An analytical model to describe nonlinear viscoelastic properties of fluids measured by Dynamic Acoustoelastic Testing

Recently a nonlinear ultrasound method called Dynamic Acoustoelastic Testing (DAET), previously developed to detect cracks in solids, has been adapted to assess the nonlinear viscoelastic properties of fluids. The DAET method is based on the interaction between a low-frequency acoustic wave to successively compress and expand the fluid, and ultrasound pulses to probe this medium. The Time Of Flight Modulations (TOFM) of the ultrasound pulses are plotted as a function of the low-frequency pressure amplitude on DAET diagrams. We propose an analytical model to relate the DAET diagrams to the variations of the longitudinal modulus M. In this approach, M is expressed as a Taylor series to quadratic and cubic order, proportional to nonlinear elastic parameters B/A and C/A, respectively. Furthermore, the modulus is considered viscoelastic, according to the Kelvin-Voigt rheological model, and so complex M*. The analytical model is first validated in Newtonian fluids with classical nonlinearities (Water, Silicone oil), then in viscoelastic gels. A good agreement is found between analytical and experimental results. Notably the hysteresis experimentally observed in some case is correctly taken into account by the model.