Gradient enhanced plasticity and damage models : adressing the limitations of classical models in softening and hardening

This thesis addresses two limitations of classical continuum models – pathological localization during softening, as well as the inability to predict size dependent behavior during hardening. A gradient enhancement is adopted and investigated to address these issues. In the latter case, the gradient formulation is derived through a newly proposed homogenization theory, using a crystal plasticity model at the fine-scale. It is well documented that classical models are mesh-dependent during strain softening. This can be avoided by adopting an "implicit" gradient enhancement, which introduces a length scale parameter into the model, characterizing the thickness of the process zone – a localized region of micro-processes during softening. However, for some material models, the implicit gradient enhancement serves only as a partial localization limiter – whereas the global response converges upon mesh refinement, localization still occurs with discontinuous strain rates. The "over-nonlocal" implicit gradient enhancement proposed in this thesis is shown to overcome the partial regularization anomaly for a linear softening von Mises model. One broad class of softening models is that of cohesive-frictional materials such as concrete. The development and calibration of these models are complicated and tedious since material responses are highly dependent on the strain path. Several models capable of predicting the experimentally observed response under different loading conditions are reported to suffer from partial regularization properties. We adopt a sophisticated plasticity-damage model for concrete and show that the proposed over-nonlocal gradient enhancement is able to fully regularize this model whereas standard nonlocal gradient, as well as integral formulations fail to do so. Another limitation of classical models stems from the fact that they are scaleindependent and thus unable to capture size effect phenomena in metals when the deformation is heterogeneous. Many rate-independent continuum models utilize the gradient of effective plastic strain to capture this size-dependent behavior. This enhancement, sometimes termed as an "explicit" gradient formulation, requires higher-order tractions to be imposed on the evolving elasto-plastic boundary and the resulting numerical framework is complicated. An implicit scalar gradient model, on the other hand, only requires boundary conditions on the external surfaces of the entire domain and its numerical implementation is therefore straightforward. However, both explicit and implicit scalar gradient models can be problematic when the effective plastic strains do not have smooth profiles. To address this limitation, a tensorial implicit gradient model is proposed based on the generalized micromorphic framework. The size effect prediction of the proposed model is shown by studying a bending problem. It is also demonstrated that both scalar and tensorial implicit gradient models give similar results when the effective plastic strains fluctuate smoothly, e.g. in flat-tip indentation. Another type of (material) size effect is observed even when the deformation is homogeneous (e.g. in tensile tests). Here, the strength of a material varies inversely with the grain size, i.e., the Hall-Petch effect. One approach to capture this phenomenon is to adopt strain gradient crystal plasticity models that account for the inter-granular resistances via non-standard interface conditions. However, this becomes computationally expensive for large problems since the discretization has to be done at a scale smaller than the average grain size. Considering uniform macroscopic shear, we propose a homogenization theory applied to a fine-scale crystal plasticity model with one slip system. The work done, the stored and dissipated energy at a (macro) point are equivalent to the corresponding average (micro) quantities within a grain in the material. When the interfacial resistances are present, the homogenized (macro) solution is able to predict additional hardening due to the micro-fluctuations. Moreover, two length scale parameters, i.e., the intrinsic length scale and the size of an average grain, naturally manifest themselves in the homogenized solution. Next, the homogenization theory is extended to a plane strain bending problem where both the non-uniform deformation and interfacial resistance contribute to the size effect. For a symmetric double slip system, the homogenized micro-force balance takes the same form as the implicit gradient equation. Using the homogenization scheme, there is now a clear physical interpretation of the kinematic variable associated with the implicit gradient equation. Moreover, the homogenized solutions match closely with those obtained from the fine-scale crystal plasticity model for two extreme cases considered (microfree and microhard boundary conditions). In addition, the study shows how the two effects and three relevant length scales propagate and interact at the macro scale. The standard formulations in a generic problem are likely to encounter both types of limitations discussed earlier – a size effect during hardening, as well as localization beyond a threshold load. Many gradient enhancements in literature are formulated with the intent to resolve only a particular type of limitation. Such models may not perform adequately when the problem also involves the other limitation. In this study, we have separately addressed the two different issues with an implicit gradient formulation. This serves as a starting point towards a unified higher order model which remedies both types of limitations in classical models.