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The exact solution of the problem given in the title is constructed by the methods of integral transforms and analytic functions. This problem has arisen in connection with the theory of the linearly-deformable foundations, characterized by a power kernel. The papers [1–6] are devoted to integral equations of the first kind. In them, one assumes that such a kernel corresponds to the elastic halfspace whose modulus of elasticity is a power function of the depth. For the determination of the exact form of the kernel one has to solve Flamant's problem (in the plane elasticity theory) and Boussinesq's problem (in the three-dimensional theory). The solution of Flamant's problem is given in [7, 2] and a partial solution of the Boussinesq problem is given in [11] (only the displacements on the boundary of the half-space are calculated). An attempt for the computation of the elastic field in the half-space can be found in a paper where the initial terms of the series, representing the solution in spherical coordinates, are computed but there is no general formula for them and the convergence is not established. ( ∗ ) In this paper we will use cylindrical coordinates; the solution is expressed in closed form in terms of known higher transcendental functions. The formulas of the solution are obtained applying A.Ia. Aleksandrov's [8] process of transforming plane problems of the theory of elasticity into axially symmetric ones and conversely, and also by applying integral transforms whose kernels are Whittaker functions.

We consider the problem of a layered-nonhomogeneous body such that the elastic constants of the isotropic medium depend exponentially on the Cartesian coordinates. Poisson's ratio is assumed to be constant. Making use of the Airy stress function and of expansions with respect to a small parameter, we construct the solution of the first fundamental boundary value problem of the plane elasticity theory for a circular domain when the circumference we are given the radial stresses as continuous functions of bounded variation of the polar angle. This problem has been investigated in [1] by the method of successive approximations with the aid of the complex representations given by the authors of [2–4]. The convergence of the successive approximation process has not been proved. In problems with nonhomogeneous elasticity the complex representation present no advantage, therefore we will apply the Airy function method in that version in which it was used in [5], in conjunction with expansions with respect to a small parameter, characterizing the nonhomogeneity. We give a recursion process in order to compute the functions which are the coefficients of the power series and we prove the convergence of this series. An example is given.

The equations of equilibrium of an isotropic nonhomogeneous medium in which the elastic moduli are nonconstant, differentiable functions of position and Poisson's ratio v has a constant value, have been studied by making use of the methods of separation of variables and integral transforms. The equation for the stress functions in isothermal coordinates has been given together with some applications. In the three-dimensional problem, conditions have been found for the existence of radial stress distributions. In particular, the case where the elastic modulus depends on a power of one of the Cartesian coordinates has been investigated. In this case, it has been established that the fundamental functions for the two-dimensional problem of a strip are certain confluent hypergeometric functions. The application of the Fourier method has been studied for three-dimensional problems — mainly in the case of axial symmetry. A method has been given for the numerical solution of the Boussinesq problem in terms of the familiar Flamant problem. In the particular case of a power law, when the Flamant problem has an exact solution, it turns out that the Boussinesq problem has an exact solution.