Factors Controlling the Difference in Brazilian and Direct Tensile Strengths of the Lac du Bonnet Granite

Abstract

Laboratory tests showed that the average direct tension strength (DTS₀) of the Lac du Bonnet granite was 6.9 MPa, and significantly less than its average Brazilian tension strength (BTS₀), i.e., 8.8 MPa. In this paper, a grain-based model (GBM) with a 2D universal discrete element code (UDEC) was used to explore factors that could control the difference in the direct tension strength (DTS) and Brazilian tension strength (BTS) of the Lac du Bonnet granite from the micromechanical point of view. Micro parameters of the grain contacts were investigated, i.e., the tensile strength (T), cohesion (c), friction angle (φ), and the ratio of the shear to normal stiffness (λ). Results show that both BTS and DTS highly depended on log(λ). With an increase in log(λ), the BTS first increased nonlinearly and decreased instead after achieving a peak while the DTS always increased nonlinearly. The BTS was sensitive to T and c, while the DTS was mainly sensitive to T. The ratio of DTS to BTS first decreased with the increase of log(λ) and then increased instead from a minimum. It was found that the GBM with UDEC can well reproduce the uniaxial compression, direct tension and Brazilian tests of the Lac du Bonnet granite simultaneously only on condition that λ = 2.51. This indicated that the shear stiffness of the grain boundary in the Lac du Bonnet granite may be several times larger than its normal stiffness. This may be the main factor controlling the difference in the BTS and DTS of the Lac du Bonnet granite.

Appendix

In Fig. 3a, the compressive force applied on the Brazilian sample F_by can be calculated as

$$ F_{\text{by}} = \frac{{w_{1} d}}{2}\sum\limits_{{i = t_{1} }}^{{i = t_{2} }} {\sigma_{\text{yy}}^{i} } $$

(2)

where, σⁱ_yy is the y-directional zone normal stress that corresponds to the ith monitoring point; w₁ the width of the loading platen in the Brazilian tension GBM; d is the thickness of the GBM which is perpendicular to the paper plane and uniformly taken as a unit thickness of 1.0 in this study.

Carneiro (1943) invented the Brazilian test for investigating the tensile strength of the concrete. The tensile stress at the center of the BT sample σ_bt can be calculated according to Eq. (3) (ASTM 2008; ISRM 1978)

$$ \sigma_{\text{bt}} = \frac{{2F_{\text{by}} }}{\pi Dd} $$

(3)

where D is the diameter of the Brazilian sample.

The uniaxial strain of the Brazilian sample can be calculated as

$$ \varepsilon_{\text{bt}} = \frac{{abs(u_{y}^{t} - u_{y}^{b} )}}{D} $$

(4)

where u^t_y and u^b_y are the y-directional displacements of the monitoring points t and b.

The BTS can be decided as

$$ {\text{BTS}} = \frac{{2P_{\text{by}} }}{\pi Dd} $$

(5)

where P_by is the peak of F_by.

From Fig. 3b, the tensile force applied on the top loading platen of the direct tension model can be acquired as

$$ F_{\text{dy}} = \frac{{w_{2} d}}{5}\sum\limits_{{i = t_{1} }}^{{i = t_{5} }} {\sigma_{yy}^{i} } $$

(6)

where w₂ is the width of the top loading platen in the direct tension GBM.

The tensile stress at the middle part of the direct tension model σ_dt can be calculated according to Eq. (7).

$$ \sigma_{\text{dt}} = \frac{{F_{\text{dy}} }}{{w_{3} d}} $$

(7)

where w₃ is the width of the middle part in the direct tension model.

If the failure occurred at the middle part of the direct tension model, the DTS can be calculated according to Eq. (8).

$$ {\text{DTS}} = \frac{{P_{\text{dy}} }}{{w_{3} d}} $$

(8)

where P_dy is the peak of F_dy.

The uniaxial strain of the direct tension sample can be calculated as

$$ \varepsilon_{\text{dt}} = \frac{{u_{y}^{t} - u_{y}^{b} }}{{h_{1} }} $$

(9)

where h₁ is the initial height of the rock sample between two loading platens.

From Fig. 3c, the compressive stress history of the uniaxial compression model can be obtained from the average of the compressive stresses monitored at the five points (t₁, …, t₅) of the top loading platen:

$$ \sigma_{n} = \frac{1}{5}\sum\limits_{{i = t_{1} }}^{{i = t_{5} }} {\sigma_{{_{yy} }}^{i} } $$

(10)

The UCS is determined as the peak value of the compressive stress history.

The axial strain can be calculated as

$$ \varepsilon_{a} = \frac{{abs(u_{y}^{t} - u_{y}^{b} )}}{{h_{2} }} $$

(11)

where h₂ is the initial height of the uniaxial compression model.

The lateral strain can be calculated as

$$ \varepsilon_{l} = \frac{{u_{x}^{\text{l}} - u_{x}^{\text{r}} }}{{w_{4} }} $$

(12)

where w₄ is the initial width of the uniaxial compression GBM; u^l_x and u^r_x are the displacement of the left and right monitoring points in the x-direction.