WIAS Preprint No. 2304, (2016)

Global existence results for viscoplasticity at finite strain


  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Rossi, Riccarda
    ORCID: 0000-0002-7808-0261
  • Savaré, Giuseppe
    ORCID: 0000-0002-0104-4158

2010 Mathematics Subject Classification

  • 74C20 74H20 35Q74 49S05


  • Viscoplasticity, gradient plasticity with hardening, multiplicative decomposition, energy-dissipation principle for, generalized metric gradient systems, left-invariant dissipation potential, non-convex energy functional




We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate and thus, depends on the plastic state variable.
The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance (EDB) and energy-dissipation-inequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.

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