Asymptotic Model Equations and Analytical Solutions for Stress-induced Phase Transitions in a Thin SMA Layer
Prof. Hui-Hui Dai (Department of Mathematics, City University, Hong K)
Thursday 1st December, 2011 14:00-15:00 325
Shape Memory Alloys (SMAs) have numerous engineering and biomedical applications due to their unique properties. Many experiments on stress-induced phase transitions in thin/slender SMA structures have been carried out and a number of interesting phenomena (e.g. some instability phenomena) have been observed. To gain insights into these phenomena, we conduct an asymptotic analysis for stress-induced phase transitions in a thin SMA layer. More specifically, we start from a constitutive relation proposed by Rajagopal and Srinivasa [1-2], which involves the introduction of two independent energy functions and an internal variable for describing the phase transition process. A methodology of coupled series-asymptotic expansions developed earlier (see [3-6]) for studying nonlinear waves and instabilities in nonlinearly elastic materials is used to derive asymptotic model equations for purely loading and purely unloading processes. For an infinitely-long layer, we manage to construct a two-phase solution, an explicit solution for the phase volume fraction and a formula for the width of the transformation front (which is in agreement with experimental observations). For a finitely-long layer, suitable boundary-value problems are studied. Based on a phase-plane analysis, the analytical solutions for both a force-controlled problem and a displacement-controlled problem are obtained. The obtained nominal stress-strain curve and the profiles of the layer at the different stages appear to capture the key experimental features. Analytical insights are provided for the instabilities leading to stress drops/jumps, as observed in experiments. We also give some further analysis on the inner loops. By using the WKB method, the explicit solutions for inhomogeneous deformations corresponding to the inner loops are obtained, which provide the information on the starting point of the reverse phase transformation. This is a joint work with my former PhD student Jiong Wang.