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New compatibility equations for shape memory alloys undergoing thermoelastic phase transitions

Zhavoronok Sergey I.

#### Abstract:

The new compatibility equations are derived for shape memory alloys undergoing thermoelastic phase transitions under varying temperature and stress state. The geometrically linearized problem statement is used as a background together with the once coupled model of thermoelastic behavior of shape memory alloys accounting for the effect of stress state on the temperatures of start and finish of the phase transitions. The once coupled problem formulation assumes the temperature to be a given spatial distribution at each point in time domain. The linear strain tensor is represented using an additive decomposition into the isotropic tensor of elastic and thermal strains, the elastic deviatoric strain, and the phase deviatoric strain corresponding to direct or inverse martensite transitions. On the other hand, the additive decomposition of the strain tensor into the accumulated strain and the small increment is introduced, where the summary accumulated strain is assumed to satisfy the compatibility equations. The small increment of the deviatoric phase strain is defined by the linear function of increments of the deviatoric stress and the martensite volume ratio used as phase constitution parameter. The effect of the phase dilatation is assumed to be negligible. Given the temperature field the analogous linear dependencies of martensite volume ratio on the deviatoric stress are derived. Thus, the obtained incremental formulation of the compatibility equations for shape memory alloys obeying the once coupled model of thermoelastic phase transitions uses only stress tensor as an unknown. The stress function tensor could be introduced for such a problem, and the appropriate boundary value problem could be formulated for the partial differential equation where the components of the increment of the stress function tensor are unknowns.

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