PROBLEM ON THE ACTION OF A CONCENTRATED FORCE ON AN ELASTIC ORTHOTROPIC HALF-PLANE - Dzundza, N. S.; Zinovieiev, I. V. (Computer Science and Applied Mathematics, 2024)
We consider the first basic boundary value problem of the linear theory of elasticity on the action of a concentrated force normal to the surface of an elastic homogeneous, continuous orthotropic half-plane under plane deformation. At infinity, the stresses tend to zero. You need to determine the stresses and displacements at an arbitrary point in the half-plane. The solution to the boundary value problem for an orthotropic half-plane is sought in the space of transformants of the one-dimensional Fourier integral transform. All the basic equations of the problem and boundary conditions are directly transformed by the one-dimensional integral Fourier transform. The solution of the formulated plane problem is based on the construction of the Fourier transform of the stress function, which satisfies the corresponding analog of the biharmonic differential equation in the transform space for the case of an orthotropic material. The appearance of the transformant of the stress function depends on the values of the elastic constants of the orthotropic material, namely, on the values of the roots of the characteristic equation obtained in the space of transformants. One of three possible cases is considered. The transformants of stresses and displacements of points of the half-plane are expressed through the transformant of the stress function, which in turn is expressed through four auxiliary functions associated with the loads on the surface of the half-plane. Two of the four auxiliary functions are found from the boundary conditions, and the connection between the other two is established from the conditions at infinity. The behavior of the stress transforms at infinity is studied, as a result of which the conditions that ensure the finiteness of stresses and displacements are obtained. After applying the inverse integral Fourier transform to the obtained expressions for the transformants, the true values of stresses and displacements at points of the orthotropic half-plane are determined. The solutions for specific cases are presented, and the numerical results are analyzed and the fulfillment of boundary conditions is monitored based on the plots of stresses and displacements in the half-plane. The obtained calculations indicate the adequacy of the results and the logical application of the chosen method for solving the problem.