A non-local constitutive model for the Mullins effect in filled elastomers

Abstract

Filled rubber-like materials are widely used in engineering applications and are well known to exhibit the Mullins effect. In this work, an established local constitutive model from the literature is extended to a non-local setting to resolve the mesh dependence inherent to the local approach. Non-local effects are incorporated using two separate approaches: (i) a Helmholtz-type equation governing a non-local soft volume fraction, and (ii) a Laplacian term introduced directly into the soft volume fraction local evolution law. In both formulations, an additional governing partial differential equation arises and is solved numerically in Abaqus using an analogy with the heat equation. The two approaches yield different results, leaving the choice between them to be guided by experimental findings. The details of the implementation, along with the code developed in this work are also provided.

Figures (6)

• Figure 1: Material-point response of the local model of qi2004constitutive: a) amplification factor $X$ evolution with soft volume fraction $\nu_s$, and b) amplified stretch $\Lambda$ evolution with effective stretch $\bar{\lambda}$ for various amplification factors $X$.
• Figure 2: Material-point response of the local model of qi2004constitutive: a) evolution of the soft volume fraction $\nu_s$ for a load/unload/reload cyclic profile, and b) the corresponding Cauchy stress vs. stretch response, for a representative material.
• Figure 3: Schematic of the boundary value problem of a rubber-like material subjected to a cyclic loading profile in uniaxial tension.
• Figure 4: Prescribed cyclic displacement-time loading profile.
• Figure 5: Finite element meshes used to assess mesh dependence near the notch: (i) coarse ($l_e = 10$ mm), (ii) medium ($l_e = 5$ mm), and (iii) fine ($l_e = 2.5$ mm), with refinement concentrated in the notch region.