation between Multiplicative Hyper-Elasto- Plasticity and Additive Hypo-Elasto-Plasticity Models
C.1 Multiplicative hyper-elasto-plasticity model of Dettmer and Reese [89] described in the current configuration
The model of type A described in Section 4.1 of Dettmer and Reese [89] is detailed herein. It can be summarized as follows:
The Mandel stress tensor is defined as
C e MC2 e (C.1) C e
The yield function is written as
2 M, χ M χ (C.2) 3 Y
1 where TTTI tr denotes the deviatoric part of a second-order tensor T ; χ is a 3
symmetric second-order tensor and Y the yield stress.
The flow rule is expressed as 215 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
M, χ Dp (C.3) M where Dp is defined in (3.10).
The kinematic hardening is governed by the following equation:
χ D χcbp (C.4) where c and b are constants; χ is expressed as
χχWχχW pp (C.5) with W p defined in (3.10).
In addition, the Kuhn-Tucker conditions are written as
0,,0,,0M χMχ (C.6)
The equations above will be now written in the current configuration. Note that BRCReeee T .
Since C e is an isotropic function of C e , the following expression holds:
CBee RRe T e (C.7) CBee
Substituting (C.7) into (C.1) yields
Be MRBR2 e Tee (C.8) Be
Be Note that isotropy of the function Be ensures that and B e commute. Consequently, Be utilizing (3.11) in (C.8) yields
216 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
MR e Te τR (C.9)
In addition, the yield function M, χ is also isotropic with respective to M and χ . Similarly
M, χ to (C.7), the following expression holds for : M
M, χ τRχR, ee T RRe Te (C.10) M τ
Letting α R χ Ree T and substituting (C.10) into the plastic flow rule (C.3) yields the plastic flow rule with respect to the current configuration:
τα, Dp (C.11) τ
Note that the definition (3.14) has been considered in obtaining (C.11). It is also evident that due to isotropy of M, χ the yield function (C.2) can be rewritten as
2 τ, ατα (C.12) 3 Y
Moreover, substituting χRαR e Te into (C.4) yields the following spatial description of the kinematic hardening law:
αDα cbp (C.13) where α , denoting an objective stress rate of α , is written as
α α Ω α α Ω (C.14) with
217 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
ΩRRRWRee Tepe T (C.15)
It can be seen that the plastic flow rule (C.11) can be described by the general expression (3.13) introduced in Section 3.3 and the kinematic hardening law (C.13) falls into the category expressed by (3.15).
C.2 Equivalent expression of the spin tensor Ω
The spin tensor Ω is expressed in (C.15) or (3.33). It is shown in the present appendix that (3.33) is equivalent to the following expression according to the basic kinematic definitions given in
Section 3.2:
ΩVLskw VVV e 1ee 1e (C.16)
Substituting FFFpe1 into the definition of W p in (3.10) yields
WFLFFFpskw e 1 e e 1 e (C.17)
D Note that the relation FFFFe 1ee 1e has been employed in obtaining (C.17). Dt
The term RWRepe T in (3.33) can therefore be written as
Rep W e Te RVL 1ee 1 Vskw e VF e T R (C.18)
Taking time derivative of the expression (3.3) yields
FVRVReeeee (C.19)
Substituting (C.19) into (C.18) yields
RRRWRVLVVVeeT epeT skw e1 e e1e (C.20) 218 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
The expression (C.16) is therefore obtained.
C.3 Alternative expression of the plastic flow rule by Simo [124]
The equation (2.19a) in [124] gives the following plastic flow rule:
1 τ,q BBee1 (C.21) 2 τ where q denotes a set of internal variables and Be denotes the Lie derivative of B e with the following expression:
BBLBBeeeeT L (C.22)
Be can be rewritten as
DD BFFBFFFFFFe1eTTp 1p TT DtDt FFFFFFepp 1p Tp Te T (C.23) 2FDFepe T
D Consider the equality FFFFpp 1pp 1 0 in (C.23). Substituting (C.23) into the plastic Dt flow rule (C.21) yield
τ,q Repe DTe 1e RVV (C.24) τ
Note that τ, q is assumed to be isotropic with respect to τ according to Equation (2.9) in [124].
τ,q in (C.24) is therefore coaxial with τ which is further coaxial with B e due to the τ
219 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
τ,q hyperelastic relation (3.11). Consequently, V e1 , and V e on the right-hand side of τ
(C.24) can commute resulting in the following expression of the plastic flow rule:
τ,q RDRepe T (C.25) τ which is consistent with the expression (3.13) introduced in Section 3.3.
C.4 Continuity of the spin tensor Ω
The spin tensor Ω is expressed by (3.57). Let us first suppose that the tensor B e in the expression
e e e e (3.57) has three distinct eigenvalues b1 , b2 and b3 (i.e. m 3). In the present appendix, the continuity of the expression (3.57) of Ω is investigated when the eigenvalues approach each other causing coalescence.
e e e Without loss of generality, let the eigenvalues b3 approach b2 while b1 is kept different from the
eeee ee ee two (i.e. bb32 , bb12 and bb13 ). In this case, the limits of the scalar coefficients cbb ,
,3 in the term WBD e , of (3.57) (see (2.34)) have been shown by Xiao et al.
[1] to be
ee ee 1 bb23 212 lim,limlim0c b23 b (C.26) e ee e ee ee 1 b3 bb 23 b 2 11lnbb ln bb 23 23
lim,lim,0c be bc ee b e b (C.27) e ee e 3 22 3 b3 bb 23 b 2
limc be , b e c b e , b e (C.28) ee 3 1 2 1 bb32
220 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
lim,,cbbcbbeeee (C.29) ee 1312 bb32
ee Similarly, the limits of the scalar coefficients c b, b ,3 in the term
WBD ep, of (3.57) (see (3.58)) can be written as
ee ee 2 bb23 222 lim,limlim0cbb23 (C.30) eeee ee ee 1 bbbb323211lnbb ln bb 23 23
lim,lim,0cbbcbbeeee (C.31) eeee 3223 bbbb3232
lim,,cbbcbbeeee (C.32) ee 3121 bb32
lim,,cbbcbbeeee (C.33) ee 1312 bb32
Note that the last equalities in (C.26) and (C.30) are obtained by applying the L'Hôpital's rule twice to each of the limits. In view of the expressions (2.34) and (3.58), the limits given in (C.26)-(C.33)
e e e imply that the spin tensor Ω is continuous when the eigenvalues b2 and b3 of B coalesce.
In addition to the cases with doubly coalescent eigenvalues of B e , the expression (3.57) of the spin tensor Ω is also continuous when the eigenvalues of B e become triply coalescent. Suppose
e e e e e e that B has two distinct eigenvalues b1 and b2 (with m 2 ). When b2 approaches b1 , the following limits of the scalar coefficients in WBD e , and WBD ep, can be written as
ee 12 lim,lim0c b12 b (C.34) ee 1 bb21 1ln
limc be , b e lim c b e , b e 0 (C.35) e e 2 1 e e 1 2 b2 b 1 b 2 b 1
221 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
ee 22 lim,lim0cbb12 (C.36) ee 1 bb21 1ln
lim,lim,0cbbcbbeeee (C.37) eeee 2112 bbbb2121
It can be inferred from the limits given in (C.34)-(C.37) that the expression (3.57) of the spin tensor
e e e e Ω is continuous when the two distinct eigenvalues b1 and b2 of B (with m 2) coalesce.
C.5 Expressions of the scalar valued functions in Equation (3.97)
In Equation (3.97), the scalar valued functions AτLFα,,,, and BτLFα,,,, are expressed as
τ, ατ ,, α , Aτ, L , F , ατDΩ ,::2:, τ L τ k ττ (C.38) τα,, :,φ , τ ,L , F α α
τ,,,,,, α pp τ α τ α B τ, L , F , α , : τ : 2 W Bk τ , τ τ τ τ τ,,,, α p τ α :,,,,,,,A τ α φ τ L F α (C.39) ατ τα,, K τα,,
τα,, Note that in obtaining the expressions (C.38) and (C.39), symmetry of the tensor has τ been utilized. In Equation (C.38), Ωk τ, L denotes the kinetic logarithmic spin, which is expressed in (3.42). In Equation (C.39), W , denotes the function defined in (3.58) and
222 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
Bkkτhτ exp2 is the kinetic left Cauchy-Green deformation tensor with hk τ defined in (3.68).
C.6 Numerical approximation of the plastic work
Both the definitions (3.108) and (3.109), in conjunction with the associative J2 plastic flow rule
(3.105) (with α 0 in the present context), suggest that the plastic work can be approximated through the following expression in either of the numerical methods adopted in Section 3.6:
ttnn11τ Wp τ: ds τ ds τ (C.40) ntt n n1 nnτ
p where Wn denotes the plastic work done by stress τ during an increment spanning the time
interval ttnn, 1 . n is the consistency parameter obtained for the same increment and τ n1 is
the approximation of stress τ at the instant tn1 .
C.7 Numerical integration of the additive hypo-elasto-plasticity model based on the modified kinetic logarithmic rate
The additive hypo-elasto-plasticity model to be numerically integrated is summarized as follows:
The rate of deformation tensor D is additively decomposed as
DDDˆˆep (C.41) where Dˆ e and Dˆ p are the elastic and plastic parts of D , respectively.
The elastic response can be written as
223 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
Dˆ eH : τ K (C.42) where H is the compliance tensor expressed in (2.30); τK is the modified kinetic logarithmic stress rate of the Kirchhoff stress τ with the following expression:
ττΩττΩK KK (C.43)
The modified kinetic logarithmic spin tensor ΩK in (C.43) is written as
ΩWWBDWBDKkkp ,, ˆ (C.44) where W is the vorticity tensor; W B Dk , and WBD kp, ˆ are skew-symmetric second order tensor valued functions expressed in (2.34) and (3.45), respectively; Bk , denoting the kinetic left
Cauchy-Green deformation tensor, is defined as
Bhkk exp2 (C.45) with the kinetic Hencky strain written as
hkH : τ (C.46)
The yield condition is expressed as
τ,, α τ α (C.47) where α is the back stress.
The plastic flow rule can be written as
η Dˆ p (C.48) η
224 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models where η τ α .
The kinematic hardening is governed by the following equation
αcb Dp α (C.49) where c and b are material parameters; α is the objective stress rate of the back stress with the following expression:
ααΩααΩ (C.50) where the spin tensor Ω can be expressed as
ΩWWBDD kp, (C.51) with the skew-symmetric second order tensor valued function WBDD kp, expressed in
(3.36).
The isotropic hardening is described as
K (C.52) with K denoting the isotropic hardening modulus.
With the hypo-elasto-plasticity model summarized in (C.41)-(C.52), the corresponding numerical integration method is then formulated. It is clear that the state variables contained in the model above are τ , α and . The objective of the numerical integration method is to obtain
approximations to the state variables at the instant tn1 (i.e. τ n1 , αn1 , n1 ) with given values ( τ n ,
αn , n ) of the state variable at the instant tn and prescribed deformation path in the time interval
225 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
ttnn, 1 . In the present appendix, the subscripts “ n ”, “ n 1” and “ n 12” mean that the
1 corresponding quantities are at the instant tn , tn1 and tt , respectively. 2 nn1
At first the rate form equations of the state variables are integrated from the instant tn to tn1 .
The equation (C.43) is rewritten as
D τQQτQQK TT (C.53) Dt where Q is an orthogonal tensor satisfying
QQTKΩ (C.54)
Substituting (C.53) into (C.42) and considering the isotropy of the compliance tenor H yield
D QDQQTeHTˆ : τQ (C.55) Dt
Integrating (C.55) from the instant tn to tn1 obtains
t QTTHTeτQQτQQDQ : n1 ˆ dt (C.56) nnnnnn111 t n where the stiffness tensor H , satisfying HHs: , is expressed in (2.31). After pre- and post-
T multiplying both sides of (C.56) by Qn1 and Qn1 , respectively, the following expression is obtained:
t τ Q τ QT H: Q n1 Q T Dˆ e Qdt Q T (C.57) n1 n n n n 1t n 1 n
226 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
H It is noteworthy that the isotropy of has been utilized in obtaining (C.57). The tensor Qn in
(C.57) is written as
T Q nnn Q Q 1 (C.58)
Substituting (C.41) into (C.57) yields
THp τQτQεεnnnnnn1 : (C.59) with
t εQQDQQ n1 TTdt (C.60) nnn 11t n
t εp Q n1 Q T D p Qdt Q T (C.61) n n11t n n
Following a procedure similar to that for obtaining (C.59)-(C.61), (C.49) can be written as
t αQ αQQQDαQQ TTpT n1 cbdsˆ (C.62) nnnnnn111 t n
where Q and Qn satisfies
QQT Ω (C.63)
T QQQnnn 1 (C.64)
In addition, integrating (C.52) from the instant tn to tn1 yields
t Kdtn1 (C.65) nn1 t n
Numerical approximations to the above integrations are then described.
In light of (C.54), Qn is approximated by the following expression:
227 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
t Q exp n1 ΩKdt (C.66) n t n
According to (C.44) and (C.48), the following expression holds
tn1 η ΩWWBτDWBτKkkdttt ,, n (C.67) t nnnnnnn n ηn with
t n1 dt (C.68) n t n
In (C.67), the approximations to tnnW and tnnD can be written as [63]
11 tnnnnnnnWFFFFFskwskw 11 (C.69)
1 t DFCCF T1 (C.70) nnnnnn 2 1
where C n and Cn1 denote the right Cauchy-Green deformation tensor at the instant tn and tn1 , respectively.
In the equation (C.60), the approximation to εn can be written as
T εn Q n1 2 t n D n 1 2 Q n 1 2 (C.71)
where Qn12 and tnnD 12 are approximated as
1 tn1 QQQ TKexp Ω dt (C.72) nnn1 211 2 t 2 n
1 t DFCCF T1 (C.73) n n1 22 n 1 2 n 1 n n 1 2
228 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
p In (C.61), the approximation to εn can be written as
p ηn1 ε nn (C.74) ηn1
Similarly to (C.66), the quantity Qn can be approximated as
t Q exp n1 Ωdt (C.75) n t n with
tn1 η ΩWWdttt BτDW Bτ kk ,, n (C.76) t n nnn nnn n ηn
The second term on the right-hand side of (C.62) is approximated as
tn1 η QQDTpT cbdscb ˆ α QQα n1 (C.77) nnnnn111t n ηn1
Substituting (C.77) into (C.62) yields
1 η αQαQ T c n1 (C.78) nnnnn1 1b nn η 1
In addition, (C.65) can be written as
nnn1 K (C.79)
After finishing numerical approximations, the quantity n is then determined as follows.
The trial state is obtained by assuming n 0 , i.e.
trial trial trial T H τn1 Q n τ n Q n : ε n (C.80)
229 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
trialtrialtrial T αQαQnnnn1 (C.81)
t r i a l nn1 (C.82) where
trialk QWWBnnnnnn exp, tt τD (C.83)
trialk QWWBnnnnnn exp, tt τD (C.84)
trialtrialtrialtrialtrialtrial If ταταnnnnnn111111,,0 , the trial state is just the actual state, i.e.
trialtrialtrial ττααnnnnnn111111,, (C.85)
trialtrialtrialtrialtrialtrial If ταταnnnnnn111111,,0 , a proper value of n is needed so that
ταταnnnnnn111111,,0 (C.86)
Considering the definition (2.31) of H and the approximations (C.71) and (C.74), the equation
(C.59) can be rewritten as
trtr23trττDnnnn11 2 t (C.87)
TT ηn1 τn1 Q n τ n Q n22 Q n 1 2 t n D n 1 2 Q n 1 2 n (C.88) ηn1
Subtracting (C.78) from (C.88) yields
c 1 12 ηX (C.89) 1b nnη 1 nn 1 with 230 Appendix C. Relation between Multiplicative Hyper-Elasto-Plasticity and Additive Hypo-Elasto- Plasticity Models
1 XQ τQQαQ TT nnnnnn 1b n (C.90) T 2 QDQnnnn1 21 21 2 t
Applying the operator to both sides of (C.89) yields
c ηX 2 (C.91) nn1 1b n
Substituting (C.91) and (C.79) into (C.86) yields
c X 20 K (C.92) nn 1b n
The equation (C.92) represents a nonlinear equation for n . The numerical solution can be obtained through the Newton-Raphson method.
After solving the equation (C.92), the solution of n can be substituted into (C.90) to obtain X .
η According to (C.89), n1 can be obtained as ηn1
η X n1 (C.93) ηXn1
The values of τ n1 , αn1 and n1 can then be evaluated through the aforementioned numerical approximations to (C.59), (C.62) and (C.65), respectively.
231 Appendix D. Hierarchical Representation of Delamination in Laminated Composites
Appendix D Hierarchical Representation of Delamination in Laminated Composites
D.1 20-node isoparametric serendipity laminated element
A solid element employed in this study is a 20-node isoparametric serendipity laminated element, which contains multiple plies in a single element. Numerical integration of the weak form is carried out over each ply to account for material heterogeneity within the element. All the plies are stacked along the same parametric coordinate axis (termed the stacking axis) and all the interfaces are planes perpendicular to the stacking axis in the parametric space. Figure D.1 shows an example of the solid laminated element in both its parametric and physical spaces. Without loss of generality, the stacking direction of the plies is assumed to be along the axis in the parametric space of the element.
In the present work, the weak form is integrated in the element domain using 33 Gauss quadrature in plane and the Simpson rule in the direction in each ply in the parametric space of the element. The integration scheme results in 27 integration points for each element ply.
232 Appendix D. Hierarchical Representation of Delamination in Laminated Composites
(a) Parametric space (b) Physical space
Figure D.1. A 20-node isoparametric serendipity laminate element in two different spaces: (a) the parametric space; (b) the physical space.
D.2 Proof of exact reproduction of shape functions of the 20-node isoparametric serendipity elements by local elements of the same type
In this appendix, we show that any global shape function of the 20-node isoparametric serendipity element can be exactly reproduced by local elements of the same type if the nodal coordinates of the local elements in the parametric space of the global element satisfy
LG iiaa10 LG iib10 b i 1,2,...,20 (D.1) LG iicc10
GGG LLL where i,, i i and iii ,, are the coordinates of the ith nodes of the global and local
elements, respectively, in the parametric space of the global element; aabbcc010101,,,,, are
constants and a111,, b c are not zero.
233 Appendix D. Hierarchical Representation of Delamination in Laminated Composites
In order to prove the above statement, the shape functions of the local element are investigated in the parametric space of the global element. At first, one can verify that the interpolation based on the shape functions of the 20-node isoparametric serendipity element results in a polynomial of the form:
Pr s,, tCC rC0123 sC t C rC222 sC tC stC trC rs 456789 (D.2) 222222 CstCs10111213141516 tCtrCt rCrsCr sCrst 222 Cr171819 stCrs tCrst
where r , s , and t are the coordinates in the parametric space of the element; Ci i 0 ,1 ,2 ,.. .,1 9 are constant coefficients.
It is evident that both the global and local elements use polynomials in the form described in
Equation (D.2) in their respective parametric spaces. Next, it will be shown that the form of the polynomial obtained through interpolation based on the local element can be described by
Equation (D.2) in the parametric space of the global element. As an isoparametric element, the local element has the following coordinate transformation from its own parametric space to the parametric space of the global element:
20 L Nii i1 20 L Nii (D.3) i1 20 L Nii i1 where ,, and ,, are coordinates in the parametric spaces of the global and local
elements, respectively; Ni denotes the standard shape function associated with the ith node of the
234 Appendix D. Hierarchical Representation of Delamination in Laminated Composites
20-node isoparametric serendipity element. After substituting Equations (D.1) into Equations
(D.3), the following expressions are obtained:
20 20 G a10 Ni i a N i ii11 20 20 G b10 Ni i b N i (D.4) ii11 20 20 G c10 Ni i c N i ii11
20 Since the shape functions Ni i 1 ,2 ,.. .,2 0 are partitions of unity, Ni in the i1 equation above is equal to one. In addition, it is evident that the global and local elements have the
G same nodal coordinates in their respective parametric spaces. i , therefore, can be replaced by
i , which denotes the coordinate of the ith node of the local element in its parametric space in
Equation (D.4). i can then be interpreted as the value of the function f at the
20 G point iii . The expression Nii in Equation (D.4) can in turn be interpreted i1 as the interpolation approximation of the function f through the local element.
According to Equation (D.2), the polynomial obtained via interpolation based on the shape functions of the 20-node isoparametric serendipity element is complete up to order 2 in the parametric space. The linear function f can thus be exactly reproduced by the
20 20 G G interpolation, i.e. Nii . Analogously, the two expressions, Nii i1 i1
235 Appendix D. Hierarchical Representation of Delamination in Laminated Composites
20 G and Nii , in Equation (D.4) can be written as and , respectively. Equation i1
(D.4) can eventually be written in the following simplified form:
1 a0 a 1 1 b0 (D.5) b1 1 c0 c1
Since Equation (D.2) describes the general form of the polynomial resulting from interpolation based on the shape functions of the local element in its parametric space, Equations (D.5) can be substituted into Equation (D.2) to obtain the general form of the polynomial in the parametric space of the global element. Herein let r , s , and t . Noticing that the coordinate transformation shown in Equations (D.5) is linear, the substitution makes each term in Equation
(D.2) become a polynomial of , , and with the same order as the original term. The resulting overall polynomial, therefore, has the same order as the original polynomial Prst . In order to prove that the resulting polynomial has the same form as the original one, one has to show that each term in the resulting polynomial has the same form as one of the terms in the original polynomial. Since the original polynomial P r s t is complete up to order 2, all the terms with orders less than 3 in the resulting polynomial must have their respective corresponding terms in the same form as in P r s t . Consequently, one only needs to pay attention to the terms with orders larger than or equal to 3 in the resulting polynomial. In fact, these terms can only result from those terms with orders 3 and 4 in the original polynomial considering the linearity of the
236 Appendix D. Hierarchical Representation of Delamination in Laminated Composites coordinate transformation. Moreover, due to the similarity among the three coordinates in
Equations (D.2) and (D.5), it is reasonable to expand and check only the three representative terms, st 2 , rst , and, r s2 t , in Equation (D.2). By substituting Equations (D.5) into the three terms, yields
2 2211 stbc 00 bc 11 (D.6) 1 2222 2 bccb0000000 cb22 c bc11
1 rst a0 b 0 c 0 a111 b c (D.7) 1 a0 b 0 c 0 a111 b c b0 c 0 c 0 a 0 a 0 b 0 a 0 b 0 c 0
2 22111 r stabc 000 abc111 222 bca000 2 (D.8) 1 22 b0 caa 000 ba00 0 c 22 a2 b c 1 1 1 222 2a0 b 0 cc 00 aa 00ba 00 0 b 0 c
In the three expressions above, the terms that have orders higher than or equal to 3 are 2 , ,
2 , 2 , and 2 all of which are included by a polynomial of , , and in the same form as shown in Equation (D.2). Consequently, the polynomial resulting from interpolation based on the shape functions of the local element has the form shown in Equation (D.2) in both its own and the global element’s parametric spaces.
It is evident that an individual shape function associated with a single node of the local element can be written in the same form as P r s t in the parametric space of the global element. The following equation can thus hold:
237 Appendix D. Hierarchical Representation of Delamination in Laminated Composites
LLLCt in (D.9) where L is the domain of the local element; L is the column matrix of all the shape functions of the local element. It can be written as
T LLLL (D.10) 1 2 20
L where i i 1 ,2 ,.. .,2 0 is the shape function associated with the ith node of the local element. t in Equation (D.9) denotes the column matrix of all the terms appearing in the polynomial in
Equation (D.2) with the variables being the coordinates, , and , in the parametric space of the global element; CL is the matrix of the specific coefficients of the terms in t . Noticing that both L and t have 20 entries, CL is a square matrix of the size 2 0 2 0 . Furthermore, if the relation in Equation (D.1) is given, the values of the entries of CL are determined.
Next it will be shown that CL is invertible. Herein reduction to absurdity is employed to prove this statement. Let us assume CL is not invertible. As a result, there exists a non-zero row matrix vL satisfying
vC0LL (D.11)
By pre-multiplying both sides of Equation (D.9) by vL , one obtains
vvL C LL t LL 0 in (D.12)
Equation (D.12) can be rewritten in the following form:
20 LLL vii 0 in (D.13) i1
238 Appendix D. Hierarchical Representation of Delamination in Laminated Composites
L L L where vi is the ith entry of the row matrix v . Without loss of generality, it is assumed that v j is one of the non-zero entries in vL . According to the Kronecker delta property of the shape
L functions, the value of the left-hand side of Equation (D.13) is v j at the jth node of the local
L element. Since v j is assumed to be non-zero, Equation (D.13) cannot hold at the jth node of the local element and therefore CL must be invertible.
It is now straightforward to prove the statement made at the beginning of the appendix. According to Equation (D.2), the shape function associated with the ith node of the global element can be written as
GGG Nii ct in1,2,...,20 (D.14) where G denotes the domain of the global element. cG is the row matrix of the coefficients of the terms in t . Since CL has been proved to be invertible, Equation (D.9) can be substituted into
Equation (D.14) yielding
GGLLG LL 1 Nii cCu in1,2,...,20 (D.15)
1 where uG , which equals to cCGL , is the row matrix of the coefficients of the shape functions of the local element. The Kronecker delta property of the shape functions requires that the jth
G G G entry u j of u is equal to the value of the global shape function Ni at the jth node of the local element. Equation (D.15) can therefore be written as
20 GGLLLLL Ni N i j, j , j j in i 1,2,...,20 (D.16) j1
This completes the proof. 239 Appendix D. Hierarchical Representation of Delamination in Laminated Composites
D.3 Relations between the nodal coordinates of the global, local and superimposed elements
Consider a 20-node isoparametric serendipity laminated element in its parametric space. The global element has n interfaces all of which are planes perpendicular to the axis in the parametric space. The interfaces are numbered from the bottom up with the axis pointing
c upwards. The interfaces are located at i .
If the ply-by-ply discretization approach is employed to model the displacement discontinuity across the interfaces, the global element is replaced by n 1 local elements one for each ply as illustrated in Figure 4.5b. The nodal coordinates of the ith local element in the parametric space of the global element can be written as
i jj i jj inj 1,2,...,1;1,2,...,20 (D.17) cccc i iiii 11 jj22
iii where jjj ,, and jjj,, denote the coordinates of the jth nodes of the ith local and global elements, respectively, in the parametric space of the global element. Note that the
c c c definition of i is detailed in Equations (D.17) by introducing 0 1 and n1 1. Equations
(D.17), which states the relation between the nodal coordinates of the local and global elements, have the form shown in Equations (4.23).
If the displacement discontinuity across the delaminated interfaces is represented by the s-method, n pairs of elements are superimposed onto the global element (see Figure 4.5d). Each pair of
240 Appendix D. Hierarchical Representation of Delamination in Laminated Composites superimposed elements compose a superimposed patch characterizing the crack in one of the interfaces. The superimposed patch has the same sequence number as its corresponding delaminated interfaces. The nodal coordinates of the lower superimposed element in the ith superimposed patch can be written as
SLowi jj SLowi jj inj 1,2,...,;1,2,...,20 (D.18) cc11 SLowi ii jj22
SLowSLowSLowiii where jjj ,, are the coordinates of the jth node of the lower element in the ith superimposed patch in the parametric space of the global element. Note that Equations (D.18) is given in the form described by Equations (4.23).
By combining Equations (D.17) and (D.18), one can obtain the relation between the nodal coordinates of the ith local element and lower element in the jth superimposed patch as follows:
ijS Low kk ij S Low kk c c c (D.19) cc i ii1 S j Low i 1 j i kkcc jj11 i1,2,..., j ; j 1,2,..., n ; k 1,2,...,20
Note that ij in the equation above so that the ith local element is within the domain of the lower element in the jth superimposed patch.
241 Appendix D. Hierarchical Representation of Delamination in Laminated Composites
Based on the proof of Equations (D.5) in Appendix D.2, the coordinate transformation from the parametric space of the lower superimposed element to that of the global element has the same form as Equations (D.18), namely
SLowi SLowi in1,2,..., (D.20) cc 11 iiSLowi 22 where SLowSLowSLowiii ,, and ,, are the coordinates in the parametric spaces of the lower element in the ith superimposed patch and the global element, respectively.
By applying the transformation in Equation (D.20) to Equation (D.19), the relation in Equation
(D.19) can be expressed in the parametric space of the lower element of the jth superimposed patch as follows:
ij S Low kk ij S Low kk (D.21) 2 cc c c c cccc 1 ij ii 1 S Low i j i11 jjii kk c c 2 j 1 1 j ij1,2,..., jn k ; 1,2,..., ; 1,2,...,20
S j Low S j Low S j Low where k,, k k are the coordinates of the kth node of the lower element in the
iii jth superimposed patch in the parametric space of the lower element; kkk ,, are the coordinates of the kth node of the ith local element in the same parametric space. Equation (D.21)
242 Appendix D. Hierarchical Representation of Delamination in Laminated Composites indicates that the lower superimposed element in the s-method and local element in the ply-by-ply discretization approach satisfy the conditions described by Equation (4.23).
Following the same procedures, one can readily show that any upper superimposed and local element satisfy the conditions described above.
243 Appendix E. Cohesive Zone Model for Inter-Ply Interfacial Behavior
Appendix E Cohesive Zone Model for Inter-Ply Interfacial Behavior
The cohesive zone model (CZM) describes decohesion by a traction-separation relation at inter- ply interfaces. The CZM model considered in the present dissertation is a combination of a bilinear cohesive zone model proposed by Song and Paulino [207] and friction-sliding model of Tvergaard
[208] in the post failure regime.
Prior to introducing the traction-separation relation, we define nine model parameters: cn , cs ,
ct , GI , GII , GIII , s , t and cr . cn , cs , and ct are mode I, II, and III interface strengths,
respectively; GI , GII , and GIII are mode I, II, and III fracture energies, respectively; s , and t
are mode II and III friction coefficients, respectively; cr is the non-dimensional critical value of effective interface separation to be subsequently defined.
A non-dimensional scalar effective separation is defined as
2 2 2 n st n 0 cn cs ct e (E.1) 22 st ,0 n cs ct
244 Appendix E. Cohesive Zone Model for Inter-Ply Interfacial Behavior
where the subscripts e and c denote effective and critical values, respectively; n , s , and t are
the separations along the normal and two tangential directions, respectively; cn , cs , and ct are single mode separations indicating totally delaminated interface along the normal and two
tangential directions, respectively. Assuming bilinear traction-separation relations, cn , cs , and
ct are computed from the corresponding fracture energies and strengths by
222GGGIIIIII cncsct;; (E.2) cncsct
Let max be the maximum value of the effective separation e throughout the deformation history.
Assuming that damage is irreversible, max serves as a state variable indicating the local state of
damage. When max 1, the interface is totally damaged forming a cohesive traction-free interface crack. In this case, the damage variable d 1. Note that the damage variable is introduced only for the purpose of post-processing.
The onset of interface damage is defined by cr . When 0 maxcr , the interface is intact and the corresponding traction-separation response is linear elastic
cnn cs s ct t tn ; tst ; t (E.3) cr cn cr cs cr ct
where tn , ts , and tt are mode I, II, and III tractions. In this case, the damage variable d 0. When
cremax , the interface is damaged. The corresponding traction-separation relation is given by
245 Appendix E. Cohesive Zone Model for Inter-Ply Interfacial Behavior
1 max 1 n cn,0 n 1 cr max cn tn (E.4) cn n ,0 n cr cn
11maxsmaxt11 ttscstct ; (E.5) 11crmaxcscrmaxct
For post-processing, it is convenient to define a damage variable as
t dmmm k1 (E.6) where the subscript “m” can be “n”, “s”, or “t”. Combining (E.6) with the definition of elastic
cm 1 stiffness km which follows from (E.3) yields crcm
d maxcr (E.7) 1 crmax
Note that when n 0 , tn exhibits linear elastic response even though the interface has been damaged already. The linear elastic response is employed herein to prevent the penetration between the contiguous plies.
When max 1 , the interface is totally delaminated and all cohesive tractions vanish. If the
interface is under tensile loadingn 0 , tttnst 0 . If the interface is under compressive
loading, ( n 0) , the traction-separation response is governed by a static friction law [208] as
cn n 100s ttnn 100 t tn t s s;0 t t t n (E.8) cr cn cs ct
246