Editing Linear elasticity (section)

===== Boussinesq–Cerruti solution - point force at the origin of an infinite isotropic half-space =====
Another useful solution is that of a point force acting on the surface of an infinite half-space.<ref name="tribonet" /> It was derived by Boussinesq<ref>{{cite book |title=Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques |last=Boussinesq |first=Joseph |author-link=Joseph Boussinesq |year=1885 |publisher=Gauthier-Villars |location=Paris, France |url=http://name.umdl.umich.edu/ABV5032.0001.001 |archive-date=2024-09-03 |access-date=2007-12-19 |archive-url=https://web.archive.org/web/20240903234441/https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV5032.0001.001 |url-status=live }}</ref> for the normal force and Cerruti for the tangential force and a derivation is given in Landau & Lifshitz.<ref name="LL" />{{rp|§8}} In this case, the solution is again written as a Green's tensor which goes to zero at infinity, and the component of the stress tensor normal to the surface vanishes. This solution may be written in Cartesian coordinates as [recall: <math>a=(1-2\nu)</math> and <math>b=2(1-\nu)</math>,  <math>\nu</math> = Poisson's ratio]:

<math display="block">G_{ik} = \frac{1}{4\pi\mu r}
\begin{bmatrix}
\frac{b r + z}{r + z} + \frac{(2 r (\nu r + z) + z^2) x^2}{r^2 (r + z)^2} &
\frac{(2 r (\nu r + z) + z^2) x y}{r^2 (r + z)^2} &
\frac{x z}{r^2} - \frac{a x}{r + z} \\

\frac{(2 r (\nu r + z) + z^2) y x}{r^2 (r + z)^2} &
\frac{b r + z}{r + z} + \frac{(2 r (\nu r + z) + z^2) y^2}{r^2 (r + z)^2} &
\frac{y z}{r^2} - \frac{a y}{r + z} \\

\frac{z x}{r^2} + \frac{a x}{r + z} &
\frac{z y}{r^2} + \frac{a y}{r + z} &
b + \frac{z^2}{r^2}
\end{bmatrix}
</math>