Differential Equations with Variable Coefficients in the Mechanics of Inhomogeneous Bodies
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erential Equations with Variable Coefficients in the Mechanics of Inhomogeneous Bodies V. I. Gorbachev Lomonosov Moscow State University, Moscow, 119991 Russia e-mail: [email protected] Received November 20, 2019; revised December 2, 2019; accepted December 15, 2019
Abstract—The paper considers differential equations in partial derivatives of an elliptic type with variables, piecewise-smooth coefficients, depending on the coordinates (initial equations). It is shown that the solution of the original equation can be represented as an integral equation through the solution of the accompanying equation with constant coefficients of the same type. This representation includes a fundamental solution to the original equation. Under the assumption that the accompanying solution is smooth, the integral equation implies the representation of the original solution in the form of a series with respect to all possible derivatives of the related solution. The coefficients of the series are called structural functions, since they are determined by the functional dependence of the coefficients of the initial equations either on coordinates, or on time, or on coordinates and time. Structural functions are identically equal to zero in the case when the initial coefficients coincide with the corresponding constant coefficients of the accompanying equation. For structural functions, systems of recurrence equations are obtained. It is shown that in the case of a plate with non-uniform thickness, the structural functions depend only on the coordinate in the thickness of the plate, and structural equations become ordinary differential equations that integrate in a general way. A scheme for solving the plate problem is considered. Keywords: mechanics of composites, differential equations with variable coefficients, averaging methods DOI: 10.3103/S0025654420030061
INTRODUCTION In the mechanics of composites, the object of study is material bodies composed of volumes of matter with various mechanical and physical properties. For the composite, the fact that the volumes of the substance constituting the body have characteristic dimensions much smaller than the characteristic dimensions of the whole body and at the same time they are much larger than the sizes of the molecules, so that the substance in each volume can be considered a continuous medium, is essential. For this reason, the processes occurring throughout the composite body are described by partial differential equations with variable coefficients. In composites with a periodic structure, the coefficients of the equations are periodic functions of Cartesian coordinates with a period along each coordinate (dispersively reinforced composites). The periodicity in two coordinates distinguishes fibrous composites. The periodicity of the coefficients in one coordinate corresponds to a layered composite. The presented classification is conditional, since many modern composites can be simultaneously assigned to two or three of the listed classes at the same time. To solve the problems of r
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