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.