Introduction to Elasticity/Flat punch indentation

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Indentation due to a Frictionless Rigid Flat Punch

Indentation by a plat rigid punch

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.

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