# Data Analysis

## Measurement of viscoelastic properties changes

During a DAET test, the low frequency palpation (LF) changes the propagation velocity and the amplitude of the ultrasound pulses (S). Yet the velocity (VUS) and viscoelasticity (M*) of the medium are explicitly related:

$V_{US}^2 = \dfrac {M ^{*}}{\rho}$ , ρ density.

Modulation of time of flight (TOFM) and relative amplitude modulation (RAM) of US pulses, during the various states of expansion/compression imposed by the LF wave, provides information on the compression modulation of viscoelastic parameters of the medium: $TOFM^{*} = - \dfrac{L}{c_{0}^2}\Delta V_{US}^* = - \dfrac{L}{2 \rho_{0} c ^3_{0}}\Delta M^*$

## Acoustical Rheogram Modeling

RheaWave has developed a non-linear viscoelastic model to estimate relevant parameters from the experimental acoustic rheograms, using the non-linearity and the viscoelasticity of the medium:

$Re(TOFM^{*}) = - \dfrac{L}{2 \rho V_{US}^3} Re[(\dfrac{B}{A} + \dfrac{j \omega \eta_{B}}{A})\Delta P + (\dfrac{C}{A}+ \dfrac{j\omega \eta_{C}}{A})\dfrac{\Delta P^{2}}{2A} + ...]$

The parameters extracted from the model are directly related to the variations of elastic parameters (B/A, C/A ...) or viscoelastic parameters (ωηB/A, ωηC/A ...) of the material. For industrial applications, they will be used as a damage index or viscoelasticity/texture score.

## What does the DAET technology measure?

In homogeneous elastic mediums, solid or liquid, the TOFM is often of linear dependence with the LF pressure. In this case, the nonlinear elastic parameters B/A and viscous ωηB/A have low values in general, which shows classical nonlinearity of the medium. These two parameters are characteristic of the composition of the medium and of its overall rigidity.

On the other hand, the TOFM and RAM responses are very different if the medium has some inhomogeneities (micro-cracks, damage and contacts between grains) or different compressibility phases (air/fluid, air/solid).

Levels of TOFM and RAM measurement are significantly higher and acoustic rheograms signatures are more complex. In order to properly describe these responses, it is necessary to add to the model nonlinear parameters of higher orders (C/A, ωηC/A ...).

In the case of an evolving medium, time tracking of viscoelastic scores or index gives information about the progress of the phenomenon. This figure quantifies the creaming of a hollow microsphere solution, by recording the first two viscoelastic parameters (B/A, ωηB/A) over time.