Experimental and numerical study of fracturing in degraded and graded crystalline rocks at the laboratory scale

Highlights

• Failure of graded and degraded rocks affected by altering factors is simulated by XFEM.

• Transgranular/intergranular enrichment functions and interaction integral are applied.

• The concept of energy release rate was used to predict the crack trajectory.

• Explicit mineral modeling reproduces a mixed mode regardless of crack situation.

Abstract

The rock masses are mineral based geomaterials which have been formed by different mineralogical compounds in terms of dimensions and mechanical properties. Geomaterials are often classified as inhomogeneous anisotropic materials due to the orientation of minerals, frequently asymmetric distribution of mechanical properties, porosity, layering, and discontinuities. Differences in deformation properties of mineral crystals under applied stresses is a potential source for crack initiation/propagation in geomaterials. Meanwhile, weathering, alteration, and thermo-chemical agents can change the distribution of mechanical properties by causing unequal effects on different minerals. Redistribution of properties changes the intensity of heterogeneity and anisotropy and ultimately changes the rock fracture mechanism. Therefore, in this study, with the use of transgranular and intergranular enrichment functions and an interaction integral for inhomogeneous materials the failure mechanism of graded and degraded rocks affected by altering factors is simulated. The concept of energy release rate was used to predict the crack initiation angle and the crack trajectory within or between mineral grains. To verify and validate this approach, the results are compared with experimental test results and those reported in the literature. The results show that the crack trajectory significantly depends on the properties of minerals and their boundaries. The smaller the ratio of intergranular to transgranular critical fracture energy is, the more likely the crack to propagate between the grains. Simulation of crack propagation in altered specimens also showed that the crack tends to preferably propagate in weaker minerals. It is also found that using homogenization techniques instead of explicit modeling of minerals may produce inaccurate results. That is because it predicts pure fracture modes at angles in which the mixed tensile-shear modes should be detected.

Introduction

Determining the physicomechanical properties of rocks is an essential part of many structural design and construction projects. While the mechanical properties of intact rock tend to have huge impacts on structural stability, discontinuities play an even greater role in this regard. However, the presence of weakness planes, cracks, joints and fractures in rocks is inevitable. Furthermore, rock masses are prefabricated mineral-based materials which consist of different mineralogical compounds in terms of dimensions and mechanical properties. Consequently, weak discontinuities between minerals can potentially be where the failure starts or can facilitate its expansion. Differences between grains in terms of thermal expansion coefficient or swelling when exposed to environmental factors such as weathering, as well as dynamic loads are the main causes of fracturing. Thus, when a rock is subjected to mechanical loading and other environmental factors, fracturing occurs by nucleation of new cracks or by extension of the tips of preexisting discontinuities. Therefore, the understanding of mineral-based geomaterial failure mechanics after alteration plays an important role in addressing a number of important engineering issues such as rock slope stability, acid fracturing, and underground excavations, nuclear waste disposal, CO₂ storage and enhanced geothermal energy.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 It should be considered that irregular layering, complex in-situ stress state, existence of joint sets and discontinuities, make rock masses intrinsically anisotropic.¹² In addition, given the asymmetric distribution of pores and grain sizes alongside the presence of different mineralogical compounds in different locations, geomaterials are often classified as heterogeneous materials. During last decade extensive researches have been carried out to study the fracture mechanics of the heterogeneous or anisotropic rock materials.^8,9,13, 14, 15, 16, 17 However, rock fracture mechanism subjected to complex physico-chemical condition such as weathering or chemical agents needs further study.

Research projects have shown that environmental phenomena and engineering activities such as weathering,¹⁸ chemical and biological effects,^7,19 and heat treatments^20,21 can alter the distribution of mechanical properties by changing the mineral characteristics and imposing new discontinuities or extension of preexisted discontinuities. This redistribution of properties may change the intensity of heterogeneity and anisotropy, where ultimately influences failure mechanism of rock materials. All of these factors can significantly alter the response of a rock structure to applied stresses by affecting its constituent minerals unequally, creating micropores and microcracks in its matrix, and changing its mechanical properties.

Environmental factors that influence rock behavior include humidity, weathering, alteration, chemical reactions in the presence of corrosive fluids and heat and also water content, which can indeed have significant impacts on its strength. By studying the effects of moisture on double torsion specimens, Nara et al.²² stated that fracture toughness decrease as contents of the moisture increase. Groundwater always has adverse impact on the rock engineering design project due to the presence of chemical solutions that potentially can change rock structure and its composition that ultimately cause alteration. Rock Alternation leads to variations in the rock mechanical properties including its compressional strength and fracture toughness. Rigopoulos et al.²³ achieved a similar conclusion on dolerite rock samples and stated that increasing the alteration of minerals, reduces the uniaxial compressive strength. Karfakis and Akram²⁴ mentioned that fracture energy, toughness and crack growth trajectory significantly changes when rocks are affected by chemical solutions. Since not all minerals are affected similarly by these environmental factors, cracks usually tend to propagate on the path with the least resistance. Therefore, it is important to consider these environmental factors while analyzing the failure mechanism of rock masses. Several theoretical models25, 26, 27, 28, 29 and experimental techniques30, 31, 32, 33, 34 were developed to investigate the mixed mode crack growth (the combination of opening and shearing modes) in rocks. However, existing theoretical models are limited to simple geometries, load conditions, and material behaviors.

Nowadays, numerical methods are mostly used in engineering and science projects. Zhang et al.³⁵ modeled the irregular shape of minerals in crystalline rocks by discrete element method using grain-based model. The proposed method is then applied to parameter studies on the mineral boundary strength influencing the fracturing of the crystalline rock by Wang et al.³⁶ Li et al.³⁷ used a grain-scale continuum–discontinuum approach to predict transgranular fracture and its influence on rock strength. Wang et al.³⁸ applied a phase-field method in modeling damage and cracking in rock-like materials by considering material heterogeneities. In this method, the effective elastic properties of rocks are determined as functions of mineral compositions by using a linear homogenization method. Li et al.³⁹ developed a local multiscale high-resolution modeling strategy, where based on the statistical meso-damage mechanical method, the meso-fracturing process simulation of rock is achieved. Mahabadi et al.⁴⁰ used the combined finite element and discrete element methods to investigate the influence of microscale heterogeneity and microcracks on the failure behavior of a crystalline rock. Failure mechanism of a poly-mineral granite rock under dynamic loading were modeled by adopting polygonal finite element method by Saksala and Jabareen.⁴¹ However, the shortcomings of existing models in recent years have led to a strong tendency to develop practical simulation tools to consider the actual behavior of mineral based geomaterials. In computational geomechanics, there are generally three main approaches for simulation: continuous-based approaches, discontinuous-based approaches, and hybrid continuous-discontinuous methods.42, 43, 44, 45, 46 While distinct minerals can be incorporated into continuum models (e.g. Finite Element Method (FEM)) with the help of special joint elements. It is practically impossible to model materials with wide mineral diversities and different mechanical properties on a large scale through this approach. For such materials, it is common to use homogenization techniques⁴⁷ to create a continuous medium with mechanical properties equivalent to those of the discontinuous materials. In fact, homogenization techniques consist of finding micro-scale parameters, which reproduce macroscale variables. This approach facilitates numerical calculations by allowing the effect of minerals to be implicitly incorporated into the chosen stress-strain constitutive equation for the material. However, it may be very difficult to choose the appropriate equivalent-continuum constitutive law for the mineral based materials.

In the discontinuous approaches (e.g. Discrete Element Method (DEM)) all components (minerals) of the geomaterial will be explicitly defined in a numerical model. Furthermore, these approaches allow the nature of minerals to be modeled more realistically with the help of Voronoi elements and probabilistic distributions. Although a realistic representation of the discontinuous nature of mineral based materials is provided, this approach has limited capacity to simulate small-scale fracture mechanisms of rocks from a geomechanical standpoint. It is also possible to use a hybrid continuous-discontinuous method (e.g. FEM/DEM) by combining the ability of continuous method to describe the intact rock behavior with the ability of discontinuous method to model the interactions of discrete parts. Despite its many advantages in the initiation, propagation and branching of cracks, this approach has a major drawback; since no adaptive remeshing operation takes place as the simulation progresses, the crack path remains bound to the initial meshing topology. Therefore, it is necessary to use a sufficiently fine unstructured mesh to minimize this tendency. Microscale modeling of minerals requires fine meshing, which leads to substantially increased degrees of freedom (DOF) and significantly more calculations. Furthermore, the mechanical response of the material remains dependent on the size of the used elements.

In addition to the aforementioned limitations, most of the previous studies in the field of mineral-based rock fracturing have been carried out using the DEM approach,^36,48, 49, 50, 51, 52, 53 which does not allow the intergranular crack modeling literally. They considered either mineral components as homogeneous and isotropic media, individually, and their combination produced heterogeneity or anisotropy on a larger scale. This is despite the fact that each of the mineral components can have heterogeneity and anisotropy properties on a micro-scale. Finally, the role of weakening of some minerals due to environmental or chemical factors on the rock fracture mechanism has not been investigated.

In order to: a) solve the problems caused by homogenization techniques, reduce related calculations and prevent strong dependence on the mesh size, b) properly implement the heterogeneity and anisotropy of minerals, and c) introduce the effects of alteration and chemical processes to numerical models, one of the most powerful numerical techniques is the Extended Finite Element Method (XFEM).54, 55, 56 The use of XFEM to model crack propagation in polycrystalline materials started with the research of Sukumar et al.57, 58, 59 They introduced enrichment functions for bimaterial interface cracks in partition of unity framework.⁵⁷ Simone et al.⁶⁰ described a method where a Heaviside function was used to describe grain boundaries. By introducing weak discontinuity and numerically determined enrichment functions in XFEM, Menk and Bordas^61,62 introduced a method without the need of adaptive meshing for polycrystals. Wu et al.⁶³ used XFEM to determine the mixed mode I/II fracture characteristics of heat-treated granite specimens, in which the homogenization technique of mineral-based samples is applied. By the use of XFEM, Song et al.⁶⁴ modeled a semi-circular bending test using hot mix asphalt based materials which were prepared with difference aggregate shapes. They showed that the sample with the round particles needs the largest failure load, followed by samples with convex, ellipse and concave particles. In XFEM, the singularity of the stress field can be reproduced by the use of appropriate asymptotic displacement functions. Therefore, it is possible to model variation of material properties, i.e. phase changes, honoring appropriate enrichment functions. This enables XFEM to simulate any type of discontinuity such as a crack, pore, or phase change. So, this study aims to improve and demonstrate these capabilities of XFEM and is organized as follows. The governing equations of XFEM and computational algorithm are explained in Sec. (2) and (3). Then, the stress intensity factors (SIFs) and crack propagation paths estimated by the proposed method and their accuracy is examined. In Sec. (5), the role of alteration on the failure mechanism of a quartz diorite specimen with different degrees of alteration is investigated and the results are compared with the results of DEM approach. Finally, the discussion and conclusion are presented in Sec. (6).