Innovative Finite Element Methods for Plates*

Finite element methods for the Reissner–Mindlin 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 Reissner–Mindlin plate equations. The Reissner–Mindlin plate equations describe the bending of a linearly elastic plate in terms of the transverse displacement, ω, of the middle plane and the rotation, φ, 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 ν, the governing differential equations, which are to hold on the two dimensional region Ω occupied by the middle plane of the plate, take the form −div C E(φ)− λt−2(gradω − φ) = 0, (1) −λt−2 div(gradω − φ) = g, (2) where t is the plate thickness, gt is the transverse load force density per unit area, E(φ) is the symmetric part of the gradient of φ, λ = Ek/2(1 + ν), and the fourth order tensor C is defined by CT = E 12(1− ν2) [(1− ν)T + ν tr(T )I] for any 2× 2 matrix T (I denotes the 2× 2 identity matrix). The load function has been normalized by a factor of t so that the solution tends to a nonzero limit as t tends to zero. The shear correction factor k is a constant often taken to equal 5/6. Solutions to this system are minimizers of the energy functional (φ, ω) 7→ J(φ, ω) := ∫ Ω [ 1 2 C E(φ) : E(φ) + 1 2 λt−2|gradω − φ| − gω ] dx, *Paper presented at the Workshop on Innovative Finite Element Methods, Rio de Janeiro, November 27–December 1, 1989. This work was supported by NSF grant DMS-89-02433. †Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802.