A novel mixed finite element for finite anisotropic elasticity; the SKA-element Simplified Kinematics for Anisotropy

A variety of numerical approximation schemes for boundary value problems suffer from so-called locking-phenomena. It is well known that in such cases several finite element formulations exhibit poor convergence rates in the basic variables. A serious locking phenomenon can be observed in the case of anisotropic elasticity, due to high stiffness in preferred directions. The main goal of this paper is to overcome this locking problem in anisotropic hyperelasticity by introducing a novel mixed variational framework. Therefore we split the strain energy into two main parts, an isotropic and an anisotropic part. For the isotropic part we can apply different well-established approximation schemes and for the anisotropic part we apply a constant approximation of the deformation gradient or the right Cauchy–Green tensor. This additional constraint is attached to the strain energy function by a second-order tensorial Lagrange-multiplier, governed by a Simplified Kinematic for the Anisotropic part. As a matter of fact, for the tested boundary value problems the SKA-element based on quadratic ansatz functions for the displacements, performs excellent and behaves more robust than competitive formulations.