We consider the geometrically nonlinear continuum theory for martensitic single crystals. (See Jim Sethna's web page for a colorful introduction to martensitic crystals). Martensitic crystals have a high-temperature phase known as austenite and a low-temperature, less symmetric phase known as martensite. We show some results for a computational model of an active martensitic thin film based on the thin film model developed by K. Bhattacharya and R. D. James.

Our goal is to investigate the functionality of a single-crystal martensitic microvalve such as the one on the following picture (Click here or on picture for animation.):

At a temperature above the transformation temperature, the film is in the austenitic state with the valve open letting the fluid pass through the inlet and out through the outlets. Note that due to a bias pressure, the film is not completely flat even though it would be if the pressure were not present.

The picture on the right shows the valve in the closed state. At a temperature below the transformation temperature, the film is stretched and effectively closes the inlet. The bias pressure below the film helps the film transform from austenite to martensite and also keep the valve closed.

Suppose the film in austenite is attached to a substrate and released on a square domain. Upon cooling the film down through the transformation temperature, the released portion of the film transforms into martensite. When certain compatibility conditions are satisfied (see Bhattacharya and James), the reference square domain can transform into a tent-like structure:

A tent such as the one pictured minimizes the thin film elastic energy for special alloys and orientation. Only two compatible variants of martensite are needed with opposing sides of the tent residing in the same martensitic well. The two variants are pictured in red and blue, respectively, while the portion which is attached to the substrate, and therefore stays in austenite, is pictured in grey. When the pressure term is neglected and an indenter of a suitable shape is used, a tent can be clearly observed (courtesy of Richard J. Cui and R. D. James of the Aerospace Engineering and Mechanics Department of the University of Minnesota).

To model the behavior of a thin film that allows the formation of a tent, while keeping the complexity of the problem low, we consider a cubic to tetragonal martensitic transformation where the three variants of martensite are given by the diagonal matrices whose diagonal elements are 1, 1, and 1+µ for a small positive constant µ. This is not representative of a wide variety of alloys; however, it is the simplest model that provides all the qualitative properties needed to study the formation of a tent.

The energy density can be, in a simplified way, pictured as we show below. The transformation temperature is set equal to 0. The energy density has, for all temperatures, two local minima, one corresponding to the identity matrix, the deformation gradient associated with the austenitic phase, and one corresponding to the matrices for the deformation gradients of the martensitic phase (see above).

At the transformation temperature (picture in the middle), the energy density has equal value (normalized to be equal to 0) for both austenite and martensite. At temperatures above the transformation temperature (picture on the left), the global minimum of the energy density corresponds to austenite, while below the transformation temperature, the global minima correspond to the martensitic variants (picture on the right).

We start with an initial approximation given by the undistorted austenite and perform continuation in pressure with a temperature fixed above the transformation temperature. The outcome might look like the image on the left, below. The grey color indicates that the film is everywhere in the austenitic well. Then we perform continuation in temperature, cooling the film down until it transforms to martensite, and warming it up until it transforms back to austenite. The picture on the right, below, corresponds to the film when it has fully transformed to martensite. Note that on most of the surface the film transforms as expected (see above). Due to the boundary conditions arising from the compatibility with the austenite attached to the substrate, in the corners of the square domain the film cannot stretch and the third variant of martensite (corresponding to stretching in the direction orthogonal to the reference square) appears. Note also that due to the pressure term, the distorted film is round, rather than pyramid-like.

Click on the image below to get a larger version of the animated simulation:

When measuring the height of the deformed film as a function of temperature, the following hysteresis loop is clearly observed (click on the image for a larger version):

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This work is based upon work supported in part by NSF DMS 95-05077, by NSF DMS-00-74043, by AFOSR F49620-98-1-0433, by ARO DAAG55-98-1-0335, and by the Minnesota Supercomputer Institute.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation, the Air Force Office of Scientific Research, the Army Research Office, and the Minnesota Supercomputer Institute.