Hysteresis in the Stress‐Cycling of Bars Undergoing Solid–solid Phase Transitions

A model is presented for the response to slow cyclic stressing of bars capable of undergoing displacive phase transitions. One‐dimensional nonlinear elasticity is used, and the sharp interface between material phases bears a strain discontinuity. As a result, the theory is dissipative, in the sense that the rate of work supplied to the bar exceeds the rate at which the stored energy increases. An essential part of the theory is a continuum‐mechanical version of a kinetic relation that controls the rate at which the phase transition takes place and therefore determines the ultimate governing dynamical system. A periodic solution of this system is constructed, from which one may determine the hysteresis loop to which the response of the bar is expected to tend after many loading cycles. The energy dissipation associated with this loop is determined for a very general class of kinetic relations. For a particular kinetic relation, the transient cycling problem is also solved explicitly, and it is shown that the response indeed tends to that for the periodic solution as the number of cycles tends to infinity. Several predictions of the model are in qualitative accord with experiments carried out on shape‐memory alloys, suggesting that transition kinetics may provide the mechanism responsible for the hysteretic behaviour observed in such materials.