Heterogeneous Thin Films of Martensitic Materials

Abstract

We study the effective behavior of heterogeneous thin films with three competing length scales: the film thickness and the length scales of heterogeneity and material microstructure. We start with three-dimensional nonhomogeneous nonlinear elasticity enhanced with an interfacial energy of the van der Waals type, and derive the effective energy density as all length scales tend to zero with given limiting ratios. We do not require any a priori selection of asymptotic expansion or ansatz in deriving our results. Depending on the dominating length scale, the effective energy density can be identified by three procedures: averaging, homogenization and thin-film limit. We apply our theory to martensitic materials with multi-well energy density and use a model example to show that the "shape-memory behavior" can crucially depend on the ratios of these length scales. We comment on the effective conductivity of linear composites, and also on multilayers made of shape-memory and elastic materials.

Variational Dimension Reduction in Nonlinear Elasticity: A Young Measure Approach

“…About one year later, Shu [21] has shown that letting the interfacial energy go to zero the variational limit coincides with that obtained by Le Dret and Raoult. Bělík and Luskin [6] have observed that with the energy considered by Bhattacharya and James [8] the deformations with finite energy cannot have sharp interfaces between compatible variants.…”

“…About one year later Shu [21] has shown that if ε and κ(ε) go to 0 then the total energy considered by Bhattacharya and James Γ -converges, in a suitable topology, to the one obtained by Le Dret and Raoult.…”

Rigidity estimate for two incompatible wells

“…Theorem 2 has interesting consequences for the scaling of the energy in thin martensitic films [3,20] which will be discussed in a forthcoming paper.…”

Shape-Memory Micropumps

“…We can show that if κ d, it costs materials more energy to form microstructure inside each grain and consequently strains can not be recovered unless the texture is exceptional. Our result 21) shows that the effective behavior of the film is obtained by averaging the properties of small grains, then passing to the two-dimensional limit. Finally, the analysis on the case of comparable κ and d is difficult but it interpolates the two extreme cases.…”

“…Thus, one can find an effective theory for this heterogeneous film in the limit when all length scales tend to zero, but with possibly different limiting ratios. Shu 21) has used the framework of Γ -convergence to show that the average behavior of the film is determined by an effective two-dimensional theory. The limiting theory implies that the overall deformation y has three components y 1 , y 2 and y 3 which depend only on the in-plane variable x 1 and x 2 .…”