Point forces and systems of point forces in the three-dimensional space and half-space

This problem was solved by Lord Kelvin (W. Thomson) in 1848 (for reference, see Thomson, 1882). Point force F is applied at the origin of coordinates. Displacement vector u and stress tensor σ at the point x = x₁e₁ + x₂e₂ + x3e₃ are $$\begin{gathered} u = \frac{1}{{16\pi G\left( {1 - v} \right)}}\left[ {\left( {3 - 4v} \right)\frac{F}{R} + \frac{{F \cdot R}}{{{R^3}}}R} \right] \hfill \\ \sigma = \frac{1}{{8\pi G\left( {1 - v} \right)}}\frac{1}{{{R^3}}}\left[ {\left( {1 - 2v} \right)\left( {IF \cdot R - FR - RF} \right) - \frac{{3F \cdot R}}{{{R^2}}}RR} \right] \hfill \\ \end{gathered} $$ where R = x, $$R = \left| {\mathbf{R}} \right| = \sqrt {x_1^2 + x_2^2 + x_2^2} $$.