Two-well linearization for solid-solid phase transitions

Abstract

In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation (SBV ). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus not being detected by the limiting deformations. By Γ-convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [32] for a model with anisotropic second-order perturbation.

Separately Global Solutions to Rate-Independent Processes in Large-Strain Inelasticity

“…Second grade materials, whose stored energy density depends also on ∇ 2 y, may model various physical phenomena, for example the flow of Korteweg fluids (depending on the Eulerian gradient of the Eulerian density field), the deformation of woven fabrics [30,14], phase transitions [3,1,52], and multisymplectic field theory [31]; see also and [55,56,25,4,44,35,13,19,51,50] for further works on non-simple continua, the list definitely not being exhaustive. We point out that for the applications described in this paper it is not necessary to incorporate the hyper-loading, an additional terms representing conservative forces, typically in a form of an edge traction or the so-called couple-stress or double force acting on the boundary (see [12,11,22,23,45,41]), for no such physical phenomena are expected to arise in our intended applications.…”

Regularity of gradient vector fields giving rise to finite Caccioppoli partitions

“…In this context, due to the frame indifference of the underlying models, the set W −1 ({0}) is typically not finite. Sometimes, however, the frame indifference is disregarded (as in the paper by Conti, Fonseca, and Leoni [4]), or the theory gives a limit with ∇u ∈ BV(Ω; A) for a finite set A ⊆ R n×n anyway (such as in recent results of Davoli and Friedrich [6,5]). In such a case, Theorem 3 and Theorem 4 are potentially useful, as they apply to the components (or other one-dimensional projections) of u.…”

Regularity of gradient vector fields giving rise to finite Caccioppoli partitions

“…In this context, due to the frame indifference of the underlying models, the set W −1 ({0}) is typically not finite. Sometimes, however, the frame indifference is disregarded (as in the paper by Conti et al [4]), or the theory gives a limit with ∇u ∈ BV( ; A) for a finite set A ⊆ R n×n anyway (such as in recent results of Davoli and Friedrich [5,6]). In such a case, Theorem 3 and Theorem 4 are potentially useful, as they apply to the components (or other one-dimensional projections) of u.…”