The spectral analysis of a PM space unit in the context of the classification of nonlinear phenomena in ultrasonic wave propagation

Experiments on diverse materials, such as e.g. rocks, soil, cement, concrete, and damaged metals, have revealed evidence for nonlinearity, hysteresis, and discrete memory in their elastic behaviour. A variety of nonlinear effects in quasi-static as well as dynamic measurements was observed, e.g. the resonance frequency downwards shift with increasing excitation amplitude, the generation of higher harmonics, the so-called slow dynamics, etc. For the simulation of these effects on the propagation of ultrasonic waves, various models have been proposed. They usually assume the presence of a large number of soft interstitial regions, which are taken to be responsible for the nonlinear and hysteretic behaviour of the material specimen. In order to simplify the treatment, a so-called "PM space" of pairs of pre-assigned interstice strain states and corresponding pressure values at which transitions from one state to the other are assumed to take place, is often considered. The relationship between the choice of the PM space and the consequent nonlinearity is, however, inferred only phenomenologicaly. Starting with the case of only one interstice, the interdependence among the parameters of the model, the input excitation, and the spectral contents of the specimen's response is derived analytically. The results are related to the strains and restoring forces as present in thin bonded interfaces and discussed with regard to the inverse problem and the classification of defects and weak bonds.