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.