Models of fractional viscous stresses for incompressible materials

Abstract

We present and review several models of fractional viscous stresses from the literature, which generalise classical viscosity theories to fractional orders by replacing total strain derivatives in time with fractional time derivatives. We also briefly introduce Prony-type approximations of these theories. Here we investigate the issues of material frame-indifference and thermodynamic consistency for these models and find that on these bases, some are physically unacceptable. Next we study elementary shearing and tensile motions, observing that some models are more convenient to use than others for the analysis of creep and relaxation. Finally, we compute the incremental stresses due to small-amplitude wave propagation in a deformed material, with a view to establish acousto-elastic formulas for prospective experimental calibrations.

Figures (7)

• Figure 1: Stress relaxation data sets show that the response of many materials follows a power-law behaviour. Reproduced from Bonfanti et al. bonfanti20 under the https://creativecommons.org/licenses/by/3.0/.
• Figure 2: Schematic representation of the fractional Kelvin-Voigt model.
• Figure 3: Exact convolution kernel (full line) and its diffusive Prony series approximation for $\alpha = 0.3$ with $N=3$ (dotted line) and $N=6$ terms (dashed lines). (a) Impulse response, (b) Fourier spectrum.
• Figure 4: Time-dependent shearing motion for Models A and B. (a) Creep function, i.e., evolution of the strain for a constant applied stress; (b) Relaxation function, i.e., evolution of the stress for a constant applied strain.
• Figure 5: Acousto-elasticity. Combination of a large static deformation and a small incremental perturbation.

Theorems & Definitions (5)

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