I'm working on ways to simulate layered elastic systems using closed
form approximations.
The idea is that there are strips of elastic material each with a
Young's modulus and a poisson ratio layered on top of each other. The
bottom one is semi infinite. A load is placed on the top layer by
specifying a stress distribution. The problem would assume radial
symmetry.
I'm coming up against 2 problems.
1) I don't understand the general form of elasticity equations very
well. I have the usual engineering mechanics of materials courses, but
my understanding of how tensors work is limited.
However, the main thing I want is stresses in the two normal
directions and the shear stress. This is 3 unknowns so I assume I'll
have 3 equations that come from static equilibrium of forces. I assume
these equations will be PDEs interrelating the stresses and their
spatial derivatives and the applied loads.
2) Even if I were trying to do something that doesn't require tensors,
like for example heat conduction in a layered system, I'm not sure
what the best way to model this kind of PDE in maxima would be.
I'll focus on the part 2 here, since it's the maxima list, but I'd
also appreciate pointers on part 1 if you have suggestions.
My idea for the maxima solutions is to use a spectral method. I've
been inspired by the reasonably well written online book:
http://www-personal.engin.umich.edu/~jpboyd/aaabook_9500may00.pdf
I'm not sure whether I should solve for stresses or strains, but I
suspect that because the surface stress distribution will be given,
the stresses are the way to go. So I need to choose a basis set for
the stresses in an elastic layer. For the finite layers I would guess
that polynomials would work fine for the z direction (where the
thickness is finite) but for the radial direction I'm not sure what to
use. Also I'm not sure how to combine the two dimensions. Should I
assume separability so that the stresses are f(z)*g(r)?? This seems
clearly easier to solve but is perhaps not reasonable for some reason
I haven't forseen?
Anyway, once I've gotten the appropriate set of PDEs and a basis set
for each layer, I assume that I'll need to express the integrated
residual over all space and minimize it subject to constraints that
strains are continuous across the boundaries. I think this part would
be a LOT easier if the spatial dimensions were separable.
The final step will be to solve the problem for several loads and
combine the solutions using linear superposition and coordinate
transforms.
Does anyone have any suggestions as to where to go from here?
One of my goals is to write up the solution with the maxima code, and
make it available as a sort of tutorial on both maxima and the
mathematics and mechanics of the method.
I appreciate any pointers any of you might be able to provide.
Thanks,
Dan
--
Daniel Lakeland
dlakelan at street-artists.org
http://www.street-artists.org/~dlakelan