LD
Leo Dostal
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1
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.
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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.
Journal article
(2024)
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Andrei B. Fărăgău, Marten Hollm, Leo Dostal, Andrei V. Metrikine, Karel N. van Dalen
A novel nonlinear 1-D gradient model has been previously proposed by the authors, combining (i) the higher-order gradient terms that capture the influence of material micro-structure and (ii) a nonlinear softening material behavior through the use of a hyperbolic constitutive model. While the previous study focused on the existence and properties of solitary-type waves, the current study focuses on the characteristics of the transient wave propagation in the proposed model. Findings show 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. The energy analysis reveals that, unlike the linear system, the nonlinear one continuously exchanges energy, in which the kinetic energy decreases over time while the potential one increases. Furthermore, the spectral (wavenumber) energy density of the nonlinear-elastic system 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. A notable feature that persists despite the presence of damping is the formation of small-amplitude waves traveling in the opposite direction to the main wave. Generalized continua, like gradient elasticity models, miss the small energy scatter by the micro-structure. This study shows that adding material nonlinearity to a homogeneous generalized continuum can capture reverse energy propagation, though at much smaller magnitudes than the main wave. These findings shed light on the characteristics of the transient wave propagation predicted by the proposed nonlinear 1-D gradient model and its applicability in, for example, predicting the seismic site response.
...
A novel nonlinear 1-D gradient model has been previously proposed by the authors, combining (i) the higher-order gradient terms that capture the influence of material micro-structure and (ii) a nonlinear softening material behavior through the use of a hyperbolic constitutive model. While the previous study focused on the existence and properties of solitary-type waves, the current study focuses on the characteristics of the transient wave propagation in the proposed model. Findings show 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. The energy analysis reveals that, unlike the linear system, the nonlinear one continuously exchanges energy, in which the kinetic energy decreases over time while the potential one increases. Furthermore, the spectral (wavenumber) energy density of the nonlinear-elastic system 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. A notable feature that persists despite the presence of damping is the formation of small-amplitude waves traveling in the opposite direction to the main wave. Generalized continua, like gradient elasticity models, miss the small energy scatter by the micro-structure. This study shows that adding material nonlinearity to a homogeneous generalized continuum can capture reverse energy propagation, though at much smaller magnitudes than the main wave. These findings shed light on the characteristics of the transient wave propagation predicted by the proposed nonlinear 1-D gradient model and its applicability in, for example, predicting the seismic site response.
Abstract
(2023)
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Andrei B. Fărăgău, Marten Hollm, Leo Dostal, Andrei V. Metrikine, Karel N. van Dalen
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.
...
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.
Abstract
(2023)
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Andrei B. Fărăgău, Marten Hollm, Leo Dostal, Andrei V. Metrikine, Karel N. van Dalen
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.
...
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.
Journal article
(2022)
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Leo Dostal, Marten Hollm, Andrei V. Metrikine, Apostolos Tsouvalas, Karel N. van Dalen
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.
...
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.
The problem of level ice interacting with compliant structures is addressed, where the ice loads can depend on the dynamical behavior of the structures. We are interested in a special type of ice-induced vibration, known as frequency lock-in, and characterized by having the dominant frequency of the ice forces near a natural frequency of the structure. It is shown that accurate estimates of the model parameters for the well-known Määttänen's model for ice-induced vibrations can be obtained from measurements of the structural vibrations and the ice velocity. Määttänen's model uses a state-dependent piecewise nonlinear function for the ice crushing strength, which leads to nonlinear negative damping in the equations of motion of the considered structure. The identification is achieved by means of an Unscented Kalman Filter using simulated noisy measurements of the structural behavior.
...
The problem of level ice interacting with compliant structures is addressed, where the ice loads can depend on the dynamical behavior of the structures. We are interested in a special type of ice-induced vibration, known as frequency lock-in, and characterized by having the dominant frequency of the ice forces near a natural frequency of the structure. It is shown that accurate estimates of the model parameters for the well-known Määttänen's model for ice-induced vibrations can be obtained from measurements of the structural vibrations and the ice velocity. Määttänen's model uses a state-dependent piecewise nonlinear function for the ice crushing strength, which leads to nonlinear negative damping in the equations of motion of the considered structure. The identification is achieved by means of an Unscented Kalman Filter using simulated noisy measurements of the structural behavior.