A C1 Incompatible Mode Element Formulation for Strain Gradient Elasticity

In the present article, a simple, yet reasonably well working 3rd order 3D tetrahedral incompatible mode finite element with C1 continuity is presented. It has been implemented into the finite element system Abaqus. To check its capabilities, it is used in elastostatic boundary value problems in conjunction with an elasticity model that includes a classical strain energy and a strain gradient energy contribution. With these examples, we discuss the regularizing effect of the strain gradient contribution on the singularities in classical elasticity in single force indentation tests and bending of an L-shaped sample with a sharp corner. We consider both, large strain gradient contributions and a mesh refinement. We further discuss the transition to the pseudorigid behavior as the strain gradient energy becomes large compared to the strain energy, which results in a material that can only undergo homogeneous deformations, also called pseudorigid. Finally, we propose an improvement of the element formulation, and compare its convergence behavior upon mesh refinement.