Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 278

Chapter VI
Finite Elements in Solid Mechanics
Finite element methods are the most widely used tools for computing the defor-
mations and stresses of elastic and inelastic bodies subject to loads. These types
of problems involve systems of differential equations with the following special
feature: the equations are invariant under translations and orthogonal transforma-
tions since the elastic energy of a body does not change under so-called rigid body
motions.
Practical problems in structural mechanics often involve small parameters
which can appear in both obvious and more subtle ways. For example, for beams,
membranes, plates, and shells, the thickness is very small in comparison with the
other dimensions. On the other hand, for a cantilever beam, the part of the bound-
ary on which Dirichlet boundary conditions are prescribed is very small. Finally,
many materials allow only very small changes in density. These various cases re-
quire different variational formulations of the finite element computations. Using
an incorrect formulation leads to so-called locking . Often, mixed formulations pro-
vide a suitable framework for both the computation and a rigorous mathematical
analysis.
Most of the characteristic properties appear already in the so-called linear
theory, i.e., for small deformations where no genuine nonlinear phenomenon oc-
curs. However, strictly speaking, there is no complete linear elasticity theory, since
the above-mentioned invariance under rigid body motions cannot be completely
modeled in a linear theory. For this reason, we don't restrict ourselves to the linear
theory until later.
§§1 and 2 contain a very compact introduction to elasticity theory. For more
details, see Ciarlet [1988], Marsden and Hughes [1983], or Truesdell [1977]. Here
we concentrate on those aspects of the theory which we need as background
knowledge. In §3 we present several variational formulations for the linear theory,
and also include an analysis of locking. Finally, we discuss membranes and plates.
In particular, we explore the connection between two widely used plate models.
We limit ourselves to those elements whose construction or analysis is based
on different approaches than the elements discussed in Chapters II and III. In
particular, we will focus on the stability of the elements.