Energetic approach to gradient plasticity

Abstract

We formulate a problem of the evolution of elasto-plastic materials subjected to external loads in the framework of large deformations and multiplicative plasticity. Our model includes gradients of the plastic strain and of hardening variables. We prove the existence of the so-called energetic solution. The stored energy density function is assumed to be quasiconvex in the elastic strain which makes our results applicable to relaxed models of shape memory materials, for instance.

Homogenization for dislocation based gradient visco-plasticity

“…Next, the weak compactness of w η in W 3,q (Ω, R 3 ) together with the convergence (35) and Rellich's theorem guarantee that…”

“…Since we assume that tr(p) = 0 (plastic incompressibility) the quadratic terms in the thermodynamic potential provide a control of the right hand side in (3). It is worthy to note that with g only monotone and not necessarily a subdifferential the powerful energetic solution concept [37,26,35] cannot be applied. In our model we face the combined challenge of a gradient plasticity model based on the dislocation density tensor Curl p involving the plastic spin, a general non-associative monotone flow-rule and a rate-dependent response.…”

Well-posedness for dislocation based gradient visco-plasticity with isotropic hardening

“…The inequality (11) expresses the fact that controlling the plastic strain sym p and the dislocation density Curl p in L 2 (Ω, M 3 ) gives a control of the plastic distortion p in L 2 (Ω, M 3 ) provided the micro-hard boundary condition is specified. It is worthy to note that with g only monotone and not necessarily a subdifferential the powerful energetic solution concept [10,16,17] cannot be applied. In this contribution we face the combined challenge of a gradient plasticity model based on the dislocation density tensor Curl p involving the plastic spin, a general non-associative monotone flow-rule and a rate-dependent response.…”

“…for some positive constant α ∈ R. In the model equations, the nonlocal backstress contribution is given by the dislocation density motivated term Σ lin curl = −C 1 B T Curl Curl p together with the corresponding micro-hard boundary condition (16). For the model we require that the nonlinear constitutive mapping…”

Untitled

“…Since in the sequel we assume that tr( p) = 0 (plastic incompressibility) the quadratic terms in the thermodynamic potential provide a control of the right-hand side in (3). It is worth noting that with g only monotone and not necessarily a subdifferential the powerful energetic solution concept [Giacomini and Lussardi 2008;Mainik and Mielke 2009;Kratochvíl et al 2010] cannot be applied. In this contribution we face the combined challenge of a gradient plasticity model based on the dislocation density tensor Curl p involving the plastic spin, a general nonassociative monotone flow-rule, and a rate-dependent response.…”