J. Cent. South Univ. Technol. (2008) 15(s1): 496-499

DOI: 10.1007/s11771-008-408-x

Experimental study on time-dependent stress and strain of in-plane shear (ModeⅡ) fracture process of rock

WANG Zhi(王 志), RAO Qiu-hua(饶秋华), XIE Hai-feng(谢海峰)

(School of Civil and Architectural Engineering, Central South University, Changsha 410075, China)

Abstract: Shear-box test with strain measurement was used to study time-dependent stress and strain of in-plane shear (Mode Ⅱ) fracture process of rock and to reveal the mechanism of Mode Ⅱ fracture. Numerical results show that the maximum shear stress τ_max at the crack tip is much larger than the maximum tensile stress σ₁ and the ratio of t_max/s₁ is about 5, which favors Mode Ⅱ fracture of rock. Test results indicate that the strain—time curve comprises three stages: the linear deformation stage, the micro-cracking stage and the macroscopic crack propagation. The strain in the direction of the original notch plane is negative, due to restraining effect of compressive loading applied to the original notch plane. Both s₁ and t_max are increased as the load increases, but the slope of t_max is larger than that of s₁ and the value of t_max is always larger than that of s₁. Therefore, t_max reaches its limited value at peak load before s₁ and results in Mode Ⅱ fracture of rock. Shear-box (i.e. compression-shear) test becomes a potential standard method for achieving the true Mode Ⅱ fracture and determining Mode Ⅱ fracture toughness of rock.

Key words: time-dependent stress and strain; Mode II fracture; strain measurement; fracture mechanism; finite element method; rock

1 Introduction

In-plane shear (ModeⅡ) fracture of rock has attracted more and more interest of researchers in mining, geological and rock engineering. A lot of test methods have been proposed to study Mode Ⅱ fracture of rock, such as four-point bending test^[1-2], punch-through shear test^[3-4], and compact double shear test^[5-6], in which all specimens are subjected to pure in-plane shear (Mode  Ⅱ) loading without any tensile or compressive loading. In this case, the new initiated crack is controlled by maximum tensile stress and propagated at an angle (about 70?-80?) deviating from its original crack plane. That is of tensile (ModeⅠ) fracture rather than true shear (Mode Ⅱ) fracture. For brittle rock, since its tensile strength is much smaller than its shear strength, the maximum tensile stress at crack tip easily reaches its critical value before the shear stress and thus results in tensile (ModeⅠ) fracture. In order to realize the true Mode Ⅱ fracture of rock, a compressive loading must be applied to the original crack plane in order to restrain the tensile stress at crack tip^[7-8].

In this study, shear-box (i.e. compression-shear) test was adopted to obtain Mode Ⅱ fracture of rock. Finite element method (FEM) and strain measurement were used to determinate stress and strain at crack tip in order

to reveal mechanism of Mode Ⅱ fracture of rock.

2 Numerical calculation

Finite element software MSC.Marc2005 was adopted to calculate stress field of specimen in shear-box test. The specimen was 50 mm×50 mm×50 mm in size and its single notch length a was 30 mm, as shown in Fig.1. The applied load p was 1 Pa and the notch inclination angle α was 70?. Eight-node, iso-parametric, hexahedral elements were adopted in the solid model. Since the distribution of the stress applied to the specimen surface was unknown, the specimen and loading set-up were modeled as a whole (Fig.2). Table 1 lists the mechanical properties of materials, sandstone for specimen and steel for loading grip.

Table 1 Mechanical properties of materials

Fig.3 illustrates typical contours of the maximum tensile stress s₁ and shear stress tress t_max for the specimen under compression-shear loading. Table 2 lists the values of s₁ and t_max as well as the ratio of t_max /s₁ at notch tip.

Fig.1 Compression-shear specimen

Fig.2 Finite element meshing as a whole

Fig.3 Stress contours for specimen: (a) Maximum tensile stress s₁; (b) Maximum shear stress t_max

It is seen that the maximum tensile stress s₁ appears on the edge of the original notch rather than at the notch tip, due to bending effect of compressive loading on the original notch. The maximum shear stress t_max exists at the notch tip. Since the compressive stress applied to the original crack plane can effectively depress the tensile stress at notch tip, t_max is much larger than s₁ and the ratio t_max/s₁ is about 5. For brittle rock, t_max can easily reach its limited value before s₁ and results in the shear (Mode Ⅱ) fracture.

Table 2 Numerical results of crack tip of specimen

3 Experiment

Fig.4(a) shows the loading set-up of shear-box tests. Specimen was tested by an Instron1346 testing machine. The loading rate was 0.05 m/s under displacement control and the notch inclination angle was 70?. In order to analyze the time-dependent stress and strain during fracture process, a rectangular strain gauge rosette, with θ=0?, 45?, 90?, was glued near the crack tip to measure strain, as shown in Fig.4(b). Fig.5 illustrates the location of the strain gauge rosettes mounted on the specimen. For the sake of clarity, a global X-Y coordinate system is adopted. Table 3 lists the angles of the three gauges in the X-Y coordinate system, j. The strain data were recorded by a multi-channel signal conditioner/amplifier system.

Fig.4 Photos of shear-box test: (a) Loading set-up; (b) Speci- men with strain rosettes

Fig.5 Arrangement of strain gauge rosettes in specimen

Table 3 Angles of strain gauge rosette in the X-Y coordinate

4 Results and analysis

4.1 Strain—time curve

Fig.6 shows the load—time curve of the strain gauge rosettes. It is seen that the load is almost linearly increased with time before its maximum value and rapidly decreased after the peak point.

Fig.6 Load—time curves of strain gauge rosette

The strain—time curve of the strain gauge rosettes is shown in Fig.7. It can be seen that the strain glued in the direction of the original notch plane, ε_90?, is negative, due to the restraining effect of compressive loading applied to the original notch plane. Because of the tiny bending effect, the strains in the directions of 45? and 135?, ε_45? and ε_135?, are tensile and compressive, respectively, which is in good agreement with the above numerical result (Fig.3(a)).

It is also found that the strain—time curves comprise three stages, 0-80 s, 80-115 s and 115-120 s. In the first stage, three strains in the three directions, positive ε_45?, negative ε_90? and negative ε_135?, are linearly increased in absolute value as the time (i.e. load) increases and the slope of the strain—time curves keep constant, meaning that the rock linearly deforms. In the second stage, the absolute value of three strains still increases with the load, and the absolute values of slopes

Fig.7 Strain—time curves in strain gauge rosette

of the strain—time curves are also increased, representing a nonlinear deformation. This is because new micro-cracks may be formed in this stage. The third stage begins when the load reaches the peak value. The absolute values of strains, ε_90? and ε_135?, increase with the load and the slopes of their strain—time curves become positive. The reason for this might be that a large number of micro-cracks are initiated, propagated and coalesced, resulting in the relaxation of the compressive strains. In essence, the change in the sign of the slopes of the strain—time curves from negative to positive indicates the beginning of the micro-crack propagation. As the micro-cracks propagate, the slopes of the strain—time curves gradually increase and abruptly reach very high levels at the peak load. At this moment a macroscopic crack is formed, resulting in the failure of the specimen.

4.2 Stress—time curve

Maximum and minimum principal stress, s₁ and s₂, and the maximum shear stress, t_max, can be calculated from the readings of the strain gauge rosettes as follows:

 (1)

This equation can be applied three times for the three values of θ of the rosette gauges. With three known values of ε_θ, i.e. ε_0?, ε_45?, ε_90?, for the three values of θ, θ=0, 45?, 90?, three strain components, ε_x, ε_y, γ_xy can be obtained.

ε_x=ε_0?, ε_y=ε_90?, γ_xy=2[ε_45?-(ε_0?+ε_90?)/2]              (2)

The stress components are expressed as

                      (3)

where

Therefore, s₁ and s₂, and t_max can be determined by the following equations:

       (4)

i.e.

    (5)

Fig.8 shows curves of s₁, s₂ and t_max vs time at the measured strain position. It is observed that s₁ and t_max are positive while s₂ is negative. Both s₁ and t_max are increased as the time (i.e. load) increases, but the slope of t_max is larger than that of s₁ and the value of t_max is always larger than that of s₁ before the load reaches its peak value. Therefore t_max reaches its limited value at peak load before s₁ and results in shear (Mode Ⅱ) fracture. That is reason why the shear fracture of rock can be successfully obtained in shear-box (i.e. compression-shear) test, rather than in four-point bending test, punch-through shear test and compact double shear test, where only pure shear loadings are

Fig.8 Stress—time curves in strain gauge rosette

applied. Therefore, the shear-box test becomes a potential standard method to achieve the true Mode Ⅱfracture and determine Mode II fracture toughness of rock.

5 Conclusions

1) Maximum tensile stress s₁ appears on the edge of the original notch rather than at the notch tip, due to tiny bending effect of compressive loading on the original notch. Since the compressive loading can effectively depress the tensile stress around the notch tip, t_max is much larger than s₁ and the ratio t_max/s₁ is about 5, which favors shear (Mode Ⅱ) fracture of rock.

2) The strain—time curve comprises three stages: the linear deformation stage, the micro-cracking stage and the macroscopic crack propagation. The strain in the direction of the original notch plane, ε_90?, is negative, due to restraining effect of compressive loading applied to the original notch plane.

3) Both s₁ and t_max are increased as the load increases, but the slope of t_max is larger than that of s₁ and the value of t_max is always larger than that of s₁. Therefore, t_max reaches its limited value at peak load before s₁ and results in shear (Mode Ⅱ) fracture of rock.

4) Shear-box (compression-shear) test becomes a potential standard method for achieving the true shear (Mode Ⅱ) fracture and determining Mode Ⅱ fracture toughness of rock .

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(Edited by CHEN Wei-ping)

Foundation item: Project(50374073) supported by the National Natural Science Foundation of China; Project(1343-77239) supported by the Graduate Education Innovation Project of Central South University, China

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: RAO Qiu-hua, Professor; Tel: +86-731-8836001; E-mail: raoqh@mail.csu.edu.cn