Analytical studies on the force-induced phase transitions in slender shape memory alloy cylinders layers
Student thesis: Doctoral Thesis
Related Research Unit(s)
Analytical studies on the force-induced phase transitions in slender shape memory alloy cylinders and layers are reported in this thesis. This thesis contains three parts. In the first part, shape memory alloy is treated as a compressible hyperelastic material with a non-convex strain-energy function. Phase transitions in a slender circular cylinder induced by axial tension is modeled based on a three-dimensional setting. Starting from the field equations and the traction-free boundary conditions at the lateral surface, the normal form equation is derived through a novel approach called the coupled series-asymptotic expansion method. Two kinds of boundary conditions are proposed at the ends of the cylinder. One is the free end boundary conditions and the other is the clamped boundary conditions. Based on the phase plane analysis, analytical solutions in terms of integrals are obtained. It can be seen that the solutions obtained can capture some key features of the experimental results. For example, the nucleation stress peak, the stress plateau, the geometrical size effect and so on. Based on the model proposed in part one, the rate-independent dissipation effect during the phase transition process is further studied in the second part. An effective one-dimensional expression for the total elastic potential energy of the cylinder is derived, which takes into account the three-dimensional effects. In a purely onedimensional setting, the mechanical dissipation functions are expressed in terms of the axial strain. The equilibrium configuration of the cylinder is then determined by using the principle of maximizing the total energy dissipation as well as minimizing the pseudo-potential energy. An illustrative example with some chosen material constants is considered. The analytical solutions obtained in this example can be used to explain the rate-independent hysteresis loops measured in the experiments. The phase fronts coalescence process is also discussed in this part. In the third part, the mechanical response of a two-dimensional shape memory alloy layer is modeled based on the constitutive assumptions of the Helmholtz free energy and the rate of mechanical dissipation functions. A phase state variable is introduced to describe the phase transition process. Starting from the two-dimensional field equations and the traction-free boundary conditions, the equilibrium equation is derived by using the coupled series-asymptotic expansion method. The evolution laws of the phase state functions for the outer loop are also derived based on the phase transition criteria and the criterion of maximum rate of dissipation. With the free end boundary conditions and some proper connection conditions at the intersection points, the analytical solutions for the outer loop are obtained explicitly, which show qualitative agreements with the experimental results. Some simple analysis have also been conducted on the inner loops.
- Shape memory effect, Shape memory alloys, Phase transformations (Statistical physics)