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On the Equivalence Between the Additive Hypo- Elasto-Plasticity and Multiplicative Hyper-Elasto- Plasticity Models and Adaptive Propagation of Discontinuities

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On the Equivalence Between the Additive Hypo- Elasto-Plasticity and Multiplicative Hyper-Elasto- Plasticity Models and Adaptive Propagation of Discontinuities 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
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