Mathematical Modeling and Analysis of Instability Phenomena in Stress-induced Phase Transitions in Thin SMA Strips

Due to the two unusual characteristics of shape memory alloys (SMAs), the shape memory effect and pseudoelasticty, they have broad applications. For example, they have been used to make satellite dampers, minimal surgery devices and snake-like robots, etc. Mathematical analysis and modelling for the behavior of this type of materials is essential for application purposes. Systematic experiments on stress-induced phase transitions in thin SMA structures reveal some very interesting instability phenomena, including a necking-type instability (associated with the stress drop), a shear-type instability (associated with the inclination of the transformation front) and an orientation instability (associated with the switch of the inclination angle). However, there are considerable mathematical difficulties to study these phenomena due to the high material nonlinearity and high-dimensional effects, and very few analytical results are available. In order to shed more light on these instabilities, in this project the researchers shall conduct analytical studies. The researchers shall consider the problem in a three-dimensional setting, which implies that one needs to study the difficult problem of the solution bifurcations of three-dimensional nonlinear partial differential equations (previously only two-dimensional problems were considered by them). By using the smallness of the maximum strain, the thickness and width of the strip, the researchers shall use a methodology of coupled series and asymptotic expansions to derive the asymptotic normal form equations (ANFE's), which can yield the leading-order behavior of the original three-dimensional field equations. It is expected that these ANFE's, which are in the form of nonlinear ODE's, are amendable of analysis of modern techniques. Both qualitative and quantitative analysis for the solutions will be conducted. The researchers shall also pay particular attention on the roles played by the geometric sizes (thickness and width). It is expected that the analytical results can help understand the main features of phase transformations in slender structures and provide mathematical explanations to the above-mentioned instability phenomena.