Generalized mixed variational principles and solutions of ill-conditioned problems in computational mechanics. Part II: Shear locking

Abstract

Although the finite element method (FEM) has been extensively applied to various areas of engineering, the ill-conditioned problems occurring in many situations are still thorny to deal with. This study attempts to provide a high-performing and simple approach to the solutions of ill-conditioned problems. The theoretical foundation of it is the parametrized variational principles, called the generalized mixed variational principles (GMVPs) initiated by Rong in 1981. GMVPs can solve many kinds of ill-conditioned problems in computational mechanics. Among them, four cases are investigated in detail: the volumetric locking, the shear locking, the inhomogeneousness and the membrane locking problems, composing four parts of the study, Part I–Part IV, respectively. This paper is Part II, wherein a GMVP specially suited to the Reissner plate theory and Timoshenko beam theory is constructed, providing a mathematical foundation for establishing FEM formulations which can automatically unlock the shear locking and produce no spurious zero-energy modes.