We propose a separation-of-variables method for the biharmonic equation and construct a complete system of orthogonal functions for constructing exact solutions in the form of non-periodic trigonometric series for two-dimensional problems of the theory of elasticity and thermoelasticity for a rectangular region.Methods of solving problems of elasticity for bounded bodies with corners have been discussed in [4][5][6]. The absence of exact closed-form solutions for elasticity problems for such bodies is connected with the separation of variables in the key biharmonic differential equations. In the present paper we propose a method of constructing exact solutions of two-dimensional problems of elasticity and thermoelasticity in the stresses for a rectangular region under prescribed forces on the boundary. The method is based on the integration of the differential equations of equilibrium, making it possible to express two of the components of the stress tensor in terms of one of the normal components and equivalently replacing the eight boundary conditions for the different components by six conditions for a single component of the normal stresses. This makes it possible to separate the variables in the key bihaxmonic equation and to give the solution of the problems of elasticity and thermoelasticity just posed in the form of expansions in nonperiodic trigonometric series using a complete system of orthogonal functions of the corresponding nonclassical spectral problem.We consider the two-dimensional quasi-static problem of thermoelasticity for a rectangular region G = {(x, y) E [-1, 1] • [-a, a]} with prescribed forces on the boundary. From the determinate system of equations that describes the thermoelastic state in a homogeneous isotropic body we take the equations of equilibrium with boundary conditions (TXlx=l---'Pl(Y), O'xix=_l--~P2(Y), xylx:l=p3(y),auly:a = ql(x), Crylu=_ a = q2(x), a yly:a = q3(x), aXy]y:_ a = q4(x).
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