On the quasi-analytic treatment of hysteretic nonlinear response in elastic wave propagation

Abstract

Microscopic features and their hysteretic behavior can be used to predict the macroscopic response of materials in dynamic experiments. Preisach modeling of hysteresis provides a refined procedure to obtain the stress-strain relation under arbitrary conditions, depending on the pressure history of the material. For hysteretic materials, the modulus is discontinuous at each stress-strain reversal which leads to difficulties in obtaining an analytic solution to the wave equation. Numerical implementation of the integral Preisach formulation is complicated as well. Under certain conditions an analytic expression of the modulus can be deduced from the Preisach model and an elementary description of elastic wave propagation in the presence of hysteresis can be obtained. This approach results in a second-order partial differential equation with discontinuous coefficients. Classical nonlinear representations used in acoustics can be found as limiting cases. The differential equation is solved in the frequency domain by application of Green's function theory and perturbation methods. Limitations of this quasi-analytic approach are discussed in detail. Model examples are provided illustrating the influence of hysteresis on wave propagation and are compared to simulations derived from classical nonlinear theory. Special attention is given to the role of hysteresis in nonlinear attenuation. In addition guidance is provided for inverting a set of experimental data that fall within the validity region of this theory. This work will lead to a new type of NDT characterization of materials using their nonlinear response.

Micro-damage diagnostics using nonlinear elastic wave spectroscopy (NEWS)

“…Any increase in the measured values of these coef®cients re¯ects an increase of the nonlinear hysteresis behaviour of the material. Finally, in agreement with wave propagation experiments, it can be shown that the presence of hysteresis does not affect the level of the even harmonics in resonance [21,32]. As in the case of wave propagation and wave modulation, it is important to note that these results are fundamentally different from a classical approach in which a nonlinear oscillator, such as the Duf®ng type oscillator, is described [37].…”

Effect of Amplitude on Wave Propagation

“…Strain amplitude effect on wave propagation investigated previously in various studies (Johnson et al 1996, Ten Cate et al 1996, Van Den Abeele et al 1997, Zinszner et al 1997, Tutuncu et al 1994, 1998a, 1998b, Ostrovsky et al 2001. However, such studies are mostly focused on wave attenuation rather than wave velocity.…”

Inherent acoustic nonlinearity of fiber reinforced composites

“…Since different laminate sequences have different resonant frequency, the frequency shift was normalized to the lowest amplitude resonant frequency f0. Several researchers [20,[36][37][38] The geometric nonlinearity considered in this work can only simulate the classic nonlinear zone as can be seen in Figure 20. The 3 rd strain zone with non-classical nonlinearity exhibiting linear frequency shifts was not observed using the model used in this study.…”

“…from Los Alamos [9][10][11][12][20][21][22][23]. Van Den Abeele, Guyer and McCall [20] used a phenomenological model to show that for a nonlinear hysteretic material, linear softening (i.e.…”