Study on the effect of micro-geometric heterogeneity on mechanical properties of brittle rock using a grain-based discrete element method coupling with the cohesive zone model

Abstract

The micro-geometric heterogeneity has a significant effect on the mechanical behavior and failure mode of brittle rocks. To study the influence of the size distribution and preferred orientation of mineral grains, a statistical grain-based discrete element method coupling with the cohesive zone model is proposed on the basis of the universal distinct element code (UDEC) in this study. First, a mixed-mode cohesive model was developed and implemented in UDEC to simulate the nonlinear behavior of grain interfaces. Then, a micro-parameter calibration procedure based on the Plackett-Burman design, central composite design, and particle swarm optimization was established to reproduce the macro-properties of the Lac du Bonnet granite. Finally, the strength characteristics, deformation response, and failure mode of rocks were numerically examined using the proposed model by varying the size distribution and preferred orientation of mineral grains. The relationship between the micro-geometric heterogeneity and the uniaxial compressive strength, the crack initiation stress, the crack damage stress, the elastic modulus, the Poisson's ratio, and the failure mode of rocks are obtained respectively.

Introduction

Rock is an aggregate of minerals with a certain structure. Due to the diversity of its microstructure, rock is a typical heterogeneous material. This heterogeneity may lead to significantly different deformation and strength characteristics of the same type of rock with similar mineral content.1, 2, 3 In the past few decades, many researchers have studied the effect of material heterogeneity on rock strength and deformation.4, 5, 6 Their research shows that this effect is mainly caused by the local stress concentration at the crystal scale, which leads to the occurrence of rock failure.⁷ This means that the inherent mechanism of rock failure needs to be studied at the grain scale. Generally, the sources of rock heterogeneity can be divided into two categories: (a) Micro-geometric heterogeneity⁸: due to the variability of micro-geometric factors, such as the grain size distribution, and the shape and preferred orientation of the grain; (b) Material properties heterogeneity⁹: due to the difference of mechanical properties among mineral grains and their contacts. In this study, we focused on the effects of micro-geometric structures on rock strength and deformation.

It has previously been observed that many mechanical parameters that are related to the rock strength, including the uniaxial compressive strength, Brazilian splitting strength, and Young's modulus, are supposed to decrease with an increasing grain size, especially when the grain size is smaller than 1 mm.10, 11, 12, 13 Rock strength is usually regarded as a function of the mineral grain size, whose relationship varies with the type of rock. For instance, for granite, the rock strength decreases linearly¹⁴ or logarithmically¹² with the increase of the average mineral grain size, while for marble, quartzite, and limestone, the rock strength usually decreases with the increase of the square root of the average grain size.^15,16 It is worth noting that there is no consensus on the influence of the grain size distribution on rock mechanical properties. Nicksiar and Martin¹⁷ point out that the ratio of crack initiation stress and uniaxial compressive strength (normalized CI stress) shows only minor variation as the heterogeneity decreases. But Liu et al.⁹ argued that the grain size distribution had a significant effect on normalized CI stress. Specifically, normalized CI stress decreases with the increase of heterogeneity. Therefore, the influence of the grain size distribution on rock mechanical behavior still needs further study. To the best of our knowledge, the mechanical behavior of rock is not only affected by the grain size distribution, but also related to the preferred orientation of grains.¹⁸ However, it is difficult to precisely control these two factors in the experiment, which may lead to inaccurate results.

In recent years, with the rapid development of numerous advanced computational methods, numerical simulations have become an alternative tool for studying rock mechanics problems. As a promising research method, numerical simulation could overcome those difficulties encountered in laboratory experiments.^19,20 In order to realistically simulate the failure process of rocks, more and more researchers^8,21, 22, 23, 24 are paying attention to the grain-based modeling methods (GBM). GBM makes it possible to perform quantitative analysis on crystals using information such as the mineral composition and size distribution inside the rock. And this modeling method has been successfully applied to reproduce the mechanical characteristics of various types of rocks.25, 26, 27 Currently, GBM has been realized in PFC and UDEC for studying rock mechanics and failure behavior. In PFC-GBMs, each mineral grain consists of several parallel-bonded circular/spherical particles, and employ smooth-joints contact to simulate the boundary properties.²⁸ One of the disadvantages of PFC-GBMs is that the particle is circular/spherical in shape, which results in a high inherent porosity and makes the method difficult to simulate low-porosity rocks. In UDEC-GBMs, polygonal or triangular blocks are used to represent mineral grains. Due to a highly interlocked blocky structure, UDEC-GBMs will not suffer from the porosity problem. Therefore, this paper uses UDEC-GBMs to study the mechanical behavior and failure process of brittle rocks. However, conventional UDEC-GBMs still have the following problems, which limit the wide use of grain-based models: (a) In the conventional UDEC-GBMs, the Coulomb-slip model is adopted as the contact model between blocks.²⁹ Although the Coulomb-slip model is widely used to study the deformation behavior and strength characteristics of rock materials, this linear contact model almost ignores the fracture process zone (FPZ) at the crack tip.³⁰ Nevertheless, it is unrealistic to ignore the size of FPZ for many geotechnical materials, such as soil, rock and concrete.³¹ Many studies have revealed that a mixed-mode cohesive zone model should be adopted to capture the non-linear characteristics of FPZ and to reproduce the possible fracturing behavior of brittle material.^32,33 (b) Traditional micro parameter calibration is through trial and error, which is usually time-consuming and cannot fully reflect the internal relationship between macro and micro parameters.

In this study, we adopt a novel grain-based discrete element model to quantitatively study the influence of the grain size distribution and the preferred orientation of grains on the macroscopic mechanical behavior of rocks. To simulate the possible fracturing behavior of brittle rock, the mixed-mode cohesive zone model was implemented in the UDEC code. Moreover, a combination calibration method, termed as PB–RSM–PSO, was proposed for the calibration of micro parameters. Based on the proposed model, the relationship between micro parameters and the macro response of rocks was studied. The rest of this paper is organized as follows: The theory of a mixed-mode cohesive zone model is introduced in Section 2; In section 3, the calibration approach for micro parameters involved in the proposed UDEC-GBM is discussed; In section 4, by performing a series of numerical tests, the influence of the grain size distribution and the preferred orientation of grains on the rock strength and deformation is studied; And finally a conclusion of the work is provided in Section 5.