Some problems in nonlinear elasticity
<p>Two separate problems are discussed: axisymmetric equilibrium configurations of a circular membrane under pressure and subject to thrust along its edge, and the buckling of a circular cylindrical shell.</p> <p>An ordinary differential equation governing the circular membra...
| Summary: | <p>Two separate problems are discussed: axisymmetric equilibrium
configurations of a circular membrane under pressure and
subject to thrust along its edge, and the buckling of a circular
cylindrical shell.</p>
<p>An ordinary differential equation governing the circular membrane
is imbedded in a family of n-dimensional nonlinear equations.
Phase plane methods are used to examine the number of solutions
corresponding to a parameter which generalizes the thrust, as well as
other parameters determining the shape of the nonlinearity and the
undeformed shape of the membrane. It is found that in any number of
dimensions there exists a value of the generalized thrust for which a
countable infinity of solutions exist if some of the remaining parameters
are made sufficiently large. Criteria describing the number of
solutions in other cases are also given.</p>
<p>Donnell-type equations are used to model a circular cylindrical
shell. The static problem of bifurcation of buckled modes from
Poisson expansion is analyzed using an iteration scheme and pertubation
methods. Analysis shows that although buckling loads are usually
simple eigenvalues, they may have arbitrarily large but finite multiplicity
when the ratio of the shell's length and circumference is rational.
A numerical study of the critical buckling load for simple eigenvalues
indicates that the number of waves along the axis of the deformed shell
is roughly proportional to the length of the shell, suggesting the possibility
of a "characteristic length." Further numerical work indicates
that initial post-buckling curves are typically steep, although the load
may increase or decrease. It is shown that either a sheet of solutions
or two distinct branches bifurcate from a double eigenvalue. Furthermore,
a shell may be subject to a uniform torque, even though one is
not prescribed at the ends of the shell, through the interaction of two
modes with the same number of circumferential waves. Finally,
multiple time scale techniques are used to study the dynamic buckling
of a rectangular plate as well as a circular cylindrical shell; transition
to a new steady state amplitude determined by the nonlinearity is
shown. The importance of damping in determining equilibrium configurations
independent of initial conditions is illustrated.</p> |
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