Dual‐mixed p and hp finite elements for elastic membrane problems

A complementary energy‐based, dimensionally reduced plate model using a two‐field dual–mixed variational principle of non‐symmetric stresses and rotations is derived. Both the membrane and bending equilibrium equations, expressed in terms of non‐symmetric mid‐surface stress components, are satisfied a priori introducing first‐order stress functions. It is pointed out that (i) the membrane‐, shear‐ and bending energies of the plate written in terms of first‐order stress functions are decoupled, (ii) although unmodified 3‐D constitutive equations are applied, the energy parts do not contain the 1/(1‐2ν) term for isotropic, linearly elastic materials. These facts mean that the finite element formulation based on the present plate model should be free from shear locking when the thickness tends to zero and free from incompressibility locking when the Poisson ratio ν converges to 0.5, irrespective of low‐order h‐, or higher‐order p elements are used. Curvilinear dual‐mixed hp finite elements with higher‐order stress approximation and continuous surface tractions are developed and presented for the membrane (2‐D elasticity) problem. In this case the formulation requires the approximation of three independent variables: two components of a first‐order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dual–mixed elements is presented and compared to displacement‐based hp finite elements when the Poisson ratio converges to the incompressible limit of 0.5. The numerical results show that, as expected, the dual–mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, for both h‐ and p‐extensions.