A Review of Pile Foundations in Viscoelastic Medium: Dynamic Analysis and Wave Propagation Modeling

Abstract

The dynamic viscoelastic theory of soil–pile interaction dominates the initial impedance calculation during the pile dynamic design and analysis. Further, it provides a firm theoretical ground for the wave propagation simulation, which could be the basis of seismic analysis and some geotechnical testing approaches. This review traces the development history and key findings of viscoelastic soil–pile interaction theory and expounds on the advantages and limitations of various theoretical advances in terms of dynamic design and wave propagation modeling. The review consists of three sub-divisions, which are the longitudinal, horizontal, and torsional viscoelastic soil-pile theories. The development and implement of multi-phase soil constitutive equations, multi-dimensional soil–pile interaction modeling methods, pile–soil–pile mutual interactions in pile groups, and the fluid–structure interaction problems in offshore piles are especially remarked and concluded. Finally, the shortcomings and deficiencies of the present development are pointed out with a view to addressing them in the future.

1. Introduction

Access to energy depends a lot on the construction of energy infrastructures. For instance, the utilization of water conservancy resources requires the construction of reservoirs and hydropower stations [1]; the exploitation of natural gas or geothermal resources involves the construction of risers and pipelines [2,3]; the utilization of wind resources depends on the installation of wind turbines [4,5,6,7], etc. Hence, the discipline of geotechnics has been closely related to energy science. The pile foundation is the most popular foundation to support energy superstructures due to its high dynamic load capacity [8,9]. Dynamic capacity or impedance is a primary consideration for the design of foundations supporting energy structures due to the fact that the production and transition of energy usually generate dynamic loads. Considering that the soil deformation is strongly nonlinear and often related to stress paths, a large amount of cumulative deformation is prone to occur if the plastic deformation of soil is permitted [10,11,12]. The plasticity of soils in recent years has not only been investigated in detail [13] but has also been analyzed in a deterministic and stochastic way [14]. Hence, for most energy structures, the allowable ultimate deformation of the foundation is small and still in the range of elasticity. For instance, the permissible pile head deflection is only 0.5 degrees for monopiles used as the foundation for offshore wind turbines [15]. As a result, the viscoelastic theory is essential and efficient for dynamic pile–soil interaction problems. From the aspect of vibration forms, the viscoelastic pile–soil interaction theory can be divided into three categories, which are longitudinal, horizontal, and torsional vibrations. This paper presents a state-of-the-art review of dynamic viscoelastic pile–soil interaction theories.

2. Longitudinal Dynamic Analysis and Wave Theory

2.1. General Progress in the Longitudinal Soil–Pile Dynamic Interaction Modeling

The longitudinal dynamic analysis of pile foundations is the most developed branch of the topic, compared to the horizontal and torsional ones. This can be attributed to the most prevalent nature of longitudinal vibration in practice. The earliest paper traced back to rigorously discussing the dynamic interaction of footings and soil under longitudinal vibrations comes from Baranov [16], who idealized the soil as a two-dimensional homogeneous medium; however, although Baranov’s solution is still in linear elasticity, its sophistication in mathematical derivation limits its accessibility to practicing engineers. Before Baranov’s work, a more straightforward and efficient method based on the Winkler foundation [17] had already been popular for decades among engineers. Compared to the rigorous elastic solution, “the pile on the Winkler foundation” model simplifies the soil as the discrete springs and dashpots. Hence, the mathematical work is significantly reduced to the boundary value problem of a one-dimensional rod function. The key to implementing “the pile on the Winkler foundation” model is the assessment of the elastic and damping coefficients of the complex discrete springs. For small-strain viscoelastic problems, these parameters are mainly derived from the plane strain model [18]. By neglecting the longitudinal component, the rigorous three-dimensional viscoelastic equations (e.g., Equation (1)) can be simplified to the plane strain ones (Equation (2)). Especially for the axisymmetric cases, the circumferential component can be overlooked at the same time (Equation (3)).

(1)$(\lambda +G)\frac{{\partial }^{2}u}{\partial z^2}+G{\nabla }^{2}u=\rho \frac{{\partial }^{2}u}{\partial t^2}$

(2)$G\frac{{\partial }^{2}u}{r^2\partial {\theta }^2}+G\frac{{\partial }^{2}u}{\partial r^2}+G\frac{\partial u}{r\partial r}=\rho \frac{{\partial }^{2}u}{\partial t^2}$

(3)$G\frac{{\partial }^{2}u}{\partial r^2}+G\frac{\partial u}{r\partial r}=\rho \frac{{\partial }^{2}u}{\partial t^2}$

where ${\nabla }^2=\frac{{\partial }^2}{\partial z^2}+\frac{{\partial }^2}{r^2\partial {\theta }^2}+\frac{{\partial }^2}{\partial r^2}+\frac{\partial }{r\partial r}$; $\lambda$ and $G$ are the lame constants of the soil; $u$ denotes the longitudinal displacement of the soil. It can be easily found that Equation (3) is a cylindrical differential function whose analytical solution can be expressed in the form of the Bessel function. Hence, the soil impedance containing the stiffness and damping can be derived [19]. In most cases, the plane strain model can be used directly instead of the Winkler model, obtaining higher calculation accuracy. For example, the complex springs in the Winkler model were put into series in the radial direction to simulate the radial inhomogeneity caused by pile installation by some scholars [20,21]. By doing this, the overall impedance of the complex springs would be smaller than any individual complex spring in the series. This can reflect the weakening of the surrounding soil after bore drilling during the installation of bored piles. In contrast, it fails to model the strengthening of the surrounding soil after the driving of precast piles. By adopting the plane strain model, the gradient soil strengthening along the radial direction can be authentically simulated by discretizing the soil into numerous circular zones and enforcing continuous boundary conditions at the interfaces of adjacent zones [22]. Figure 1 illustrates the advantages of modeling the radial inhomogeneity of the soil with the plane strain model over the Winkler model. Besides being more reliable and accessible, another advantage of applying the plane strain model to derive the elastic and damping coefficients in the Winkler model is that the plane strain model can take the inertia effect of the soil into account. The inertia effect is one of the key differences between static and dynamic equations. The utilization of the plane strain model effectively distinguishes the dynamic analysis from the static analysis. Except for the high computational efficiency, the Winkler model also has the features of good adaptivity and editability. For instance, after certain modifications, it can also model the slippage at the soil–pile interface when encountering large strain deformation [23,24].

However, the dynamic Winkler model is still an approximation to the rigorous answers [25]. Compared to the rigorous ones, the dynamic Winkler model has the following drawbacks:

• Incapable of simulating the stress or strain wave propagation inside the soil.

• Incapable of modeling the multi-phase nature of the soil.

• Incapable of modeling the soil plug inside the pipe pile.

The derivation of some analytical solutions to the rigorous 3D continuum model enriches the knowledge of the longitudinal vibration of piles embedded in the sand, saturated marine clay, unsaturated clay, etc. The one-phase 3D continuum model is a preliminary update to the Winkler model. For most axisymmetric problems, the governing equations of the one-phase soil can be written as Equation (4).

(4)$\{\begin{array}{c}G\left({\nabla }^2-\frac{1}{r^2}\right)u_r+\left(\lambda +G\right)\frac{\partial e}{\partial r}=\rho \frac{{\partial }^2{u}_r}{\partial t^2}\\ G{\nabla }^2{u}_z+\left(\lambda +G\right)\frac{\partial e}{\partial z}=\rho \frac{{\partial }^2{u}_z}{\partial t^2}\end{array}$

where ${\nabla }^2=\frac{{\partial }^2}{\partial z^2}+\frac{{\partial }^2}{\partial r^2}+\frac{\partial }{r\partial r}$; $\lambda$ and $G$ are the lame constants of the soil; $u_r$ and $u_z$ denote the radial and longitudinal displacement of the soil, respectively. Nogami and Novak [26] further simplified the mathematical work by overlooking the radial displacement of the soil and derived the analytical solution to the dynamic response of a pile embedded in homogeneous anisotropic viscoelastic one-phase soil. Subsequently, it was found that the mathematical effort would not significantly increase if appropriate differential operators could be found to decouple the rigorous equations, as shown in Equation (4) [27]. With two scalar potentials (shown in Equations (5) and (6)), Senjuntichai and Rajapakse [28] decoupled Equation (4) into two Laplace functions (shown in Equations (7) and (8)) in the complex number field and derived the analytical solution to the response of a circular cavity in a semi-infinite viscoelastic medium.

(5)$u_{rj}(r,z)=\frac{\partial {\varphi }_j(r,z)}{\partial r}+\frac{{\partial }^2{\phi }_j(r,z)}{\partial z\partial r}$

(6)$u_{zj}(r,z)=\frac{\partial {\varphi }_j(r,z)}{\partial z}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {\phi }_j(r,z)}{\partial r}\right)$

(7)$\left({\lambda }_j+2G_j\right)\cdot {\nabla }^2{\varphi }_j(r,z)=\rho s^2{\varphi }_j(r,z)$

(8)$G_j{\nabla }^2{\phi }_j(r,z)=\rho s^2{\phi }_j(r,z)$

Wu et al. [29] further investigated the influence of radial displacement of the soil on the dynamic response of the pile under axisymmetric longitudinal vibration cases. In his study, an analytical solution to the pile dynamic response utilizing mathematical techniques, including the Laplace transform, partial differential equation decoupling, variable separation, and the inverse Fourier transform, is derived. Under the same mathematical framework, studies investigating the hysteretic damping and soil plug effects were successively reported [30,31]. The necessity of employing the 3D continuum model instead of the plane strain model is that it can reveal the resonance frequencies in the low-frequency domain. As shown in Figure 2, the most rigorous solution can genuinely reveal the first and second resonance frequencies at $a\approx 0.05$ and $a\approx 0.1$. Once the radial displacement of the soil is overlooked, the first resonance frequency cannot be distinguished. As for the plane strain model, it is not capable of revealing either the first or the second resonance frequency; however, the calculated results in the higher frequency domain coincide well. The good news about the plane strain model is that the calculated results in the low-frequency domain are all slightly smaller than those calculated by rigorous 3D solutions. Hence, due to the underestimation of the pile stiffness, the stiffness decrease at the resonance frequencies is mitigated. Except for the calculation related to the resonance frequencies, the utilization of the plane strain model is biased toward safety.

In nature, the soil usually behaves as a multi-phase medium, whereas the above-mentioned studies all treated the soil as a solid material. As the first theory capable of coupling the solid and pore fluid inside the saturated soil, Biot’s poroelastic theory is the most popular theoretical basis for poroelastic medium [32,33]. One of the most classic analytical solutions to the dynamic consolidation equations given by Biot is the displacement function method proposed by McNamee and Gibson [34]. Although Biot had established the 3D poroelastic equations for the saturated soil medium, deriving the true 3D analytical solution to the problem remains a great challenge so far. McNamee and Gibson [34] simplify the equations into the plane strain and the axisymmetric cases, which is capable of modeling almost all the horizontal and longitudinal consolidation and wave propagation problems. As the 3D problems degenerated into the plane strain and the axisymmetric problems, the utilization of the Fourier and Hankel transforms to solve the Bessel function problems became straightforward. Ai and Wang [35] derived an efficient analytical solution to the axisymmetric Biot’s consolidation problem using the Hankel and Laplace transform. As for the theoretical answers to saturated soil–pile interaction, there are generally two mainstream methods: one is decoupling the ‘model’ [36,37], and the other one is decoupling the ‘equations’. The differences between these two methods are evident. The method of decoupling the ‘model’ tried to reduce the soil–pile interaction problems back to the elastic problems of infinite half-space, which can be solved under the framework of Green’s function. To some extent, the soil–pile interaction problems can be regarded as the response of the infinite half-space with a cylindrical area (pile) being specially strengthened. Hence, the problem can be reduced by decoupling the soil–pile interaction model into an infinite half-space and a fictitious bar, and the properties of the fictitious bar are equal to the properties of the pile minus the ones of the soil. Then, by enforcing the superposition principle in elasticity, the displacement fields of the bar (pile) and the soil are compatible. Utilizing this approach, Zeng and Rajapakse [36] investigated the influence of the nondimensional parameter $b^{\ast }$, which equals the fluid viscosity divided by the soil permeability, on the pile dynamic impedance. A reproduction of their findings is plotted in Figure 3. As they pointed out, $b^{\ast }$ has a more significant influence on the damping (imaginary part of impedance) than on the stiffness (real part of impedance), and ‘the poroelastic effect is more dominant in clays rather than in sands’ [36]. The derivation of this method involves considerable mathematical effort, and the involvement of integral functions significantly limits the computational efficiency of the program. As for the other approach, “decoupling the equations” inherits the thought used in the derivation of one-phase soil, whose aim is to decouple the differential equations into the PDEs that have general analytical solutions. For instance, Liu et al. [38] introduced the potential functions to decouple the axisymmetric Biot’s poroelastic equations. They found out that the increase in the permeability coefficients would result in larger oscillation amplitudes of both the dynamic stiffness and damping and increased resonance frequencies, as shown in Figure 4. The above-mentioned studies significantly enrich the knowledge of the dynamic saturated soil–pile interactions. However, the unsaturated soil, containing the soil skeleton, pore fluid, and air bubbles, is more prevalent in nature. It is reported that the air bubble inside the pore will generate considerable matrix suction, increasing the effective stress and stiffness of the soil [39,40,41]. Based on the Van Geluchten (V–G) model, Shan et al. [42] and Ye and Ai [43] established the unsaturated soil–pile dynamic interaction model, and Ma et al. [44] extended the study to the pile group case. As the saturation of soil decreases, the matrix suction becomes stronger. As a result, both the dynamic stiffness and damping increase as the saturation decreases, which is demonstrated in Figure 5. Hence, utilizing the saturated soil–pile model to investigate the pile in unsaturated soil would underestimate the dynamic impedance of the pile. To sum up, as shown in Figure 6, the theory of the longitudinal soil–pile dynamic interaction has experienced a development process from the Winkler model to the plane strain model to the three-dimensional one-phase continuum model to the multi-phase continuum model.

2.2. Progress in the Pile End Soil Modeling and Its Application in Strain Wave Modeling

The strain wave propagation across the soil–pile system is an important issue in the discipline of earthquake engineering, structural dynamics, and structural health monitoring. The multi-phase continuum theory is fully capable of modeling the wave propagation inside the soil medium, whereas the reflection boundaries of the pile can significantly influence the simulation results of the wave reflection. Numerous studies have proven that the reflection of the strain wave is usually triggered by the alternation of the pile cross-section’s impedance [45,46]. The most signification alternation of the impedance usually happens at the interface of the pile bottom and the soil. Hence, the reflection at the interface of the pile bottom and pile end soil is usually significant. The simplest approach to simulate this wave reflection boundary is setting the pile bottom fixed (simply regarding the displacement of the pile bottom as zero) [45,46,47,48,49]. The results calculated by the fixed pile bottom model [45,46,47,48,49] can be easily distinguished from other results, for the reflected wave shows an opposite oscillation direction compared to the incident wave. However, in practice, engineers would find that the oscillation direction of the reflected signal is usually in accordance with the incident wave. This is because, in most cases, the soil layer or bedrock at the bottom of the pile is not hard enough to prohibit the small deformations from occurring and thus usually leads to the reflected waves sharing the same direction as the incident wave. As an extension of the Boussinesq solution inside the semi-infinite medium, Mindlin [50] provided a rigorous solution to the internal stress distribution under a point load inside a semi-infinite space. On the basis of the superposition principle in linear elasticity, Poulos and Davis [51] dispersed the pile into several continuous rigid bodies and integrated the soil–rigid body interactions from the pile bottom to the pile head to calculate the overall static settlement of the pile. However, as for dynamic analysis, this rigorous answer becomes extremely mathematically sophisticated, making it only suitable for academic research [52]. In practice, the Voigt model [53,54], which uses discrete springs and dashpots to simplify the viscoelastic interaction at the pile bottom, has gained much popularity. The Voigt model is mathematically convenient and easy to implement, despite the disadvantages of vague spring and dashpot coefficients for the model. The values for the springs and dashpots in the Voigt model are generally experiment-based, and few rigorous or simplified theoretical equations have been established to assess the elastic and damping coefficients in the model. To overcome this limitation, Wu et al. [55,56] proposed the fictitious soil pile model, in which a ‘fictitious soil pile’ is applied at the bottom of the real pile and utilized to model the soil–pile interaction at the pile bottom. The ‘fictitious soil pile’ properties equal the soil’s properties, avoiding the shortage of vague parameter values in the Voigt model. Subsequently, a series of the “fictitious soil pile model” was developed to consider the saturation [42,57,58,59,60,61] and stress diffusion [62,63,64] at the pile bottom. The fictitious soil pile shows excellent mathematical efficiency and acceptable accuracy, and a category of this model is summarized in Figure 7. In addition, Figure 8 compares the calculated soil stresses and displacements by the fictitious soil pile model and the rigorous elastic theory, the differences of which are within the tolerance. Because of the advantages of efficiency and simplicity, the fictitious soil pile model is subsequently introduced to many other kinds of soil–pile interaction problems, such as the soil–necking pile interactions [65,66] and the soil-stepped pile interactions [67,68].

2.3. Progress in the 3D Soil–Pile Wave Propagation Theory

Besides the wave reflection at the pile bottom, it is found the reflection of the transverse wave at the pile shaft could be significant as the dimension of the pile increases. Many studies reported intensive interferences when conducting low-strain integrity tests for pile foundations [69,70,71,72]. The formation mechanism of this so-called “high-frequency interference” was not theoretically revealed until the last decade. The 1-D rod theory is no longer capable of revealing the “high-frequency interference” caused by transverse wave propagation. Hence, utilizing viscoelastic continuum theory, Ding et al. [73,74] established a soil–pile interaction model capable of simulating the 3D strain wave propagation inside the pipe pile. Zheng et al. [75,76,77] investigated the high-frequency interference under axisymmetric loading conditions, where the high-frequency interference is mainly caused by the radially propagated strain wave. Subsequently, Zheng et al. [78] extended their studies to the non-axisymmetric cases and found the high-frequency interference caused by the circumferentially propagated strain wave more significant than that caused by the radially propagated strain wave. Dai et al. [79] studied the three-dimensional wave scattering of the soil–pile system induced by the vertical P wave, and derived an analytical solution to the equivalent Winkler model for vibrations induced by the seismic P wave. Meng et al. [80] discovered that the impedance of the pile would vary radially. In detail, the most significant dynamic stiffness would appear at the center of the pipe pile, whereas the most significant dynamic damping would appear at the outer radius of the pipe pile, as shown in Figure 9. Their mathematical model also illustrates the formation mechanism of the radial transverse wave interference and the guided wave [81]. As illustrated in Figure 10, for axisymmetric loading, radially propagated Rayleigh waves induced by the incident wave would propagate between the center axis and the edge of the pile, forming high-frequency strain wave signals interfering with the signal captured at the pile head. The arrival of the incident wave at the pile head would vary significantly according to the radial positions. However, as the guided wave forms beneath the pile head, the incident strain wave would subsequently propagate in the form of a surface wave, under which circumstance the arrival of the signal starts to converge. A spectrum describing the wave propagation during non-axisymmetric loading is presented by Zhang et al. [82]. As shown in Figure 11, for non-axisymmetric loading conditions, the circumferentially propagated transverse wave dominantly governs the formation of the high-frequency interferences. The circumferentially propagated radial wave is the source of the interferences. In addition, an analogous phenomenon is observed that the differences in the arrival time of the incident wave start to mitigate as the depth increases, which is the side evidence of the formation of the guided wave. Lu et al. [83] studied the 3D wave effect of the soil–pile system under non-axisymmetric excitations and discovered that the wave signal captured at the location with 90 degrees to the excitation place is least influenced by the high-frequency interference. Based on the wave theory they established, Lu et al. [84] investigated the performance of flexural waves in identifying cracks in pipe piles. The use of continuum mechanics to model the dynamic behavior of the pile enables the reveal of high-frequency interferences. However, the mathematical complexity hinders its application in real-life engineering. Hence, in pursuit of a mathematically simpler solution, the Rayleigh–Love rod model gains popularity [85,86]. However, the original Rayleigh–Love rod model can only model the transverse inertia effect instead of the transverse wave effect. After a 3D upgrade, a modified Rayleigh–Love rod model can realize the modeling of both transverse inertia and wave effects [87]. For pipe piles, scholars also found a new engineering phenomenon, that is, the speed of stress wave propagation is closely related to the properties of the soil plug, and the higher the height of the soil plug, the smaller the wave speed [86,87]. After the work of Wu et al. [88,89], Liu et al. proposed the general additional mass model to simulate the interaction among pipe pile, soil plug, and pile surrounding soil [90], and applied the proposed model to interpret the results of low strain detection signals of pipe piles [91,92]. In order to facilitate engineers to reasonably set the test wave velocity in the low-strain detection of pipe piles, Wu et al. also proposed a new method to calculate the apparent phase velocity of open-ended pipe piles [93,94].

2.4. Progress in Studies Associated with the Pile Group Effect

In reality, pile foundations usually appear more in the form of pile groups, while the use of single pile foundation is quite rare [95,96,97]. During the vibration of pile groups, the impedance of each pile would be influenced by other piles, which is described as the “pile group effect”. To account for this effect, Kaynia and Kausel [98], and Dobry and Gazetas [99] introduced the concept of ‘the dynamic interaction factor ${\alpha }_v$’ to describe the relationship between the displacements of the pile induced by external loads and other pile foundations. In their model, the piles in the pile groups are classified as the active pile and passive pile: the former type of pile endures the external loads and influence the displacement of other piles, whereas the latter type of pile experience additional displacement caused by the active one. An evident defect of ‘the dynamic interaction factor ${\alpha }_v$’ given by Dobry and Gazetas [99] is that it is applicable only if each pile in the pile group is subjected to equivalent external loads. To consider the ‘wave diffraction’ induced by different external loads, Mylonakis and Gazetas [100] utilized the Winkler model to simulate the soil–passive pile interactions. Zhang et al. [101] subsequently extended the study to the saturated soil cases. The above-mentioned ‘dynamic interaction factor ${\alpha }_v$’ ignores that the additional displacement of the passive pile would, in turn, exert influence on the active pile, resulting in the additional displacement of the active pile [102]. Luan et al. [103] further took the geometry of the pile cross-section and the secondary wave effect into account and derived a more rigorous solution to the pile group response. A detailed comparison of the referred ‘dynamic interaction factor ${\alpha }_v$’ is given in Table 1.

3. Horizontal Dynamic Analysis and Wave Theory

3.1. General Progress in Horizontal Soil–Pile Dynamic Interaction Modeling

Horizontal dynamic analysis of pile foundations is not as common on land as longitudinal dynamic analysis. However, as the exploitation of energy develops into the deep sea, the scenarios of horizontal vibrations of the pile, for instance, offshore wind turbines and oil and gas platforms, become prevalent [104]. The dynamic Winkler model is the most popular theory utilized for the horizontal dynamic analysis of pile foundations [105,106,107,108,109]. Considering the implement of the dynamic Winkler model in the horizontally vibrating cases is quite similar to that in the longitudinally vibrating cases, only a brief discussion focusing on the model parameters is given. Similar to the longitudinal dynamic analysis, the elastic and damping coefficients in the Winkler model can be derived from the plane strain model [110]. Except for the plane strain model, the coefficients of the dynamic Winkler model can also be derived from the formulas given by Biot [111], Vesic [112], Klopple and Glock [113], Selvadurai [114], etc. The subgrade reactions calculated by these formulas vary significantly, and a detailed comparison of these formulas is summarized by Prendergast and Gavin [115]. It should be noted that the impedance calculated from the Winkler model can only be regarded as the initial viscoelastic complex stiffness for the subgrade reaction and can only be used for small-strain problems. For cases where the nonlinearity of the soil must be included, the p–y method can make up for the deficiencies in the nonlinear part of the soil–pile interaction. Since the scope of this review is within the viscoelastic problems, the development of the p–y method will not be discussed. However, a review of the p–y method summarized by Bouzid [116] is encouraged for reference if interested. As mentioned before, the results calculated from different elastic and damping coefficients in the Winkler model could vary significantly because many of the formulas for the coefficients in the Winkler model are either empirical or based on experimental data. Hence, the call for rigorous theoretical answers promotes the development of viscoelastic pile–soil interaction models under horizontal loads. Similar to the development of the longitudinal pile–soil dynamic interaction models, the horizontal one also experiences a developing process from the Winkler model to the simplified 3D continuum model [117,118] and then to the multi-phase 3D continuum model [119,120,121,122]. A reproduced figure, originating from Zhang et al. [119], is presented to illustrate the influence of soil saturation on the pile’s dynamic horizontal impedance. As shown in Figure 12, when soil saturation increases from 0.7 to 0.999, the dynamic stiffness (real part of the impedance) could increase by 50%. In other words, the impedance of the pile could be significantly overestimated if adopting the saturated soil model to simulate the unsaturated soil. It should be noticed that this trend is opposite to that found in the longitudinal vibration situation, in which the impedance would decrease as the saturation of soil increases. For longitudinal vibrations, the utilization of the two-phase soil model is biased toward safety, whereas it turns out to be unsafe for horizontal vibrations.

3.2. Influence of the Vertical Loads on the Horizontal Dynamic Performance of Pile

The development of the viscoelastic theory for the horizontally vibrating piles is quite similar to that of the longitudinal one: fulfillment from the Winkler model to multi-phases continuum theory, the consideration of the active and passive piles in the pile group effect, etc. Hence, in this section, some interesting findings or opposite conclusions to those drawn in longitudinal vibrations will be discussed.

One interesting phenomenon is observed when vertical loads are subjected together with the horizontal loads. According to the P-delta effect, the vertical loads would cause additional moments during the bending of the pile, whereby exacerbating the deflection of the pile. This theory is supported by several analytical studies utilizing classic Biot’s poroelasticity to investigate the dynamic response of pile foundations induced by combined loads [123,124,125]. As one of the most representative examples, Ding et al. [123] reported that as the vertical load subjected at the pile head increases, the initial impedance decreases, and both the deflection and rotation angle would be exacerbated. However, some experimental studies as shown in Figure 13 indicated completely opposite results [126,127]. As an example, Lu and Zhang [126] conducted a model test and found that the horizontal displacement would decrease with the increase in the vertical load applied at the pile head. The theory and experiment seem to be the opposite regarding this phenomenon. Some scholars have attempted to elucidate the reasons for the differences between the theoretical and experimental results. For instance, Lu and Zhang [126] attributed this phenomenon to the strengthened pile–soil interface as the vertical load increases. Li et al. [128] thoroughly investigated the mechanics behind it and deduced that the change in the mean effective stress level could be responsible for the increased initial stiffness and capacity. However, it should be specially noted that the initial stiffness or capacity does not necessarily increase as the vertical load applied at the pile head increases. From the perspective of the authors, whether the initial impedance of the pile weakens or enhances highly relies on which effect is more significant, the P-delta effect or the enhanced pile–soil interface, and the mean effective stress level. If the P-delta effect is more dominant, the initial impedance of the pile will decrease. On the other hand, if the soil strength is significantly enhanced to overcome the adverse effect brought about by the P-delta effect, the initial impedance would increase. The slenderness ratio also plays a significant role in the results: for piles with smaller slenderness ratios, the initial impedance is more likely to increase with the increase in vertical loads applied at the pile head.

3.3. Progress in Coupled Soil–Pile–Water Modeling for Offshore Engineering

As lateral vibration often occurs at the pile foundations used offshore, the hydrodynamic pressure acting on the pile shaft drew the interest of the researchers. The most referred research investigating the hydrodynamic forces acting on the pile foundation came from Morison [129]. The Morison equation is the sum of two components: one is the inertia load associated with the acceleration of the wave, and the other is the drag load related to the instantaneous flow velocity. Since the inertia and dragging coefficients influence the calculated wave forces, a number of studies focus on the correction of these two coefficients [130,131,132,133]. Recently, Beji [134] extended the scope of the Morison equation by taking the geometry of the pile and the wave kinematic into consideration. Zan and Lin [135] pointed out the deficiency of the Morison equation in the internal solitary wave and proposed a modified empirical equation. Li et al. [136] further investigated the wave formulas of the interfacial periodic gravity waves in a two-layer fluid. It has already become common sense that the wave forces acting on a single pile in a pile group would increase as the gap between the piles decrease, whereas the Morison equation is not capable of modeling this phenomenon. To account for the pile group effect on wave scattering, Mindao et al. [137] introduced the interference coefficient $K_g$ and shelter coefficient $K_z$ to account for the wave scattering under side-by-side and tandem arrangement, respectively. Subsequently, a comprehensive study investigating the wave loads acting on the randomly arranged pile group was conducted by Bonakdar et al. [138]; the pile group effect coefficient $K_G$ is summarized and plotted in Figure 14.

Besides the Morison equation, the continuum equation overlooking the shear stress, which is also known as the radiation wave theory, is also a popular approach to estimating the hydrodynamic effect on pile foundation [139]. This theory originated from the hydrodynamic analysis of a dam, and was subsequently used to analysis the hydrodynamic pressure on piles [140,141,142,143,144]. An analytical solution to linear radiation wave theory is accessible, making it widely applied in inertia and kinematic (seismic) analysis of the laterally loaded piles [145,146]. Radiation wave theory still involves significant simplifications and is far away from a rigorous solution. Based on the Reynolds-averaged Naiver–Stokes (RANS) equations and the standard $k-\epsilon$ model, Zhao et al. [147] established a rigorous numerical model, capable of modeling the pore pressure response caused by seawater and the consolidation during the motion. For numerical efficiency, Zhao et al. [147] utilized a quasi-static soil model instead of a dynamic soil model. To account for the inertia effect of the soil, Sui et al. [148] upgraded the model to a dynamic one and investigated the liquefication around the pile caused by the wave. The mathematical complexity hinders their application in practice, no matter the radiation wave theory or the FEM model based on RANS equations. The Morison equations or simply utilizing “added mass” to model the hydrodynamic loads remains the most popular approach in practical engineering [143,149]. It is interesting to find that most of the theories established or phenomena that happened for the lateral foundation are more suitable or prone to occur for slender piles instead of stubby piles. For instance, the Morison equation, the P-delta effect, and the plane section assumption. In fact, researchers have long found the deficiency of the Euler–Bernoulli beam (EB) theory, especially for stubby piles [150,151]. Due to the shear deformation being overlooked in the EB model, the calculated deflection could be underestimated. With the adoption of the Timoshenko beam theory, the deflection and impedance of stubby piles can be better predicted [152,153,154].

3.4. Progress in Seismic Performance of Pile Foundations

Compared to the dynamic loads subjected at the pile shaft, known as the inertia response, the horizontal vibrations induced by seismic loads are more significant [155,156,157,158]. Hence, numerous scholars have investigated the seismic performance of pile foundations caused by the vertically propagated S-wave. An essential simplification for the seismic response analysis in the viscoelastic medium is given by Gazetas [155], who assumed that the pile responses induced by the external loads and the seismic waves could be decomposed or superimposed. With this simplification, the horizontal seismic response of the pile foundation can be regarded as the vibrations induced by the motions of far-field soils [159]. Based on the beam-on-dynamic-Winkler foundation (BDWF) model, Torshizi et al. [160] investigated the kinematic bending strains at the pile head of the pile groups. Álamo et al. [161] studied the seismic tangential interactions between the soil and pile and verified the capability of the BDWF model by comparing it with the rigorous continuum model. The reproduced results are shown in Figure 15. As is shown, the BDWF’s results are generally in accordance with the continuum model, while deviations can be observed at the soil layers’ interfaces, which can be attributed to the neglect of the coordinated deformation between soil layers in the BDWF model. With the combination of the BDWF model and the finite element method (FEM), Dezi et al. [162] studied the seismic response of the pile group in layered soils.

In pursuit of a more rigorous answer, elastic or poroelastic theories were also implemented for seismic analysis of piles in viscoelastic soil. For instance, Kaynia [163] utilized Green’s function for layered media to model the dynamic response of the soil under the seismic S-wave; Zheng et al. [164] derived an analytical solution to this seismic response problem with the adoption of Biot’s poroelastic equations to simulate the viscoelastic behavior of the soil; Dai et al. [165] investigated the influence of the radial inhomogeneity on the seismic response of the pile foundation. Although the acquisitions of these rigorous answers avoided the coefficients assessment in the Winkler model, they lost the simplicity, versatility, and extensibility of the Winkler model. For instance, the BDWF model can conveniently take the nonlinearity of the soil into consideration with the introduction of the p–y curve profiles of the intended soil [166], whereas the constitutive soil equations for rigorous analytical solutions are limited in linear elasticity. Usually the earthquake-induced dynamic response usually involves large strain deformation and strong nonlinearity, making these rigorous viscoelastic solutions inapplicable. To overcome the drawback that the continuity of the soil deformation is overlooked in the BDWF model, some ‘two-parameter’ subgrade reaction models, such as the modified Vlasov model [167,168,169,170] and the Pasternak model [171,172], are proposed by some researchers. The ‘two-parameter’ subgrade reaction model established the deformation relations between the adjacent springs and dashpots in the BDWF model to simulate the continuous deformation in the soil while only resulting in a limited increase in computational efforts. During the discussion of the research adopting the Winkler model, the authors pointed out that the neglect of the continuity of the spring and dashpots could cause some evident deviations at the interfaces of the soil layers compared to the continuum model, as shown in Figure 15. However, with the consideration of the continuity of the Winkler model, the deviation can be significantly reduced, as shown in Figure 16, which is a reproduction of the work by Ke et al. [170]. Although the application of FEM [173,174,175] or BEM [176] methods continued to expand in the soil–pile dynamic interaction problems, the BDWF model and some other subgrade reaction methods remain the most efficient and versatile ones in seismic analysis.

4. Torsional Dynamic Analysis and Wave Theory

Compared to longitudinal and horizontal dynamic vibrations, the torsional vibration is the rarest one in practical engineering. The torsional vibration of the pile is often caused by machinery loads or unbalanced horizontal loads [177,178]. After decades of research, the torsional vibration theory of pile foundation in the viscoelastic medium had been established on the basis of the plane strain model [179,180,181,182], one-phase 3D continuum model [183,184,185], and multi-phase 3D continuum model [186,187,188], successively. For torsional vibrations, the results calculated from the plane strain model and the 3D continuum model are highly consistent, as shown in Figure 17. However, the saturation of the soil still has an unneglectable influence on the impedance of the pile. As shown in Figure 18, for a pipe pile, the saturation of the outer soil significantly influences the dynamic stiffness and damping of the pile. In detail, as the saturation of soil decreases, the dynamic stiffness of the pile would increase dramatically, whereas the dynamic damping would decrease slightly. It is also observed that the influence of the saturation of the outer soil is more pronounced than that of the inner soil. Hence, the most efficient and comparably accurate approach for torsional soil–pile modeling is adopting the multi-phase plane strain model.

Due to the fact that the shear wave velocity is almost half of the compressive wave velocity, some scholars pointed out that the utilization of the shear wave can effectively reduce the detection blind zone of the compressive wave if adopting the shear wave for low strain integrity test for pile foundations [190]. Considering the torsional wave signal is very rare in nature, it can hardly be interfered with by other ambient dynamic impulses, making it an optimal choice for the incident wave input. Liu et al. [191] pioneered the research of the torsional low-strain integrity test and demonstrated the advantages of having a smaller detection blind zone. Zhang et al. [192] investigated the three-dimensional torsional wave propagation and found that the high-frequency interferences during the torsional low strain integrity test are highly controllable compared to those that occurred during the longitudinal low strain integrity test. In addition, the torque can be subjected at any location of the pile shaft instead of only at the pile head, making it more versatile than the longitudinal test, especially when evaluating the existing pile foundations [193]. In summary, compared with the longitudinal wave, the torsional wave has the following advantages in structure health detection:

• Smaller detection blind zone.

• Less significant high-frequency interferences.

• More versatile in existing structure health detecting.

However, it also has the following disadvantages:

• Faster stress wave dissipation.

• More complicated incident wave input equipment.

• Higher requirements for sensor accuracy.

Inspired by the 3D wave effect of the torsional low strain test, it can be deduced that, for large-diameter pile foundations, the torsional impedance at the cross-section of the pile head could also vary in the radial direction. Zhang et al. [194] reported that the dimension of the pile and the differences in the elastic modulus between the inner and outer soil could significantly influence the impedance distribution across the cross-section.

5. Conclusions and Future Work

The theory of the “pile dynamics in viscoelastic medium” enables the initial impedance calculation, resonance frequency identification, and strain wave propagation modeling of the pile foundation. The calculated initial impedance and the resonance frequencies are critical parameters in the dynamic design of energy structures, while the strain wave propagation modeling is the key to the seismic response analysis and the structure health monitoring. This paper provides a state-of-the-art review of the development of longitudinal, horizontal, and torsional dynamic soil–pile interaction modeling techniques. Generally, during the decades of development, the dynamic soil–pile interaction problems within the discipline of viscoelasticity solutions are becoming more and more refined, from simplified subgrade reaction models (e.g., the Winkler, Vlasov, and Pasternak models) to rigorous multi-phase poroelastic models (e.g., the Biot’s poroelastic and unsaturated soil models). For longitudinal and torsional vibration cases, the decrease in the saturation of soil shows a significantly positive influence on the dynamic impedance of the pile, indicating that the matrix suction can evidently increase the dynamic stiffness of the soil. However, for the horizontal vibration case, the decrease in the saturation of soil would decrease the dynamic impedance in the opposite case. In addition, for large diameter piles, researchers in the longitudinal and torsional vibrations both observed obvious transverse wave interference during the 3D strain wave propagation modeling, whereas the shear deformation effect was reported significant once the slenderness ratio of the pile was small during horizontal vibrations. In brief, the viscoelastic soil–pile interaction theories enable a quick and reliable calculation of the initial impedance, the deviation of which, compared to the test results, fluctuates no more than 5% in most cases [89,96].

To the best knowledge of the authors, the development of pile dynamic theories in a viscoelastic medium still has the following insufficiency. For starters, the friction and relative sliding at the soil–pile interface need further investigation since most of the present studies assumed the soil and pile maintain perfect contact during the vibrations [195]. Secondly, the effective length theory for horizontally vibrating piles needs further development. For the long piles, the entire pile length can hardly all participate in the resistance of vibrations, while only a portion of the pile length can be effective for the exerting of reactions of the foundation. Thirdly, there is a lack of research on applying the rigorous viscoelastic theory solution as the initial impedance to the nonlinear subgrade reaction solutions (e.g., p–y methods). Lastly, the theoretical answers regarding the pile group effect for torsional vibrations are scarce. For the pile group subjected to torsions, every single pile in the pile group could undergo different vibration modes, including horizontal, torsional, and superposition modes. Hence, the complexity of this problem results in very few mature theories capable of modeling this phenomenon.

Author Contributions

Conceptualization, W.W.; formal analysis, W.W. and Y.Z.; writing—original draft preparation, W.W. and Y.Z.; writing—review and editing, W.W.; supervision, W.W.; funding acquisition, W.W. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures

Figure 1. Advantages of plane strain model for modeling of the radial inhomogeneous problem: (a) tandem spring model [20]; (b) Voigt model in series [21]; (c) plane strain model [22].

Figure 2. Pile impedance calculated by plane strain, simplified continuum model, and 3D continuum model: (a) dynamic stiffness; (b) dynamic damping (reproduced from Zheng et al. (2015) [30]).

Figure 3. Influence of fluid viscosity and permeability of soil on pile impedance (reproduced from Zeng and Rajapakse (1999) [36]).

Figure 4. Influence of permeability of soil on pile impedance: (a) dynamic stiffness; (b) dynamic damping (reproduced from Liu et al. (2014) [38]).

Figure 5. Influence of saturation of unsaturated soil: (a) matric suction; (b) dynamic longitudinal stiffness; (c) dynamic longitudinal damping (reproduced from Ye and Ai (2022) [43]).

Figure 6. Development of the dynamic soil–pile interaction model.

Figure 7. Category of the fictitious soil pile model: (a) Ref. [56]; (b) Ref. [63]; (c) Ref. [64]; (d) Ref. [65]; (e) Refs. [67,68]; (f) Ref. [57]; (g) Ref. [42].

Figure 8. Stress and displacement of soil calculated by the fictitious soil pile model and the rigorous elastic theory: (a) principle stress; (b) displacement.

Figure 9. Variation of the pile head impedance at the cross-section of the pile (reproduced from Meng et al. (2020) [80]): (a) dynamic stiffness; (b) dynamic damping.

Figure 10. Formation of the high-frequency interference caused by radial transverse wave (reproduced from Meng and Su (2022) [81]).

Figure 11. Formation of the high-frequency interference caused by circumferential transverse wave: (a) circumferentially propagated Rayleigh wave; (b) guided wave (reproduced from Zhang et al. (2022) [82]).

Figure 12. Influence of saturation of unsaturated soil on the dynamic horizontal impedance of the pile (reproduced from Zhang et al. (2019) [119]).

Figure 13. Influence of vertical loading on the lateral displacement of pile (reproduced from Li et al. (2022) [128]).

Figure 14. Pile group effect coefficients for different pile group arrangements (reproduced from Bonakdar et al. (2015) [138]).

Figure 15. BDWF-based FEM results compared to the continuous model (reproduced from Álamo et al. (2018) [161]).

Figure 16. Results derived from the Vlasov model compared to the continuous model (reproduced from Ke et al. (2019) [170]).

Figure 17. Comparisons between the plane strain and 3D continuum model for torsional soil–pile interaction modeling (reproduced from Wang et al. (2008) [189]): (a) dynamic stiffness; (b) dynamic damping.

Figure 18. Influence of the saturation of soil on the dynamic torsional impedance of pile (reproduced from Ma et al. (2022) [188]): (a) dynamic stiffness with the variation of both the outer and inner soil’s saturations; (b) dynamic damping with the variation of both the outer and inner soil’s saturations; (c) dynamic stiffness with the variation of the inner soil’s saturations; (d) dynamic stiffness with the variation of the inner soil’s saturations; (e) dynamic stiffness with the variation of the outer soil’s saturations; (f) dynamic damping with the variation of the outer soil’s saturations.

Table 1

Comparison of the referred ‘dynamic interaction factor ${\alpha }_v$’ for consideration of pile group effect.

Source	Passive Pile Deformation Caused by Active Pile	Wave Diffraction Effect	Multi-Phase Nature of Soil	Active Pile Deformation Caused by Passive Pile
Kaynia and Kausel [98]	√
Dobry and Gazetas [99]	√
Mylonakis and Gazetas [100]	√	√
Zhang et al. [101]	√		√
Luan et al. [102,103]	√		√	√

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