On the Strains of an Elastic Solid with a Superficial Layer due to a Statica Force applied to the Surface

Abstract

The problems concerning to the Statics of the semi-infinite elatic body have been discussed by many authors. Recently, Dr. H. Honda and Mr. T. Miura have introduced the solutions of this problem by Fourier's theorem of double integral and discussed some applications to the seismic problem. Mr. F. J. W. Whipple discussed the problem of internal nucleus of strain in the case of the semi-infinite elastic body. But when there is a surface layer on the semi-infinite elastic body, the mathematical treatment becomes too complex to have the general solution. Mr. G. Nisimura treated of this problem, but his discussions can be applied only to the special case in which the thickness of the surface layer is very small. The present author treats of the axial-symmetric case by the complex integral expanding the exponential terms. The approximate expression of the normal displacement at the surface becomes in cylindrical coordinates Fz(r)=function of the applied statical force, uz_??_z=0=the displacement at the surface.
(z-component), μ, μ'=elastic constants, H=thickness of surface layer.
In the above expression the axis of z directs to internal part of the body, perpendicularly to the surface, and r in horizontal axis, and Lame's constants λ, μ; λ', μ' are defined by λ=μ, λ'=μ'. The solution of this equation is separated in two terms, one shows the solution when there is no surface layer and the other is a correction term of the layer.
As an example, let the applied surface traction be and μ'/μ=2, then the pole exists only one at k=1.45/H. Now let the pole be expressed by and the solution is
The first term shows the same solution as that which was introduced by Dr. H. Honda and Mr. T. Miura in the case of no surface layer, and the second term is a correction te_??_m of the layer.