Elasticity/Distributed force on half plane

Distributed force on a half-plane

Distributed force on a half plane

• Applied load is ${\displaystyle p(\xi )\,}$ per unit length in the ${\displaystyle x_{2}\,}$ 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 ${\displaystyle p(\xi )\delta \xi \,}$.

At the point ${\displaystyle P\,}$

${\displaystyle u_{2}=-{\frac {(\kappa +1)}{4\pi \mu }}\int _{A}p(\xi )\ln |x-\xi |~d\xi }$

As ${\displaystyle x\rightarrow \infty \,}$, ${\displaystyle u_{2}\,}$ is unbounded. However, if we are interested in regions far from ${\displaystyle A\,}$, 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

${\displaystyle {\frac {du_{2}}{dx_{1}}}=-{\frac {(\kappa +1)}{4\pi \mu }}\int _{A}{\frac {p(\xi )}{x-\xi }}~d\xi }$

If the point ${\displaystyle P\,}$ is inside ${\displaystyle A\,}$, then the integral is taken to be the sum of the integrals to the left and right of ${\displaystyle P\,}$.