Equations of the Dynamic Problem of Thermoelasticity in Stresses in a Three-Orthogonal Coordinate System
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EQUATIONS OF THE DYNAMIC PROBLEM OF THERMOELASTICITY IN STRESSES IN A THREE-ORTHOGONAL COORDINATE SYSTEM R. S. Musii and H. B. Stasyuk
UDC 539.3
By using the system of source equations including the equations of motion, Cauchy relations, generalized Hooke’s law, and Saint-Venant compatibility equations for strains, we deduce the system of defining equations for the dynamic problem of thermoelasticity in stresses in an arbitrary three-orthogonal curvilinear coordinate system. This system is reduced to a system of successively coupled wave equations in which the equation for the first invariant of the stress tensor is independent. The initial conditions are presented for the resolving functions.
The problems of mechanics of deformable bodies are formulated either in displacements or in stresses [1– 5]. If the problems of elasticity theory with bulk forces and surface loads [3–8] are solved by using the equations in displacements, then the evaluation of stresses requires the procedure of numerical differentiation (or numerical differentiation) of polynomials, which decreases the accuracy of the solution by an order of magnitude [1, 9]. In the literature, one can find the well-known equations of the quasistatic problems of elasticity [1, 3, 4, 6, 7, 10] and thermoelasticity [11–13] in stresses written, as a rule, in Cartesian coordinate systems. In [5, 8, 13– 16], the systems of source equations of motion and compatibility for the dynamic problem of thermoelasticity in stresses are obtained in the invariant form both in a Cartesian coordinate system and in an arbitrary curvilinear coordinate system. The duality of statements of the problems of elasticity theory in displacements and stresses is proved in the tensor form in [2]. The systems of source equations for six components of the stress tensor are presented for three-, two-, and one-dimensional dynamic problems of thermoelasticity in Cartesian [17], cylindrical [17, 18], spherical [19, 20], and elliptic cylindrical [21] coordinates with regard for the existing coupling between the equations of motion and compatibility in stresses. Without introducing auxiliary functions, these systems of six coupled equations for the six components of the stress tensor are reduced to systems of six successively coupled wave equations for the following key functions: the first invariant of the stress tensor, certain linear combinations of the normal components of stresses, and tangential components of stresses. In what follows, we deduce the equations of the dynamic problem of thermoelasticity in stresses in an arbitrary three-orthogonal curvilinear coordinate system and reduce these equations to a system of wave equations in which the key role is played by the equation for the first invariant of the stress tensor. Statement of the Problem Consider a homogeneous isotropic body referred to a three-orthogonal curvilinear coordinate system ( x1 , x2 , x3 ) [6]. The body is subjected to the action of a nonstationary bulk force F( x1, x2 , x3 , t ) = ( F x1 , F x2 , F x3 ) and a non
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