Finite element analysis of the behavior of shape memory alloys and their applications

Abstract

A nonlinear finite element procedure is developed which incorporates a thermodynamically derived constitutive law for shape memory alloy material behavior. The constitutive equations include the necessary internal variables to account for the material transformations and are utilized in a one-dimensional finite element procedure that captures the unique shape memory alloy responses of pseudoelasticity and of the shape memory effect at all temperatures, stress levels and loading conditions. Detailed material properties for the alloy used are necessary for the analysis. The solution of the geometrically and physically nonlinear problem is achieved by application of a Newton's method in which a sequence of linear problems is numerically solved. Due to consistent linearization, a quadratic rate of convergence is obtained.Several test cases are presented to illustrate the potential of the finite element procedure. Cases simulating the stress-strain behavior of a bar of shape memory alloy under simple uniaxial loading as well as restrained recovery responses at different temperatures compare well with experimental and analytical results. Two further generalized applications are examined : the use of a shape memory alloy ring as a pipe connector and eigenfrequency tuning of a composite beam with embedded shape memory wires. The results of these analyses correlate well with analytical results and the methodology for use of the finite element procedure in general cases is demonstrated. The finite element procedure is thus shown to be a powerful tool for studying various applications of shape memory alloys.