Elasticity/Flat punch indentation

Indentation due to a frictionless rigid flat punch

 * Start with uneven surface profile $$u_0(x_1)\,$$.
 * Unsymmetric load $$F\,$$, but sufficient for complete contact over the area $$A\,$$.

Displacement in $$x_2\,$$ direction is

u_2 = - u_0(x_1) + C_1 x_1 + C_0\, $$ where $$C_0\,$$ is a rigid body translation and $$C_1 x_1\,$$ is a rigid body rotation.

Rigid body motions can be determined using a statically equivalent set of forces and moments
 * $$\begin{align}

\int_A p(\xi)~d\xi & = -F \\ \int_A p(\xi)\xi~d\xi & = -F~d \end{align}$$

The displacement gradient is

-\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 $$p(\xi)\,$$.If we change the variables so that

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

\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 $$p(\theta)\,$$ and $$du_0/d\phi\,$$ as
 * $$\begin{align}

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{align}$$ and do some algebra, we get
 * $$\begin{align}

p_0 & = -\frac{F}{\pi a} \\ p_1 & = -\frac{F d}{\pi a^2} \\ p_n & = -\frac{4\mu u_n}{(\kappa+1)a} ~; n > 1 \end{align}$$

Flat punch with symmetric load

 * $$u_0 = C\,$$

In this case,

\frac{du_0}{d\phi} = 0 \Rightarrow u_n = 0 ~; n = 1 {\infty} $$ Also, $$d = 0\,$$ (origin at the center of $$A\,$$), hence $$p_1 = 0\,$$. Therefore,

p(x) = \frac{p_0}{\sin\phi} = -\frac{F}{\pi \sqrt{a^2 - x^2}} $$ At $$x = \pm a$$, the load is infinite, i.e. there is a singularity.