The authors have previously introduced a novel gradient elasticity model for seismic wave predictions. The said model combines (i) the higher-order gradient terms that capture the influence of small-scale soil heterogeneity and/or microstructure and (ii) the nonlinear softening soil behaviour through the use of the hyperbolic soil model. The current study presents an in-depth analysis of the proposed model. The findings indicate that as nonlinearity increases, the bulk of the wave slows down, and its shape becomes more distorted in comparison to the response of the linear system. Furthermore, the wavenumber spectrum of the nonlinear-elastic response presents peaks at large wavenumbers. However, these are eliminated when a small amount of linear viscous damping is added indicating that they are not physically relevant. One model feature that does not disappear with the presence of damping is the formation of small-amplitude waves travelling in the opposite direction to the main wave. These findings shed light on the characteristics of the proposed nonlinear gradient elasticity model and its applicability for predicting the seismic site response.@en

A novel nonlinear gradient elasticity model for predicting seismic wave behavior was proposed by the authors in previous studies, introducing higher-order gradient terms to account for small-scale soil heterogeneity and micro-structure. This work delves into various characteristics of the proposed model, focusing on the system’s response under initial conditions that induce different levels of nonlinearity. The results reveal that high nonlinearity in the initial conditions can lead to distinctive wave shapes and a non-zero plateau trailing behind the propagating wave. The greater the initial nonlinearity, the more pronounced this plateau becomes. These findings provide valuable insights into the behavior of the proposed nonlinear gradient elasticity model.@en

The prediction of the so-called seismic site response (i.e., the response of the top soil layers induced by seismic waves) is important for designing structures in areas prone to earthquakes. For seismic loads that induce large soil strains, accounting for the nonlinear behaviour of the soil can be of importance for accurate predictions. In Ref. [1], the authors propose a nonlinear gradient elasticity model for predicting the seismic site response. In the said model, the nonlinear constitutive behaviour of the soil is governed by the hyperbolic soil model, in which the secant shear modulus is dependent on the shear strain through a non-polynomial (hyperbolic) relation. Moreover, the classical wave equation was extended to a nonlinear gradient elasticity model to capture the effects of small-scale heterogeneity/micro-structure. Compared to the classical continuum, higher-order gradient terms are introduced into the equation of motion, which lead to dispersive effects [2] prohibiting the formation of un-physical jumps in the response. The aforementioned model is used in this work too, in which a Gaussian pulse is imposed as an initial condition and the solution is determined using a novel finite difference scheme (see Ref. [1]). This work investigates the behaviour of the proposed model for different levels of initial nonlinearity (i.e., induced by the initial conditions). More specifically, we focus on explaining and studying the appearance of a non-zero plateau trailing behind as the initial shape propagates away. It is shown that the higher the initial nonlinearity, the more pronounced the plateau, indicating that the non-zero plateau is a characteristic of the system’s nonlinearity. The in-depth investigation of the proposed model's characteristics can be helpful when using it to accurately predict the seismic site response.@en

This paper aims at investigating the existence of localized stationary waves in the shallow subsurface whose constitutive behavior is governed by the hyperbolic model, implying non-polynomial nonlinearity and strain-dependent shear modulus. To this end, we derive a novel equation of motion for a nonlinear gradient elasticity model, where the higher-order gradient terms capture the effect of small-scale soil heterogeneity/micro-structure. We also present a novel finite-difference scheme to solve the nonlinear equation of motion in space and time. Simulations of the propagation of arbitrary initial pulses clearly reveal the influence of the nonlinearity: strain-dependent speed in general and, as a result, sharpening of the pulses. Stationary solutions of the equation of motion are obtained by introducing the moving reference frame together with the stationarity assumption. Periodic (with and without a descending trend) as well as localized stationary waves are found by analyzing the obtained ordinary differential equation in the phase portrait and integrating it along the different trajectories. The localized stationary wave is in fact a kink wave and is obtained by integration along a homoclinic orbit. In general, the closer the trajectory lies to a homoclinic orbit, the sharper the edges of the corresponding periodic stationary wave and the larger its period. Finally, we find that the kink wave is in fact not a true soliton as the original shapes of two colliding kink waves are not recovered after interaction. However, it may have high amplitude and reach the surface depending on the damping mechanisms (which have not been considered). Therefore, seismic site response analyses should not a priori exclude the presence of such localized stationary waves.@en