4 Transient response of inclusion
The dynamic disturbance in the practical project is a non-periodic transient disturbance, which is different from the simple harmonic wave. The transient wave can be decomposed into the superposition of simple harmonic wave with different frequencies by Fourier transform. The steady-state response has been obtained in the previous section, and the dynamic response of the transient disturbance to the system can be represented as:
where χ(ω) denotes steady-state response of the system under the incidence of simple harmonics, namely, the steady-stateαDSCF determined in the previous section,F(ω) denotes the distribution function of the transient disturbance in frequency domain.
In order to calculate the dynamic stress concentration around the elliptical inclusion under practical project disturbance, seismic waves were introduced as the transient disturbance model. In the study of seismic wave forward modeling, seismic wavelet was the basic unit of seismic wave, which can express the basic characteristics of the wave source. Seismic data can be obtained by combining seismic wavelet with convolution model.27-29 It was generally believed that what a single seismic source excites was a sharp pulse with a short time. When seismic waves propagated in the stratum, due to the attenuation and dispersion effected of the viscoelastic stratum on the high-frequency components, the waveform was elongated and seismic wavelets were formed. Based on the difference of wavelet phase spectrum delay, seismic wavelet can be divided into zero phase wavelet, constant phase wavelet, minimum phase wavelet and mixed phase wavelet. In this study, the Ricker wavelet in the zero-phase seismic wavelet was selected as the transient disturbance model to simulate the disturbance caused by the earthquakes. The distribution of the Ricker wavelet in the time domain and frequency domain was shown in Fig. 13.
The Ricker wavelet is symmetric in the time domain, which can be represented as30:
where ωp denotes the dominant frequency of the Ricker wavelet.
Fourier transform requires the distribution of the Ricker wavelet in the frequency domain, F(ω) can be expressed as:
The distribution of the Ricker wavelet in the time domain and the frequency domain is mainly determined by the ωp . The Ricker wavelet in the time domain has a main lobe and two side lobes, and the duration is short and the convergence is fast. The ratio of the amplitude of the main lobe to the amplitude of the side lobes is 0.5e1.5, which is approximately equal to 2.241.31 The Ricker wavelet in the frequency domain always takes the maximum value at the ωp . The larger the ωp , the wider the frequency domain of the Ricker wavelet. In this study, three Ricker wavelets with differentωp were selected for transient incidence.
Substituting Eqs (31) and (34) into Eq (32), and the frequency variablesω was canceled by the integration, then a time-dependent functionut was obtained. By selecting different timest , the system response at different times in the process of transient wave incident was obtained. Due to the mathematical difficulties in direct integration, trapezoidal approximation was used to determine the integration result. In the practical project, we were more interested in the system response when the incident wave reached the peak, so the case selected the system response at that moment for analysis.