Analytical solution of the problem of bending of a round plate made of shape memory alloy | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 26 > №1 / 2020 / Pages: 74-97

Analytical solution of the problem of bending of a round plate made of shape memory alloy


In this paper, we consider the problem of the solids mechanics about the bending of a circular plate made of a shape-memory alloy (SMA) during a direct thermoelastic martensitic phase transformation under the action of a constant in magnitude and uniformly distributed transverse load radius. The problem of relaxation in a similar plate during direct phase transformation has also been solved. As the second problem, a normal load uniformly distributed over the radius is applied to the plate surface in the austenitic phase state. Next, the plate material is cooled through the temperature range of direct thermoelastic martensitic transformation. It is required to determine how the uniformly distributed load should decrease during such a transition so that the deflection of the plate remains unchanged. During the work, rigidly and articulated plates were investigated. The solution was obtained in the framework of the Kirchhoff-Love hypotheses. To describe the behavior of the plate material, we used the well-known model of linear deformation of SMA during phase transformations. The solution was obtained under the assumption that the phase composition parameter at each moment of the process under consideration is uniformly distributed over the plate material, which corresponds to non-coupled statement of the problem for the case of uniform distribution of temperature over the material. The possibility of structural transformation in the plate material is not taken into account. It neglects the variability of the elastic moduli during the phase transition, as well as the property of the SMA diversity resistance. To obtain an analytical solution to all the equations of the boundary value problem, the Laplace transform method with respect to martensite volume fraction parameter was used. After transformation in the space of images, an equivalent elastic problem is obtained. As a result of solving the equivalent elastic problem, the Laplace images of the desired quantities are obtained in the form of analytical expressions, which include operators that are Laplace images of elastic constants. These expressions are fractional rational functions of the Laplace image of the phase composition parameter. After returning to the original space, which is carried out analytically by decomposing the expressions for the desired quantities in the image space into simple factors, the desired analytical solutions are obtained.