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Re: Higher order derivatives in elasticity.



Hi Folks,

This is embarrassing. When I took these notes circa. 1969, I supposed
that I would never forget what the variable symbols stood for. Almost 30
years later it's all Greek to me. I never used elasticity theory.
(No one would support research on drums and violins.) I'll just write
out some equations (e-mail style) for the buckling of a thin shell (could
be circular) with load, f(x,y), normal to unstressed position (horizontal
say) and stresses in the plane of the shell, p(s), normal to the edge.
It's a two-dimensional problem despite the displacement in the
z-dimension = w(x,y). In the linear theory, the compatibility condition
for phi(x,y) is del^4 phi = 0, where phi satisfies sigma_xx = phi_yy,
sig_xy = - phi_xy, sig_yy = phi_xx. Sigma is stress - I hope. Phi
obviously is some dual type variable. Phi and sigma are like the
conjugate variables in analytic plane fluid flow, in which theory the
baseball can't curve, which, in turn, gave scientists a bad rap due to a
stupid misunderstanding. (In the nonlinear case, compatibility is del^4
phi = E(w^2_xy -w_xx*w_yy), where _xx means second partial w.r.t. x.)

For normal stress, p, uniform, f(x,y)=0 (I think), we get

del^4 w + h*p/D = 0
s.t. w=0 and del^2 w = 0 on bddy,

where D = Eh^3/[12(1- nu^2)] , h is shell thickness, E is Young's modulus
or some damn thing that I didn't bother to identify, and I have no idea
what nu is.

Does anyone out there know? Not a clue in my notes. It must have
been like my first name then and I thought I would never forget what it
meant when I wrote it. The (uniform) buckling pressure is applied at
the boundary (normal). The shell can be loaded by f(x,y) at every point
on the shell. I think f(x,y) has to be continuous.

This is an eigenvalue problem for the buckling pressures, the
eigenvalues corresponding to various modes of buckling. Back then we did
not even use the word *bifurcation*, but Stoker has a nice bifurcation
drawing (for the nonlinear case, which is due to von Karman and Foppl (as
close as I can read my writing) ) showing three branches followed by K.O.
Friedrichs and himself. (I guess they had von Karman and Foppls'
solution at their disposal.) The trivial solution, then a turning point
to three solutions, then a turning point to five solutions. The first
branch goes to infinity.

This may be more than anyone wanted to know. It may be the answer to
the wrong question even. But, clearly, the fourth derivative is
important in this application.

Regards / The Amateur