General fractional derivative viscoelastic models applied to vibration elastography

In this work we derive general fractional-derivative-based viscoelastic models built from one, two, or three basic elements called viscoelastic springs. The basic viscoelastic spring used has a stress-strain relationship where stress is the fractional derivative of the strain. Fractional derivative models extend traditional Maxwell and Kelvin-Voigt viscoelastic models to allow for fractional powers of frequency in the Fourier domain. Combining these basic viscoelastic elements in series and in parallel results in increasingly complex models for the modulus as a function of frequency. We show how these models can be applied in the frequency domain to shear modulus data acquired using vibration elastography. Shear modulus data for curve fitting in the frequency domain were acquired using 10% and 15% bovine-gel mixtures (with added graphite particles) that were vibrated at 200 to 600 Hz in 100 Hz steps. The vibrations were detected using a 3.5 MHz ultrasound transducer. Shear modulus data for the same materials were also acquired using the dynamic mechanical analyzer (DMA 2980 from TA Instruments, Inc.) in a frequency range from 10Hz to 200Hz. Weighted least-squares fitting was used to determine the model parameters for two and three-element models applicable to viscoelastic solids. The results show that a two-element parallel combination of viscoelastic springs (the Kelvin-Voigt fractional derivative model) can somewhat explain the modulus data, though perhaps a three-element model generalizing the standard linear solid is more accurate over a wider frequency range.