Finite-Element Formulation with Special Orthogonal Group SO (3). 3rd Report, Study on Formulation with SO (3).

Abstract

In the previous papers, the relationship between the stress rate and the tangent siffness with the special orthogonal group SO (3) for finite element formulation was shown. The stress rates used were Truesdell stress rate, Jaumann stress rate, Neo-Green stress rate and Ishihara stress rate. And, stiffness elements of tangent stiffnesses of beam elements with SO (3) were shown. They were material stiffness ΔδΠm, geometric stiffness of rigid rotation ΔδΠrg, geometric stiffness of stretch of stress direction ΔδΠsg, stress, geometric stiffness of stretch of perpendicular line of area ΔδΠsg, area and geometric stiffness of stretch of deformation rate ΔδΠsg, deform. In this paper, the constitutive equation of the new Ishihara stress rate will be considered. And it will be shown that although the stress rate and the strain rate are nonsymmetric, the material stiffness matrix will be symmetric. And it will be proved that geometric stiffness of rigid rotation will also be able to be symmetric with 2nd order convergent rate. And it will be shown that the formulation with SO (3) will be different from the conventional formulation with infinitesimal rotation.