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Variational Formulations and FE Active‐Set Strategies for Rate‐Independent Nonlocal Material Response
Author(s) -
Welschinger Fabian,
Miehe Christian
Publication year - 2008
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200810475
Subject(s) - dissipative system , isotropy , discretization , dissipation , boundary value problem , statistical physics , mathematics , boundary (topology) , finite element method , field (mathematics) , classical mechanics , physics , mathematical analysis , quantum mechanics , pure mathematics , thermodynamics
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. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)