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In this work, through introducing the first and second derivatives of strain gradient into the strain energy density function that depends only on the classical strain in conventional continuum mechanics theory to describe the microscopic interactions, the elastic wave equations including rotational deformation and strain gradient based on the Aifantis’s strain gradient theory is derived. The elastic wave equations including rotational deformation and strain gradient with consideration of the scale effects caused by the microscopic interactions will make seismograms appear obvious changes, which are reflected in amplitude attenuation and dispersion, especially P-wave and S-wave propagate in a dispersive manner based on this theory. The results of numerical modelling and theoretical analysis verify this conclusion. The closer the wavelength is to the characteristic scale of the media (lattice size and interparticle distance), the macroscopic response which results from the microscopic interactions becomes more prominent. To further study the effects of microstructure interactions, we compare the results of numerical modelling with the real data when high-speed train passed the piers in Dingxing County in both time and frequency domains, some conclusions are drawn from the comparison.

In this work, we simplify a moving high-speed train as combination of a series of the axe loads from the front and rear bogies of cars. We build a simplified bridge pier model, as a result, train source expressions with cars moving on land and bridges are given respectively. Considering that the piers are inserted into the ground at a depth of a few tens meters and reach the bedrock. When the high-speed train passes the piers, the piers will appear oscillating motion which is subjugated by the ground, and this oscillating motion is relative to the microstructure interactions of subsoil and bedrock in the frame of the generalized continuum mechanics theory. Hence, we derive the elastic wave equations based on the modified couple stress theory to perform numerical modelling for a simplified bridge pier model under high-speed train passage over the bridge. Based on the wave equation and the train source expressions, a theoretical seismogram of a moving high-speed train with 16 cars on the bridge are obtained. Compared with the real data recorded when a high-speed train passed the piers in Dingxing County, some conclusions are drawn.

The staggered-grid finite-difference (SGFD) method is one of the most popular means used in seismic numerical modelling because of its computational efficiency and it is easy to implement. However, it will lead to serious numerical errors while using coarse grids or wavelet with high-frequency. In this work, we proposed an optimal spatial SGFD method based on a new improved particle swarm optimization (PSO) algorithm. Then the dispersion of numerical solution of the first-order spatial derivatives is analyzed. Meanwhile, numerical modelling of elastic wave equations and elastic wave equations including rotational deformation are performed with the optimal spatial SGFD method based on the improved PSO algorithm. Numerical dispersion analysis and numerical modeling results indicate that the optimal spatial SGFD method based on the improved PSO algorithm has high modeling accuracy and can efficiently suppress the numerical dispersion.