# Relation between the Friction Angle of Sand at Triaxial Compression and Triaxial Extension and Plane Strain Conditions

## Abstract

**:**

## 1. Introduction

## 2. Stress-Dilatancy Relationship for Sand

_{R}= I

_{D}(10 − ln p′) − 1 is the relative dilatancy index defined by Bolton [4], I

_{D}is the density index and p’ is in units of kilonewtons per square meter. Equations (6)–(8) are correct for 0 ≤ I

_{R}≤ 4. Therefore, $-1.2\le {\left(\delta {\epsilon}_{\upsilon}/\delta {\epsilon}_{a}\right)}_{c}={\left(\delta {\epsilon}_{\upsilon}/\delta {\epsilon}_{a}\right)}_{b}\le 0$. These ranges of strain increment ratios are taken later for calculation.

${J}_{2}=\frac{1}{2}{s}_{ij}{s}_{ij}$, | ${J}_{3}=\frac{1}{3}{s}_{ij}{s}_{jk}{s}_{ki}$, | ${s}_{ij}={\sigma}_{ij}^{\prime}-{p}^{\prime}{\delta}_{ij}$, |

${J}_{\epsilon 2}=\frac{1}{2}\delta {e}_{ij}^{p}\delta {e}_{ij}^{p}$, | ${J}_{\epsilon 3}=\frac{1}{3}\delta {e}_{ij}^{p}\delta {e}_{jk}^{p}\delta {e}_{ki}^{p}$, | $\delta {e}_{ij}^{p}=\delta {\epsilon}_{ij}^{p}-\frac{1}{3}\delta {\epsilon}_{\upsilon}{\delta}_{ij}$, |

_{ε}= π/6:

_{ε}= − π/6:

_{b}< 19°) and depends on the contact conditions on the confining platens, initial porosity, and the height-to-width ratio of the sample [18,21,22]. In later calculations, it is assumed θ

_{b}= 15°. The stress ratio at failure is calculated from Equation (13) with use of Equations (16)–(18).

## 3. Relation between the Friction Angle of Sand at TXC, BXC, and TXE

_{1}, I

_{2}, I

_{3}are invariants of the effective stress tensor. For assumed values of friction angle for TXC, the friction angles for BXC and TXE can be simply calculated and the relation between them specified.

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Relationship between the friction angle and strain increment ratio: (

**a**) Triaxial compressing testing, (

**b**) Biaxial testing, (

**c**) Triaxial extension testing.

**Figure 2.**Relationship between the excess of friction angle for biaxial and triaxial compression: (

**a**) ${\varphi}^{\xb0}={\varphi}_{c\upsilon}^{\prime}={30}^{\xb0}$, (

**b**) ${\varphi}^{\xb0}={\varphi}_{c\upsilon}^{\prime}={33}^{\xb0}$, (

**c**) ${\varphi}^{\xb0}={\varphi}_{c\upsilon}^{\prime}={36}^{\xb0}$.

**Figure 3.**Relationship between excesses of friction angles for triaxial extension and triaxial compression: (

**a**) ${\varphi}^{\xb0}={\varphi}_{c\upsilon}^{\prime}={30}^{\xb0}$, (

**b**) ${\varphi}^{\xb0}={\varphi}_{c\upsilon}^{\prime}={33}^{\xb0}$, (

**c**) ${\varphi}^{\xb0}={\varphi}_{c\upsilon}^{\prime}={36}^{\xb0}$.

**Figure 4.**Difference between the friction angle for biaxial testing or triaxial extension testing and the excess of friction angle for triaxial compression testing: (

**a**) $\left({\varphi}_{b}^{\prime}-{\varphi}_{c}^{\prime}\right)-\left({\varphi}_{c}^{\prime}-{\varphi}^{\xb0}\right)$, (

**b**) $\left({\varphi}_{e}^{\prime}-{\varphi}_{c}^{\prime}\right)-\left({\varphi}_{c}^{\prime}-{\varphi}^{\xb0}\right)$.

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**MDPI and ACS Style**

Szypcio, Z.
Relation between the Friction Angle of Sand at Triaxial Compression and Triaxial Extension and Plane Strain Conditions. *Geosciences* **2020**, *10*, 29.
https://doi.org/10.3390/geosciences10010029

**AMA Style**

Szypcio Z.
Relation between the Friction Angle of Sand at Triaxial Compression and Triaxial Extension and Plane Strain Conditions. *Geosciences*. 2020; 10(1):29.
https://doi.org/10.3390/geosciences10010029

**Chicago/Turabian Style**

Szypcio, Zenon.
2020. "Relation between the Friction Angle of Sand at Triaxial Compression and Triaxial Extension and Plane Strain Conditions" *Geosciences* 10, no. 1: 29.
https://doi.org/10.3390/geosciences10010029