Dispersion Analysis of Generalized Wave Equations Under the Single-Parameter Second-Order Strain Gradient Theory

In the field of seismic exploration, scholars have been working to conduct wave propagation models that are close to physical reality. Researches for high-speed rail seismology show that the microstructural interactions by different scales will trigger the heterogeneous response of the medium, which in turn has an impact on the mechanical behavior of macro-scales. The generalized wave equations enhance the ability to reflect the heterogeneity of the medium by introducing the higher derivative of displacement and the characteristic scale parameters related to the microstructural properties of the medium. In this paper, we introduce the generalized wave equations under the single-parameter second-order strain gradient theory by considering the nonlocal effects, give the decoupled generalized wave equations using the Helmholtz decomposition theorem, and derive the expression of the phase-velocity of the P- and S-wave. Then, we investigate the dispersion characteristics of seismic wave propagation by considering the microstructural interactions in the medium utilizing theoretical dispersion analysis and numerical experiments which can provide a new approach for the establishment and interpretation of wave propagation models under actual medium.