Interfaces and Free Boundaries, 4 : 2002, 71-88 A TOTAL-VARIATION SURFACE ENERGY MODEL FOR THIN FILMS OF MARTENSITIC CRYSTALS

Martensitic thin films have applications in actuators, sensors, and micromachines because of their large work output/(cycle · volume) [22]. Single-crystal martensitic thin films have recently been grown in the laboratory [12] and theoretically offer even larger work output/(cycle · volume) [5]. Bhattacharya and James [5] have rigorously derived a thin film variational principle from the three-dimensional elastic energy for martensite with the surface energy modeled by κ ∫ Ωh |∇2u|2 dx, where κ is a small positive strain-gradient coefficient, Ωh is the reference configuration of a crystal with thickness h, and ∫ Ωh |∇2u|2 dx is the square of the L2–norm of the matrix of all the second derivatives of the deformation u. Unless we set κ = 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. Our total-variation model allows the use of continuous, piecewise linear approximations of the deformation. Conforming numerical approximations of the Bhattacharya-James thin film energy require higher-order finite element approximations [8]. We also note that although mixed finite element methods can be used to approximate a plate problem (and presumably the BhattacharyaJames thin film model) with piecewise linear deformations, the mixed variational formulation transforms the primal energy-minimization problem to a more computationally challenging saddle-point problem [11]. For this reason, we give a derivation of an alternative thin film variational principle in which the interfacial energy is modeled by a term of the form κ ∫ Ωh |D(∇u)|, where κ again denotes a small positive strain-gradient coefficient and ∫ Ωh |D(∇u)| denotes the total variation of the deformation gradient ∇u in Ωh. 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 deformations for martensitic crystals with small surface energy.