Summary: INNOVATIVE FINITE ELEMENT METHODS FOR PLATES*
DOUGLAS N. ARNOLD
Abstract. Finite element methods for the Reissner­Mindlin plate theory are discussed. Methods in which
both the tranverse displacement and the rotation are approximated by finite elements of low degree mostly
suffer from locking. However a number of related methods have been devised recently which avoid locking
effects. Although the finite element spaces for both the rotation and transverse displacement contain
little more than piecewise linear functions, optimal order convergence holds uniformly in the thickness.
The main ideas leading to such methods are reviewed and the relationships between various methods are
clarified.
1. The Reissner­Mindlin plate equations. The Reissner­Mindlin plate equations
describe the bending of a linearly elastic plate in terms of the transverse displacement, ,
of the middle plane and the rotation, , of the fibers normal to the middle plane. This
model, as well as its generalization to shells, is frequently used for plates and shells of
small to moderate thickness. Assuming that the material is homogeneous and isotropic
with Young's modulus E and Poisson ratio , the governing differential equations, which
are to hold on the two dimensional region occupied by the middle plane of the plate,
take the form
- div C E() - t-2
(grad - ) = 0,(1)
-t-2

Source: Arnold, Douglas N. - School of Mathematics, University of Minnesota

Collections: Mathematics