Variational Formulations and FE Active‐Set Strategies for Rate‐Independent Nonlocal Material Response

Abstract

We propose a variational framework for the description of rate-independent nonlocal materials with microstructure and outline details of its numerical implementation. On the theoretical side, we focus on a multifield approach to microstructural dissipative mechanisms. To this end, the current state of the evolving microstructure is described by global fields of microscopic order parameters and their gradients, which together with the macroscopic deformation field define the multifield character of the problem under consideration. Focussing on rate-independent standard dissipative materials, the constitutive response is governed by an energy storage and a non-smooth dissipation function. For this scenario, we outline an incremental variational framework whose Euler equations define the macroscopic equilibrium along with the non-smooth global evolution of the order parameters. The formulation features fully-coupled macroscopic and microscopic field equations and associated boundary conditions. The key difficulty on the numerical side is the implementation of the global non-smooth evolution of the order parameters. Here, we outline a new active set strategy for the global finite element discretization of the multifield problem, where inequality constraints on the microscopic fields are taken into account by active nodal sets. An important consequence of the proposed variational approach is the symmetry of the coupled algebraic system. The proposed setting is specified for model problems of isotropic damage mechanics and crystal plasticity. The applicability of the proposed approach to nonlocal solids is highlighted by means of representative numerical examples. Basic Kinematics and Constitutive Functions for Continua with MicrostructureThis work focusses on a multiscale approach to describe rate-independent inelastic material response of a solid with microstructure, where dissipative effects are related to microstructural mechanisms. Besides the small-strain displacement field u(x, t), generalized internal variable fieldsũ(x, t) are introduced to capture microstructural mechanisms, see CAPRIZ [1]completed with the Dirichlet-type boundary conditionsū(, ideas of local standard dissipative materials are generalized and two constitutive functionals are introducedin terms of the reduced energy storage function ψ and dissipation function φ, satisfying a priori material frame invariance. For rate-independent processes φ is a homogeneous function of degree one. By restriction to the case where φ depends onu only, we evaluate the principle of maximum dissipation, which states that the current thermodynamical micro-force β maximizes the dissipation in comparison to all alternative admissible states bounded by the elastic domainin terms of the indicator function ϕ(β) for the elastic domain. Thus the dissipation function in its primary representation φ(u) is defined by a Legendre-Fenchel-transformation of its conjugate φ (β) and reflects the principle of maximum dissipation. Incremental Variational Formulation and Active-Set ...