Bhattacharya and James [3] have rigorously derived a thin film variational principle for single-crystal martensitic alloys, starting from the three-dimensional elastic energy form with the surface energy modeled by the product of a small positive strain-gradient coefficient and the square of the L²-norm of the matrix of all the second derivatives of the deformation over the reference configuration of a crystal with thickness h. Unless the strain-gradient coefficient equals 0, deformations with finite energy for this thin film model cannot have sharp interfaces between two compatible variants of martensite or between austenite and martensite, and the variational principle requires the use of high-order finite element approximations for the deformation [4].

For this reason, we give a derivation of an alternative thin film variational principle in which the interfacial energy is modeled by the product of a small positive strain-gradient coefficient and the total variation of the deformation gradient over the reference configuration of the crystal of thickness h. Deformations of finite energy can have sharp interfaces with this model, and it can be seen that the interfacial energy is concentrated along the surfaces separating regions of constant deformation gradient. Hence, when continuous, piecewise linear finite elements are used for numerical simulations, the interfacial energy is concentrated along the edges of the finite element triangulation. Not only does this property make this model computationally attractive, but we believe it also better models energy-minimizing deformations for martensitic crystals with small surface energy.

We remark that a similar approach to the interfacial energy has been studied in [5] and that a rigorous analysis of the relation between the two models for the interfacial energy has been given in [6-8] for some scalar models. Related work can also be found in [10-12]. See also [1,2,9] for scaling arguments relating the surface energy to the fineness of microstructures in martensitic crystals.

In our work, we first describe the three-dimensional model with the total-variation interfacial energy for thin films with finite thickness. Then we present some known properties of functions of bounded variation and prove two lemmas needed for the derivation of the thin film theory. Using these results, we next show that for any positive thickness of the film there exists a minimizer of the bulk energy. Then we analyze the behavior of the minimizers as the thickness of the film tends to zero. We show that there exists a convergent subsequence of these minimizers, a limiting two-dimensional energy, and a two-dimensional minimum principle allowing one to characterize and numerically compute the limiting deformations.

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