Plane stress analysis of a shape memory annular plate subject to edge pressure

Abstract

We present a boundary value problem for the equilibrium stress-strain fields in an annular plate composed of a shape memory material. The plate is subject to uniform inner and outer pressures. At a constant temperature above A r , the plate is everywhere in a state of Austenite without loads. An isothermal plane stress analysis is performed using a constitutive model that incorporates a single internal variable for the Martensite phase fraction. This phase fraction evolves with the effective stress as determined by a simple phase transformation kinetic. An issue is the partitioning of the plate into annular regions of pure Austenite, pure Martensite, or Austenite/Martensite mixture. Although closed form solutions are available in certain special cases, the general case requires a numerical analysis of a pair of governing ordinary differential equations. Analysis of these equations provides the pressure values that trigger an Austenite-to-Martensite transformation. This transformation initiates at the inner boundary, giving rise to an initiation front that moves toward the outer boundary as the loads are increased. Even higher loads trigger a similar completion front that also moves from the inner boundary to the outer boundary under increasing load. The phase partitioning in the plate as a function of the applied pressures is then concisely described in terms of structure maps that follow from numerical analysis of the governing equations. Such information would generally be useful for systems involving device control by means of shape memory alloy tubes.