Elasticity/Distributed force on half plane

Distributed force on a half-plane

File:Distributed force on half plane.png 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 $P$

$u_2=-\frac{(\kappa +1)}{4\pi \mu }\underset{A}{\int }p(\xi )\ln |x-\xi |d\xi$

As $x\to \mathrm{\infty }$, $u_2$ 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

$\frac{d{u}_2}{d{x}_1}=-\frac{(\kappa +1)}{4\pi \mu }\underset{A}{\int }\frac{p(\xi )}{x-\xi }d\xi$

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 $P$.

Template:Subpage navbar