Variational Theorem for the Dynamical Problem of Coupled Mechanothermodiffusion in Inhomogeneous Anisotropic Shells with Distortions

Abstract

The variational principles are extensively used in the construction of new mathematical models and approximate methods for the solution of boundary-value problems of the elasticity theory [2, 10]. In the works by Biot [1], the variational principles of isothermal elasticity were generalized to the theory of coupled thermoelasticity for three-dimensional problems [6, 7, 9] and the problems of the theory of shells [4, 5, 12]. However, in the variational statement, the dynamical problems were formulated in these works without initial conditions. The approaches to the construction of variational principles proposed by Gurtin [8] made it possible to pose the boundary-value dynamical problems of thermoelasticity in the variational form as problems with initial conditions [11]. However, in the theory of shells, these approaches have not been widely applied yet. In the present work, we propose a variational statement of the dynamical problem of mechanothermodiffusion with initial conditions for inhomogeneous anisotropic shells with distortions. Consider a shell of constant thickness 2h . We refer the points of the shell in the three-dimensional space to a normal curvilinear coordinate system x = {x α ,z} , α =1,2 , where z = 0 is the middle surface of the shell G bounded by the contour g . It is assumed that the material of the shell is an inhomogeneous anisotropic two-component solid solution with a single symmetry plane at any point (for the physicomechanical properties of the shell). Suppose that, at the initial time, the shell is in the natural state at a temperature T 0 . Starting from the time τ > 0 , the shell suffers the action of an external surface force load q = {q α ,q 3 } ,