Elasticity/Distributed force on half plane

Distributed force on a half-plane

 * Applied load is $$p(\xi)\,$$ per unit length in the $$x_2\,$$ direction.
 * We already know the stresses and displacements due to a  concentrated force. The stresses and displacements due to the   distributed load can be found by superposition.
 * The Flamant solution is used as a Green's function, i.e., the distributed load is taken as the limit of a set of point   loads of magnitude $$p(\xi)\delta\xi\,$$.

At the point $$P\,$$

u_2 = - \frac{(\kappa+1)}{4\pi\mu} \int_A p(\xi)\ln|x - \xi|~d\xi $$ As $$x \rightarrow \infty\,$$, $$u_2\,$$ is unbounded. However, if we are interested in regions far from $$A\,$$, we can apply the distributed force as a statically equivalent concentrated force and get displacements using the concentrated force solution.

The avoid the above issue, contact problems are often formulated in terms of the displacement gradient

\frac{du_2}{dx_1} = - \frac{(\kappa+1)}{4\pi\mu} \int_A \frac{p(\xi)}{x - \xi}~d\xi $$ If the point $$P\,$$ is inside $$A\,$$, then the integral is taken to be the sum of the integrals to the left and right of $$P\,$$.