A new locking-free equilibrium mixed element for plane elasticity with continuous displacement interpolation Original Research Article

With reference to plane elastic problems, we first show that using the Johnson and Mercier (JM) element [C. Johnson, B. Mercier, Numer. Math. 30 (1978) 103] for the stresses and an H1 piecewise linear interpolation for the displacements leads to a severe locking when the Lamé coefficient λ tends to infinity, i.e. in the case of an incompressible material. Motivated by the need to have a continuous displacement field that emerges in several engineering applications, we use an eigenvalue argument to prove that locking is due to the presence of the (element-average) pressure among the degrees of freedom. We therefore relax the incompressibility constraint by introducing a quadrilateral macroelement basically consisting of two adjacent JM triangles sharing the same average pressure, in a sense switching from a (locking) T1P0 approximation to a locking-free Q1P0. Numerical tests on the pathological Cook cantilever are performed to assess the validity of the proposed approach in which comparisons between the newly devised element and the original JM element with L2 displacements are presented.