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Inverse incremental constitutive relations and compatibility equations for a shape memory alloy undergoing structure transitions

Zhavoronok S.I.

#### Abstract:

New inverse incremental constitutive relations and compatibility equations are derived for a shape memory alloy undergoing the stress-induced isothermal structural transition in the entirely martensite phase state. The once coupled model of thermoelastic phase and structure transitions in shape memory alloys together with the geometrically linear statement of the solid mechanics problem is used as a background. The initial state of the alloy under vanishing stresses is assumed be entirely twinned martensite. The structural transition consists in the untwining of the crystalline structure of entirely martensite phase constitution (so-called martensitic inelasticity phenomenon). To derive the constitutive equations and the appropriate compatibility equations, first the additive decomposition of the linear strain tensor into a sum of the elastic strain tensor and the structural deviatoric strain introduced; secondly the additive decomposition of the aforementioned tensors into sums of accumulated strains and some small increments is used assuming that the summary accumulated strain satisfies the compatibility equations. The structural deviatoric strain increment is defined as linear function of the deviatoric stress increment while the summary dilatation is resulted by only elastic strain. The obtained new compatibility relations are linearized with respect to both strain and stress increments and similar to the Hookean law for anisotropic elastic media where the instantaneous anisotropy is resulted by the structural compliance tensor being a bilinear function of covariant deviatoric tensors of accumulated stress. The new compatibility equations for the stress tensor increment are obtained accordingly to the Beltrami form. Finally, the stress function tensor is introduced and the appropriate formulation of the Beltrami equations for the stress function increment are derived.

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