Lower semicontinuity and existence of minimizers in incremental finite‐strain elastoplasticity

Abstract

We study incremental problems in geometrically nonlinear elastoplasticity. Using the multiplicative decomposition Dϕ = F el F pl we consider general energy functionals of the formwhich occur as the sum of the stored energy and the dissipation in one time step.Here G(F pl ) is the dislocation tensor which takes the form 1 detImposing the usual constraint det F pl ≡ 1 and suitable growth and polyconvexity conditions on U we show that the minimum of I is attained in the natural Sobolev spaces. Moreover, we are able to treat multiple time steps by controlling the stored and dissipated energies. We also address the relation of the incremental problem to the time-continuous energetic formulation of elastoplasticity.

Global Existence for Rate-Independent Gradient Plasticity at Finite Strain

“…On the one hand we may use the physically more desirable growth rate r = 1 (cf. However, it remains an open problem to generalize our result to the case treated in [MiM06], where only the term G = (curl P )P T is used for regularization. At the moment the best we can do in this direction is to use a regularizing term in the form Ω κ 1 |(curl P )P T | r + κ 0 |∇P | r dx with 0 < κ 0 ≪ κ 1 .…”

“…In [Mie04] it is shown that (1.4) can be solved for arbitrarily large K even without the regularizing terms (∇P, ∇p), but under severe restrictions on W and D. In [MiM06], where the term (curl P )P T is used for regularization, again the solvability of (1.4) is established for more general W and D. Here, we use full regularization via (∇P, ∇p) which is also common in engineering models, cf. [FlH97, Gur00, Gur02].…”

Curl bounds Grad on SO(3)

“…Let X ∈ C 1 (Ω, M 3×3 ) with X 1 , X 2 , X 3 the rows of X. Then, for the first arrangement, Curl is defined row wise as in [21,31] …”

“…Here and subsequently we let o(x), O(x) denote the Landau symbols, respectively, i.e., 20) we conclude with (3.19) and (3.20) that 21) such that…”

Variational formulations and functional approximation algorithms in stochastic plasticity of materials

“…In contrast to the infinitesimal, the finite strain theory [101,24,54,161,40] deals with several issues such as the existence of the non-unique stress and strain measures, the problem of finding the physically acceptable decomposition of the total deformation into elastic and plastic parts, as well as non-objective material time derivatives of the spatial variables. In order to overcome these issues several different specifications of the rate equations have been considered over time and studied in many papers and books, such as [10,24,69,159,101,161,128]. Accordingly, the finite strain theory is considered to be controversial even though the computational analysis offers the numerical algorithms capable to solve the highly complex problems.…”