Transmission of elastic P-wave across one fracture with improved nonlinear normal deformation behaviors

The improved elastic nonlinear normal deformation constitutive model under the quasi-static condition which generalizes the BB model and the classical exponential model can describe the extent of nonlinearity quantitatively. Without the loading-rate effect considered, the improved constitutive relationship is extended to be under the dynamic condition, and then a nonlinear displacement discontinuity model for normally incident P-wave propagation across a dry fracture is established in an elastic half-space. By introducing the Lemaitre hypothesis, the approximate analytical solutions of transmission and reflection coefficients TI MP and RI MPare obtained. Furthermore, using one-dimensional wave equation characteristic method, the time-domain numerical difference scheme of reflected and transmitted particle velocities is proposed, and computational programs are compiled to obtain semi-numerical solutions of TI MPand RI MP, transmission and reflection energy e tra and er ef, and delay time Tdel. Parameter studies are conducted to get an insight into the effects of the extent of nonlinearity ξ and the incident wave maximum amplitude vi nc on transmitted wave amplitude and energy attenuation, waveform and spectral aberration, and time delays, in terms of fracture initial stiffness Kn,i, the ratio of given closure to the maximum allowable closure γ=dn/dmax, and frequency f. In addition, the phenomenon of exceeding is explained.