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Institute for Structural Mechanics, University of Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany
The locking phenomenon, in the context of finite elements, gives rise to underestimated displacements and oscillating, parasitic stresses depending on a critical parameter, for instance, the slenderness of a beam. In the last eleven years, Long et al. (2012), Echter et al. (2013), and Oesterle et al. (2017) discovered that the transverse shear locking characteristics of a finite element can be avoided by reparametrizing the underlying kinematic equations. Later, Bieber et al. (2022) extended the idea of reparameterization of the kinematic equations to avoid membrane locking for the special case of curved beam formulations. Even though this technique aids in achieving a membrane-locking-free finite element formulation on a theoretical level, several difficulties are detected in terms of the practical application of the concept, the handling of boundary conditions like for the simply supported case due to the reparameterization of the primary variables, the extension of the theory to shell structures, and considering the case of large deformations.
Revisiting the idea of the discrete strain gap method (Bletzinger et al., 2000) within a variational framework, Bieber et al. (2018) developed the mixed displacement (MD) method, which includes additional degrees of freedom fulfilling a chosen kinematic law. This method can not only help to remove shear locking effects intrinsically but also to eliminate membrane locking in finite elements. In spite of having simpler implementational aspects, the MD method carries the challenge of handling certain additional constraints that are to be imposed on the additional degrees of freedom.
In this work, an overview of the above-mentioned two strategies to mitigate the locking characteristics of finite elements on a theoretical level is provided. This will be followed by a discussion of recent investigations on the applicability of the methods, considering the treatment of constraints and the handling of boundary conditions. Numerical examples demonstrating the locking-free characteristics of the proposed methodology will be addressed as well. The main focus will be on alleviating the membrane locking phenomenon on a theoretical level through these novel ideas.
