A numerical approach to enforce the orientation-preserving condition

In the last decades, many attempts have been made to solve non linear problems with the finite element method. But some numerical topics related to hyperelasticity are still questionable. For example, the Newton-Raphson algorithm diverges if the orientation-preserving condition does not hold. An original algorithm ensuring the orientation-preserving condition for hyperelastic materials is presented in this paper. This algorithm is based on the analysis of the deformation gradient matrix eigenvalues. It has been actually demonstrated that a lack of the orientation-preserving condition results from numerical occurrence of negative eigenvalues. The proposed approach is applicable to any geometries, any boundary conditions, any applied loads and any energy densities. It was implemented in the university finite element code FER developed by the Mechanical Laboratory of Evry University. This implementation was successfully tested by using several examples with classical energy densities as the Blatz-Ko, the Ogden and the Gent models.