The helicoidal modeling in computational finite elasticity. Part III: Finite element approximation for non-polar media

The helicoidal modeling of the continuum, as proposed in Part I, is applied to finite elasticity analyses of simple materials unable of couple-stressing. First, the non-polar medium is introduced via a constitutive postulate and results in a sort of constrained medium, having the axial vector of the Biot stress tensor as a primary unknown field and the statement of polar decomposition of the deformation gradient as a governing equation. Next, the variational formulation is accommodated to the non-polar case, and the ensuing principle is approximated by the finite element method. The nonlinear finite elements have the nodal oriento-positions as degrees-of-freedom and are based on the multiplicative interpolation developed in Part II. The interpolation and an analysis methodology based on the multiplicative updating of the kinematical unknowns, ensure frame-invariant and path-independent solutions. Several examples, with either linear or nearly incompressible Neo-Hookean elastic materials, attest the performance of the proposed modeling in high deformation problems with large three-dimensional rototranslations.