View source for Elasticity/Flat punch indentation

== Indentation due to a frictionless rigid flat punch ==

{|align="center" 
|[[image:indentation_by_flat_punch.png|thumb|400px|Indentation by a plat rigid punch]]
|}
   
*Start with uneven surface profile <math>u_0(x_1)\,</math>.  
*Unsymmetric load <math>F\,</math>, but sufficient for complete contact  over the area <math>A\,</math>.  

Displacement in <math>x_2\,</math> direction is
:<math>
u_2 = - u_0(x_1) + C_1 x_1 + C_0\,
</math>
where <math>C_0\,</math> is a rigid body translation and <math>C_1 x_1\,</math> is a rigid body rotation.

Rigid body motions can be determined using a statically equivalent 
set of forces and moments
:<math>\begin{align}
\int_A p(\xi)~d\xi & = -F \\
\int_A p(\xi)\xi~d\xi & = -F~d 
\end{align}</math>

The displacement gradient is
:<math>
-\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
</math>

Integral is a [[w:Cauchy singular integral|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 <math>p(\xi)\,</math>.If we change the variables so that

<math>x = a \cos\phi\,</math> and <math>\xi = a \cos\theta\,</math>, then 
:<math>
\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
</math>

If we write <math>p(\theta)\,</math> and <math>du_0/d\phi\,</math> as
:<math>\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}</math>
and do some algebra, we get
:<math>\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}</math>


===  Flat punch with symmetric load ===
: <math>u_0 = C\,</math> 
In this case,
:<math>
\frac{du_0}{d\phi} = 0 \Rightarrow u_n = 0 ~;~~ n = 1  {\infty}
</math>
Also, <math>d = 0\,</math> (origin at the center of <math>A\,</math>), hence <math>p_1 = 0\,</math>. Therefore,
:<math>
p(x) = \frac{p_0}{\sin\phi} = -\frac{F}{\pi \sqrt{a^2 - x^2}}
</math>
At <math>x = \pm a</math>, the load is infinite, i.e. there is a singularity.


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[[Category:Elasticity]]