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Influence of Kelvin-Voigt Viscoelasticity on Transient Pore Pressure Response of Cylindrical Cavities in Unsaturated Soil

This paper extends analytical solutions for cylindrical cavities in unsaturated soil by incorporating a Kelvin-Voigt viscoelastic skeleton, revealing that while displacement and stress remain largely unaffected, pore pressure is significantly attenuated—particularly under pulse loading—due to bulk viscoelasticity acting as a low-pass filter on the fast compressional wave.

Original authors: Xiong Hao

Published 2026-10-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xiong Hao

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The ground beneath our feet is rarely a simple, solid block. In many places, especially near the surface, the soil is a complex mixture of solid grains, water, and air trapped in the tiny spaces between them. When this kind of earth is disturbed by a sudden event—like a pile being driven into the ground, an underground explosion, or the rapid excavation of a tunnel—the forces ripple through the soil in a way that is difficult to predict. Engineers have long relied on mathematical models that treat the soil's solid skeleton as perfectly elastic, meaning it snaps back instantly and predictably after being squeezed, much like a stiff spring. However, real-world soils, particularly soft clays and organic silts, behave differently. They are "viscoelastic," a property where the material resists motion not just because of its stiffness, but also because of its internal friction, causing it to react more slowly and absorb energy over time. Understanding how this time-dependent behavior changes the pressure of the water and air trapped inside the soil is critical for designing safe underground structures that can withstand sudden shocks.

A researcher at Taizhou University has recently refined our understanding of this phenomenon by updating the standard models to include this viscous behavior. The study focuses on what happens inside a cylindrical hole in the ground when it is subjected to a sudden radial load, simulating conditions like an internal blast or a rapid excavation. By applying a specific mathematical framework known as the Kelvin-Voigt model, which accounts for the rate at which the soil deforms, the researcher was able to calculate how the pressure of the pore fluids changes over time. The work builds upon existing solutions for elastic soil but introduces a crucial correction: the realization that the soil skeleton does not react instantly. Instead, the internal friction within the soil grains acts as a damper, slowing down the transmission of force and altering the pressure waves that travel through the fluid and gas within the pores.

The most striking discovery from this research is that while the movement of the soil itself and the stress on the solid grains remain almost exactly the same as predicted by older, simpler models, the pressure of the water and air inside the soil drops significantly. When the soil is modeled as viscoelastic, the peak pressure generated by a sudden load can be reduced by as much as forty-three percent compared to the elastic prediction, depending on how quickly the load is applied. If the force is applied gradually, like a step function, the reduction is smaller, around two percent. But if the load is applied as a sharp, triangular pulse, the reduction is dramatic. This happens because the viscous nature of the soil skeleton resists the rapid squeezing that generates high fluid pressure. The solid framework absorbs some of the energy through internal friction before it can be transferred to the fluids, effectively acting as a filter that dampens the pressure spike.

The study also reveals that not all types of soil stiffness contribute equally to this effect. The reduction in pressure is driven almost entirely by the soil's resistance to changes in volume, known as bulk viscosity. The resistance to shape changes, or shear viscosity, plays a negligible role because the pressure in the pores is generated by the soil being squeezed together, not by it being twisted or sheared. This distinction is vital for engineers, as it means they only need to focus on measuring the volume-related viscous properties of the soil to accurately predict pore pressure, rather than characterizing every possible way the soil might deform. Furthermore, the effect is most pronounced in soils that are stiff and highly permeable, allowing the fluid to move freely, and in conditions where the soil is moderately to highly saturated with water. In soft, low-permeability clays, the viscous correction is minimal, suggesting that the old elastic models are still quite accurate for those specific materials.

To verify these findings, the researcher used advanced numerical techniques to solve the complex equations governing the soil's behavior, ensuring the results were precise to a very high degree. The analysis confirmed that the primary wave carrying the energy through the soil—the fast compressional wave—is the one most affected by the soil's viscosity, while the slower, diffusive waves remain largely unchanged. This selective damping explains why different types of loads produce different levels of pressure reduction; loads with high-frequency components, such as blasts, are filtered out more effectively than slower loads. The study concludes that while traditional elastic models are reliable for predicting how much the ground will move or how much stress the soil structure will bear, they tend to overestimate the dangerous fluid pressures in viscoelastic ground. For projects involving sudden, high-energy events in sandy silts or similar soils, ignoring this viscous damping could lead to overly conservative designs, whereas accounting for it provides a more realistic picture of the forces at play.

Technical Summary: Influence of Kelvin-Voigt Viscoelasticity on Transient Pore Pressure Response of Cylindrical Cavities in Unsaturated Soil

Problem Statement
Existing analytical solutions for the transient dynamic response of cylindrical cavities in unsaturated soil rely on the assumption of a linear elastic solid skeleton. However, natural porous media such as soft clays, organic silts, and peat exhibit rate-dependent deformation that elastic models cannot capture. While previous studies have addressed unsaturated poroelasticity (e.g., Li et al. [27]) and viscoelastic wave propagation in simpler configurations (e.g., Guo et al. [30]), a comprehensive analytical solution for the transient response of a cylindrical cavity in a three-phase (solid-fluid-gas) unsaturated medium with a viscoelastic skeleton has been lacking. This paper addresses that gap by extending the elastic baseline solution to incorporate Kelvin-Voigt (KV) viscoelasticity, specifically investigating how skeleton viscosity influences pore pressure, displacement, and stress under transient loading.

Methodology
The study employs an analytical approach based on the following framework:

Governing Equations: The problem is formulated using the continuum theory for three-phase porous media (Wei and Muraleetharan [15]), incorporating the Van Genuchten soil-water characteristic curve. The system is described in dimensionless terms using the $u_s$ - $w_f$ - $w_a$ formulation (solid, fluid, and gas relative displacements).
Viscoelastic Extension: The Kelvin-Voigt constitutive model is applied to the soil skeleton. Using the elastic-viscoelastic correspondence principle in the Laplace domain, the elastic moduli are substituted with frequency-dependent complex moduli: $\lambda^*(s) = \lambda^*(1 + t^*_r s)$ and $\mu^*(s) = 1 + t^*_r s$ , where $t^*_r$ is the dimensionless relaxation time.
Solution Structure: The correspondence principle allows the retention of the original elastic solution structure. The Helmholtz decomposition reduces the coupled system to decoupled Helmholtz-type equations. The general solution is expressed using modified Bessel functions ( $K_n$ ) for the three compressional wave modes (P1, P2, P3).
Numerical Implementation: The boundary value problem (permeable and impermeable conditions) is solved in the Laplace domain. Time-domain solutions are obtained via the Den Iseger numerical inverse Laplace transform algorithm, which utilizes a vertical contour in the complex plane and Gaussian quadrature. This method was chosen over the Stehfest algorithm (used in prior elastic studies) because the viscoelastic coefficients are complex-valued.
Verification: The solution is verified against the elastic limit ( $t^*_r = 0$ ), the saturated limit ( $S_r \to 1$ ), and independent frequency-domain dispersion analyses (Guo et al. [30]).

Key Contributions and Results
The paper provides a benchmark analytical solution and quantifies the specific effects of skeleton viscoelasticity:

Pore Pressure Attenuation: The most significant finding is that skeleton viscoelasticity strongly attenuates pore pressure. Depending on the loading type and relaxation time, peak pore pressure is reduced by 2% (step loading) to 43% (triangular pulse loading) at $t^*_r = 2.0$ .
Invariance of Displacement and Stress: In contrast to pore pressure, radial displacement and circumferential stress remain nearly unchanged by skeleton viscoelasticity. The relative change is less than $10^{-6}$ , indicating that elastic solutions remain accurate for predicting total stress and displacement fields even in viscoelastic grounds.
Bulk vs. Shear Contribution: A decomposition analysis reveals that pore pressure attenuation is driven almost exclusively by bulk (volumetric) viscoelasticity ( $\lambda$ ). Shear (deviatoric) viscoelasticity ( $\mu$ ) contributes negligibly because pore pressure generation couples to volumetric strain, not deviatoric strain.
Frequency-Domain Mechanism: The viscoelastic effect acts as a low-pass filter. The transfer function $H(s)$ , representing the ratio of viscoelastic to elastic pore pressure, decreases monotonically with frequency. This explains why high-frequency transient loads (like triangular pulses or blasts) experience greater attenuation than low-frequency sustained loads (step functions).
Wave Mode Selectivity: Wave mode analysis confirms that the fast P1 compressional wave (skeleton-dominated) is the primary target of viscoelastic damping, with its attenuation length decreasing by a factor of 10–30. The slow P2 and P3 diffusion modes remain largely unaffected as they are governed by transport coefficients rather than skeleton moduli.
Parameter Sensitivity: The magnitude of viscoelastic correction is highest in stiff ( $\lambda^*$ high), highly permeable ( $b^*_f$ low), and moderately to highly saturated ( $S_r > 0.6$ ) soils. In soft, low-permeability clays, the correction is minimal (< 2%).

Significance and Claims
The paper claims that its primary significance lies in providing a rigorous analytical benchmark for poroviscoelastic dynamics and correcting the overestimation of pore pressure inherent in elastic models for viscoelastic ground.

Practical Implication: Elastic solutions are non-conservative for pore pressure prediction in viscoelastic soils, particularly under transient blast-type loading. The degree of overestimation can be substantial (up to 43%), whereas predictions for displacement and total stress remain reliable.
Methodological Efficiency: The study demonstrates that extending elastic solutions to viscoelasticity does not require re-deriving the complex boundary value problem; it can be achieved through algebraic parameter substitution in the Laplace domain.
Experimental Guidance: The results suggest that for pore pressure estimation, experimental characterization of volumetric viscoelastic parameters is sufficient, as shear viscosity has a negligible impact on the pore pressure response.

The study concludes that while the KV model captures rate-dependent stiffening, future work could extend this to standard linear solid or fractional derivative models to capture stress relaxation, and could incorporate the stiffening effects of elastic tunnel linings.

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