Elasticity/Flat punch indentation

• Start with uneven surface profile ${\displaystyle u_{0}(x_{1})\,}$.
• Unsymmetric load ${\displaystyle F\,}$, but sufficient for complete contact over the area ${\displaystyle A\,}$.

Displacement in ${\displaystyle x_{2}\,}$ direction is

${\displaystyle u_{2}=-u_{0}(x_{1})+C_{1}x_{1}+C_{0}\,}$

where ${\displaystyle C_{0}\,}$ is a rigid body translation and ${\displaystyle C_{1}x_{1}\,}$ is a rigid body rotation.

Rigid body motions can be determined using a statically equivalent set of forces and moments

${\displaystyle {\begin{aligned}\int _{A}p(\xi )~d\xi &=-F\\\int _{A}p(\xi )\xi ~d\xi &=-F~d\end{aligned}}}$

The displacement gradient is

${\displaystyle -{\frac {du_{0}}{dx_{1}}}+C_{1}=-{\frac {(\kappa +1)}{4\pi \mu }}\int _{-a}^{a}{\frac {p(\xi )}{x-\xi }}~d\xi ~;~~-a<x<a}$

Integral is a Cauchy Singular Integral that appears often and very naturally when the problem is solved using complex variable methods.

Note that the only thing we are interested in is the distribution of contact forces ${\displaystyle p(\xi )\,}$.If we change the variables so that

${\displaystyle x=a\cos \phi \,}$ and ${\displaystyle \xi =a\cos \theta \,}$, then

${\displaystyle {\frac {1}{a\sin \phi }}{\frac {du_{0}}{d\phi }}+C_{1}=-{\frac {(\kappa +1)}{4\pi \mu }}\int _{0}^{\pi }{\frac {p(\theta )\sin \theta }{\cos \phi -\cos \theta }}~d\theta ~;~~0<\phi <\pi }$

If we write ${\displaystyle p(\theta )\,}$ and ${\displaystyle du_{0}/d\phi \,}$ as

${\displaystyle {\begin{aligned}p(\theta )&=\sum _{0}^{\infty }{\frac {p_{n}\cos(n\theta )}{\sin \theta }}\\{\frac {du_{0}}{d\phi }}&=\sum _{1}^{\infty }u_{n}\sin(n\phi )\end{aligned}}}$

and do some algebra, we get

${\displaystyle {\begin{aligned}p_{0}&=-{\frac {F}{\pi a}}\\p_{1}&=-{\frac {Fd}{\pi a^{2}}}\\p_{n}&=-{\frac {4\mu u_{n}}{(\kappa +1)a}}~;~~n>1\end{aligned}}}$