Reports of the Academy of Sciences of the USSR
1970. Volume 194, No. 5

UDC 539.311

THEORY OF ELASTICITY

A. P. SHMAKOV

ON THE LAWS OF REFLECTION OF BODY FORCES IN THE THEORY OF ELASTICITY THROUGH A PLANE BOUNDARY

(Presented by Academician L. I. Sedov, 15 VI 1970)

It is known that some boundary-value problems of the theory of elasticity can be solved by superposition of Kelvin solutions. These include the problems of Boussinesq, Cerruti \(\left({}^{1}\right)\), and Mindlin \(\left({}^{2}\right)\). The most general of them is Mindlin’s problem on the action of a concentrated force inside an elastic half-space whose boundary is free of external actions. In the present paper a further development and generalization of Mindlin’s method is given by realizing a simple and general idea. This idea is based on the analytic continuation of solutions of the equations of the theory of elasticity through a plane boundary. Knowledge of the laws of analytic continuation (reflection) makes it possible to construct solutions of boundary-value problems in a general form and to obtain reflection laws for body forces. The problem of analytic continuation through a plane boundary was considered in works \(\left({}^{3,4}\right)\).

In work \(\left({}^{4}\right)\), analytic continuation was used to solve problems of the theory of elasticity with mixed boundary conditions. Some of the results of that work are presented in the monograph \(\left({}^{5}\right)\).

The main results of the present paper are formulated in the following theorem.

Theorem. Let in the half-space \(x_3>0\), on whose boundary \(x_3=0\) either a) displacements or b) stresses are absent, body forces \(\dot F_i\) act, with \(\dot F_i(x_1,x_2,x_3)=0\) for \(x<0\) \((i=x_1,x_2,x_3)\). Then the displacements that arise in the elastic half-space \(x_3>0\) from the action of these forces can be regarded as the solution of an elastic problem for the whole space, in which the following system of body forces acts:

\[ \begin{aligned} \text{a)}\quad F_i={}&(\dot F_i-\dot F_i') +\frac{2(1-\nu)}{(1-2\nu)(3-4\nu)}\,x_3 \left[ 2\,\frac{\partial \dot F_3'}{\partial x_i} -\frac{1}{2(1-\nu)}\,x_3 \times \right.\\ &\left. \times \frac{\partial}{\partial x_i}(\operatorname{div}\dot{\mathbf F})' \right] +\frac{2}{(1-2\nu)(3-4\nu)}\,\delta_{i3} \left[\dot F_3'-2\nu x_3(\operatorname{div}\dot{\mathbf F})'\right] \quad (i=x_1,x_2,x_3); \end{aligned} \]

\[ \begin{aligned} \text{b)}\quad F_i={}&(\dot F_i+\dot F_i') -\frac{4(1-\nu)}{1-2\nu}\,x_3\frac{\partial \dot F_3'}{\partial x_i} +4(1-\nu)\frac{\partial}{\partial x_i} \int_{x_3}^{+\infty}\dot F_3'\,dx_3 \\ &+\frac{1}{1-2\nu}x_3^2\frac{\partial}{\partial x_i} (\operatorname{div}\dot{\mathbf F})' +2(1-\nu)x_3\frac{\partial}{\partial x_i} \int_{x_3}^{+\infty}(\operatorname{div}\dot{\mathbf F})'\,dx_3 \\ &-2(1-\nu)\int_{x_3}^{+\infty}x_3(\operatorname{div}\dot{\mathbf F})'\,dx_3 \quad (i=x_1,x_2), \end{aligned} \]

\[ F_3=\dot F_3-\frac{1+2\nu(3-4\nu)}{1-2\nu}\dot F_3' -\frac{4(1-\nu)}{1-2\nu}x_3\frac{\partial \dot F_3'}{\partial x_3} +\frac{4\nu}{1-2\nu}x_3(\operatorname{div}\dot{\mathbf F})' +\frac{1}{1-2\nu}x_3^2\frac{\partial}{\partial x_3}(\operatorname{div}\dot{\mathbf F})' +2\nu\int_{x_3}^{+\infty}(\operatorname{div}\dot{\mathbf F})'\,dx_3 . \]

Here \(\nu\) is Poisson’s ratio, \(\delta_{i3}\) is the Kronecker symbol, and the prime operation denotes replacing \(x_3\) by \((-x_3)\). We note that the integrals may turn out to be divergent in the ordinary sense; therefore they should be understood in the sense of Hadamard or, equivalently, in the sense of the theory of generalized functions \((^6)\).

Moscow State University
named after M. V. Lomonosov

Received
6 VI 1970

REFERENCES

\(^1\) A. Lyav, Mathematical Theory of Elasticity, Moscow, 1935.
\(^2\) R. D. Mindlin, Physics, 7, 195 (1936); Collected Translations: Mechanics, 4, 118 (1952).
\(^3\) R. J. Duffin, J. Rat. Mech. and Analysis, 5, No. 5, 939 (1956).
\(^4\) E. I. Obolashvili, Rev. Roumaine Math. pures et appl., 11, No. 8, 965 (1966).
\(^5\) V. D. Kupradze, T. G. Gegelia et al., Three-Dimensional Problems of the Mathematical Theory of Elasticity, Tbilisi, 1968.
\(^6\) Ch. Solver, in: Problems of Mechanics, 3, Moscow, 1961, p. 7.