Constitutive Equation and Mixed Variational Principle of Hyperelasticity Using Rotationless Strain

Summary

The present paper proposes a new constitutive equation for rubber-like material through the strain energy function using strain measure called ‘rotationless strain’. This strain is defined as an accumulated strain obtained by integrating strain rate excluding from the stretching tensor the effects of rotation entering through the polar decomposition of the deformation gradient tensor. The proposed constitutive equation takes the form of a nonlinear stress-strain relation, and include an isotropic linear relation for metal alloy as a special case, where the objective stress rate is taken as the Green and Naghdi rate in both of the constitutive equations. The validity of the present constitutive modeling is confirmed by good agreements with the previous experimental results. Particularly, the present model cam predict accurately the simple shear response, which can not be be explained by the previous model. The present paper is also concerned with the theoretical treatment of volume constraint arising from nearly incompressible response of materials. By applying the method of Lagrange multipliers with respect to the internal work related to volumetric change, derived are three-field Hu-Washizu and two-field Hellinger-Reissner variational principles. The mixed variational principle proposed here is proved to hold exactly the conditions of equilibrium in rate form, which has been bypassed in the previous works.