A pseudo–elastic model for the Mullins effect in filled rubber

TLDR

The Mullins effect describes stress softening in rubber, where reloading after unloading requires less stress than the initial loading up to the maximum stretch, and the proposed model applies to multiaxial stress–strain states. This paper proposes a simple phenomenological model to account for the Mullins effect in filled rubber elastomers. The model extends incompressible isotropic elasticity by introducing a single damage parameter that governs the strain‑energy function on unloading and submaximal loading, with a damage function dependent on the parameter and the unloading point, and is illustrated using a two‑constant form coupled to standard strain‑energy functions for tension and shear. The pseudo‑elastic model predicts energy dissipation during a primary loading‑unloading cycle and qualitatively captures Mullins effect behavior in simple tension and pure shear.

Abstract

When a rubber test piece is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved on the initial loading. This stress softening phenomenon is referred to as the Mullins effect. In this paper a simple phenomenological model is proposed to account for the Mullins effect observed in filled rubber elastomers. The model is based on the theory of incompressible isotropic elasticity amended by the incorporation of a single continuous parameter, interpreted as a damage parameter. This parameter controls the material properties in the sense that it enables the material response to be governed by a strain–energy function on unloading and subsequent submaximal loading different from that on the primary (initial) loading path from the virgin state. For this reason the model is referred to as pseudo-elastic} and a primary loading-unloading cycle involves energy dissipation. The dissipation is measured by a damage function which depends only on the damage parameter and on the point of the primary loading path from which unloading begins. A specific form of this function with two adjustable material constants, coupled with standard forms of the (incompressible, isotropic) strain–energy function, is used to illustrate the qualitative features of the Mullins effect in both simple tension and pure shear. For simple tension the model is then specialized further in order to fit Mullins effect data. It is emphasized that the model developed here is applicable to multiaxial states of stress and strain, not just the specific uniaxial tests highlighted.