Series expansion of the fundamental solution of three-dimensional elastostatics

where are base vectors in the Cartesian coordinates. In the new system, we have

where . The derivatives with respect to these variables satisfy

with which is written as

The functions and (See (F.9)) satisfy relations given by

and the functions and (See (F.9)) satisfy relations given by

Using above relations, (D.1), (D.2), (D.3), (D.7), (D.8) and (D.11), we differentiate (F.8) using (G.1) to obtain the derivatives of as

Another way to obtain this result is to use an expansion of given by[88]

With (G.3) we can derive (G.2) as follows:

From (G.2) we obtain the second order derivatives of as

where is the second order unit tensor. The fundamental solution of the equation of elastostatics is expressed as

Using (G.3), (G.4) and (G.5), we obtain a series expansion of given by

where

Also, we can derive (G.6) in another way. We first rewrite then
fundamental solution (G.5) as

Substituting (G.3) into (G.7), we obtain

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