Alemayehu Arara

Shape memory alloy (SMA) materials are used in wide areas ranging from micro- and nano-machinery, automotive, aerospace technology, oil exploration, self repairing shielding devices, biomedical implants, medicine. A particularly nice application of shape memory alloys materials is the fabrication of valve mechanisms for rice cooking machines. The situation is rather simple: The valve feels large threshold temperatures and opens. Then the hot vapors escape the boiling room and the local (close to the valve) temperature goes down. Then the valve remembers its initial position, and consequently, it closes. Based on Lagrange and Hamilton principles, we derive a mathematical model for a valve mechanism made of two springs: a shape memory alloy (SMA) spring and a conventional (bias) spring. The resulting shape memory alloy problem is a one-dimensional system of nonlinear hyperbolic-parabolic equations with a free boundary (defining the position of the valve). The research presented in this thesis focusses on the reduced version of the problem - the so-called fast-temperature-activation limit problem. We review the results concerning the existence and uniqueness of weak solutions and complement these results with new ones on the stability with respect to data and material parameters (especially w.r.t. those entering Falk's model for shape memory alloys). Finally, we approximate numerically the solution of the original shape memory alloy problem using a central finite differences scheme. We illustrate the behavior of the valve and of the displacements in the shape-memory alloy spring for a few sets of parameters.