Editorial Type:
Article Category: Research Article
 | 
Online Publication Date: 03 Mar 2025

ADVANCED MULLINS DAMAGE MODELING AFTER MULTIPLE CYCLIC LOADING IN VULCANIZATES. PART 1: THEORY AND PARAMETER IDENTIFICATION

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Page Range: 124 – 141
DOI: 10.5254/rct.24.00047
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ABSTRACT

This study explores the deformation behavior of carbon black–filled elastomeric components under multiple cyclic loading conditions. Understanding the effects of repeated cyclic deformation is crucial for accurate finite element method simulations. We present an approach that captures both initial and subsequent cycles, converging to an equilibrium state. Experimental evidence from cyclic deformation and relaxation tests indicates that an equilibrium state is achieved within five deformation cycles. Our method analytically describes the relaxed or equilibrium state after multiple cyclic deformations by combining a relaxation model with a material model. We used a combination of a relaxation approach and the modified extended tube model (METM), allowing us to distinguish between polymer network effects and filler properties, thereby establishing a direct correlation between elastomer parameters and mechanical properties. The multiple cyclic deformation behavior can be analytically described using the advanced Mullins damage modeling (AMDM) approach, validated through experimental data. The AMDM uses two damage functions for the increasing and decreasing stress phases of the cycle via the relaxed METM approach. The feasibility of this approach is demonstrated through three-dimensional finite element simulations, confirming the validity of AMDM under cyclic deformation. The combination of the relaxed METM and AMDM approaches provides a robust framework for predicting the cyclic deformation behavior of filled elastomers, with significant applications in engineering and material science.

Fig. 1.
Fig. 1.

Stress–strain measurement and multiple cyclic deformation of C3 with increasing deformation.

Fig. 2.
Fig. 2.

Stress–strain measurement and stepwise relaxation measurement.

Fig. 3.
Fig. 3.

Comparison of the maximum stresses from the hysteresis and the relaxation experiment for the compound sample C3.

Fig. 4.
Fig. 4.

Relaxation parameter r∞ and γc as a function of the volume fraction of the filler.

Fig. 5.
Fig. 5.

Analytical evaluation of multiple cyclic deformation at various deformation of the sample C3.

Fig. 6.
Fig. 6.

Analytical evaluation of multiple cyclic deformation at various deformation of the sample C1.

Fig. 7.
Fig. 7.

Parameters of the damage function as a function of the volume fraction of the filler.

Fig. 8.
Fig. 8.

Stress–strain curves, that is, experimental vs FE simulation, for stress relaxation and Mullins damage (AMDM) in uniaxial deformation of C3 vulcanizate. (a) One damage cycle up-down at 154% strain; (b) two damage cycles up-down-up-down at 75 and 154% strain.

Fig. 9.
Fig. 9.

Von Mises stress distribution in FE models for uniaxial tension loading of C3 vulcanizate at identical deformation state: (a) single element and (b) discretization of testing specimen.

Contributor Notes

*Corresponding author. Ph. +41 78 4062263; email: claus.wrana@compounds.ch
Received: 13 Aug 2024
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