The mixed displacement method to avoid shear locking in problems in elasticity

Abstract The mixed displacement (MD) method was initially developed to mitigate geometrical locking effects in beams, plates, and shells with the intention of having intrinsically locking‐free characteristics while using equal‐order interpolation for all degrees of freedom. In other words, it is an unlocking scheme that works independent of the element shape, polynomial order, and discretization scheme. It includes additional degrees of freedom that adhere to a carefully designed differential relation that can be interpreted as a kinematic law, incorporated in a mixed sense. Certain constraints are to be enforced on these additional degrees of freedom to obtain a well‐posed system of equations. In this work, the MD method is extended for problems in solid mechanics. We present the underlying variational formulation, followed by its application to 2D solid elements. Additionally, we showcase an idea to enforce the additional constraints in a general sense. Various numerical examples, within the framework of the finite element method and isogeometric analysis, are outlined to demonstrate the performance of the MD method in the geometrically linear and geometrically nonlinear cases.