# Deformation/fracture of material on impact

1. Oct 4, 2011

### ITAmember

I want to calculate the deformation of a plate of a material upon being struck by a mass moving along the plate's normal, and if the impact force is sufficient, if the plate will break or not. The given information can be assumed to be: mass, velocity, and contact area of the mass, and the thickness of the plate along with necessary material constants (e.g. Young's modulus).

For the sake of an example, let's assume a 100 kg hammer moving at 10 $\frac{m}{s}$ with a contact area of 0.01 $m^{2}$ strikes a 0.1 m thick steel plate, of which Young's modulus is equal to 200 GPa.

Some equations:

Stress
$\sigma=\frac{F}{A}$

Strain
$\tau=\frac{\Delta{L}}{L_{0}}$

Hooke's law
$E=\frac{stress}{strain}=\frac{\sigma}{\tau}=\frac{{F}L_{0}}{{A}\Delta{L}}$

To calculate $\Delta{L}$ I need force, which AFAIK is non-trivial to calculate. I found this equation

$F=m\frac{v^{2}}{2\Delta{L}}$

Solving Hooke's law for F, substituting, and solving for $\Delta{L}$ yields the following

$F=\frac{EA\Delta{L}}{L_{0}}$
$\Delta{L}=mL_{0}\frac{v^{2}}{2EA}$

This finds the deformation of the material using the given information, and with substituting the example the deformation is 0.0005m, which sounds reasonable for a 0.1m steel plate. However, I noticed that as the thickness of the plate increased, so did the deformation. I suppose this makes sense given the equation for strain, but seems counter-intuitive and leads me to wonder if I am doing this wrong. In addition, I have no idea where to start with the breaking point of the plate and how much force that requires.

All help is appreciated.