Study of scale effect on intact rock strength using particle flow modeling

Abstract

Based on the extensive review of the UCS versus specimen size test data and the various empirical relations between the UCS and the specimen size, a new expression is proposed to describe the dependence of the UCS on specimen volume. The proposed new relation can fit the UCS versus specimen size test data of different rocks very well. Then, a numerical study of the scale effect on UCS is conducted using a numerical model in which the intact rock is represented by particles bonded to each other at contact points, with the contact bonds having both normal and shear strength components. The bond can break if the normal or shear contact stress exceeds the corresponding bond strength. To simulate the initial micro-fractures (flaws or cracks) in the rock, the smooth-joint contact model is used. The fractures are considered to be randomly orientated and located disks. The size and number of fractures are described by an exponential expression derived using fractal theory. The numerical model is calibrated using the test stress–strain curves of 80 mm×40 mm×40 mm prism Yamaguchi marble samples. Then, the calibrated model is used to predict the UCS of Yamaguchi marble samples at different sizes. The predicted UCS values are in good agreement with the experimental values. The numerical simulations show that to capture the scale effect on UCS of intact rock, initial fractures with sizes increasing faster with the specimen size must be considered in the modeling.

Introduction

The uniaxial, or unconfined, compressive strength (UCS) of intact rock is an important and widely used parameter in rock mechanics and rock engineering. In order to determine UCS, laboratory and/or in situ compression tests are usually conducted. It is long recognized that the UCS of intact rock depends on the size of specimen involved in the tests. The variation of UCS with the specimen size is called scale effect or size effect. The scale effect on UCS of intact rock has been verified by compressive strength tests on different types of rocks [1], [2], [3], [4], [5], [6], [7], [8]. For accurate determination of UCS, it is important to take the scale effect into consideration.

Extensive experimental research has been conducted on the scale effect on UCS of intact rock. Mogi [1] tested rectangular prisms of Yamaguchi marble with length to width ratio of 2:1 and found that the UCS decreased about 11% as the specimen length was increased from 0.04 to 0.12 m. Bieniawski [3] tested cubic specimens of coal ranging in size from 0.02 to 2.0 m and found that the UCS decreased by a factor of 7 with increasing size until the UCS approached 4.3 MPa. Pratt et al. [5] performed laboratory and in situ uniaxial compression tests on quartz diorite and granodiorite and found that the UCS decreased significantly, by a factor of 10 as the right triangular prism length was increased from 0.3 to 2.7 m. Liu [6] tested limestone and mica schist with cubic specimens and found that the UCS decreased as the specimen length was increased from 5 to 30 mm. Natau et al. [7] tested yellow limestone with length to diameter ratio of 2:1 and found that the UCS decreased with increasing size until the UCS approached 3.63 MPa. Jackson and Lau [8] tested 56 specimens of Lac du Bonnet gray granite with length to diameter ratio of 2:1 and found that the UCS decreased as the specimen diameter was increased from 63 to 294 mm. Based on the test results, researchers have proposed different empirical relations between the UCS and the specimen size (see Section 2 for detailed discussion).

Theoretical research has also been conducted to interpret the phenomenon of scale effect. The theoretical research can be divided into two general categories: the one using statistical theory (also often called weakest link theory) and the other based on fracture mechanics. The formal formulation of the scale effect problem as a statistical phenomenon was introduced by Weibull [9]. He assumed that the probability of failure of a solid body for a given applied stress is a function of its volume and that a solid body is made of small units and the chance of its survival under the applied stress is equal to the multiplication of chances of survival of these units. The latter assumption means that the survival of different parts of the body is considered as independent events, which, although correct for an ideal brittle material, is not valid for a quasi-brittle material like rock [10], [11], [12]. Many researchers have used the statistical theory to explain the scale effect on UCS of intact rock. Bieniawski [3] assumed that the scale effect on the UCS of coal is due to the various (micro) fractures (weaknesses) or flaws such as cracks in it. The UCS is a statistical value depending on the number and types of fractures present in the coal. In smaller specimens, the probability of finding flaws is smaller and thus the UCS is higher. Pretorius and Se [10] showed the correlation between weaknesses in rock and proposed a method to account for the weakness correlation in explaining the size effect. Bažant et al. [13], [14] presented modifications for the quasi-brittle material to match the basic Weibull ideas, and then studied the size effect based on results derived from liner elastic fracture mechanics, in which the strength of self-similar blocks is proportional to the square root of the block size [11], [12].

The theoretical investigation of scale effect based on fracture mechanics was first performed by Griffith [15]. He argued that due to the presence of flaws or cracks, the strength of larger specimens are smaller than that of smaller specimens. Adey and Pusch [16] studied the scale effect on rock strength based on linear elastic fracture mechanics, by quantifying the impact of the proximity of the sample boundary to the fractures in the sample and the conditions necessary for the inter fracture interaction. Exadaktylos and Stavropoulou [17] slightly modified the mathematical model of Bažant et al. [18] to relate the UCS to the specimen height and the fracture toughness of the rock necessary for the formation of new model-I cracks. Researchers such as Carpinteri and his colleagues [19], [20], [21] studied the scale effect on strength of brittle materials using the fractal theory to consider the fractal distribution of (micro) cracks. It is noted that many researchers [22], [23], [24], [25], [26], [27] have studied the fractal characteristics of cracks in rock. The research results show that the fractal features exist in both mechanical and geometrical crack properties such as the mode of crack branching, crack density, the shape of crack surfaces, and the distribution of cracks. Although some of the studies did not involve the scale effect, they paved the way for studying the scale effect, which is closely related to the fractal characteristics of (micro) cracks in rock.

In this paper, a systematic study of the scale effect on UCS of intact rock is presented. First, in Section 2, an extensive review of the UCS versus specimen size test data and the various relations between UCS and specimen size is performed. Based on the review, a new equation is proposed to describe the dependence of the UCS on the specimen volume. The proposed new relation can fit the UCS versus specimen size test data of different rocks very well. Then, in Section 3, a numerical study of the scale effect on UCS is conducted using the particle flow modeling. For the numerical modeling of scale effect, it is important to include the initial (micro) fractures (or defects) in the intact rock. To do that, the smooth-joint contact model in the three dimensional Particle Flow Code (PFC3D) is used to simulate the initial micro-fractures. The fractures are considered to be randomly oriented and located disks in the rock. The size and number of fractures are described by an exponential expression derived using the fractal theory and statistical method. After the numerical model is calibrated using the experimental stress–strain curves of rock samples of a specific size, it is used to predict the UCS of the same type of rocks at different sizes. The predicted UCS values are then compared with the test UCS values to evaluate the capacity of the numerical modeling for simulating the scale effect on UCS of intact rock. To check the importance of properly considering the initial micro-fractures in the numerical simulations, two other fracture size patterns are also considered in the simulations. Finally, the conclusions are presented in Section 4.