# Study of Frictional Effects of Granite Subjected to Quasi-Static Contact Loading

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## Abstract

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## 1. Introduction

## 2. Elastoplastic Damage Model

## 3. Finite Element Model

## 4. Frictional Study Results and Discussions

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$d$ | Cohesion, Equation (1) |

$F$ | Yield function, Equation (1) |

$G$ | Flow potential, Equation (3) |

$m$ | Weibull modulus, Equations (6)–(8) |

$n,N$ | Test number and total number of tests, Equation (10) |

$p$ | Pressure, Equations (1)–(3) |

${P}_{F}$ | Failure probability, Equation (5) |

$q$ | Von Mises equivalent stress, Equations (1) and (3) |

${S}_{0}^{m}/{\lambda}_{0}$ | Weibull scale parameter, Equation (6) |

${Z}_{eff}$ | Effective volume, Equations (5) and (7) |

$\beta $ | Friction angle, Equation (1) |

$\Gamma $ | Euler function of the second kind, Equation (8) |

${\dot{\epsilon}}_{a}^{i},\text{}{\dot{\epsilon}}_{v}^{i}$ | Inelastic axial strain rate and inelastic volumetric strain rate, Equation (4) |

${\lambda}_{t}$ | Initiation density, Equations (5) and (6) |

$\mu $ | Frictional coefficient |

$\mathbf{\sigma}$ | Cauchy stress tensor, Equation (2) |

${\sigma}_{F}$ | Maximum tensile stress, Equations (5)–(7) |

${\sigma}_{i}$ | Principal stress, ($i=1,2,3$) |

${\sigma}_{w}$ | Average failure stress, Equation (8) |

${\sigma}_{1}$ | Maximum principal stress, Equation (7) |

$\Psi $ | Dilation angle, Equations (3) and (4) |

${\omega}_{i}$ | Damage parameter, ($i=1,2,3$) |

$\langle \xb7\rangle $ | Macaulay brackets |

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**Figure 1.**Dilation angle ($\Psi $) as a function of pressure ($p$) based on quasi-oedometric (QO) compression test results.

**Figure 2.**Schematic illustration of an indentation test where $P$ is the contact force, $h$ the penetration depth, $a$ the contact radius and $R$ the radius of the indenter.

**Figure 3.**Weibull plot for frictionless contact [25].

**Figure 4.**Force-penetration (P-h) response from numerical simulations for frictionless contact (using Table 1 parameters) as well as experimentally obtained responses.

**Figure 5.**(

**a**) Damage variable ${D}_{1}$ corresponding to a penetration depth of 0.5 mm Figure 1. (

**b**) Only the damaged elements are shown.

**Figure 6.**Finite element mesh with (

**a**) the rock specimen geometry and the rigid indenter geometry. (

**b**) 3D view, and (

**c**) top view.

**Figure 8.**(

**a**) Indented surface of two rock samples loaded up to large load-drops. The fracture patterns; (

**b**) for frictionless contact [25] and (

**c**) for frictional contact ($\mu =0.4$).

**Figure 9.**(

**a**) Equivalent plastic strain field from numerical simulations for frictionless contact using DP model, and (

**b**) for frictional contact ($\mu =0.4$).

**Figure 10.**The fracture patterns (

**a**) for friction coefficient μ = 0.1 and (

**b**) for friction coefficient μ = 0.5.

**Table 1.**Material parameters based on [25] in which the frictional effects were neglected.

Material Parameter | |
---|---|

$E\text{}\left(\mathrm{GPa}\right)$ | $52$ |

$\nu $ | $0.25$ |

$\rho \text{}\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | $2630$ |

$\beta \text{}{(}^{\xb0})$ | $51.7$ |

$d\text{}\left(\mathrm{MPa}\right)$ | 153.3 |

$\Psi \text{}{(}^{\xb0})$ $m$ ${\sigma}_{w}\text{}\left(\mathrm{MPa}\right)$ ${Z}_{eff}\left({\mathrm{mm}}^{3}\right)$ | Figure 1 24 120 1 |

**Table 2.**Weibull parameters used in DFH model. The friction coefficient was assumed equal to 0.4. The elastoplastic material parameters are the same as in Table 1.

Material Parameter | |
---|---|

$m$ | 12 |

${\sigma}_{w}\text{}\left(\mathrm{MPa}\right)$ | 75 |

${Z}_{eff}\left({\mathrm{mm}}^{3}\right)$ | 2 |

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**MDPI and ACS Style**

Shariati, H.; Saadati, M.; Weddfelt, K.; Larsson, P.-L.; Hild, F. Study of Frictional Effects of Granite Subjected to Quasi-Static Contact Loading. *Lubricants* **2020**, *8*, 106.
https://doi.org/10.3390/lubricants8120106

**AMA Style**

Shariati H, Saadati M, Weddfelt K, Larsson P-L, Hild F. Study of Frictional Effects of Granite Subjected to Quasi-Static Contact Loading. *Lubricants*. 2020; 8(12):106.
https://doi.org/10.3390/lubricants8120106

**Chicago/Turabian Style**

Shariati, Hossein, Mahdi Saadati, Kenneth Weddfelt, Per-Lennart Larsson, and Francois Hild. 2020. "Study of Frictional Effects of Granite Subjected to Quasi-Static Contact Loading" *Lubricants* 8, no. 12: 106.
https://doi.org/10.3390/lubricants8120106