On the Modelling and Numerics of Gradient-Regularized Plasticity Coupled to Damage
This thesis is primarily concerned with issues that arise in conjunction with the modelling of localization phenomena in solids using gradient-regularized plasticity with coupling to damage. The thermodynamic framework for a larger class of models is established, whereby gradients of the internal variables are included as arguments of the free energy. Small as well as large deformation formulations are discussed. In particular, the dissipation inequality provides constraints on the choice of boundary conditions for the gradient-regularized variables. A perturbation analysisis is presented that establishes the localization condition for a homogeneous thermodynamic state.
For the numerical investigations a prototype model is chosen that is characterized by nonlinear local hardening, linear gradient hardening and (local) scalar damage, while the local format of the von Mises yield criterion is adopted. The incremental format of the constitutive equations is obtained upon using the implicit (exponential) Backward Euler algorithm to integrate the evolution equations in a fashion that is similar to that of local theory. The appropriate mixed variational format of the constitutive problem, which is a full-fledged boundary value problem, is proposed. The corresponding mixed finite element algorithm is discussed in some detail. An adaptive finite element strategy is also developed (in the small deformation setting), > whereby the underlying a > posteriori error computation is carried out in the energy norm, that is naturally associated with the mixed variational format of the constitutive problem.
A brief comparison of the nonlocal (averaging) and gradient theories of plasticity is presented, whereby a thermodynamically consistent theory of the nonlocal format is developed in the same spirit as for gradient theory. The comparison concerns both formulation and numerics.
Numerical examples of a one-dimensional tension bar and a two-dimensional plate in plane stress or plane strain are examined to show the performance of the proposed models and algorithm(s). The results are found to be insensitive to the mesh design, even for unstructured meshes, and some analytical predictions of the shear band orientation are confirmed.