[Paper Review] Physics-based Constitutive Equation for Thermo-Chemical Aging in Elastomers based on Crosslink Density Evolution

Maryam Shakiba, Aimane Najmeddine|arXiv (Cornell University)|Apr 19, 2021

Elasticity and Material Modeling42 references4 citations

TL;DR

This paper proposes a physics-based constitutive equation for thermo-chemical aging in elastomers by modifying the Arruda-Boyce eight-chain model to account for crosslink density evolution via equilibrium swelling experiments. The model predicts mechanical responses of aged elastomers with high accuracy using only unaged material parameters and crosslink density data, eliminating the need for mechanical testing of aged samples.

ABSTRACT

This paper presents a physics-based constitutive equation to predict the thermo-chemical aging in elastomers. High-temperature oxidation in elastomers is a complex phenomenon. The macromolecular network of elastomers' microstructures undergoes chain scission and crosslinking under high temperature and oxygen diffusion conditions. In this work, we modify the network stiffness and the chain extensibility in the Arruda-Boyce well-known eight-chain constitutive equation to incorporate the additional Helmholtz free energy due to network changes in elastomers' microstructures. The effect of network evolution due to aging in changing the shear modulus and the number of chain monomers is considered. The modification is based on chemical characterization tests, namely the equilibrium swelling experiment to measure the crosslink density evolution. The developed constitutive equation predicts the mechanical responses of thermo-chemically aged elastomers independent of any mechanical tests on aged samples. The proposed constitutive equation is validated with respect to a comprehensive set of experimental data available in the literature that were designed to capture thermo-chemical aging effects in elastomers. The comparison showed that the constitutive equation can accurately predict the intermittent tensile tests based on crosslink density evolution input. The developed constitutive equation is physics-based, simple, and includes minimal material parameters.

Motivation & Objective

• To develop a thermodynamically consistent constitutive equation that captures the coupled thermo-chemo-mechanical aging of elastomers.
• To link macroscopic mechanical behavior directly to chemically measurable quantities like crosslink density, avoiding reliance on phenomenological assumptions.
• To eliminate the need for mechanical testing of aged specimens by using only unaged material parameters and crosslink density evolution data.
• To provide a minimal-parameter, physically grounded model that accurately predicts aging-induced changes in shear modulus and chain extensibility.
• To enable integration with diffusion-reaction models for full multiphysics simulation of aging in elastomers.

Proposed method

• Modify the Arruda-Boyce eight-chain model by incorporating an additional Helmholtz free energy term due to network changes from crosslinking and scission.
• Use equilibrium swelling experiments to measure crosslink density evolution as the primary chemical input for the model.
• Relate changes in crosslink density to variations in shear modulus and number of chain monomers via a physics-based formulation.
• Formulate the stored energy function as a function of crosslink density, enabling a one-to-one mapping between chemical and mechanical variables.
• Validate the model against experimental tensile data from multiple studies on filled and unfilled SBR and NR under various aging conditions.
• Ensure thermodynamic consistency by deriving the constitutive model from the Clausius-Planck inequality and energy balance principles.

Experimental results

Research questions

• RQ1 How can the mechanical response of thermo-chemically aged elastomers be predicted without conducting mechanical tests on aged samples?
• RQ2 What is the role of crosslink density evolution in altering the shear modulus and chain extensibility in aging elastomers?
• RQ3 Can a physics-based, minimal-parameter constitutive model accurately predict the aging response across different elastomer types and aging conditions?
• RQ4 How does the proposed model compare to existing phenomenological or multi-parameter models in terms of accuracy and simplicity?
• RQ5 To what extent can crosslink density serve as a sufficient chemical descriptor for predicting macroscopic mechanical behavior during aging?

Key findings

• The proposed constitutive equation accurately predicts intermittent tensile responses of thermo-chemically aged elastomers using only unaged material parameters and crosslink density evolution data.
• Average prediction errors across multiple datasets (SBR and NR) ranged from 4.68% to 38.9%, with most errors below 15% for unaged and short-term aged states.
• For temperatures up to 100 °C and aging durations up to 16 days, crosslink formation was the dominant network evolution mechanism, and the model captured the response with high fidelity.
• The model achieved lower average errors (e.g., 5.48% at 90 °C for 35-day aging) compared to existing models that rely on multiple empirical parameters.
• The model’s physical basis allows direct integration into diffusion-reaction frameworks, enabling future full multiphysics simulations.
• The approach bypasses the need for complex, multi-parameter assumptions by directly linking chemically measured crosslink density to mechanical energy storage.