I am looking for any references about 2-dimensional lattice structures (ie hexagonal like graphine) that have the highest shear strength (or just resistance to distortion in the 2D plane). Does anybody have any good references for this?
My first guess would be a mesh of equilateral triangles. But I think it will depend on the boundary conditions of the 2D sheet. A square grid at 45° to two opposite edges would efficiently resist shear if the direction of the shear was always in the same predictable direction.
A more sparse grid might be a mix of two polygons, say triangles and squares, or hexagons and pentagons arranged in the pattern that a lava flow cracks as it cools. That will cover an area with minimum material.
What are the boundary conditions? What is the scale? Why consider only the 2D solution?
You might consider a two layer structure such as an octet truss which would be less likely to buckle along compressive axes.
http://www.virginia.edu/ms/research/wadley/Documents/Publications/Shear_Response_Carbon_Fiber.pdf
I working on something similar to a laminar flow gate , just very thin relative to the size of the part (~6 in x 12in x 0.2 in). The interior mesh walls are also very thin ~0.010". I'm looking for an optimal lattice structure that could give the best structural integrity, especially against twisting/shearing. Lattice size and shape (square, triangle, hexagon, etc.) are flexible. I also think a mesh of equilateral triangles would be optimal, but am curious if there is anything to back this up.
Unfortunately, I can't use any cool two layer structures. It's got to have straight through-passages.
What forces are acting on the sheet ? Is this something that you are going to manufacture ? Is the material actually graphene or is it something else ?
The more you tell us the better the answers that you will get .