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Analytical solution of the problem of a toroidal, ellipsoidal and spherical tank made of shape memory alloy under internal pressure

Bobok D.I.

#### Abstract:

The paper focuses on solving the problem of mechanics of a solid deformable body about a toroidal tank made of shape memory alloy (SMA) under internal pressure during a direct thermoelastic martensitic phase transformation under constant pressure. As special cases, a shell with a circular cross-section, as well as an ellipsoidal and spherical tank are considered. For a spherical tank, the problem of relaxation under direct transformation was solved, where it was necessary to determine the necessary decrease in the uniformly distributed load during cooling during direct thermoelastic phase transformation so that the deflection of the shell remained unchanged. The behavior of the shell was described in the framework of the model of linear deformation of the SPF under phase transformations and the theory of thin isotropic shells. Also, the problem was solved within the framework of an unrelated problem statement, that is, the distribution of the phase composition and temperature parameter over the shell material at each moment of time was assumed to be uniform. Similarly, the possibility of structural transformation in the shell material, the variability of the elastic modules during the phase transition, and the property of the SPF resistivity were neglected. To obtain an analytical solution to all the equations of the boundary value problem, the Laplace transform method was used in terms of the volume fraction of the martensitic phase. After the transformation in the image space, an equivalent elastic problem is obtained. In solving this problem, the Laplace images of the desired quantities are obtained in the form of analytical expressions that include operators that are Laplace images of elastic constants. These expressions are fractional-rational functions of the Laplace image of the phase composition parameter. To return to the original space, the expressions for the desired values in the image space are decomposed into simple fractions. As a result of the inversion of these fractions, the desired analytical solutions are obtained.

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