# A Method to Estimate Dynamic Pore Water Pressure Growth of Saturated Sand-Gravel Materials
## 1 Introduction
First, the method for determining liquefaction \(N_{\text{L}}\) for most pore water pressure models is not standardized, even though most use the number of cycles required to cause  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Pore Water Pressure Model** & **Equation** & **Explanation** & **Reference** \\ \hline Stress model & \(\frac{\mu_{d}}{\
u_{\infty}}=\frac{1}{2}+\frac{1}{n}\text{arcsin}\left[2\left( \frac{N}{N_{\text{L}}}\right)^{\frac{1}{3}}-1\right]\) & \(u_{\text{d}}\) is the dynamic pore water pressure; \(\sigma_{\text{3c}}\) is the initial confining pressure; \(N_{\text{L}}\) is the number of cycles required to cause liquefaction; \(\theta\) is the experimental constant related to the type of material. & Seed [6] \\ \hline Strain model & \(\left\{ \begin{array}{ll}\Delta u=\overline{E}_{r}\Delta e_{\text{rl}}& \text{or}\\ \Delta e_{\text{rl}}=c_{1}(\gamma-c_{2}e_{\text{rl}})+\frac{c_{3}e_{\text{rl} }^{2}}{\gamma+c_{\text{eff}}}&\text{intial vertical effective stress;}\\ c_{1},c_{2},c_{3},c_{4}\text{ are test constants.}\end{array} \right.\) & \(\overline{E}_{r}\) is the tangent modulus of the one-dimensional unloading curve at a point corresponding to the initial vertical effective stress; \(c_{1},c_{2},c_{3},c_{4}\) are test constants. & [PERSON] et al. [11] \\ \hline Transient model & \( \begin{array}{ll}u(t+\Delta t)=u(t)+\Delta u\\ =u(t)+\Delta u_{\sigma}+\Delta u_{0}+\Delta u_{\text{T}}&\text{in pressure without dissipation at}\\ =u(t+\Delta t)*+\Delta u_{\text{T}}&(t+\Delta t);u(t+\Delta t)\text{ is the dissipation}\\ \end{array} \) & \( \begin{array}{ll}\Delta u_{\sigma},\Delta u_{0},\Delta u_{\text{T}}\\ \text{ pore water pressure during }\Delta T;\\ \text{ }u(t+\Delta t)^{\frac{1}{3}}\text{ is the pore water}&\text{in pressure without dissipation at}\\ =u(t+\Delta t)*+\Delta u_{\text{T}}&(t+\Delta t);u(t+\Delta t)\text{ is the dissipation}\\ \end{array} \) & \( \begin{array}{ll}\Delta u(t+\Delta t)^{\frac{1}{3}}\text{ is the dissipation energy; }K\text{ and}\\ \beta\text{ are test constants.}\end{array} \) & \( \begin{array}{ll}\Delta u(t+\Delta t)^{\frac{1}{3}}\text{ cano}\\ \end{array} \) \\ \hline Endochronic theory & \(\frac{\Delta u}{\
u_{\pi}}=\frac{A}{B}\ln(1+Bk)\) & \( \begin{array}{ll}K\text{ is the damage parameter; the}\\ \text{variable includes the shear strain}\\ \text{amplitude and number of cycles. }A\\ \text{ amplitude and number of cycles. }A\\ \text{ and }B\text{ are test parameters.}\end{array} \) & Finn [14] \\ \hline Effective stress path model & \(\left\{ \begin{array}{ll}q/p^{\prime}\geq\left(q/p^{\prime}\right)_{\text{max }},\text{loading}\\ q/p^{\prime}{<}(q/p^{\prime})_{\text{max }},\text{ unloading}\\ \end{array} \right.\) & \(\begin{array}{ll}\text{Under loading conditions,}\\ \text{ additional plastic shear}\\ \textliquefaction \(N_{\rm L}\) to normalize the cyclic loading numbers \(N\). Secondly, compared with sand, there is not sufficient liquefaction test data on sand-gravel materials, and the pore water pressure growth model needs to be further verified. Studying the liquefaction characteristics of sand-gravel materials is, therefore, necessary to provide a basis for the analysis of the liquefaction resistance of dam foundations and dam bodies. This paper used the GCTS large-scale dynamic triaxial apparatus to carry out a series of undrained cyclic triaxial tests on sand-gravel materials at different initial confining pressures \(\sigma_{\rm 3c}\) and consolidation stress ratios \(K_{\rm c}\). The paper proposes a method for determining the number of cycles required to cause liquefaction \(N_{\rm L}\) that considers the relative density \(D_{\rm r}\). The pore water pressure growth model of saturated sand-gravel materials was established and its applicability was verified.