The objective of this article is to lay down the proper mathematical foundations
of the two-dimensional theory of linearly elastic shells. To this end,
it provides, without any recourse to any a priori assumptions of a geometrical
or mechanical nature, a mathematical justification of two-dimensional
linear shell theories, by means of asymptotic methods, with the thickness as
the ‘small’ parameter.
A major virtue of this approach is that it naturally leads to precise mathematical
definitions of linearly elastic ‘membrane’ and ‘flexural’ shells. Another
noteworthy feature is that it highlights in particular the role played by two
fundamental tensors, each associated with a displacement field of the middle
surface, the linearized change of metric and linearized change of curvature
tensors.
More specifically, under fundamentally distinct sets of assumptions bearing
on the geometry of the middle surface, on the boundary conditions, and
on the order of magnitude of the applied forces, it is shown that the three-dimensional
displacements, once properly scaled, converge (in H1, or in L2,
or in ad hoc completions) as the thickness approaches zero towards a ‘two-dimensional’
limit that satisfies either the linear two-dimensional equations
of a ‘membrane’ shell (themselves divided into two subclasses) or the linear
two-dimensional equations of a ‘flexural’ shell. Note that this asymptotic analysis
automatically provides in each case the ‘limit’ two-dimensional equations,
together with the function space over which they are well-posed.
The linear two-dimensional shell equations that are most commonly used
in numerical simulations, namely Koiter's equations, Naghdi's equations, and
‘shallow’ shell equations, are then carefully described, mathematically analysed,
and likewise justified by means of asymptotic analyses.
The existence and uniqueness of solutions to each one of these linear two-dimensional
shell equations are also established by means of crucial inequalities
of Korn's type on surfaces, which are proved in detail at the beginning
of the article.
This article serves as a mathematical basis for the numerically oriented
companion article by Dominique Chapelle, also in this issue of Acta Numerica.