Existence and time-discretization for the finite-strain Souza–Auricchio constitutive model for shape-memory alloys

Abstract

We address the analysis of a constitutive model for the evolution of a shape-memory alloys at finite strains. The model has been presented in Evangelista09 [20] and corresponds to a suitable finite-strain version of the celebrated Souza-Auricchio model for SMAs Auricchio-Petrini02,souza98 [4, 46]. We reformulate the model in purely variational fashion under the form of a rate-independent process. Existence of suitably weak (energetic) solutions to the model is obtained by passing to the limit within a constructive time-discretization procedure.

Existence and linearization for the Souza-Auricchio model at finite strains

“…Finite strains: constitutive model. A finite-strain version of the original small strains Souza-Auricchio model was proposed in [22,23] and further studied from the analytical viewpoint in [30]. Let F := ∇y = I + ∇u (in components F ij = ∂y i /∂x j ) be the deformation gradient, assumed to be defined almost everywhere on Ω with values in GL + (3).…”

“…)R , and we can express the elastic energy solely in terms of the Cauchy-Green tensors [30,34,35] ψ el (C e ) = ψ el (C…”

“…sym → R be the yield function associated with R, namely, let T → φ(C tr , T ) be the Legendre conjugate ofĊ tr → R(C tr ,Ċ tr ) given by [30,34]…”

“…At finite strains the model retains its variational structure, although in a geometrically nonlinear setting, and the energy and the flow rule are checked to be pointwise converging to the original Souza-Auricchio model as deformations are small [22,23]. A first analytical investigation of the model is in [30] where the constitutive relation, corresponding indeed to a strongly nonlinear ODE for tensors is proved to admit suitably weak solutions. Specifically, incremental time-discretizations of the constitutive relation are proved to converge to energetic solutions [51].…”

“…The aim of this note is twofold. Firstly, we extend the finite-dimensional analysis of [30] by combining the constitutive material relation with the quasistatic equilibrium system and moving to the full three-dimensional problem. This is again accomplished within the frame of energetic solvability and, in particular, extends the results by Mielke & Petrov [50] to the finite-strain case.…”