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Three-dimensional contact problems on the action of an absolutely rigid punch, in the form of a strip in plan, on a transversely isotropic elastic half-space, when the isotropy planes are perpendicular to the boundary of the half-space are investigated. In view of the fact that the elastic rigidity of the boundary of the half-space, characterizing the normal displacement under the action of a given concentrated force, depends considerably on the chosen direction on this boundary, two cases of the arrangement of the punch are considered: along the first or second axis of a Cartesian system of coordinates on the body boundary (Problems A and B). The normal displacement of the body boundary under the action of a specified concentrated normal force, after using a double Fourier transformation, is obtained in a form that is free of quadratures, which enables the stiffness of the boundary in different directions to be determined without difficulty, as well as the direction of the extremal stiffness. Assuming that the function, which describes the form of the base of the punch, can be represented by a Fourier series, one-dimensional integral equations of contact Problems A and B, the symbols of the kernels of which are independent of the number of the term of the Fourier series, are obtained. For a special approximation of the symbol of the kernel, a closed solution of the contact problem is derived in terms of Mathieu functions using Rvachev's method, which is a closed solution of the analogous contact problem of the action of a strip punch on an isotropic elastic half-space. To solve the integral equations of contact problems, regular and singular asymptotic methods are used with the introduction of a dimensionless geometrical parameter λ, which represents the ratio of the value of the period of the wavy base of the punch to the thickness of the contact strip. Using the method of orthogonal functions, the integral equations are also reduced to an infinite system of linear algebraic equations, to solve which the reduction method is employed for any values of λ. |
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