Structure testing of wave propagation models used in identification of viscoelastic materials

In this paper, matrix perturbation theory is applied to test the structure of wave propagation models used to identify the complex modulus of a viscoelastic material. The analysis is based on a data matrix, containing the measured data from a number of independent experiments. The key observation is that if the structure of the model is correct then the unperturbed (noise-free) matrix is rank deficient of a known order. This means that the noise corrupted matrix will have a known number of singular values that diverge from zero only due to the measurement noise. A test quantity based on the distribution of these perturbed singular values is used, assuming that the signal-to-noise ratio is large and that measurement noise is white and Gaussian distributed. If the magnitudes of the smallest singular values are too large to be explained by the measurement noise only, the model is rejected. Data from two different types of experimental setups is explored; longitudinal wave propagation in a slender bar and the non-equilibrium SHPB procedure. It is shown that the model can be accepted in the first case, but should be rejected in the second.