ON THE INCREMENTAL CONSTITUTIVE RELATIONS AND COMPATIBILITY EQUATIONS FOR THIN SHAPE MEMORY ALLOY SHELLS UNDERGOING NON-ISOTHERMAL PHASE TRANSITIONS

Abstract

A new formulation of the constitutive equations for thin shape memory alloy shells is obtained on the background of the once coupled phenomenological model of "smeared" non-isothermal thermoelastic phase transitions. The shell is considered under the canonical Kirchhoff assumptions and defined on a two-dimensional manifold corresponding to its mid-surface. The inverse incremental relations express small increments of tangent and bending strain tensors through small tensor increments of tangent forces, bending couples, and martensite volume ratio while the thermodynamic temperature is a given scalar field. Contrarily to the extrinsic problem statement, such constitutive equations do not require either complex analytical inversion or numerical inversion at every point of the deformation pattern. The appropriate compatibility equations for small increments of the tangent forces and bending couples' tensors are derived, and the intrinsic incremental formulation of the geometrically linear theory of thin-walled shape memory alloy shells is proposed.