Mathematical Modeling of Finite Deformations in Shape Memory Materials

Within the framework of the formalized approach, which was proposed in our previous works to construct mathematical models describing the behavior of complex media under finite deformations, we developed the constitutive equations for materials undergoing the austenitic—martensitic phase transition (shape memory alloys—SMA and ferromagnetic shape memory alloys—FSMA) and for polymers undergoing the relaxation transition from a highly elastic state to a glassy state (shape memory polymers—SMP). For the inelastic strains, the model considers the strains that take place in alloys during the direct phase transition in reliance with the temperature, as well as on stresses and the magnetic field applied (in the case of ferromagnetic alloys), or the frozen strains developed in polymers with a specific structure in the process of relaxation transition from a highly elastic state to a glassy state. The mathematical model constructed was numerically implemented by a finite element method using the FEnicCS package. The last one, has been used to solve some of the boundary value problems on the accumulation and removal of inelastic strains in the process of phase and relaxation transitions. In this work, a mathematical model was used to solve the problem of azimuthal torsion of a long hollow cylinder made of a shape memory polymer. The obtained solution has been compared with the result of azimuthal torsion of the same cylinder made of Heusler alloy.