Elasticity/Distributed force on half plane

Distributed force on a 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 $P\,$.

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