FUNDAMENTAL SOLUTIONS OF THE DEFINING SYSTEM OF DIFFERENTIAL EQUATIONS OF THE MATHEMATICAL THEORY OF PLATES - Zelenskiy, A. G. (Computer Science and Applied Mathematics, 2018)
Studies of the local load of thin-walled elements were carried out by a number of authors. Fundamental solutions of boundary value problems were constructed using the Fourier transform integral transform method, which was applied directly to the system of differential equations of equilibrium. This approach led to complex mathematical calculations. In these works, differential equations for boundary domains, the equation with allowance for the deformation of the transverse shear were used. But in the area of local load there is, in essence, a spatial deformed state, and physical theories in principle can not describe it sufficiently accurately. In our study, fundamental solutions to the problems of bending of non-thin trans-versal-isotropic plates with skew-symmetric loading relative to the median plane are obtained. For this purpose, a variant of the mathematical theory was used, which is based on the position of the use of the three-dimensional theory of elasticity. In accordance with this theory, all components of the displacement (as functions of three independent variables) are approximated by Fourier-Legendre decompositions in a transverse coordinate. Other components of the stress-strain state and boundary conditions on the lateral surface of the plate are also dependent on three coordinates and are also presented in the form of series with the aid of Legendre polynomials. The transition from three-dimensional boundary value problems to two-dimensional ones is carried out by means of the Reissner variational principle. Unlike other variants of mathematical theory, the method of obtaining the basic equations is constructed in such a way that the conditions on boundary planes in any approximations are perfectly accurate. This is a significant advantage over other theories; which increases the accuracy of the solution of boundary problems. The distribution of stresses and displacements along the thickness of the plate depends on the Legendre polynomials. With an increase in the order of the polynomials, the accuracy of determining the stresses and displacements increases. A method is developed that enables to construct fundamental solutions of the defining system of inhomogeneous differential equations (the transformed system of differential equations of equilibrium) of a variant of the mathematical theory of transversally isotropic plates of arbitrary constant thickness. The stress-strain state of a local load is described by systems of non-uniform differential equations. In this paper we consider the defining systems of equations in the second and third approximations in rows for displacements in the case of skew-symmetric loading relative to the median plane. In the second (third) approximation, the system of differential equations has an eighth (twelfth) order in relation to two (three) displacement functions. By the method of operator transformations, they are reduced to convenient solvable inhomogeneous equations of the eighth (twelfth) order in relation to new functions. In these equations, the left-hand sides are the same, and the right-hand ones have the following form: , where – mechanical and geometric parameters plates, – Laplace operator, F – local transverse force acting at the origin of coordinates, – two-dimensional Dirac function. The search for the fundamental solution of each obtained inhomogeneous equation of high order is reduced to the definition of operators from a linear combination of fundamental solutions of nonhomogeneous differential equations of low order, namely: Poisson equation, inhomogeneous biharmonic equation and two inhomogeneous Helmholtz equations (Poisson equation, inhomogeneous biharmonic equation and four inhomogeneous Helmholtz equations). The right-hand sides of these equations are the same and equal to . Fundamental solutions to each equation of low order are obtained from partial solutions of similar equations, the right-hand side of which is , where is a uniformly distributed load, – the circle load radius. Partial solutions are sought by Hankel’s integral transformation method. The fundamental solutions of the defining system of equations, which are generalized for a pole at an arbitrary point, are constructed. The proposed method for the search for fundamental solutions can be used for other systems of differential equations with partial derivatives of the considered class. The obtained fundamental solutions of the defining equations of the mathematical theory of the plates allow us to determine partial solutions of the equations in bending problems of plates of arbitrary thickness at any transverse loads.