A phenomenological model for coupled Mullins effect and permanent set in rubber-like hyperelastic materials

Rubber-like hyperelastic materials exhibit stress softening (Mullins effect) and irreversible residual deformation (permanent set) during cyclic loading. This work proposes a simplified phenomenological two-phase softening model that simultaneously captures both phenomena in isotropic incompressible elastomers. The formulation extends the classical pseudo-elastic framework by introducing a residual stress contribution associated with deformation-induced evolution of a secondary network, thereby enabling prediction of permanent deformation. Coupled with the Gent hyperelastic model, the proposed framework is validated against published experimental data under uniaxial tension, equibiaxial deformation, pure shear, and transverse vibration loading. Using only two additional material parameters, the model accurately reproduces stress softening and permanent set while maintaining a simple constitutive structure. The proposed approach offers a physically motivated and computationally efficient framework for modeling cyclic deformation of rubber-like materials.