Some gradient theories in linear visco-elastodynamics towards dispersion and attenuation of waves in relation to large-strain models

TL;DR

The work analyzes spatial-gradient extensions of classical linear viscoelastic rheologies (Kelvin-Voigt, Maxwell, Jeffreys) in 1-D to characterize wave dispersion and attenuation, and then situates these gradient models within large-strain nonlinear variants to reveal connections between linear wave propagation and nonlinear modeling. It derives dispersion relations, phase velocities, and attenuation measures for multiple gradient implementations (dissipative, conservative, mixed) across KV, Maxwell, and Jeffreys frameworks, including inertial-gradient and nonmonotone dispersion cases. Key findings show that gradient terms can induce normal or anomalous dispersion, introduce cutoff wavelengths, or yield nondispersive limits, thereby offering mechanisms to tune wave propagation and attenuation through internal length scales. The results provide a bridge between gradient-enhanced linear theories and nonlinear large-strain formulations, with implications for microstructured, porous, or complex media and for rigorous well-posedness in nonlinear extensions.

Abstract

Various spatial-gradient extensions of standard viscoelastic rheologies of the Kelvin-Voigt, Maxwell's, and Jeffreys' types are analyzed in linear one-dimensional situations as far as the propagation of waves and their dispersion and attenuation. These gradient extensions are then presented in the large-strain nonlinear variants where they are sometimes used rather for purely analytical reasons either in the Lagrangian or the Eulerian formulations without realizing this wave-propagation context.The interconnection between these two modeling aspects is thus revealed in particular selected cases.

Figures (10)

• Figure 1: Schematic 1D-diagrammes of 3 basic rheologies considered in this paper.
• Figure 2: Dependence of the velocity (left) and the Q-factor (right) on the angular wavelength $\lambda=1/k$ illustrating the normal, the anomalous, and the general dispersion for the Kelvin-Voigt (with $D=3$), Maxwell (with $D=7$), and Jeffreys (with $D_1=3$ and $D_2=7$) rheologies, respectively, for ${C}$=1 and $\varrho=1$.
• Figure 3: Schematic 1D-diagrammes of some gradient-enhanced rheologies from Figure \ref{['fig1']}; the enhanced elements are depicted in wider lines. Some other gradient enhancements from Sections \ref{['sec-Brenner']} and \ref{['sec-kinematic-grad']} do not bear such a figuration and thus are not included in this diagram.
• Figure 4: A comparison of the (normal) dispersion and Q-factor due to the "hyper" viscosity (solid line) with the standard viscosity in the Kelvin-Voigt viscoelastic model as in Figure \ref{['KV-dispersion-']} (dashed line). For the same $\lambda_\text{\sc crit}$ from (\ref{['crit']}), the dispersion due to hyper-viscosity can be less visible than the dispersion due to usual simple viscosity and facilitates easier wave propagation due to a higher Q-factor.
• Figure 5: Normal (solid line) and anomalous (dotted line) dispersion which both can be obtained by the model (\ref{['dispersion+']}) as well as the nondispersive variant (dashed line); $\ell=1$, ${C}$=1 and $\varrho=1$.

Theorems & Definitions (3)

• Remark 1: Standard solid: Poynting-Thomson-Zener rheology.
• Remark 2: Energetics for $\ell\to0$.
• Remark 3: Energetics for $\ell\to0$.