The microscopic origin of K₀ on granular soils: the role of particle shape

Abstract

The at-rest coefficient of lateral pressure, $K_0$, is a critical macroscopic parameter for evaluating stress transmission in granular media for engineering practice. This paper revisits the microscopic origin of $K_0$ and its corresponding underlying physics based on micromechanics-based theories and numerical simulations, with a focus placed on the effect of particle shape. Two typical kinds of distortion (elongation and blockiness) in particle shape are considered, modeled by ellipsoids and superballs in the discrete element method. One-dimensional compression tests are performed on numerical specimens with dense and loose initial states for different particle shapes. An analytical relationship between $K_0$ and anisotropy of fabric measures (i.e., contact normal, contact force, and shape-related anisotropy) is established within the stress-force-fabric framework. It is found that the analytical $K_0$ is consistent with the measured one directly from the simulation regardless of particle shape, verifying a well-established relationship between $K_0$ and fabric. It is further found that contact force partition of $K_0$ plays the most prominent role in $K_0$ compared to contact normal and shape-related partitions. Results also reveal the different influence of shape distortion on $K_0$ and the corresponding mechanical properties related to $K_0$, such as void ratio, mobilized friction angle, and coordination number. More specifically, $K_0$ has a ‘W’-shape relationship with elongation for dense ellipsoids and an ‘M’-shape relationship with blockiness for superballs.

Appendices

Appendix A: Tensors of stress-force-fabric relationship

By analogy to formula of stress tensor [15, 19], integral form of Eq. (4) can be determined as:

$$\begin{aligned} \sigma _{ij} = \frac{N_{c}}{V} \int _{ {{\varvec{n}}}} E({{\varvec{n}}})\bar{f_{i}}({{\varvec{n}}}) \bar{d_{j}}({{\varvec{n}}})\,d{{\varvec{n}}} \end{aligned}$$

(A.1)

where $N_c$ is the total contact number, $E({{\varvec{n}}})$ is the probability distribution function (PDF) of contact normal, $\bar{f_{i}}({{\varvec{n}}})$ and $\bar{d_{j}}({{\varvec{n}}})$ are directional distribution function of contact force vector and branch vector.

The second-order approximation of $E({{\varvec{n}}})$ takes the expansion form [13]:

$$\begin{aligned} E({{\varvec{n}}}) = \frac{1}{4\pi } [1+G_{ij}^{u}n_{i}n_{j}] \end{aligned}$$

(A.2)

Second-order tensor $G_{ij}^{u}$ is determined as:

$$\begin{aligned} G_{ij}^{u} = \frac{15}{2} {\varPhi _{ij}'} \end{aligned}$$

(A.3)

where ${\varPhi _{ij}'}$ is the deviatoric part of ${\varPhi _{ij}}$ and can be found as ${\varPhi _{ij}} - \delta _{ij}{\varPhi _{kk}}/3$ (in which $\delta _{ij}$ is the Kronecker delta).

$$\begin{aligned} {\varPhi _{ij}} = \frac{1}{N_{c}} \sum _{c \in V} n_{i}n_{j} \end{aligned}$$

(A.4)

The second-order approximation of $\bar{f_{n}}({{\varvec{n}}})$ or $\bar{d_{n}}({{\varvec{n}}})$ can be rewritten as:

$$\begin{aligned} \bar{f}_{n}({{\varvec{n}}})&= \bar{f_{0}}[1+G_{ij}^{f_{n}}n_{i}n_{j}]{{\varvec{n}}} \end{aligned}$$

(A.5a)

$$\begin{aligned} \bar{d}_{n}({{\varvec{n}}})&= \bar{d_{0}}[1+G_{ij}^{d_{n}}n_{i}n_{j}]{{\varvec{n}}} \end{aligned}$$

(A.5b)

And second-order approximation of $\bar{f_{i}^{t}}({{\varvec{n}}})$ or $\bar{d_{j}^{t}}({{\varvec{n}}})$ are:

$$\begin{aligned} \bar{f}_{t, i}({{\varvec{n}}})&= \bar{f_{0}}[G_{ik}^{f_{t}}n_{k}-(G_{kl}^{f_{t}}n_{k}n_{l})n_{i}] \end{aligned}$$

(A.6a)

$$\begin{aligned} \bar{d}_{t, j}({{\varvec{n}}})&= \bar{d_{0}}[G_{jk}^{d_{t}}n_{k}-(G_{kl}^{d_{t}}n_{k}n_{l})n_{j}] \end{aligned}$$

(A.6b)

Anisotropic fabric tensors of normal and tangential contact force ($G_{ij}^{f_{n}}$ and $G_{ij}^{f_{t}}$) or branch vectors ($G_{ij}^{d_{n}}$ and $G_{ij}^{d_{t}}$) are determined by:

$$\begin{aligned} G_{ij}^{f_{n}}&= \frac{15}{2}\frac{K_{ij}^{'f_{n}}}{\bar{f_{0}}} \,\,\,\,\, G_{ij}^{f_{t}} = \frac{15}{3}\frac{K_{ij}^{'f_{t}}}{\bar{f_{0}}} \,\,\,\,\, \bar{f_{0}} = K_{ii}^{f_{n}} \end{aligned}$$

(A.7a)

$$\begin{aligned} G_{ij}^{d_{n}}&= \frac{15}{2}\frac{K_{ij}^{'d_{n}}}{\bar{d_{0}}} \,\,\,\,\, G_{ij}^{d_{t}} = \frac{15}{3}\frac{K_{ij}^{'d_{t}}}{\bar{d_{0}}} \,\,\,\,\, \bar{d_{0}} = K_{ii}^{d_{n}} \end{aligned}$$

(A.7b)

Corresponding tensor $K_{ij}^{*}$ (superscript * stands for $f_n$ or $d_n$) are given by:

$$\begin{aligned} K_{ij}^{f_{n}}&= \frac{1}{N_{c}} \sum _{c \in V} \frac{\bar{f}_{n, i}n_{j}}{1+G_{kl}^{u}n_{k}n_{l}} \end{aligned}$$

(A.8a)

$$\begin{aligned} K_{ij}^{f_{t}}&= \frac{1}{N_{c}} \sum _{c \in V} \frac{\bar{f}_{t, i}n_{j}}{1+G_{kl}^{u}n_{k}n_{l}} \end{aligned}$$

(A.8b)

$$\begin{aligned} K_{ij}^{d_{n}}&= \frac{1}{N_{c}} \sum _{c \in V} \frac{\bar{d}_{n, i}n_{j}}{1+G_{kl}^{u}n_{k}n_{l}} \end{aligned}$$

(A.8c)

$$\begin{aligned} K_{ij}^{d_{t}}&= \frac{1}{N_{c}} \sum _{c \in V} \frac{\bar{d}_{t, i}n_{j}}{1+G_{kl}^{u}n_{k}n_{l}} \end{aligned}$$

(A.8d)

Substitutes Eq. (A.2), Eq. (A.5) and Eq. (A.6) into Eq. (A.1) leads to:

$$\begin{aligned} \begin{aligned} \sigma _{ij}&= \frac{N_{c}\bar{f_{0}}\bar{d_{0}}}{4\pi V} \int _{ {{\varvec{n}}}} (1+G_{ij}^{u}n_{i}n_{j}) \\&\quad \times (n_{i}+G_{ik}^{f_{t}}n_{k}+(G_{kl}^{f_{n}}-G_{kl}^{f_{t}})n_{k}n_{l}n_{i})\\&\quad \times (n_{j}+G_{jk}^{d_{t}}n_{k}+(G_{kl}^{d_{n}}-G_{kl}^{d_{t}})n_{k}n_{l}n_{j})\, d({{\varvec{n}}}) \end{aligned} \end{aligned}$$

(A.9)

We decompose stress tensor as a sum of isotropic term and anisotropic term, Eq. (A.9) is rewritten as:

$$\begin{aligned} \begin{aligned} \sigma _{ij}&= \frac{N_{c}\bar{f_{0}}\bar{d_{0}}}{4\pi V} \int _{ {{\varvec{n}}}} (E_{ij}^{I}+E_{ij}^{A}) (f_{i}^{I}+f_{i}^{A})(d_{j}^{I}+d_{j}^{A})\, d({{\varvec{n}}}) \end{aligned} \end{aligned}$$

(A.10)

Isotropic and anisotropic terms are given:

$$\begin{aligned} E_{ij}^{I}&= 1 \,\,\,\,\,\,\,\, E_{ij}^{A} = G_{ij}^{u}n_{i}n_{j} \end{aligned}$$

(A.11a)

$$\begin{aligned} f_{i}^{I}&= n_{i} \,\,\,\,\,\,\,\, f_{i}^{A} = G_{ik}^{f_{t}}n_{k}+(G_{kl}^{f_{n}}-G_{kl}^{f_{t}})n_{k}n_{l}n_{i} \end{aligned}$$

(A.11b)

$$\begin{aligned} d_{j}^{I}&= n_{j} \,\,\,\,\,\,\,\, d_{j}^{A} = G_{jk}^{d_{t}}n_{k}+(G_{kl}^{d_{n}}-G_{kl}^{d_{t}})n_{k}n_{l}n_{j} \end{aligned}$$

(A.11c)

Appendix B: Decomposition of stress tensor

Finally, partitions of stress tensor can be determined as:

$$\begin{aligned} \sigma _{ij} = C(\sigma _{ij}^{0}+\sigma _{ij}^{u}+\sigma _{ij}^{f}+ \sigma _{ij}^{d}+\sigma _{ij}^{ud} +\sigma _{ij}^{uf}+\sigma _{ij}^{df}+\sigma _{ij}^{udf}) \end{aligned}$$

(B.1)

where

$$\begin{aligned} C&= \frac{N_{c}\bar{f_{0}}\bar{d_{0}}}{4\pi V} \end{aligned}$$

(B.2a)

$$\begin{aligned} \sigma _{ij}^{0}&= \int _{{{\varvec{n}}}} n_{i}n_{j}\, d{{\varvec{n}}} \end{aligned}$$

(B.2b)

$$\begin{aligned} \sigma _{ij}^{u}&= \int _{{{\varvec{n}}}} E_{ij}^{A}n_{i}n_{j}\, d{{\varvec{n}}} \end{aligned}$$

(B.2c)

$$\begin{aligned} \sigma _{ij}^{f}&= \int _{{{\varvec{n}}}} f_{i}^{A}n_{j}\, d{{\varvec{n}}} \end{aligned}$$

(B.2d)

$$\begin{aligned} \sigma _{ij}^{d}&= \int _{{{\varvec{n}}}} n_{i}d_{j}^{A}\, d{{\varvec{n}}} \end{aligned}$$

(B.2e)

$$\begin{aligned} \sigma _{ij}^{ud}&= \int _{{{\varvec{n}}}} E_{ij}^{A}n_{i}d_{j}^{A}\, d{{\varvec{n}}} \end{aligned}$$

(B.2f)

$$\begin{aligned} \sigma _{ij}^{uf}&= \int _{{{\varvec{n}}}} E_{ij}^{A}f_{i}^{A}n_{j}\, d{{\varvec{n}}} \end{aligned}$$

(B.2g)

$$\begin{aligned} \sigma _{ij}^{df}&= \int _{{{\varvec{n}}}} f_{i}^{A}d_{j}^{A}\, d{{\varvec{n}}} \end{aligned}$$

(B.2h)

$$\begin{aligned} \sigma _{ij}^{udf}&= \int _{{{\varvec{n}}}} E_{ij}^{A}f_{i}^{A}d_{j}^{A}\, d{{\varvec{n}}} \end{aligned}$$

(B.2i)

Substitutes Eq. (A.11), Eq. (B.2) into Eq. (B.1) we link up the $\sigma _{ij}$ and $G_{ij}^{(*)}$ tensors (superscript * stands for u, $f_{n}$, $f_{t}$, $d_{n}$ and $d_{t}$).

Isotropic and anisotropic terms are individual now, and stress tensors is derived by these two terms (third-order terms are neglected):

$$\begin{aligned} \sigma _{ij}^{0}&= C\,\delta _{ij} \end{aligned}$$

(B.3a)

$$\begin{aligned} \sigma _{ij}^{u}&= C\,\frac{2}{5}G_{ij}^{u} \end{aligned}$$

(B.3b)

$$\begin{aligned} \sigma _{ij}^{f}&= C\,\left( \frac{2}{5}G_{ij}^{f_{n}}+\frac{3}{5}G_{ij}^{f_{t}}\right) \end{aligned}$$

(B.3c)

$$\begin{aligned} \sigma _{ij}^{d}&= C\,\left( \frac{2}{5}G_{ij}^{d_{n}}+\frac{3}{5}G_{ij}^{d_{t}}\right) \end{aligned}$$

(B.3d)

$$\begin{aligned} \sigma _{ij}^{ud}&= C\,\left( \frac{2}{5}G_{ik}^{u}G_{jk}^{d_{t}}+ \frac{4}{35}G_{ik}^{u}G_{jk}^{D}\right. \nonumber \\&\quad \left. +\frac{4}{35}G_{ik}^{D}G_{jk}^{u}+\frac{2}{35} G_{kl}^{u}G_{kl}^{D}\delta _{ij}\right) \end{aligned}$$

(B.3e)

$$\begin{aligned} G_{ij}^{D}&= G_{ij}^{d_{n}}-G_{ij}^{d_{t}} \end{aligned}$$

(B.3f)

$$\begin{aligned} \sigma _{ij}^{uf}&= C\,\left( \frac{2}{5}G_{ik}^{u}G_{jk}^{f_{t}} +\frac{4}{35}G_{ik}^{u}G_{jk}^{F}\right. \nonumber \\&\quad \left. +\,\frac{4}{35}G_{ik}^{F}G_{jk}^{u}+ \frac{2}{35}G_{kl}^{u}G_{kl}^{F}\delta _{ij}\right) \end{aligned}$$

(B.3g)

$$\begin{aligned} G_{ij}^{F}&= G_{ij}^{f_{n}}-G_{ij}^{f_{t}} \end{aligned}$$

(B.3h)

$$\begin{aligned} \sigma _{ij}^{df}&= C\,(G_{jk}^{d_{t}}G_{ik}^{f_{t}}+\frac{2}{5}G_{jk}^{D}G_{ik}^{f_{t}}+\frac{2}{5}G_{jk}^{d_{t}}G_{ik}^{F}\nonumber \\&\quad +\frac{4}{35}G_{ik}^{D}G_{jk}^{F}+\frac{4}{35}G_{ik}^{F}G_{jk}^{D}+\frac{2}{35}G_{kl}^{D}G_{kl}^{F}\delta _{ij}) \end{aligned}$$

(B.3i)