Elasticity of a drilled plate

Question

If I take a large thin plate and I uniformly drill a very large number of holes (diameter "d") very close together (minimum spacing "s"), what will be the stiffness of this new plate? When I say "stiffness" I mean the derivative of average macroscopic displacement with respect to stress. With no holes it would be Young's modulus.

Here is a way to visualize how the drills are to positioned, i.e. close packed circles with space in between.

This is the question I want to answer. Of course, the stiffness will depend on which direction the stress is applied - for example, if I am pulling along the "grain" it will be stiffer than if I turn the plate 45 degrees and pull.

I thought about doing some sort of analytic model (found a few papers but they are hard to understand). Or a finite element model. However, I am interested in how the parameters "d" and "s" (see above) change the stiffness for each type of circle packing pattern. I am stuck now, what would you recommend as the best way to approach this problem?

Also, if this problem has already been solved (wouldn't be surprised) and someone could point me to a reference that would work too.

Answer 1

A simple first order approach would be to treat the plate like a composite material, with the holes acting as a medium with no modulus. The rule of mixtures , treating the "holes" as fibers with 0 modulus, would yield a modulus of 0. So, the Semi-Emperical Halpin Tsai would be better:

$$ \eta = \frac{\frac{E_f}{E_m} -1}{\frac{E_f}{E_m}+1} $$

$$ E_c = \frac{E_m(1+\eta f)}{(1-\eta f)}$$

Since $ E_f = 0 $, we have $ \eta = -1 $, so:

$$ E_c = E_m(\frac{1-f}{1+f}) $$

In the case of the square packing arrangement, the circles occupy 78.54% of the area. So, the "composite" would be ~12% of the original modulus. Again, this would be a first order approximation to save you from running 50 finite element models. Then, run your FEA to watch out for the stress concentrators for final design.