A Linear Time-Variant Rheological Model for Frictional Aging, Stress Relaxation, and Creep

Abstract

Most materials undergo aging, leading to time-dependent evolution of their mechanical properties. This aging is reflected in their mechanical response to external strain and stress, which often exhibits logarithmic stress relaxation and power-law creep. Such responses are typically described using complex phenomenological models, including fractional viscoelastic formulations. While these approaches successfully reproduce experimental trends, they typically provide limited insight into the physical origin of aging and its connection to material parameters. We introduce jerk-elasticity, a linear time-variant rheological model in which the constitutive response incorporates the time evolution of stress-rate dynamics through time-dependent material parameters. The framework is motivated by the physics of interfacial stick-slip dynamics underlying frictional aging, together with thermodynamic considerations. An asymptotic correspondence is found between the jerk-elasticity model and the rate-and-state friction law, thereby linking rheological aging with interfacial frictional aging. The proposed model reproduces the Guiu-Pratt law of logarithmic stress relaxation and Andrade's power-law creep. It further provides a framework for interpreting different creep regimes through the evolution of material parameters, without invoking distributed relaxation spectra or nonlinear constitutive assumptions. The governing parameters admit interpretation in terms of thermodynamic quantities, including activation volume, whose evolution provides a physically interpretable measure of aging. In appropriate asymptotic limits, the model recovers behaviors analogous to classical viscous and fractional Maxwell models, while also approaching Mittag-Leffler-type relaxation and Lomnitz-type creep in a specific limit. Jerk-elasticity provides a LTV framework that links frictional aging to creep and relaxation.