Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 326

By Korn's inequality, the bilinear form a( ∇ w, ∇ v) is H 2 -elliptic on H 0 () .
For a conforming treatment, the variational problem

1
2 a( ∇ v, ∇ v) − (f, v) −→

min ! ( 5 . 9 )

H 0 () v ∈

for the clamped Kirchhoff plate requires C 1 elements, which is computationally
expensive. In this respect, the numerical treatment of the Mindlin plate appears
at first glance to be simpler since the problem (5.7) is obviously H 1 -elliptic for
(w, θ) ∈ H 0 () × H 0 () 2 . However, as we shall see, the Mindlin plate contains
a small parameter.
Finally, we would like to mention the so-called Babuska paradox ; see, e.g.,
Babuska and Pitkaranta [1990]. If we approximate a domain with a smooth bound-
ary by polygonal domains, then the solutions for Kirchhoff plates on these domains
usually do not converge to the solution for the original domain. This holds for the
clamped plates as well as for some other boundary conditions.

Note on Beam Models

While plate models refer to elliptic problems in 2-space, the beam models lead to
boundary-value problems with ordinary differential equations. The beam with the
Kirchhoff hypothesis is called the Bernoulli beam , and the beam which corresponds
to the Mindlin plate is the Timoshenko beam . If we eliminate the Lame constants,
we obtain the energy of the Timoshenko beam by a reduction of (5.7) to the
one-dimensional case:

b b fwdx.

t − 2
2

1
2

(θ ) 2 dx + (w − θ) 2 dx −

(θ, w) : =

0 0

Here θ,w ∈ H 0 ( 0 ,b) where b denotes the length of the beam.
We have already considered this model in §3 when illustrating shear locking.
It should be emphasized that the computation and the analysis of plates is much
more involved and cannot be understood as a simple generalization; cf. Problem
5.13.

Mixed Methods for the Kirchhoff Plate

Nonconforming and mixed methods play an important role in the theory of Kirch-
hoff plates. We begin with mixed methods since we will also use them for the
analysis of nonconforming elements. In the following, a( · , · ) always denotes the
H 1 -elliptic bilinear form on H 0 () 2 defined in (5.5).