Computational simulation of strain localization: From theory to implementation

Strain localization in the form of shear bands or slip surfaces has widely been observed in most engineering materials, such as metals, concrete, rocks, and soils. Concurrent with the appearance of localized deformation is the loss of overall load-carrying capacity of the material body. Because the deformation localization is an important precursor of material failure, computational modeling of the onset and growth of the localization is indispensable for the understanding of the complete mechanical response and post-peak behavior of materials and structures. Simulation results can also be used to judge the failure mechanisms of materials and structures so that the design of materials and structures can be improved. Although the mechanisms responsible for localized deformation vary widely from one material to another, strain softening behavior is often observed to accompany the deformation localization in geotechnical materials. In this dissertation, a rate-independent strain softening plasticity model with associated flow rule and isotropic softening law is formulated within the framework of classical continuum mechanics to simulate the strain localization. A stress integration algorithm is developed to solve the nonlinear system of equations that comes from the finite element formulation of the incremental boundary value problem for linear strain softening plasticity. Two finite element programs, EP1D and EPLAS, are developed to simulate strain localization for 1-D and 2-D problems. Numerical examples show that the developed strain softening model and computer programs can reproduce well the occurrence and development of strain localization or shear band localization. Because the classical strain softening model does not contain a material length scale, the finite element simulation suffers from pathological mesh dependence. To regularize the mesh dependence of a classical strain softening model, gradient plasticity theory or nonlocal plasticity theory has to be used. To provide correct boundary conditions for higher-order differential constitutive equations with regard to internal state variables, a comparison of boundary conditions for gradient elasticity with gradient plasticity is carried out to show that the Dirichlet boundary condition is the correct boundary condition to force the strain to be localized into a small region and to remove the mesh-dependence. A nonlocal plasticity model with C0 finite elements is proposed to simulate strain localization in a mesh independent manner. This model is based on the integral-type nonlocal plasticity model and the cubic representative volumetric element (RVE). Through a truncated Taylor expansion, a mathematical relationship between an integral-type nonlocal plasticity model and a gradient plasticity model is established, which makes it possible to use the C0 elements to approximate the internal state variable field. Variational formulae and Galerkin\u27s equations of the two coupled fields, displacement field and plastic multiplier field, are developed based on the C0 elements. An algorithm consisting of nonlocal elements and moving boundary technique is proposed to solve the two coupled fields. A numerical example shows the ability of the proposed model and algorithm to achieve mesh-independent simulation of strain localization