Editorial Type: research-article
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Online Publication Date: 24 Sept 2025

EXPLORING THE RHEOLOGICAL COMPLEXITIES OF FILLED RUBBERS: INSIGHTS INTO THE LINEAR-NONLINEAR DICHOTOMY

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Article Category: Research Article
Page Range: 513 – 526
DOI: 10.5254/rct.25.00018
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ABSTRACT

Under large-amplitude oscillatory shear, filled rubbers usually exhibit a rheological behavior characterized by a linear-nonlinear dichotomy. Specifically, while the amplitude of the stress deviates significantly from the linear dependence on strain, the time-dependent stress response remains linear and sinusoidal. This study seeks to unravel the origin of this complex rheological phenomenon, focusing on the evolution of the linear-nonlinear dichotomy under various dynamic and static strain conditions. Our results reveal that the linear-nonlinear dichotomy in rheological responses of filled rubbers is not due to a mismatch between filler network recovery time and dynamic perturbation time, as commonly believed. We have observed that even at extremely low frequencies (10−5 Hz), which far exceed the typical recovery time frame of the broken filler network in a polymer matrix, there is no sign of a transition in the stress response from sinusoidal to non-sinusoidal behavior. These results challenge the conventional understanding of the dichotomy in filled rubber rheology, suggesting that it is far more complex than previously thought.

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Fig. 1.
Fig. 1.

Test protocol used in the measurements to provide information on the effect of γ0 on the dynamic moduli G′ and G′′ and on the stress nonlinearity I3/I1.

Fig. 2.
Fig. 2.

Dependence of dynamic moduli G′ and G′′ on γ0. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s and T = 25 °C.

Fig. 3.
Fig. 3.

Normalized harmonic amplitude In/I1 of the stress response as a function harmonic number n. The inset shows the time dependence of the stress response of the material to a forced oscillatory shear. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s, T = 25 °C, and γ0 = 10%.

Fig. 4.
Fig. 4.

Low-strain storage modulus measured following an abrupt change of strain amplitude from 10 to 0.01%. The arrow marks where the deformation changes. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s and T = 25 °C.

Fig. 5.
Fig. 5.

Degree of recovery S of the broken filler network and the restoration rate dS/dt as a function of time t. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s, T = 25 °C, and γ0 = 0.01%.

Fig. 6.
Fig. 6.

Dependence of the dynamic moduli G′ and G′′ and the stress nonlinearity I3/I1 on test frequency f. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s, T = 25 °C, and γ0 = 10%.

Fig. 7.
Fig. 7.

Test protocol used in the measurements of the static strain effect on the γ0 dependence of dynamic moduli G′ and G′′ and on the stress nonlinearity I3/I1.

Fig. 8.
Fig. 8.

Effect of 10% static strain offset on the dynamic moduli G′ and G′′ as a function of γ0. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s and T = 25 °C.

Fig. 9.
Fig. 9.

Effect of 10% static strain offset on the normalized harmonic amplitude In/I1 of the stress response as a function harmonic number n. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s, T = 25 °C, and γ0 = 10%.

Fig. 10.
Fig. 10.

Static shear torque M and oscillation torque M^ as functions of static shear time t during the equilibrium period. The measurements were conducted under a static strain of 10%, supplemented by a small oscillatory shear amplitude of 0.001%. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s and T = 25 °C.

Fig. 11.
Fig. 11.

Recovery rate Ṡ as a function of static shear time t during the equilibrium period. The measurements were conducted under a static strain of 10%, supplemented by a small oscillatory shear amplitude of 0.001%. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s and T = 25 °C.

Fig. 12.
Fig. 12.

Effect of equilibration time on the dynamic moduli G′ and G′′ as a function of γ0. The measurements were conducted under a static strain of 10%, supplemented by a small oscillatory shear amplitude of 0.001%. The test material is CB-filled NR. The test conditions are ω = 3.14 rad/s and T = 25 °C.

Fig. 13.
Fig. 13.

Dependence of the dynamic moduli G′ and G′′ and the stress nonlinearity I3/I1 on test frequency f. The measurements were conducted under a static strain of 10%, supplemented by an oscillatory shear amplitude of 10%. The test material is CB-filled NR. The test temperature is T = 25 °C.

Contributor Notes

*Corresponding authors: Ph: +1 (330) 696-9218; email: chris168robertson@gmail.com; Ph: 011-86-15221255921; email: xrwang@tongji.edu.cn
Received: 23 Mar 2025
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