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In the generalized continuum mechanics (GCM) theory framework, asymmetric wave equations encompass the characteristic scale parameters of the medium, accounting for microstructure interactions. This study integrates two theoretical branches of the GCM, the modified couple stress theory (M-CST) and the one-parameter second-strain-gradient theory, to form a novel asymmetric wave equation in a unified framework. Numerical modeling of the asymmetric wave equation in a unified framework accurately describes subsurface structures with vital implications for subsequent seismic wave inversion and imaging endeavors. However, employing finite-difference (FD) methods for numerical modeling may introduce numerical dispersion, adversely affecting the accuracy of numerical modeling. The design of an optimal FD operator is crucial for enhancing the accuracy of numerical modeling and emphasizing the scale effects. Therefore, this study devises a hybrid scheme called the dung beetle optimization (DBO) algorithm with a simulated annealing (SA) algorithm, denoted as the SA-based hybrid DBO (SDBO) algorithm. An FD operator optimization method under the SDBO algorithm was developed and applied to the numerical modeling of asymmetric wave equations in a unified framework. Integrating the DBO and SA algorithms mitigates the risk of convergence to a local extreme. The numerical dispersion outcomes underscore that the proposed SDBO algorithm yields FD operators with precision errors constrained to 0.5‱ while encompassing a broader spectrum coverage. This result confirms the efficacy of the SDBO algorithm. Ultimately, the numerical modeling results demonstrate that the new FD method based on the SDBO algorithm effectively suppresses numerical dispersion and enhances the accuracy of elastic wave numerical modeling, thereby accentuating scale effects. This result is significant for extracting wavefield perturbations induced by complex microstructures in the medium and the analysis of scale effects. [ABSTRACT FROM AUTHOR]

With the continuous research in seismology, the influence of the heterogeneity caused by microstructure interactions of the medium on seismic wave propagation has been paid more attention. Wang et al. (2020) proposed to describe the complex microstructures of the medium in the framework of the generalized continuum mechanics theory, and derived the asymmetric elastic wave equations containing the characteristic length scale parameter of the medium. In addition, scale effects of seismic wave propagation will appear when considering microstructures of the medium. In this work, in order to better analyze the scale effects and extract the new components of wave fields owing to the microstructure interactions, we propose an optimized finite-difference (FD) method based on the improved bat swarm optimization (BSO) algorithm. Then the optimized FD method is used to perform numerical modeling for the asymmetric elastic wave equations considering microstructures of the medium. Numerical dispersion analysis indicate that the optimized FD method has high accuracy. According to the numerical modeling results obtained by the optimized FD method, scale effects of seismic wave propagation can be clearly observed when considering microstructures of the medium. [ABSTRACT FROM AUTHOR]

The second-order strain gradient wave equation is based on the generalized continuum mechanics theory, and it can enrich the content of classical continuum mechanics theory with the incorporation of the second-order spatial derivative term of displacement. Furthermore, this equation incorporates scale parameters of media characteristics to bridge the gap between micromodels and classical continuum mechanics to refl ect the microstructural characteristics in the media. Wang used a nonlocal theory to derive a constitutive equation of single-parameter secondorder strain gradient theory and provided a mathematical expression for the second-order strain gradient asymmetric elastic wave equation combined with geometric equation and differential equation of motion. The second-order strain gradient theory can reflect the small-scale effect of seismic wave propagation within the neighborhood with an average particle diameter of l of the medium. Given the relatively weak spatial-scale effect, the numerical dispersion generated when the difference operator approximates the differential operator can suppress the scale effect. Thus, the finite-difference operator must be optimized accurately to describe and analyze scale effects. In this paper, an improved black widow optimization algorithm was proposed to obtain optimized finite-difference coefficients, and numerical modeling based on the second-order strain gradient wave equation was performed. Numerical modeling results reveal the well-suppressed numerical dispersion, and seismograms display the clear extraction of small spatial-scale effects. [ABSTRACT FROM AUTHOR]

The macroscopic mechanical characteristics of a material can be influenced by the complex microstructural interactions within the medium, which trigger scale effects and deformation localization. The impact of these microstructural interactions on seismic wave propagation can be described by the higher-order derivatives of state variables and the scale parameter related to the microstructural properties of the material, as outlined in the second-order strain gradient (SSG) theory. Research shows that microstructural interactions cause new perturbations to emerge in the P- and S-wavefields. Rayleigh waves, which propagate near the surface, are generated by the interference of P- and S-waves, and their dispersive characteristics can provide notable insights into the shallow geological structure. Therefore, investigating the propagation characteristics of seismic waves, particularly the scale effects of Rayleigh waves, under the SSG theory, is necessary. This study first introduces the nonlocal effects into the constitutive relationship of the conventional wave equations, derives expressions for the nonlocal strain/stress, and formulates the generalized wave equations under the SSG theory. Then, the study presents the free-surface conditions for the generalized wave equations and conducts numerical experiments to explore the scale effects on the propagation of surface and body waves. Results of these numerical experiments show that the heterogeneity introduced by the complex microstructural properties within the medium leads to observable scale effects in the propagation of P-, S-, and Rayleigh waves. Moreover, the dispersive characteristics of Rayleigh waves are modified when considering the nonlocal strain gradient effects, which are dependent on frequency and scale parameters. [ABSTRACT FROM AUTHOR]