The present paper is concerned with the theoretical treatment of volume constraint arising from nearly incompressible response of rubberlike material in the previous hyperelastic modeling of Green, as well as the hyperelastic one using rotationless strain proposed by the present author. The constitutive equation is based on the modification of the deformation gradient tensor of Flory, which can decompose volumetric and dilatational parts of the utilized strain precisely. By applying the method of Lagrange multipliers with respect to internal work related to volumetric change, three-field Hu-Washizu and two-field Hellinger-Reissner variational principles are systematically derived. The mixed variational principle proposed here is proven to hold exactly the condition of equilibrium in rate form, which has been bypassed in previous works.