Locking-free finite elements for the Reissner-Mindlin plate

Two new families of Reissner-Mindlin triangular finite elements are analyze d. One family, generalizing an element proposed by Zienkiewicz and Lefebvre , approximates (for k greater than or equal to 1) the transverse displaceme nt by continuous piecewise polynomials of degree k + 1, the rotation by con tinuous piecewise polynomials of degree k + 1 plus bubble functions of degr ee k + 3, and projects the shear stress into the space of discontinuous pie cewise polynomials of degree Ic. The second family is similar to the first, but uses degree k:rather than degree k + 1 continuous piecewise polynomial s to approximate the rotation. We prove that for 2 less than or equal to s less than or equal to k + 1, the L-2 errors in the derivatives of the trans verse displacement are bounded by Ch(s) and the L-2 errors in the rotation and its derivatives are bounded by Ch(s) min(1, ht(-1)) and Ch(s-1) min(1, ht(-1)), respectively, for the first family, and by Ch(s) and Ch(s-1), resp ectively, for the second family (with C independent of the mesh size h and plate thickness t). These estimates are of optimal order for the second fam ily, and so it is locking-free. For the first family, while the estimates f or the derivatives of the transverse displacement are of optimal order, the re is a deterioration of order h in the approximation of the rotation and i ts derivatives for t small, demonstrating locking of order h(-1). Numerical experiments using the lowest order elements of each family are presented t o show their performance and the sharpness of the estimates. Additional exp eriments show the negative effects of eliminating the projection of the she ar stress.