Consequences of Linear Time-Variant Rheology for Aging, Relaxation, and Creep

TL;DR

The work tackles aging-related relaxation and creep in solids, where conventional LTI rheology struggles to capture logarithmic relaxation and power-law creep. It proposes jerk-elasticity, a linear time-variant constitutive element defined by $\dot{\sigma}(t)=\lambda(t)\varepsilon(t)$ with $1/\lambda(t)=\xi+\theta t$, in parallel with a spring, yielding a time-evolving elastic response. The model yields Guiu-Pratt logarithmic relaxation and Andrade-like creep, with $\alpha=\frac{1}{E\theta}$ and $\tau_{\sigma}=\frac{\xi}{\theta}$, connects to the Mittag-Leffler function as $\alpha\to0^+$, and shows viscous and fractional Maxwell limits. Moreover, the activation volume $V^*$ provides a physical handle on aging, and the framework unifies the three creep stages while offering a thermodynamically consistent alternative to distributed-relaxation or nonlinear models.

Abstract

Most materials age, and their properties change over time. The aging of materials is reflected in their mechanical responses to external stress and strain, which exhibit logarithmic relaxation and universal power-law creep. Those responses are typically described using complex phenomenological models, including fractional viscoelastic models. While successful at reproducing experimental trends, such approaches often obscure the underlying rheological mechanism and its connection to material parameters. Their physical interpretation remains debated. We introduce jerk-elasticity, a linear time-variant model whose constitutive relations are motivated by thermodynamic principles and experimental observations of the stick-slip-induced friction. The model reproduces the Guiu-Pratt law of logarithmic stress relaxation, Andrade's power-law creep, and a unified description of the three stages of creep, without invoking distributed relaxation times or nonlinear constitutive laws. The rheological parameters of jerk-elasticity are linked with the thermodynamic variables and activation volume. The evolution of activation volume emerges as a physically interpretable measure of aging. Interestingly, viscous and fractional Maxwell responses appear as limiting cases of the jerk-elastic response, thereby offering a unified constitutive interpretation of fractional rheology. Besides, the Mittag-Leffler function gains physical interpretation. The findings are validated by established experimental observations.

Figures (3)

• Figure 1: The missing rheological link between $\dot{\sigma}\left(t\right)$ and $\varepsilon\left(t\right)$ is completed by the property of jerk-elasticity, which is shown as a parallel combination of a jerk element and a spring. In contrast, the fractional Maxwell model is represented by a series combination of a spring and a fractional dashpot. The relaxation test on the jerk-elasticity model gives Guiu-Pratt's logarithmic law of stress relaxation. For $0<\alpha\ll1$, the logarithmic stress relaxation scales as the Mittag-Leffler function, which is inherent in the relaxation from the fractional Maxwell model. The creep test on the jerk-elasticity model gives Andrade's temporal power-law, which is approximately the same as that from the fractional dashpot.
• Figure 2: (Color online) A schematic plot of the time-dependent creep (continuous curve), $\varepsilon\left(t\right),$ and the corresponding creep-compliance rate (dashed curve), $\dot{J}\left(t\right)$, which is proportional to the creep rate under constant applied stress. The creep behaviour is separated into three stages: transient, steady-state, and accelerating creep. The creep-compliance rate first decreases in the transient stage, followed by an almost constant phase in the steady-state phase. Finally, the creep compliance rate accelerates sharply in the tertiary stage until the material fails, indicating the stage is thermodynamically inconsistent. The transient creep is also the creep from the fractional dashpot. The steady-state creep corresponds to the creep from the classical Maxwell model.
• Figure 3: (Color online) An almost perfect match between the creep compliances of Lomnitz's law (continuous curve) and jerk-elasticity creep (dashed curve). The values used to obtain the plots are $\alpha\approx0.009\text{ }\left(0.01\right)$ and $\tau_{\varepsilon}\approx0.0006\text{ }\left(0.001\right)$, in which the values inside the parentheses were obtained by Lomnitz from creep tests on igneous rocks Lomnitz1956.