Nonlocal continuum models for plasticity and damage

Abstract

Nonlocal interactions of material points play a vital role in modelling certain important aspects of inelastic phenomena such as plasticity and damage in solids. For plasticity problems, nonlocal interactions allow characterizations of size-dependence and energetic hardening. In the case of damage, nonlocality describes energetically favourable conditions for propagation as well as branching of cracks. The nonlocal description of inelastic phenomenon introduces certain internal length scales representative of the material micro-structure. A geometric perspective of the kinematics of inelastic deformation induces certain interesting attributes in the form of a non-trivial metric, curvature, etc. to the mathematical model. Towards realizing a unified and rational modelling setup, it is important to trace the geometric origins of the kinematics underlying nonlocal interactions.

The first part of the thesis dwells on modelling of visco-plasticity and damage in metals by introducing gradients of plasticity and damage variables to capture the size-dependent plastic response and the nonlocal aspects of damage. We also try to account for dislocation inertia affecting the yield strength at high strain rates. In addition, the nonlocal flow rule also encapsulates energetic hardening. We describe temperature evolution, which is thermodynamically consistent and accounts for the heat dissipated. The coupled visco-plastic damage model is numerically implemented through peridynamics (PD) and validated via the simulations of adiabatic shear band propagation and shear plugging failure. The nonlocal terms can be accorded a geometric meaning using the concepts of gauge theory and differential geometry. We therefore focus on a geometric characterization of brittle damage via the gauge theory of solids. The local configurational changes in the manifold are captured using a non-trivial affine connection, called gauge connection. The resulting manifold is equipped with the gauge covariant quantities like gauge torsion and gauge curvature. Consequently, this theory serves as a natural device to model different aspects such as stiffness degradation, tension-compression asymmetry and microscopic inertia. The model is again numerically implemented using PD, and validated through the simulations of dynamic fracture instabilities and dynamic crack propagation. Similar to damage, the geometric underpinnings of plastic deformation are unveiled using ideas from differential geometry, e.g. the postulate that a plastically deforming body is a Riemannian manifold endowed with a metric structure and a non-trivial connection. The geometric approach provides a rational means of modelling several important features of plastic deformation, e.g. the free energy of defects, yielding and energetic hardening; and results into a nonlocal flow rule. The model is validated through the numerical simulations of homogeneous visco-plastic deformation and Taylor impact test.

The brittle damage in materials undergoing small deformation typically correspond to small strain. The symmetry principles of gauge theory are also used to obtain a brittle damage model in the linearized setting that is invariant with respect to local or inhomogeneous transformations. The efficacy of the model is established through PD based quasi-static simulations and investigation of blast-induced fracture in rocks.

The applied loads causing deformation may be of thermomechanical origin, rather than being purely mechanical. In the second part of thesis, brittle damage modelling under thermomechanical loading is undertaken. The deformation due to thermal and mechanical loads is coupled via Duhamel's postulate. The heat equation considers radiative and conductive heat transfer, temperature fluctuations due to thermomechanical effect and local temperature rise at crack tip. PD reformulation of this model involves a scalar entropy flux to incorporate nonlocal thermal interactions. The correspondence relations for entropy flux and other PD states, are derivable through energy and entropy equivalence. Numerical simulations include transient heat flow in a silica tile and its coupled thermomechanical analysis, and the temperature change study in Kalthoff's problem.

The damage mechanism of certain materials like ceramics is sensitive to the rate of applied loading. The third part of the thesis develops a damage model for ceramics based on micro-mechanical considerations to account for its strain rate dependent behavior. PD is used to reformulate the equations in the integro-differential form, considering the discontinuities and fragmentation at high strain rates. Numerical studies include spherical cavity expansion problem, impact induced damage in a ceramic target and a composite ceramic target.