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The time-dependent behaviour of saturated soils under static and dynamic loading is generally attributed to the flow-dependent and viscous behaviour of pore fluid. However, the intrinsic energy dissipative effects from the flow-independent viscoelastic behaviour of solid skeleton are not always considered. In this study, the effect of flow-independent viscoelastic behaviour on the seismic amplification of ground soil in vertical and horizontal directions is studied based on a two-phase poroviscoelastic model. A generalized Kelvin–Voigt model is used to define the effective stress in the soils, and the compressibilities of both solid skeleton and pore fluid are considered. The seismic-induced dynamic displacements are analytically derived and are shown to depend on soil layer thickness, soil properties, and ground motion parameters. The formulation neglecting the viscoelastic behaviour of solid skeleton could overestimate both the vertical and horizontal motion amplifications at the surface of ground soil. In addition, the seismic responses of viscoelastic soils are demonstrated to be closely related to the saturation state of surface soil.

The characteristics of seismic ground motions at a specified site are significantly affected by site conditions such as soil properties and topography [

In general, the seismic waves propagating in the isotropic Earth are associated with the vibrating direction of the substrate bedrock. So far, a great number of models which are based on existing seismic records have been proposed for horizontal site response. However, the study on the amplification effect of vertical motion is quiet limited. This is perhaps due to the fact that the engineering structures are sufficiently resistant to the vertical earthquake action, which has smaller magnitude but becomes more obvious in high frequency range [

It is widely accepted that the two-phase porous theory presented by Biot [

In the present paper, the effective stress of soil skeleton is characterized by using a Kelvin–Voigt model, which can be represented by a spring and damper arranged in parallel. The arising two-phase viscoelastic model could describe both the flow-independent and flow-dependent dissipative phenomena. The various response variables in the cases of vertical and horizontal seismic motion are obtained analytically. A set of parametric analysis is then carried out to analyze the influence of critical parameters on the seismic wave propagation and the motion amplification in the viscoelastic soil layer.

In general, soil is a so-called damping material as a part of energy of the wave which will dissipate during the propagation [

Based on the continuity of flow condition, the constitutive equation for compressible fluid in the pores is

In the case of high saturation (e.g., Sr > 95%), one may treat the air-water mixture as a homogeneous pore fluid by assuming that the air exists in the form of bubbles. _{r} is the saturation degree,

Neglecting the body forces, the overall equilibrium equation for a unit total volume could be expressed as

The equilibrium equation for the liquid phase is^{2}. It is noted that the third term at the right side of equation (

Then, the governing equations for the poroviscoelastic medium could be derived as

In this section, the steady-state displacement excitations are assumed to act at the bottom of the soil layer in vertical and horizontal directions, respectively. As shown in Figure

A viscoelastic soil layer subjected to vertical/horizontal earthquake excitation at rigid base.

When the vibration of bedrock is vertical, the dynamic motion of particles in the soils could be simplified as a one-dimensional problem. Then, equations (

Then the displacement of soil skeleton could be further obtained as

The seismic response for the steady-state excitation is assumed to take the complex form as

With aid of equation (

The roots for four-order equation (_{p1} and ±_{p2}, and satisfy |Re(_{p1})| < |Re(_{p2})|, where Re() denotes the real part. Here _{p1} and _{p2} are the wave numbers for the two compressional waves, namely, P1 wave and P2 wave in the poroviscoelastic soils.

Without losing generality, the general solution for

The average relative fluid displacement

The boundary conditions, as shown in Figure

In this case, the seismically induced pore pressure is assumed to be zero, since no compression is generated when the plastic deformation is ignored. Following similar procedures as equation (

Similarly, equation (

The roots for equation (_{s}, which indicate the complex wave number for the shear wave (namely, S wave) in the soils. The displacement of solid skeleton takes the form

This case has the boundary conditions expressed as

In this way, the coefficients _{j} (

By choosing

So far, a general theoretical formulation for the dynamic analysis of vertical and horizontal amplification in the soils is briefly presented. In this section, we carry out detailed investigations on the influence of soil properties in various situations. The physical parameters of the viscoelastic soils, which are chosen for numerical calculation, are given in Table

Soil properties used in theoretical analysis.

Parameter, notation (unit) | Value |
---|---|

Porosity, | 0.47 |

Density of solid grain, _{s} (kg/m^{3}) | 2650 |

Density of water, _{f} (kg/m^{3}) | 1000 |

Permeability, | 10^{−6} |

Absolute fluid pressure, | 100 |

Bulk modulus of solid skeleton, _{b} (Pa) | 4.36 × 10^{7} |

Bulk modulus of solid skeleton, _{s} (Pa) | 3.6 × 10^{10} |

Bulk modulus of water, | 2.0 × 10^{9} |

Shear modulus of solid skeleton, | 2.61 × 10^{7} |

The effect of soil viscosity on the wave speeds of P1, P2, and S waves in the frequency range below 30 Hz is depicted in Figure

Variation of wave speeds of P1, P2, and S waves with frequency. (a) P1 wave. (b) P2 wave. (c) S wave.

Variation of attenuation coefficients of P1, P2, and S waves with frequency. (a) P1 wave. (b) P2 wave. (c) S wave.

The existing experimental results have revealed that partial saturation may significantly reduce the velocity of the P-wave [_{r} is taken to be 95%, 99%, 99.9%, and 100%, respectively. It is shown that even a slight amount of air could significantly reduce the bulk modulus of fluid in terms of stiffness and absolute pore pressure, as given in equation (_{r} = 95%. For S waves, as expected, the effect of saturation degree is negligible. As shown in Figure

Variation of wave speeds of P1, P2, and S waves with damping coefficient

Figure

Variation of attenuation coefficients of P1, P2, and S waves with damping coefficient

In this section, the effects of soil viscosity on the seismic vertical/horizontal amplification in a soil layer are further investigated. The seismic excitation at the surface of rigid bedrock is assumed to be vertical and horizontal, respectively. The distributions of amplification factor with depth for three specified saturation degrees (_{r} = 99%, 99.9%, and 100%) in the case of vertical motion are given in Figure

Distributions of amplification factor with depth for three specified saturation degrees (vertical motion).

Distribution of amplification factor with depth (horizontal motion).

Figure _{r} = 99% and

Variation of amplification factor at surface with frequency at three cases of different saturation degrees (vertical motion).

Variation of amplification factor at surface with frequency (horizontal motion).

For the earthquake loading, the frequencies of interest are usually not high. For example, the predominant frequency in vertical earthquake recorded at the array site is around 5 Hz [_{r} = 100%. For the vertical amplification, the influence of damping coefficient on the variation of amplification factor at surface is more remarkable at higher frequency. In addition, the effect of damping coefficient is negligible when

Variation of amplification factor at surface with damping coefficient

In this paper, an analytical study is presented to identify viscosity effect of soil skeleton on dynamic behaviour of the soil layer under the action of vertical/horizontal earthquake excitation at the underlying rigid base. The Kelvin–Voigt stress-strain relationship is incorporated into the governing equations. The seismic-induced displacements depending on the soil properties, thickness of soil layer, and seismic frequency are derived. The main conclusions are summarized as follows:

The soil viscosity has noticeable impact on the wave speed of P2 wave, while the influence is insignificant on the P1 wave. The wave speed of S wave increases with damping coefficient, which is independent of saturation degree.

The seismic waves attenuate faster in the soil with higher viscosity. The attenuation coefficients of P1 and S waves, which consider the effect of soil viscosity, are several orders of magnitude greater than that in elastic medium.

The distribution of vertical amplification factor with depth is demonstrated to depend on the soil viscosity considered in this paper only in the unsaturated soil layer. The fluctuation of horizontal amplification factor with respect to soil depth becomes more obscure as the damping coefficient increases.

The peak frequency for the vertical amplification factor at soil surface is substantially shifted to the low frequency end even if the saturation degree is only slightly below 100%. The peak value of amplification factor decreases markedly with the increasing of soil viscosity, though the soil viscosity will not affect the peak frequency.

The influence of damping coefficient on the variation of vertical amplification factor at soil surface is more remarkable at higher frequency. Both the horizontal and vertical amplifications at surface approach 1.0 when the damping coefficient is high enough.

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of the article.

This research was supported by the National Natural Science Foundation of China (Grants nos. 41877243 and 41502285), Natural Science Foundation of Jiangsu Province (Grant no. BK20150952), Nanning Science Research And Technology Development Plan (Grant no. 20173160-6), and the China Postdoctoral Science Foundation (Grant no. 2017M623294XB).