The relationship between stress rates and tangent stiffness values with special orthogonal group SO (3) for finite-element formulation was shown in Part 1*(1), and stiffness elements of finite beam element using SO (3) were shown in Part 2*(2). Those stress rates are the Truesdell stress rate, the Jaumann stress rate, the Neo-Green stress rate and the Ishihara stress rate, a nonsymmetric stress rate that will be defined by this author in this report. In this study, the constitutive equation of the Ishihara stress rate will be considered. It will be shown that although the stress rate and the strain rate are nonsymmetric, the material stiffness matrix is symmetric. Next, it is shown that although the geometric stiffness of a rigid rotation is nonsymmetric, even if only the symmetric part of the stiffness is used, the second-order convergent rate determined by the Newton-Raphson method is conserved. Finally, it will be shown that the tangent stiffness formulated with SO (3) is different from that of the conventional formulation with linear rotation.
