Avoiding locking in Reissner-Mindlin plates with high-order hierarchic bubble plus finite elements

In this paper we prove that special finite elements of suitable degree do not exhibit any kind of locking phenomenon when used for the numerical solution of Reissner-Mindlin plate bending problems. Despite its simple approach the discretization of the Reissner-Mindlin model is not straightforward. The inclusion of transverse shear strain effect in standard finite element models introduces undesirable numerical effects. The approximate solution is very sensitive to the plate thickness and, for small thickness, it is very far from the true solution. The phenomenon is known as locking of the numerical solution. The finite element we present is constructed by adding bubble functions to standard Serendipity elements. The new elements, referred to as bubble plus elements, from the degree three onwards avoid the numerical locking. The mathematical proof of such a good behavior is based on a deep relation between the C0 bubble plus element of degree three for Reissner-Mindlin plates and the C1 element for Kirchhoff plates known as Bogner-Fox-Schmit element.