The objective of Volume II is to show how asymptotic methods, with
the thickness as the small parameter, indeed provide a powerful
means of justifying two-dimensional plate theories. More
specifically, without any recourse to any "a priori" assumptions of
a geometrical or mechanical nature, it is shown that in the linear
case, the three-dimensional displacements, once properly scaled,
converge in "H"1 towards a limit that satisfies the well-known
two-dimensional equations of the linear Kirchhoff-Love theory; the
convergence of stress is also established.
In the nonlinear case, again after "ad hoc" scalings have been
performed, it is shown that the leading term of a formal asymptotic
expansion of the three-dimensional solution satisfies well-known
two-dimensional equations, such as those of the nonlinear
Kirchhoff-Love theory, or the von Karman equations. Special
attention is also given to the first convergence result obtained in
this case, which leads to two-dimensional large deformation,
frame-indifferent, nonlinear membrane theories. It is also
demonstrated that asymptotic methods can likewise be used for
justifying other lower-dimensional equations of elastic shallow
shells, and the coupled pluri-dimensional equations of elastic
multi-structures, i.e., structures with junctions. In each case,
the existence, uniqueness or multiplicity, and regularity of
solutions to the limit equations obtained in this fashion are also
studied.
General
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