Nonlocal plasticity models for localized failure

In quasi-brittle materials such as concrete, failure is preceded by highly localized deformation patterns. Localized failure is caused by strain softening, which cannot be described by classical local models, because they lack a length scale. Mathematically, the use of such models leads to ill-posed boundary value problems. If properly formulated, nonlocal models can provide a sound description of strain softening and model the complete failure process of quasi-brittle materials. A bifurcation analysis is performed for a number of nonlocal plasticity models of the gradient and integral type, and the load-displacement response up to complete failure is computed. The nonlocal models are constructed mainly as enhancements of the standard smallstrain, rate-independent, isotropic local plasticity model. Most nonlocal models respond in a physically appropriate manner at initial bifurcation, but for some models, the plastic zone widens at later stages of the loading process, and stresses remain at a spuriously high level. For the integral format of two nonlocal models selected for further development, the VermeerBrinkgreve model and the ductile damage model, the bifurcation analysis is extended to hardening and the two-dimensional, plane strain setting. For both models, the enhancement is based on a modified hardening law, which depends in a specific way on the local hardening variable and its nonlocal counterpart. The analysis determines the range of material parameters for which the solution is regular and provides an analytical expression for the initial distribution of the rate of plastic strain. The use of the Vermeer-Brinkgreve model is restricted to softening, while the ductile damage model can be used for hardening and softening. An efficient and robust iterative numerical algorithm for the stress return of integral-type nonlocal models that use the local and nonlocal hardening variable in the yield function is developed and implemented. Due to weighted averaging, the stress return is a coupled nonlinear complementarity problem. It is solved by a Jacobi-like iterative technique, derived in a rigorous manner for a special case and generalized based on a physical interpretation of the resulting equations. In view of the numerical modeling of concrete, the nonlocal enhancement is applied to plasticity models with pressure-dependent yield functions and non-associated flow rules. Simulations of laboratory tests demonstrate that the nonlocal model regularizes localization due to softening and non-associated plastic flow. Neither the load-displacement response nor the propagation of the process zone depend in a spurious manner on the computational grid. For eccentric compression of a prismatic column under plane strain conditions, the trend for the size effect on structural strength and post-peak response is correctly predicted.