A Theory of Shape-Memory Thin Films with Applications
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INTRODUCTION Shape-memory alloys have the largest energy output per unit volume per cycle of known actuator systems [1]. Unfortunately, they are temperature activated and hence, their frequency is limited in bulk specimens. However, this is overcome in thin films; and hence shape-memory alloys are ideal actuator materials in micromachines[1]. The heart of the shape-memory effect lies in a martensitic phase transformation and the resulting microstructure. It is well-known that microstructure can be significantly different in thin films as compared to bulk materials. In this paper, we report on a theory of single crystal martensitic th" elms. We show that single crystal films of shape memory material offer interesting possibilities for producing very large deformations, at small scales. A THEORY OF THIN FILMS Consider a single crystal film with surface S and thickness h: 0 h = S x (0, h). Let x be a typical point in the film and let :ý(x) be any deformation of the film. The total energy stored in the film is given by (1) {aIV2SV ± wb(Vy)} dx. Eh where the first term represents interfacial energy (a is a constant) while the second is the elastic energy. Because the thickness is small, it is important to consider some form of interfacial energy. We choose the above because of simplicity. The elastic energy density Wb has a multi-well structure in martensitic materials as explained below. Minimizing this total energy (1) over all possible deformations subject to suitable boundary conditions gives rise to martensitic microstructure[2]. We are interested in studying this microstructure in very thin films. In [3] we study the behavior of the minimizers of the energy (1) as h -- 0 and obtain a limiting theory, which we describe below. It turns out that this is exactly a special Cosserat membrane theory [4]. Consider a film occupying a region S E !M 2 . Let z = (z1, z2) be a typical point on the film. The deformation of the film is characterized by two three-dimensional vector fields, y(z) and b(z) as shown in Figure 1. y describes the deformation of the base while b describes the deformation of the film relative to the base. To prevent tearing, y is assumed to be continuous, but b can jump at interfaces. The total energy of the film in the limiting theory is given by
Ely,b]
=
W(YY,21b)dz
(2)
where W = Wb/h is the stored energy per unit reference area. The notation A = (a, Ja2 lb) means that the columns of the 3 x 3 matrix A are the vectors a1 , a2 and b; and y,i Oy/dzi. 311 Mat. Res. Soc. Symp. Proc. Vol. 459 01997 Materials Research Society
hb(z) z
y::(z)Y (1)
Figure 1: The deformation of a film is characterized by two vector fields: the point z in the reference film (left) goes to y(z) in the deformed film (right) while a vector drawn through the thickness at z goes to the vector hb(z). In martensitic materials, Wb and consequently W has a multi-well structure. Above the transformation temperature, the material is in the austenite state. We choose this as the reference, and hence the austenite is described by the
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