Four-node tetrahedral elements for gradient-elasticity analysis

Computational analyses of gradient-elasticity often require the trial solution to be C-1 yet constructing simple C-1 finite elements is not trivial. This article develops two 48-dof 4-node tetrahedral elements for 3D gradient-elasticity analyses by generalizing the discrete Kirchhoff method and a relaxed hybrid-stress method. Displacement and displacement-gradient are the only nodal dofs. Both methods start with the derivation of a C-0 quadratic-complete displacement interpolation from which the strain is derived. In the first element, displacement-gradient at the mid-edge points are approximated and then interpolated together with those at the nodes whilst the strain-gradient is derived from the displacement-gradient interpolation. In the second element, the assumed constant double-stress modes are employed to enforce the continuity of the normal derivative of the displacement. The whole formulation can be viewed as if the strain-gradient matrix derived from the displacement interpolation matrix is refined by a constant matrix. Both elements are validated by the individual element patch test and other numerical benchmark tests. To the best knowledge of the authors, the proposed elements are probably the first nonmixed/penalty 3D elements which employ only displacement and displacement-gradient as the nodal dofs for gradient-elasticity analyses.