Constitutive relations of the model of nonlinear deformation of shape memory alloys, resolved with respect to stress increments | Mekhanika | kompozitsionnykh | materialov i konstruktsii

Constitutive relations of the model of nonlinear deformation of shape memory alloys, resolved with respect to stress increments

Abstract:

A well-known variant of the constitutive relations of the model of nonlinear deformation of shape memory alloys (SMA) expresses the increments of strains through increments of stresses, the parameter of phase composition and temperature, as well as through the values of stress, strains, the parameter of phase composition and temperature themselves. However, for the development of numerical algorithms for analyzing the thermomechanical behavior of elements from the SMA in the form of the finite element method (displacement method), it is necessary to have constitutive relations resolved with respect to stress increments. It is this form of constitutive relations that allows us to obtain expressions for the incremental stiffness matrix of the finite element method for SMA. In a number of works, such an inversion was obtained numerically. Such a numerical procedure significantly slows down the solution process. In this paper, we propose an analytical inversion algorithm that uses specific features of the structure of the original system of constitutive relations, namely, the fact that the linear operator by which the increments of the components of the stress deviator enter the right part of the original system of constitutive relations is degenerate (the case of direct transformation or inverse, but without a structural transition), or the increments of the components of the stress deviator enter the right part of the original systems in the form of two terms, defined in terms of two different degenerate operators (the case of an inverse transformation together with a structural transition). The inversion of the constitutive relations is obtained for a coupled statement of boundary value problems, in which the increments of the components of the stress deviator are included in the right part of the original system of constitutive relations not only through the equations for the increments of the components of the deformation deviator due to elastic deformation, phase and structural transitions, but also due to the differential relations used for the phase composition parameter. When considering, the phase change, the volume effect of the phase transition reaction, as well as the influence of the variability of elastic modules (both volume and shear) on the increment of the phase composition parameter are taken into account.

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