Numerical modelling of the crack-pore interaction and damage evolution behaviour of rocklike materials with pre-existing cracks and pores

Abstract

The combined effect of pre-existing cracks and pores on the damage evolution behaviour and mechanical properties of rocklike materials under uniaxial compression was numerically studied. Simulations of cracks and pores alone showed that increasing crack length and pore diameter decrease uniaxial compressive strength (UCS) and elastic modulus. Subsequent simulations considered two types of combinations of pre-existing cracks and pores – two cracks either side of a centric pore, and two pores either side of a centric crack – and the distance between cracks and pores was changed. In the case of two cracks at either side of the pore, UCS increased only slightly when the distance between the cracks and pore was increased. This was attributed to the more profound effect of the presence of the pore on UCS, and was confirmed by the progressive crack development characteristics and the major principal stress distribution patterns, which showed that the cracks initiated from the tips of the two pre-existing cracks made little or no contribution to the ultimate macroscopic failure. In contrast, models with two pores at either side of a centric crack showed a marked dependency of UCS on the distance between the pores and the crack. Cracks propagating from pre-existing pores made a greater contribution to the ultimate macroscopic failure when the pores were close to the centric crack and the effect gradually diminished with increasing space between pre-existing pores and the centric crack. Major principal stress distributions showed an asymmetric mobilisation of compressive stresses at the right and left sides of the two pores, favouring macroscopic shear failure when they were close to the centric crack which had led to a lower UCS. Overall, this study presents some critical insights into crack-pore interaction behaviour and the resulting mechanical response of rocklike materials to assist with the design of rock structures.

Keywords

Cracks and pores microcracking damage evolution stress distribution

Introduction

Rocks contain various types of discontinuities, ranging from grain-scale microcracks to large-scale faults. These discontinuities can be ‘crack-like’ (i.e. cracks) and/or ‘pore-like’ (i.e. pores); the former refers to discontinuities with tips at which stress can be concentrated and the latter refers to open pores of various shapes (Huang et al., 2017). In general, since all types of discontinuities are weaker elements than the intact rock bridges they intersect, their presence generally weakens the mechanical integrity of the rock mass (Hoek, 1983; Terzaghi, 1965). Disturbing the natural stress state of a rock mass results in a stress redistribution, which often leads to the creation of stress concentration localities around discontinuities (Fan et al., 2018). Microcracking and damage initiate from these high-stress nuclei leading to brittle failure (Deng and Nemat-Nasser, 1994; Duriez et al., 2016; Martini et al., 1997; Nesetova and Lajtai, 1973; Wasantha et al., 2016).

Cracks/joints are common manifestations in many types of rock. They may have various sizes and properties (Wasantha et al., 2014), and the mechanical behaviour of strong rocks is greatly influenced by these cracks/joints (Singh and Rao, 2005; Singh and Singh, 2008; Usefzadeh et al., 2013; Zhu et al., 2003). Pores/voids are also commonly encountered in many rocks, particularly sedimentary rocks, which can have porosities as high as 50%, caused mainly by the pores (Wong and Baud, 2012). Extrusive igneous rocks such as basalt and pumice often contain visible pores known as vesicles. Cracks and pores can also be present in rocks simultaneously. For instance, pore system characterizations of various shale rocks have revealed various types of micro-scale pores and cracks caused by shrinkage in clay minerals (c.f. Yang et al., 2015; Loucks et al., 2009; Chalmers et al., 2012). In addition, underground excavations such as tunnels, wellbores, and shafts in jointed rocks can be viewed as creating pore spaces in a medium containing cracks/joints (Cao et al., 2018; Fan et al., 2018; Sagong et al., 2011). The mechanical responses of different types of discontinuities (e.g. cracks and pores) to stress changes are fundamentally different, leading to marked differences in the global mechanical response of rock masses. As Kachanov (1999) explained, the elastic interactions between different defects such as cracks and pores may either increase or decrease their individual contributions to overall compliance as dictated by the stress redistribution characteristics. Therefore, in addition to the independent effects of cracks and pores, a detailed understanding of their interaction behaviour and the resulting global mechanical behaviour of rock masses is critical for reliable rock characterizations. While numerous previous studies have explored the independent effects of cracks and pores on rock mechanical behaviour, their combined effects have rarely been investigated to date. This study numerically investigates the crack-pore interaction behaviour, stress redistribution and damage evolution characteristics, and compressive strength variation under uniaxial compression for various combinations of crack and pore configurations.

Literature review

Many early attempts to develop a physical theory to describe the failure of brittle materials considered an extension of the Griffith theory (Griffith, 1920) and assumed that the development of macroscopic failure is due to the extension of a single microcrack with the critical geometry (Baud et al., 2014). However, numerous previous studies suggest that the macroscopic brittle faulting process involves initiation, propagation and coalescence of multiple micro-cracks, and the modified Griffith theories may be valid for the initiation of individual micro-cracks and the onset of dilatancy (Paterson and Wong, 2005). Krajcinovic et al. (1991) suggested a micromechanics-based phenomenological damage theory for the brittle response of solids. The most commonly used damage evolution models suggest that the wedging effect due to the frictional sliding on the pre-existing cracks leads to open tension cracks kinking in the maximum compression direction (Krajcinovic et al., 1991). Also, Krajcinovic et al. (1991) describes that the assumption of material homogeneity, in which case the growth of a Griffith crack when the homogenous tensile field becomes critical cannot be arrested, is not realistic in the case of real materials and short cracks. The internal energy barriers such as grain boundaries may arrest the growth of a sufficiently short crack. Crack initiation, propagation and coalescence behaviours of cracks and pores have been widely investigated by researchers.

Crack initiation from pre-existing cracks

Two types of cracks initiating from pre-existing crack tips have generally been observed: (1) wing cracks, which are tensile cracks, which appear first and propagate in a stable manner parallel to the major principal stress direction, and (2) secondary cracks, which are shear cracks initiating from pre-existing crack tips and propagate coplanar to or at an oblique angle with the pre-existing crack (Cao et al., 2015; Dyskin et al., 2003; Park and Bobet, 2009; Shen et al., 1995; Wong et al., 2001) (see Figure 1). An analytical expression for crack initiation principal stresses for a pre-existing inclined crack is given in equation (1) (Ashby and Hallam, 1986; Nemat-Nasser and Horii, 1982).

σ_{1} = [\frac{\sqrt{1 + μ^{2}} + μ}{\sqrt{1 + μ^{2}} - μ}] σ_{3} + [\frac{\sqrt{3}}{\sqrt{1 + μ^{2}} - μ}] \frac{K_{I C}}{\sqrt{π a}}

(1)

where

σ_{1}

and

σ_{3}

are the major and minor principal stresses, respectively,

μ

is the coefficient of friction acting across the flaw faces,

K_{I C}

is the fracture toughness of the material through which the new crack propagates, and

2 a

is the length of the original inclined flaw.

In addition to the basic categorisation of wing and secondary cracks as illustrated in Figure 1, numerous experimental and numerical studies of rock and other brittle materials (e.g. polymethylmethacrylate (PMMA), glass, and cement mortar) have reported some other types of cracks generated in pre-cracked media subject to various properties of the pre-existing crack (e.g. length, inclination angle). For example, Lajtai (1974) observed three types of fractures initiating from the tips of flaws (tensile, normal shear, inclined shear) in plaster of Paris specimens with a single flaw for various flaw inclination angles and lengths, whereas Wong and Einstein (2009) observed seven different types of tensile and shear cracks based on the geometry and propagation mechanism of pre-existing cracks embedded in moulded gypsum and Carrara marble specimens.

In the case of multiple pre-existing cracks, the propagation of wing and secondary cracks from different pre-existing cracks can coalesce leading to ultimate instability, and this has been investigated by many researchers. When two pre-existing cracks are present, characteristics such as crack length, crack inclination angle, crack arrangement, rock bridge length, and rock bridge angle affect crack initiation, propagation and coalescence behaviour and finally the strength of rocks or rocklike materials (Huang et al., 2016; Huang and Yang, 2019; Wong and Chau, 1998; Yang et al., 2013; Zhao et al., 2016, 2018). As the number of pre-existing cracks increases, the density of initiating wing cracks increases, which results in the formation of a macroscopic shear band leading to failure. The sliding wing crack model analytically expresses the principal stresses evolving with progressive damage for a given pre-existing crack density (Sammis and Ashby, 1986). Strong interactions between wing cracks initiated from pre-existing cracks have been experimentally and numerically observed in previous studies, depending on the geometry and the number of pre-existing cracks (c.f. Sagong and Bobet, 2002; Zhou et al., 2014; Bahaaddini et al., 2013; Prudencio and Van Sint Jan, 2007; Yang et al., 2019).

Crack initiation from pre-existing pores

Crack initiation from pores is fundamentally different from that from cracks. Kirsch equations (Kirsch, 1898) can be used to understand the stress and strain distributions around a circular opening due to far-field stresses. Stresses around a circular opening under biaxial stress conditions as shown in Figure 2(a) are expressed by equations (2) to (4).

σ_{l l} = \frac{σ_{1}}{2} [(1 + K) (1 - \frac{r^{2}}{l^{2}}) - (1 - K) (1 - 4 \frac{r^{2}}{l^{2}} + 3 \frac{r^{4}}{l^{4}}) \cos 2 θ]

(2)

σ_{θ θ} = \frac{σ_{1}}{2} [(1 + K) (1 + \frac{r^{2}}{l^{2}}) + (1 - K) (1 + 3 \frac{r^{4}}{l^{4}}) \cos 2 θ]

(3)

σ_{l θ} = \frac{σ_{1}}{2} [(1 - K) (1 + 2 \frac{r^{2}}{l^{2}} - 3 \frac{r^{4}}{l^{4}}) \sin 2 θ]

(4)

Under uniaxial stress conditions (i.e. $K = 0$ ), $σ_{θ θ}$ at the top (i.e. $θ = π / 2$ ) and right side (i.e. $θ = 0$ ) of the opening can be simplified from equation (3) as follows:

σ_{θ θ (θ = π / 2)} = - σ_{1} (negative indicates tension)

(5)

σ_{θ θ (θ = 0)} = 3 σ_{1} (this indicates a compression)

(6)

It can be seen from Figure 2(b) and equation (5) that tensile stresses exist in the medium although the far-field stresses are compressive. Since rocks are generally a few times weaker in tension than compression, microcracking generally initiates at the top and bottom of the opening (Wong and Peng, 2020). The results of previous experimental and numerical studies generally show that type of crack initiation and propagation behaviour from a single pore (e.g. Gui et al., 2017; Wang et al., 2012; Wong and Peng, 2020; Zhao et al., 2011). Pores also weaken rock integrity; Wong and Peng (2020) observed that the UCS and elastic modulus of the numerical model with a centric pore of diameter 15.44 mm were 58.6% and 56.4% lower, respectively, than those of the intact model. It has also been found that strength decreases with increasing pore diameter (Wong et al., 2006; Wong and Peng, 2020).

In relation to the presence of multiple pores, the pore-emanated cracking model of Sammis and Ashby (1986) describes the micromechanics of brittle failure in porous solids with randomly- distributed equal-sized pores in unconfined compression. According to this model, with the increase of applied stress wing cracks propagate from pores and coalesce with each other leading to instability. Zhu et al. (2010) derived an analytic approximation for UCS ( $σ_{u}$ ) following this pore-emanated cracking model (equation (7)):

σ_{u} = \frac{1.325}{Φ^{0.414}} \frac{K_{I C}}{\sqrt{π r}}

(7)

where

Φ

is the total porosity.

According to equation (7), $σ_{u}$ decreases with increasing porosity, and scales with the fracture toughness and inverse square root of the pore radius. Previous experimental and numerical studies that considered multiple pores have generally shown behaviours consistent with equation (7). For example, higher porosities have been observed to decrease UCS (e.g. Gui et al., 2017; Lin et al., 2015) and Heap et al. (2014) reported a decrease in UCS with increasing porosity of same-sized pores and increasing pore diameter for the same porosity. Heap et al. (2014) also found that microcracking behaviour is mainly governed by the largest pores in the case of varying pore sizes, and brittle strength is closer to that of the largest pore end-member.

While numerous studies have investigated the independent effects of cracks and pores, their combined effects on rock mechanical behaviour have rarely been explored. Sarfarazi and Haeri (2016) studied the crack propagation behaviour of a pre-existing microcrack with a micropore in the vicinity using the two-dimensional finite element code FRANC2D/L and found that the stress concertation at the crack tip is higher when there is a nearby pore space leading to greater crack extension length and the effect is more pronounced when the pore is closer to the crack tip and located directly in front of the crack tip. Fan et al. (2018) experimentally investigated various cracking patterns of sandstone with embedded cracks and holes: sandstone specimens with two pre-existing pores and a single crack, and two pre-existing cracks and a single pore revealed strong interactions between cracks and pores that varied with the inclination angle of the pre-existing crack. Due to the lack of comprehensive studies, spatio-temporal microcracking and damage evolution behaviour, and the resulting mechanical responses of rocks containing both cracks and pores remain poorly understood to date.

Numerical simulation program

Universal distinct element code (UDEC) software by Itasca Pty Ltd., which is two-dimensional and distinct element method-based, was used for the numerical simulation program. UDEC is ideally suited for the simulation of the response of discontinuous materials subjected to either static or dynamic loading. Blocks and contacts are the two main elements of UDEC models (UDEC manual, Itasca Consulting Group, Inc., 2013); blocks are indivisible, whereas contacts can be broken (i.e. cracks initiate) when the maximum tensile stress exceeds the tensile strength of the contacts or the maximum shear stress exceeds the shear strength of the contacts. Hence, both tensile and shear cracking can be simulated using UDEC models. We used a Voronoi tessellation random block generator to create an assemblage of randomly-shaped blocks in the models, as shown in Figure 3. The Voronoi blocks were made deformable by discretising them to finite-difference zones (triangular zones).

We first simulated the intact gypsum specimens tested under uniaxial compression in the experimental work of Wong (2008). Hence, we selected a model size of 100 mm (height) ×50 mm (width) to duplicate the specimen size of the experimental study. More details of the validation procedure is given in Dai et al. (2018). After validating the model for intact material, models with cracks and pores were simulated; first, models with a single crack (crack lengths of 10 mm, 20 mm, 30 mm and 40 mm) and a single pore (pore diameters of 4 mm, 8 mm, 12 mm and 16 mm) were simulated. Cracks in all these cases were oriented at an unchanged angle of 45° to the loading direction.

In order to understand the interaction behaviour of cracks and pores under uniaxial stress conditions, different combinations of cracks and pores were considered. Hence, we simulated models with (1) two cracks either side of a centric pore (Figure 4(a)), and (2) two pores either side of a centric crack (Figure 4(c)). Furthermore, models with only two cracks (Figure 4(b)) and two pores (Figure 4(d)) (i.e. without the centric crack/pore) were also simulated to assess the effect of the presence of centric cracks/pores on the strength and damage evolution behaviour of the models. In each case, the crack length of 10 mm, the inclination angle of 45° to the loading direction, and the pore diameter of 8 mm were unchanged, whereas the distance between the centric crack/pore and the crack/pore either side of that (i.e. ‘d’) was varied. The selection of crack length, inclination angle and pore diameter was arbitrary and an investigation of their effects in parameter space was beyond the scope of this study. Mechanical properties, stress redistribution and microcracking behaviour for various model configurations were monitored and analysed.

Results and discussion

The mechanical behaviours of the intact UDEC model and those of the gypsum specimen in the Wong (2008) study were first compared. Figure 5 shows the axial-stress versus axial-strain curves and Table 1 shows the various mechanical parameters for both the UDEC numerical model and the gypsum specimen of Wong’s (2008) experimental work. Both Figure 5 and Table 1 show good consistency between the mechanical properties of the UDEC models and the experimental specimens.

Table 1.

Various parameters determined in experimental study and UDEC simulations.

Parameter	Physical experiment	UDEC
Density (g/cm³)	1.54^a	1.54
Elastic modulus (GPa)	5.96^b	5.79
Poisson’s ratio	0.15^b	0.15
UCS (MPa)	33.85^a	33.5
Tensile strength (MPa)	3.2^b	4.2

Determined by Wong (2008).

Determined by Bobet (1997).

Single crack and single pore cases

UCS and damage evolution behaviours were monitored in the models with single crack and single pore. Figure 6 shows the axial stress versus axial strain curves for all the different crack lengths and pore diameters studied, and Figure 7 shows the variations of normalized UCS (i.e. the ratio of the UCS of any model to the UCS of the intact model) and normalized elastic modulus (i.e. the ratio of the elastic modulus of any model to the elastic modulus of the intact model) against crack length and pore diameter. According to Figure 6(a), models with shorter cracks show more brittle behaviour with clear peak stress. Similar behaviour can be observed for the varying pore diameter cases with increasing pore diameter (Figure 6(b)). Figure 7 clearly shows that increasing crack length and pore diameter decrease the UCS and elastic modulus in agreement with the analytical models expressed in equations (1) and (7), respectively, and the results of other experimental and numerical studies discussed in Section 2. In fact, crack length shows a dramatic weakening compared to pore diameter within the ranges of crack length and pore diameter considered in this study.

We then analysed the progressive cracking and subsequent damage evolution of the models. Figures 8 and 9 show the progressive crack initiation and propagation characteristics of the models containing pre-existing cracks and pores until failure (only two representative cases are shown for each). Both wing cracks and secondary cracks can be observed in Figure 8 and in the case of longer cracks (Figure 8(b)) the wing and secondary cracks developed from the pre-existing crack tips contribute substantially to the ultimate macroscopic failure. In contrast, microcracks develop in a more diffuse manner in the shorter crack case, indicating the less significant effect of the pre-existing crack on overall failure (Figure 8(a)). Models with pores show tensile fracture initiation and propagation from the top and bottom of the pores, consistent with the Kirsch (1898) model (Figure 9). Further microcracks also developed in both cases. In case of the smaller pore diameter, a shear band and some major cracks developed disconnected to the pre-existing pore can be observed, which indicates the lower significance of the pore on the overall failure mechanism (Figure 9(a). On the other hand, many major cracks contributing to the ultimate macroscopic failure initiate from the pre-existing pore in the larger pore diameter case, displaying the greater effect of the pore on failure (Figure 9(b)). Overall, the mechanical properties and damage evolution behaviour showed by the models with single cracks and single pores are consistent with the relevant analytical models and the results of comparable previous studies.

Two cracks either side of a centric pore

Models with two pre-existing cracks either side of a centric pre-existing pore space showed important behaviours. The variations of normalized UCS against the distance between the pore and cracks (case ‘A’) and the distance between cracks (case ‘B’) are shown in Figure 10. According to Figure 10, normalized UCS is quite low and does not vary considerably with distance, d, for the case of two cracks and pore. In addition, the normalized UCS of all the four distances considered in case A of Figure 10 are only slightly lower than that when there is only a pore 8 mm in diameter. Figure 10 also displays a noteworthy difference of normalized UCS values between the cases with and without pores (the scattered normalized UCS values around the fitted curve at lower distances of case B may be a product of the greater interaction between the crack segments when they are positioned closer). These observations indicate that the presence of the pore space has a prominent influence on strength and the two cracks considered have only a minor impact on strength. Hence, the effect of distance to cracks is insignificant in models already weakened significantly by pores.

Progressive crack development behaviours for different distances between the pre-existing cracks and pore are shown in Figure 11. The figure shows an approximately simultaneous crack initiation from pre-existing cracks and pore which later coalesce for all cases of distances (Figure 11). A shear band forms eventually to complete macroscopic failure in all cases, which is more distinct when the cracks and pore are closely positioned. Therefore, in general, the crack initiation and propagation characteristics are consistent with the behaviours discussed above using normalized UCS values.

We also studied the stress distribution characteristics within models with increasing external loading. Figure 12 shows the evolution of major principal stress surrounding cracks and pores with increasing external loading for the shortest and longest distances considered (i.e. d = 10 mm and d = 40 mm). It can be clearly seen that the stress fields overlap from the early stages of loading when the cracks and pore are closely positioned, showing a greater interaction between them. In addition to the tensile fractures initiating and propagating from the top and bottom of the pore, compressive stresses at the right and left sides of the pore keep accumulating. At higher applied stresses, cracks develop due to these compressive stresses in both cases and contribute substantially to the formation of macroscopic failure patterns. In both cases, cracks intiating from pre-existing cracks do not possess a considerable fraction of the ultimate macroscopic failure pattern. More importantly, the compressive stresses on the right and left sides of the pore initially mobilise in a more or less symmetrical manner and microcracking later disturbs the symmetry. These stress redistribution characteristics are consistent with the less significnat role of the distance between the pre-existing cracks and pores on the UCS of the models.

Two pores either side of a centric crack

The variations of normalized UCS against the distance between the pre-existing pores and crack (case ‘A’) and the distance between pores (case ‘B’) are shown in Figure 13. In contrast with the two cracks one pore case, Figure 13 shows a noteworthy increase of UCS with increasing distance between pores and crack. In addition, UCS increases in a similar fashion with increasing distance between the two pre-existing pores when the crack segment is absent (i.e. case ‘B’) and UCS drops only slightly due to the presence of crack segment at all distances. When the pores are farthest from the crack the UCS is nearly equal to that when there is only a crack of 10 mm. These observations indicate the more profound effect of pores on UCS than the crack segment used in the models.

Progressive cracking behaviours for all the distances considered are shown in Figure 14. When the pores and crack are closer, microcracks initiating from both pre-existing pores and crack clearly contribute to ultimate macroscopic failure. When the pores are positioned away from the centric crack, their effect on failure gradually diminishes. Figure 14(d) clearly displays this, where cracks initiating from the pores are isolated from the macroscopic shear band. This also explains the higher UCS observed for the case in Figure 14(d), which is also close to that of the model with only the pre-existing crack segment of 10 mm.

The distribution of major principal stress with increasing external loading is shown in Figure 15 for the distances between pores and crack of 10 mm and 40 mm. When the pores are closer to the centric crack segment the compressive stress accumulation at the right and left sides of the pores is highly asymmetric; greater stress is mobilised at the right side of the top pore and the left side of the bottom pore. Microcracking initiates from these positions due to the high compressive stresses can easily form macroscopic shear failure when pores and crack are closely positioned, as Figure 15(a) shows. The arrangement of pores and cracks in the models facilitates this easier formation of shear failure. In contrast, when the pores are positioned away from the crack no initial interaction between them is apparent. The ultimate failure is due to tensile and shear cracks initiated from the pre-existing crack segment with minimal contribution from the two pre-existing pores. These changes in stress distribution patterns result in a lower strength when the pores and crack are closer compared to when they are away from each other, as Figure 13 shows.

Conclusions

The mechanical behaviour of rocklike materials containing cracks and pores was studied using a distinct element method-based numerical simulation program. First, the effects of pre-existing cracks and pores were studied independently, followed by simulations with various combinations of cracks and pores. Increasing crack length and pore diameter were found to decrease the uniaxial compressive strength (UCS), consistent with the relevant analytical models and previous experimental and numerical studies. Crack-pore interactions and the resulting mechanical behaviour were studied under two crack-pore arrangements: two cracks either side of a centric pore, and two pores either side of a centric crack, and the distance between the cracks and pores was varied. When there were two pre-existing cracks on either side of a pore, the distance between the cracks and centric pore had only a minor effect on UCS, and the tensile cracks developed from the pre-existing cracks were observed to make only a minor contribution to macroscopic failure, as evident from the progressive crack development of the models. In contrast, the UCS of models with two pre-existing pores either side of a centric crack showed an increase with increasing distance between the pores and centric crack. Greater interaction between pores and crack was observed when pores and crack were closely positioned. The evolution of major principal stress distribution indicates an asymmetric stress accumulation between the left and right sides of pores, favouring macroscopic shear failure when the pores are positioned close to the centric crack, which leads to the observed lower UCS at lower distances between pores and centric cracks. Overall, the critical insights presented by this study into the mechanical behaviour of rocklike materials containing cracks and pores are expected to assist with the design of rock structures. As this study only considered selected crack lengths, crack inclination angles and pore diameters, future studies need to investigate their effects on the observed behaviours in parameter space.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the National Science Foundation of China (51811530312).

ORCID iD

PLP Wasantha

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