1 Introduction

Pre-existing cracks and pores in natural rocks and concretes may reduce their mechanical properties and strength [1]. The mechanical behavior of rocks and rock-like materials (concrete) is affected by the micromechanical behaviors of flaws and soft inclusions. Investigating the crack propagation process of various rocks and rock-like samples can be a critical task to employ for the rock fracture engineers for better design of most geomechanical projects. Therefore, it may be important to have a clear understanding of the crack propagation and cracks coalescence phenomena in pre-cracked rock-like materials (or specially made concrete samples with cylindrical holes where the holes can induce some pre-existing cracks) under various loading conditions. More specifically, the pre-existing holes and their geometry have been considered as very vital structures in controlling the strength of brittle materials because, due to the local stress concentration around the holes, some tensile cracks may be induced and cause the brittle failure of the material [2, 3].

In some recent experimental and numerical works the mechanism of brittle rocks failures has been studied [4–13]. One of the most suitable experiments for determining the tensile strength of rocks and rock-like materials is the Brazilian disc test. The disc specimens may or may not contain central pre-existing cracks or holes. The specimens with cracks or holes are well suited to study the crack initiation, propagation path, and cracks coalescence of brittle solids such as rocks [2, 3, 14–21]. This testing procedure is used extensively to measure the tensile strength, fracture toughness and mixed mode breakage process in the pre-cracked or pre-holed and intact disc specimens of various brittle materials such as rocks and concrete under compressive line loadings [22–31]. In the Brazilian disc specimens, the crack propagation and failure process of the specimens may happen very soon due to the low tensile strength of rocks and rock-like materials (concrete). Although splitting of the disc specimen containing pre-cracks or holes into two halves is expected, other rupture modes may be occurring due to the inclination, number, and size of these internal discontinuities compared to the loading conditions and specimen size.

Several researchers have conducted various kinds of experiments and analytical and numerical works to study the complex behavior of brittle solids such as rocks and concrete considering different circumstances happen naturally or occur during the design and performance stages of engineering structures [19–31]. For example, Al-Shayea [31] performed some experiments to study the crack propagation paths in the Central Straight through Crack Brazilian Disk (CSCBD) specimens of brittle limestone with different crack inclination angles under mixed Mode I/II loading. The influence of confining pressure and temperature on the crack initiation and propagation of the rock samples are also investigated and the experimental results were compared with theoretical predictions of crack initiation angle.

Also, Sammis and Ashby [4] carried out a set of uniaxial compression tests on plate specimens containing a single hole of the same size or array of holes with different diameters to experimentally investigate the interaction of propagating cracks with surfaces of the specimen. In addition, Mellor and Hawkes [14] conducted some tests to study the cracks initiation and propagation in ring specimens under diametrical compression. Various numerical methods such as the finite element method (FEM), the boundary element method (BEM), and the discrete element method (DEM) have been developed for the simulation of crack propagation in brittle materials [32]. Meanwhile,Tang and Hudson [12] numerically investigated the crack propagation patterns of specimens containing a single hole with different sizes and array of holes with different diameters. They used a 2D finite element code (the \(\hbox {RFPA}^{2\mathrm{D}}\) code) to model a number of numerical simulations.

Among the classical fracture mechanics criteria are (1) the maximum tangential stress (\(\sigma \)-criterion) [33], (2) the maximum energy release rate (G-criterion) [34], and (3) the minimum energy density criterion (S-criterion) [35]. The \(\sigma \)-criterion has been used more effectively to study the crack propagation of brittle solids such as rocks and concrete. It should be noted that, in some cases, the modified form of these criteria (e.g., F-criterion (a modified energy release rate criterion) proposed by Shen and Stephansson [36]) were also used for crack analyses. Several computer codes such as: FROCK code [8], Rock Failure Process Analysis \((\hbox {RFPA}^{2\mathrm{D}})\) code [12, 32], and 2D Particle Flow Code \((\hbox {PFC}^{2\mathrm{D}})\) [12, 13] have been developed and used to model the breaking mechanism of brittle materials.

In the present study, the cracks propagation mechanism in the disc and rectangular specimens of rock-like materials containing either single thick cylindrical hole or multi-holes are being studied both experimentally and numerically. The concrete specimens prepared from PCC, fine sands, and water were tested in a compression testing apparatus (under uniaxial and diametrical compressive loading) in a rock mechanics laboratory. The breaking load and the distribution of the induced lateral stress in the disc specimens containing thick cylindrical holes with different sizes are measured. The propagation and coalescence mechanism of cracks through the specimens and in the bridge area (the area in between the two holes in the specimens containing multi-holes) have been studied. Then these specimens are simulated numerically by a modified higher order displacement discontinuity method and the crack propagation and cracks coalescence are studied based on linear elastic fracture mechanics (LEFM) principles by computing the Mode I and Mode II stress intensity factors (SIFs). The maximum tangential stress criterion (\(\upsigma \)-criterion) is used to analyze the tensile cracks produced around the cylindrical holes within the specimens prepared from rock-like materials (a special type of concrete). Considering a center crack of length 2b in an elastic solid, an iterative method has been implemented in the displacement discontinuity code to investigate the crack propagation direction and its path after each successive crack extension, \(\Delta b=0.1{b}\).

Comparing the experimental and numerical results, it is concluded that there is a very good agreement between the crack propagation and cracks coalescence paths obtained experimentally and numerically, which in turn validates the accuracy and applicability of the present crack analyses. Therefore, the proposed numerical simulation makes the necessary flexibility in the analyses so that it is readily possible to investigate many different cases related to the fracture mechanics of brittle substances such as rocks and concrete.

2 Experimental analyses

Some disc and rectangular specimens are prepared from rock-like materials (a special type of concrete) in a rock mechanics laboratory. These specially prepared concrete specimens include: (1) Brazilian disc type specimens containing single holes with different sizes (ring type specimens), (2) Brazilian discs containing multiple parallel cylindrical holes with different arrangements, and (3) rectangular specimens containing three cylindrical holes with different arrangements. Then, some experimental compression (under diametrical and uniaxial compression) testing is carried out to obtain the breaking stresses and also to visualize the crack propagation paths and the cracks coalescence in these pre-holed rock-like specimens.

2.1 Testing of Brazilian disc specimen

The rock-like disc specimens with diameters, 100 mm and thickness of 30 mm are specially prepared by mixing the Portland Pozzolana Cement (PPC), fine sands, and water. The mechanical properties of these rock-like (concrete) specimens are obtained by testing some of the specimens in the rock mechanics laboratory before inserting the holes. All these specimens are assumed to be elastic, homogeneous, and isotropic.

The important mechanical properties used in the present analysis are: compressive strength, \({\sigma }_{\mathrm{c}}=28\,\hbox { MPa}\); Young’s modulus, \(E=15\) GPa, Brazilian tensile strength, \({\sigma }_{\mathrm{t}}=3.81\,\hbox { MPa}\); Poisson’s ratio, and \(\nu =0.21\).

The geometry and loading condition of a pre-holed specimen is illustrated in Fig. 1.

Fig. 1
figure 1

Schematic view of a Brazilian disc specimen containing a single cylindrical hole loaded under diametrical compression

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Some ring tests are conducted on the rock-like specimens containing holes with different ratios of internal radius to external radius i.e., \(r/R=0\), 0.12, 0.17, 0.2, 0.3, 0.37, 0.47, and 0.65. These cylindrical holes are created axially by inserting one thin steel ring in the molds (before casting the specimens). The compressive line loading, F, was diametrically applied and the loading rate was kept constant at 0.5 MPa/s during the tests.

2.1.1 Breaking loads of the pre-holed disc specimens (PHDS)

The breaking load analysis of the disc specimens containing parallel cylindrical holes with different sizes (pre-holed disc specimens or PHDS) is used to study the mechanical behavior of the brittle materials. Figure 2 describes a variation of the normalized breaking load for the PHDS. The failure load of the pre-holed disc specimens is normalized by the average failure load of the intact specimens (disc specimens with no holes). The average failure load of the intact specimens is about 18 kN. It is obvious that the pre-holed rock-like disc specimens have a lower strength than those specimens having no holes (intact disc specimens). Therefore, the normalized breaking loads for the holed disc specimens are usually less than one because the pre-existing hole may decrease the final strength of specimen (Fig. 2). In the pre-holed specimens, breakage loads at different stages of crack propagation process are decreasing for \(r/R=0-0.7\) (Fig. 2).

Fig. 2
figure 2

Normalized failure load versus different r / R ratios for concrete disc specimens

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2.1.2 Stress distribution in ring type (single holed) disc specimens

The critical tensile stress normal to the loading diameter at the intersection of the loading diameter with the hole can be expressed [12]:

$$\begin{aligned} \sigma _x =-\omega \frac{F}{\uppi \!RB}, \end{aligned}$$
(1)

where, \(\sigma _x \) is the distribution of the initial lateral (horizontal) stress expressed in MPa, F is the compressive load at failure in Newtons, B is the thickness (length) of the disk in mm, and \(\omega \) are the non-dimensional coefficients depending on the specimen geometry and experimental condition, which can be defined as [12]:

$$\begin{aligned} \omega =[6+38(r/R)^{2}]/\!\uppi . \end{aligned}$$
(2)

As can be seen in Eq. (1), the distribution of the induced lateral stress, \(\sigma _x \) is affected by the specimen geometry such as the internal radius (r) and external radius (R). In analytical solutions, thickness (B) of the disc is essential to estimate the distribution of the induced lateral stress. The value of the thickness is assumed to be 30 mm. Variations of \(\omega \) for the tested ring specimen are illustrated in Fig. 3 considering different r / R ratios. As shown in this figure, \(\omega \) increases monotonically with increasing r / R ratio.

Fig. 3
figure 3

Variation of the geometrical factor, \(\omega \), for ring type disc specimens with different r / R ratios

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Figure 4 describes a variation of the normalized lateral stress for single ring specimens. In these specimens, the lateral stress of the pre-holed disc specimens is normalized by the average lateral stress of the intact specimens (specimens containing no holes). The average lateral stress of the intact specimens is about 7.29 MPa. Therefore, the normalized lateral stress for \(r/R =0\) is larger than the normalized lateral stress for other r / R ratios. The horizontal stresses at different stages of the crack propagation process are decreasing for \(r/R =0\)–0.37 but increasing gradually for \(r/R =0.47\)–0.65, as shown in Fig. 4.

Fig. 4
figure 4

Normalized lateral stress for ring type disc specimens with different r / R ratios

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2.1.3 Crack propagation process of PHDS

The experimental investigation of concrete (rock-like) Brazilian disc specimens is accomplished considering the two cases: (1) specimen containing a single cylindrical hole (ring type disc specimens) and (2) specimens containing multi-holes (PHDS).

2.1.3.1 Specimens containing a central hole (ring type disc specimens)

Some experiments have been carried out to study the mechanism of crack initiation and crack propagation in ring type disc specimens with different r / R ratios. In this study, it was observed that the radial cracks initiated and propagated around the internal hole, starting their extension from the initiated cracks, the wing cracks are produced and continue their growth in a direction (approximately) parallel to the direction of the maximum compressive stress (as shown in Fig. 5a–c). These figures show the breaking process of the single hole specimens and also the effect of cylindrical hole size (i.e., for \(r/R=0.12\), 0.17, 0.2) on the final crack propagation process, which is experimentally observed during the tests. It should be noted that, in these cases, a crack can start its propagation from the internal hole and propagate mainly in the form of tensile cracks toward the direction of loading. Then, wedges near the specimen’s surfaces also formed and the ring specimen fails into halves.

Fig. 5
figure 5

Experimental tests showing the cracking patterns in the single-holed Brazilian discs under diametrical compression with different r / R ratios: a \(r/R=0.12\). b \(r/R=0.17\). c \(r/R=0.2\). d \(r/R= 0.3\). e \(r/R=0.37\). f \(r/R=0.47\) and g \(r/R= 0.65\)

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When the size of the internal hole is increased considerably e.g., for the case shown in Fig. 5d (for \({r/R}=0.3\)), the secondary crack initiates only at one edge (left or right edge) and propagates horizontally towards the surface of the central hole. In this case, more than two tensile cracks are produced at the wall of the internal hole. Finally, for the cases shown in Fig. 5e–g, i.e., when \(r/R= 0.37\), \(r/R=0.47\) and \(r/R=0.65\), the effect of central hole become more and more dominant. The symmetrical cracks are produced around the central hole and propagate toward the wall (i.e., two vertical and two horizontal cracks). The two horizontal cracks are produced due to stress concentration (tensile stress) at the side wall of the central hole shown in Fig. 5e–g.

2.1.3.2 Brazilian discs containing multiple parallel cylindrical holes

Considering a disc specimen with multiple parallel cylindrical holes in its central part, the experiments show that the cracks propagation and cracks coalescence phenomena may occur simultaneously because during the test the multiple pre-existing holes combine due to propagation of tensile and/or shear cracks. Although the size of the holes (with internal radius to external radius ratios \(r/R=0.006\)) in Fig. 6 is exactly the same, the current experiments show that the tensile cracks are instantaneously initiated. The development and coalescence of tensile cracks in the bridge area (i.e., the area in between the two pre-existing holes) may be the main cause of failure in the pre-holed rock-like disc specimens.

Figure 6a–e, show some of the experiments carried out in this research, where, the rock-like specimens containing more than one cylindrical hole are being tested under diametrical compressive loading in a rock mechanics laboratory.

The holes are driven parallel to the axis of the specimen and arranged in different patterns as shown in Fig. 6a–e. Then the diametrically loaded specimens are being compressed from the top and bottom of the specimens.

Fig. 6
figure 6

Experimental tests showing the cracking patterns in the disc specimens containing multi-cylindrical holes under diametrical compression. a Vertical arrangements of three holes in disc specimens with \(r/R=0.006\). b Diagonal arrangements of three holes in disc specimens with \(r/R=0.006\). c Uniform arrangements of five holes in disc specimens with \(r/R=0.006\). d Two parallel arrangements of six holes in disc specimens with \(r/R=0.006\). e Two parallel arrangements of four holes in disc specimens with \(r/R=0.006\)

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Fig. 7
figure 7

Geometry and loading condition for the three holes in the three models of pre-holed rock-like specimens. a Horizontal arrangement with a diameter of 15 mm. b Vertical arrangement with a diameter of 15 mm. c Diagonal arrangement with a diameter of 15 mm

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It is observed that for the case shown in Fig. 6a (three vertically arranged holes) the cracks initiated toward and downward from the holes area first and then the specimen might fail due to cracks coalescence. For the case shown in Fig. 6b (three diagonally arranged holes), the cracks propagated toward and downward only from the hole that was located at the centerline of the specimen and, also, no propagation might occur from the other holes that were located at the external line of compressive loading. For the case shown in Fig. 6c (the arrangement of holes (5 holes) is uniform), the cracks propagated toward and downward only from the hole that was located at near centerline of the specimen and the propagated cracks from holes coalesced each other in the bridge area also. No propagation might occur from the other remote holes that were located at the external line of compressive loading (of course the size and distance of holes is important but in this study to make the experiments more concise, they are kept constant). In the specimen with six holes, the cracks initiated from the surfaces of the central holes that were located at left side of specimen’s central line and then the cracks coalesced with each other at the propagating crack tips (Fig. 6d), but for the case shown in Fig. 6e with four holes, the cracks may start to initiate at the left or right sides of the central holes first and then the specimen may fail in the direction of the crack propagation paths originating from either of the two holes.

2.2 Testing of rectangular specimens

The rock-like rectangular specimens with height, 170 mm, width, 100 mm, and thickness, 40 mm, are specially prepared in a rock mechanics laboratory. Some uniaxial compression tests are conducted on the rock-like specimens containing holes with different arrangements. The uniaxial compressive stress, \(\upsigma \), was uniformly applied and the loading rate was kept constant at 0.5 MPa/s during the tests. The geometry and loading condition of specimens containing three pre-holes in different hole arrangements are shown in Fig. 7a–c, respectively. Three modes of holes arrangements are used: a) Horizontal arrangement (Fig. 7a), b) Vertical arrangement (Fig. 7b), and c) Diagonal arrangement (Fig. 7c). The hole arrangements are selected in such a manner that the hole sizes of each of three holes are the same as that schematically shown in Fig. 1. It should be noted that all holes are of 15 mm diameter.

2.2.1 Breakage analysis of the pre-holed rectangular specimens

It has been shown that the pre-holed rock-like rectangular specimens have a lower strength compared to the specimens having no holes. The breaking stresses of the pre-holed specimens containing three holes with different arrangements (i.e., diagonal, vertical, and horizontal arrangements) can be used to study the behavior of the brittle materials. It has been shown that these specimens are breaking due to the propagation of cracks emanating from the pre-existing holes (defects). Therefore, the stresses initiating the wing cracks from the top and bottom of the three holes (hole I, hole II, and hole III), and also the final breaking stresses for these three cases of hole arrangements are presented in Table 1. The average uniaxial compressive strength of the intact specimens is about 16 MPa (i.e., \({\sigma }_{\mathrm{c}}=16\,\hbox { MPa}\)). The results given in Table 1 show that a tensile (wing) crack may initiate first from the boundary of the holes in the diagonal arrangement compared to those of other arrangements (i.e., vertical and horizontal arrangements). The final breaking stress of the specimens with diagonally arranged holes is about 5.3 MPa, which seems to be the lowest stress for the breaking stress of these three specimens shown in Fig. 7.

Table 1 The initiation stress for wing cracks originating from the boundary of the three holes (hole I, hole II, hole III) within the specimens and the final breaking stress for the three cases of hole arrangements shown in Fig. 7
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2.2.2 Crack propagation process of the pre-holed rectangular specimens with different hole arrangements

Considering the cases shown in Fig. 7a–c, several experiments of these three test models were carried out in a rock mechanics laboratory and the results are illustrated in Fig. 8a–c. The horizontal arrangement of parallel holes (Fig. 8a), illustrates that the wing cracks are originating from all holes propagating toward the applied compressive load. The development of wing cracks originating from the boundary of hole 1 (right hole) and hole 3 (left hole) may be the main cause of the fracture paths in this type of specimens. For the case of a vertical arrangement of parallel holes (Fig. 8b), the wing cracks firstly initiated from the boundary of all internal holes and then the cracks coalesced with each other in the bridge area. The diagonal arrangement of pre-existing holes was also tested under uniaxial compressive loads and the results are shown in Fig. 8c. The wing cracks are also initiated from the boundary of internal holes and propagated toward each other in a curved path until they coalesce with the other holes. Therefore, no coalescence might occur at the tips of propagating cracks. In this case, the wing cracks may instantaneously develop when another wing crack is nearby.

Fig. 8
figure 8

Experimental results showing the propagating of wing cracks for pre-holed specimens with different arrangements of three parallel holes; i.e., a Horizontal arrangement. b Vertical arrangement. c Diagonal arrangement

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3 Indirect boundary element analyses

The indirect boundary element method used in this study implements the cubic variation of displacement discontinuities along each boundary elements to solve two-dimensional problems in elastoestatic bodies. This higher order displacement discontinuity method (HDDM), which is an indirect form of the dual boundary element method (DBEM) proposed by Chen and Wong [37] is employed to simulate all the pre-holed specimens tested (under compression) in this study [38–43].

It has been shown that the higher order displacement discontinuity method gives accurate results of normal displacement discontinuity (crack opening displacement) and shear displacement discontinuity (crack sliding displacement) near the crack tips and holes. The Mode I and Mode II SIFs for crack analysis of brittle solids can be formulated based on these discontinuities using the LEFM principles [44]. This method gives great accuracy when the special crack tip elements can be used to account for the singularities of stress and displacement fields near the crack tips and holes in the pre-holed disc specimens of rock-like materials (i.e., for the problem under consideration in this research).

3.1 Numerical modeling of pre-holed specimens by a higher order displacement discontinuity method

In this research, the displacement discontinuities along the boundary of the problem, which can be achieved more accurately by using higher order displacement discontinuity (DD) elements (e.g., quadratic or cubic DD elements) are used for the numerical analyses of breaking mechanism of the pre-holed discs and rectangular concrete specimens.

In the higher order displacement discontinuity modeling of the pre-holed rock-like specimens, the cubic (third order) variation of DD is assumed along each boundary element. A cubic DD element (\(D_k (\varepsilon )\)) is divided into four equal sub-elements such that each sub-element contains a central node for which the nodal DD is evaluated numerically. The opening displacement discontinuity \(D_y \) and the sliding displacement discontinuity \(D_x \) can be formulated as [41]:

$$\begin{aligned} D_k (\varepsilon )=\sum _{i=1}^4 {\varPi _i (\varepsilon )D_k^i }, \quad k=x,y, \end{aligned}$$
(3)

where \(D_k^1\, (\mathrm{i.e.,} D_x^1 \;\mathrm{and}\; D_y^1 )\), \(D_k^2\, (\mathrm{i.e.,} D_x^2 \;\mathrm{and}\; D_y^2 )\), \(D_k^3\, (\mathrm{i.e.,} D_x^3 \;\mathrm{and}\; D_y^3 )\) and \(D_k^4\, (\mathrm{i.e.,} D_x^4 \;\mathrm{and}\; D_y^4 )\) are the cubic nodal displacement discontinuities and,

$$\begin{aligned} \begin{aligned} \varPi _1 (\varepsilon )&={-\left( {3a_1^3 -a_1^2 \varepsilon -3a_1 \varepsilon ^{2}+\varepsilon ^{3}} \right) }\Big /{\left( {48a_1^3 } \right) }, \\ \varPi _2 (\varepsilon )&={\left( {9a_1^3 -9a_1^2 \varepsilon -a_1 \varepsilon ^{2}-\varepsilon ^{3}} \right) }\Big /{\left( {16a_1^3 } \right) }, \\ \varPi _3 (\varepsilon )&={\left( {9a_1^3 +9a_1^2 \varepsilon -a_1 \varepsilon ^{2}-\varepsilon ^{3}} \right) }\Big /{\left( {16a_1^3 } \right) }, \\ \varPi _4 (\varepsilon )&={-\left( {3a_1^3 +a_1^2 \varepsilon -3a_1 \varepsilon ^{2}-\varepsilon ^{3}} \right) }\Big /{\left( {48a_1^3 } \right) } \end{aligned} \end{aligned}$$
(4)

are the cubic collocation shape functions using \(a_1 =a_2 =a_3 =a_4 \). A cubic element has four nodes, which are the centers of its four sub-elements as shown in Fig. 9

Fig. 9
figure 9

Cubic shape function showing the variation of higher order displacement discontinuities along an ordinary boundary element

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The potential functions f(x,y), and g(x,y) for the cubic case can be found from:

$$\begin{aligned} \begin{aligned} f(x,y)&=\frac{-1}{4\!\uppi (1-\nu )}\sum _{i=1}^4 {D_x^i F_i (I_0 ,I_1 ,I_2 ) }, \\ g(x,y)&=\frac{-1}{4\!\uppi (1-\nu )}\sum _{i=1}^4 {D_y^i F_i (I_0 ,I_1 ,I_2 ) }, \end{aligned} \end{aligned}$$
(5)

in which the common function \(F_i \), is defined as

$$\begin{aligned} F_i (I_0 ,I_1 ,I_2 ,I_3 )= & {} \int {\varPi _i (\varepsilon )\ln \left[ {(x-\varepsilon )+y^{2}} \right] } ^{\frac{1}{2}}\mathrm{d}\varepsilon ,\nonumber \\&{i}= 1, \hbox { to } 4, \end{aligned}$$
(6)

where the integrals \({I}_{0}, {I}_{1}, {I}_{2}\) and \(I_{3}\) are expressed as follows

$$\begin{aligned} \begin{aligned} I_0 (x,y)&=\int {\ln \left[ {(x-\varepsilon )^{2}+y^{2}} \right] } ^{\frac{1}{2}}\mathrm{d}\varepsilon , \\ I_1 (x,y)&=\int \limits _{-a}^a {\varepsilon \ln \left[ {(x-\varepsilon )^{2}+y^{2}} \right] } ^{\frac{1}{2}}\mathrm{d}\varepsilon , \\ I_2 (x,y)&=\int \limits _{-a}^a {\varepsilon ^{2}\ln \left[ {\left( {x-\varepsilon } \right) ^{2}+y^{2}} \right] } ^{\frac{1}{2}}\mathrm{d}\varepsilon , \\ I_3 (x,y)&=\int \limits _{-a}^a {\varepsilon ^{3}\ln \left[ {\left( {x-\varepsilon } \right) ^{2}+y^{2}} \right] } ^{\frac{1}{2}}\mathrm{d}\varepsilon . \end{aligned} \end{aligned}$$
(7)

Since the singularities of the stresses and displacements near the holes may reduce their accuracies, special crack tip elements are used to increase the accuracy of the DDs near the crack tips [46]. As shown in Fig. 10, the DD variation for three nodes can be formulated using a special crack tip element containing three nodes (or having three special crack tip sub-elements).

$$\begin{aligned} D_k (\varepsilon )&=[\varGamma _{C1} (\varepsilon )]D_k^1 (a)+[\varGamma _{C2} (\varepsilon )]D_k^2 (a)\nonumber \\&\quad +[\varGamma _{C3} (\varepsilon )]D_k^3 (a), \quad k=x,y, \end{aligned}$$
(8)

where, the crack tip element has a length \(a_1 =a_2 =a_3 \). Therefore, considering a crack tip element with the three equal sub-elements \((a_1 =a_2 =a_3 )\), the shape functions, \(\varGamma _{C1} (\varepsilon )\), \(\varGamma _{C2} (\varepsilon )\), and \(\varGamma _{C3} (\varepsilon )\) can be obtained as

$$\begin{aligned} \varGamma _{C1} (\varepsilon )= & {} \frac{15\varepsilon ^{\frac{1}{2}}}{8a_1^{\frac{1}{2}} }-\frac{\varepsilon ^{\frac{3}{2}}}{a_1^{\frac{3}{2}} }+\frac{\varepsilon ^{\frac{5}{2}}}{8a_1^{\frac{5}{2}} }, \nonumber \\ \varGamma _{C2} (\varepsilon )= & {} \frac{-5\varepsilon ^{\frac{1}{2}}}{8a_1^{\frac{1}{2}} }+\frac{3\varepsilon ^{\frac{3}{2}}}{2\sqrt{3}a_1^{\frac{3}{2}} }-\frac{\varepsilon ^{\frac{5}{2}}}{4\sqrt{3}a_1^{\frac{5}{2}} },\\ \varGamma _{C3} (\varepsilon )= & {} \frac{3\varepsilon ^{\frac{1}{2}}}{8\sqrt{5}a_1^{\frac{1}{2}} }-\frac{\varepsilon ^{\frac{3}{2}}}{2\sqrt{5}a_1^{\frac{3}{2}} }+\frac{\delta ^{\frac{5}{2}}}{8\sqrt{5}a_1^{\frac{5}{2}} }\nonumber \end{aligned}$$
(9)
$$\begin{aligned} F_C (x,y)=\frac{-1}{4\uppi (1-\nu )}\int \limits _{-a}^a D_k (\varepsilon )\ln \left[ {(x-\varepsilon )^{2}+y^{2}} \right] ^{\frac{1}{2}}\mathrm{d}\varepsilon .\nonumber \\ \end{aligned}$$
(10)
Fig. 10
figure 10

Special crack tip element with four equal sub-elements

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Inserting the common displacement discontinuity function, \(D_k (\varepsilon )\) Eq. (8) in Eq. (10) gives:

$$\begin{aligned}&F_C (x,y)=\frac{-1}{4\!\uppi (1-\nu )}\nonumber \\&\quad \times \left\{ \left\{ \int \limits _{-a}^a \varGamma _{C1} (\varepsilon )\ln \left[ {(x-\varepsilon )^{2}+y^{2}} \right] ^{\frac{1}{2}}\mathrm{d}\varepsilon \right\} D_k^1 \right. \nonumber \\&\quad +\left\{ \int \limits _{-a}^a {\varGamma _{C2} (\varepsilon )\ln } \left[ {(x-\varepsilon )^{2}+y^{2}} \right] ^{\frac{1}{2}}\mathrm{d}\varepsilon \right\} D_k^2 \nonumber \\&\quad +\left. \left\{ \int \limits _{-a}^a {\varGamma _{C3} (\varepsilon )\ln } \left[ {(x-\varepsilon )^{2}+y^{2}} \right] ^{\frac{1}{2}}\mathrm{d}\varepsilon \right\} D_k^3 \right\} . \end{aligned}$$
(11)

Inserting the shape functions, \(\varGamma _{C1} (\varepsilon )\), \(\varGamma _{C2} (\varepsilon )\), and \(\varGamma _{C3} (\varepsilon )\) in Eq. (11) after some manipulations and rearrangements the following three special integrals are deduced:

$$\begin{aligned} I_{C1} (x,y)= & {} \int \limits _{-a}^a {\varepsilon ^{\frac{1}{2}}\ln \left[ {(x-\varepsilon )^{2}+y^{2}} \right] ^{\frac{1}{2}}\mathrm{d}\varepsilon },\nonumber \\ I_{C2} (x,y)= & {} \int \limits _{-a}^a {\varepsilon ^{\frac{3}{2}}\ln \left[ {(x-\varepsilon )^{2}+y^{2}} \right] ^{\frac{1}{2}}\mathrm{d}\varepsilon }, \\ I_{C3} (x,y)= & {} \int \limits _{-a}^a {\varepsilon ^{\frac{5}{2}}\ln \left[ {(x-\varepsilon )^{2}+y^{2}} \right] ^{\frac{1}{2}}\mathrm{d}\varepsilon } \nonumber . \end{aligned}$$
(12)

Based on the linear elastic fracture mechanics (LEFM) principles, the Mode I and Mode II stress intensity factors \(K_\mathrm{I}\) and \(K_{\mathrm{II}}\), (expressed in MPa \(\cdot \hbox { m}^{1/2}\)) can be written in terms of the normal and shear displacement discontinuities obtained for the last special crack tip element near the hole as [47–52]:

$$\begin{aligned}&K_\mathrm{I} =\frac{\mu }{4(1-\nu )}\left( {\frac{2\!\uppi }{a_1 }} \right) ^{\frac{1}{2}}D_y (a_1 ),\hbox { and }\nonumber \\&K_{\mathrm{II}} =\frac{\mu }{4(1-\nu )}\left( {\frac{2\!\uppi }{a_1 }} \right) ^{\frac{1}{2}}D_x (a_1 ), \end{aligned}$$
(13)

where \(\mu \) is the shear modulus and \(\nu \) is Poisson’s ratio of the brittle material.

3.2 Numerical analysis

Cubic elements formulation of two-dimensional displacement discontinuities along with three special crack tip elements is used to develop a computer program for the solution of plane elasticity crack problems. This method uses four collocation points for each boundary element (i.e., two sub-elements on either side of the center of the element in question), and is modified so that it is able to model elastostatic and crack problems in finite, infinite, and semi-infinite planes. A third order (cubic) distribution of displacement discontinuities is assumed along each boundary element with only two degrees of freedom. This approach more efficiently improves the accuracy of the conventional displacement discontinuity method. Although using three special elements for the treatment of each crack tip is somewhat complicated, it will greatly increase the accuracy of the displacement discontinuity variations near these singular ends.

The LEFM concept is used to compute the Mode I and Mode II SIFs and the \(\upsigma \)-criterion is also implemented in the computer code to predict the possibility of crack propagation and estimate the crack initiation direction.

The crack propagation paths are estimated by an incremental crack extension (of length \(\Delta b=0.1{b}\)) in the predicted direction by using a standard iterative. In the present analyses, to investigate the crack propagation and cracks coalescence phenomenon, single holes with different diameters and multiple small holes in the disc and rectangular specimens are considered at the central part of a disc specimen and loaded diametrically (under compression). The crack propagation path of the emanating cracks of each hole has been estimated by the iterative method and the coalescence of the cracks and holes in specimens containing multiple holes has been observed. In this iterative method, the cubic displacement discontinuity elements (i.e., using a relatively smaller number of elements but larger number of nodes) gives accurate results for the Mode I and Mode II stress intensity factors.

3.2.1 Cracking boundaries for ring discs and rectangular specimens

In the simulation of the ring discs and rectangular specimens, the discretization of the cracking boundaries has been accomplished by using 14 cubic elements along Brazilian discs, 20 cubic elements along rectangular specimens,and ten cubic elements along each internal ring (Fig. 11a, b). In addition, in the propagation process of the cracks, the crack propagation angle \(\theta \) for each crack has been calculated in different steps by incrementally extending crack length in the direction of \(\theta \) for about 1–2 mm in each step. Two cubic elements are taken along each crack increment and three crack tip elements are also added to the last crack increment. In the numerical modeling, the ratio of crack tip length, L to the crack length, b is \(0.2\, (L/b =0.2)\) and three special crack tip elements are used.

Fig. 11
figure 11

Discretization of the cracking boundaries for. a Ring type disc specimen (Brazilian disc specimen). b Rectangular specimen

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3.2.2 Numerical simulation of the experimental works

The same rock-like disc and rectangular specimens containing either a single thick cylindrical hole or multi-holes (as studied experimentally in Sects. 2.1.3, 2.1.3 and 2.2.2) are also simulated based on the modified higher order displacement discontinuity method and solved by a computer code (iterative method code). The proposed code has also the potential of predicting the wing cracks extension and the breaking stress of the specimens. The mechanical properties (i.e., input parameters) of the simulated specimens are the same as those already given in the experimental work.

The cracks propagation and cracks coalescence in rock-like brittle materials (e.g., rocks and concrete) is numerically studied and these results are compared with the corresponding results observed experimentally in Fig. 5a–g,  6a–e and 8a–c. The disc specimens containing single cylindrical holes (already shown in Fig. 5a–g) are numerically modeled by the proposed higher order displacement discontinuity method and the crack propagation paths are graphically shown in Fig. 12a–e. The five different disc specimens containing multi-cylindrical holes (already shown in Fig. 6 a–e) are also simulated by the proposed numerical method and the results are graphically shown in Fig. 13a–e. Finally, the pre-holed rectangular specimens containing different arrangements of holes as shown in Fig. 8a–c are numerically simulated by the modified code. The corresponding numerical results are illustrated in Fig. 14a–c.

Fig. 12
figure 12

Numerical simulation of the crack propagation path for cylindrical specimens with an axial hole (ring type disc specimen) under diametrical compression

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Fig. 13
figure 13

Numerical simulation of the crack propagation paths and cracks coalescence for Brazilian discs with multiple holes (or PHDS) under diametrical compression. a Three holes with vertical arrangement and \(r/R=0.006\). b Three holes with diagonal arrangement and \(r/R=0.006\). c Five holes with uniform arrangement and \(r/R=0.006\). d Six holes with two parallel arrangement and \(r/R=0.006\). e Four holes with two parallel arrangement and \(r/R=0.006\)

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Fig. 14
figure 14

Numerical simulation of the crack propagation paths and cracks coalescence for rectangular specimens with different arrangements of holes under uniaxial compression. a Horizontal arrangement of three holes with a diameter of 15 mm. b Vertical arrangement of three holes with a diameter of 15 mm. c Diagonal arrangement of three holes with a diameter of 15 mm

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In the present numerical simulations, the Mode I and Mode II SIFs proposed by Irwin (1957) are estimated based on the LEFM approach [44]. A boundary element code is provided using the maximum tangential stress criterion given by Erdogan and Sih [33] in a stepwise procedure so that the propagation paths of the propagating radial cracks are estimated [41, 42]. These simulated propagation paths are in good agreement with the corresponding experimental results (already explained in Figs. 5, 6, 8) and are presented in Figs. 12, 13 and 14 for comparison.

4 Discussion

The mechanism of crack propagation, cracks growth path, cracks coalescence in pre-holed disc and rectangular specimens of rock-like materials (i.e., a type of fine grained concrete) have been studied both experimentally and numerically. The experimental and numerical results given in this research are in good agreement with each other. These results are also compared with those already cited in the literature [12], which validates the applicability and accuracy of the present research approach for the crack analyses of brittle rocks and rock-like materials such as concrete.

The center slant crack problem (Fig. 15) has been used by many researchers to verify many fracture mechanics works [41–43, 45, 46, 53, 54]. In the present work, the same problem has been numerically solved by the proposed approach and the results are compared with the results cited in the literature by using the standard extended finite element method (XFEM) XFEM and EFEM+SC Map.

Fig. 15
figure 15

A center slant crack in an infinite body under uniform tension parallel to the y axis

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The normalized stress intensity factor, \(K_\mathrm{I} /(\sigma \sqrt{b\!\uppi })\), has been numerically evaluated by the proposed HDDM. These numerical results are compared with the results obtained by Natarajana et al. [54], using the standard XFEM and EFEM+SC Map (Fig. 16).

Fig. 16
figure 16

The normalized stress intensity factor, \(K_\mathrm{I} /(\sigma \sqrt{b\!\uppi })\), for the center crack problem (obtained by the proposed HDDM method and XFEM methods is given by Natarajana et al. [54]

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The normalized stress intensity factor \(K_\mathrm{I} /(\sigma \sqrt{b\!\uppi })\) for this particular problem can be estimated from the analytical results given by Scavia [55] as 1.000. This comparison illustrates the high accuracy of the results obtained by the indirect boundary element method by using only a small number of elements compared to the extended finite element method results.

It should be noted that boundary element code (where applicable) is much faster as this method essentially reduces one dimension of the problem and alternatively reduces the mesh size sharply and makes the discretization of the problem simpler and quicker. As can be seen in Fig. 16, XFEM needs more than 6400 nodes to present an accurate solution but the HDDM method using cubic elements has reached the same level of precision and even with better accuracy by using about 100 nodes.

In addition, Tang and Hudson [12] have numerically (using a finite element code) presented the solution for the problem shown schematically in Fig. 1 considering a rock-like specimen with diameter of 150 mm containing a single cylindrical hole (i.e., a ring type rock like specimen) under diametrical compressive loading. Also, Tang and Hudson [12] have numerically investigated the crack propagation patterns of ring type specimens for different r / R ratios, \(r/R =0.1, 0.3, 0.5\). They have used the \(\hbox {RFPA}^{2\mathrm{D}}\) code (a 2D finite element code) to conduct a number of numerical simulations. Table 2 shows the mechanical properties of rock specimens used in their simulations.

Table 2 Mechanical properties of a rock specimen [12]
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Figure 17 illustrates the numerical results presented by Tang and Hudson [12] using \(\hbox {RFPA}^{2\mathrm{D}}\) simulations of the crack propagating patterns in the pre-holed specimens with different r / R ratios, \(r/R =0.1, 0.3,\) and 0.5 respectively.

Fig. 17
figure 17

\(\hbox {RFPA}^{2\mathrm{D}}\) simulation of the propagating paths in pre-holed specimens with different r / R ratios (\(r/R =0.1, 0.3, \mathrm{and }\, 0.5\)) [12]

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In this study, the same problem is numerically solved with the proposed indirect boundary element method. The numerical results obtained by the boundary element simulation of pre-holed disc specimens are shown in Fig. 18. Comparing Figs. 17 and 18 illustrates that the crack propagation paths shown in Fig. 18 are in good agreement with the numerical results given by Tang and Hudson [12] in Fig. 17. Therefore, comparing the results graphically shown in Figs. 17 and 18 clearly demonstrate the accuracy and validity of the boundary element results presented in this study. It should be noted that the boundary element code is much faster, and it is quite easy to work with it because the boundary element method essentially reduces one dimension of the problem and alternatively reduces the mesh size sharply and makes the discretization of the problem simpler and quicker so that the computation time is reduced considerably. It should be noted that different crack propagation increments (steps) are used in the numerical analysis of the present problem.

Fig. 18
figure 18

Boundary element analysis of crack propagation process in pre-holed disc specimens (based on the mechanical properties given in Table 2)

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5 Conclusions

The subject of cracks propagation and cracks coalescence in brittle solids such as geo-materials (rocks and concrete) has gotten much attention in recent years. Further research may be devoted to investigating the mechanism of the crack propagation process in rocks and rock-like materials. In this research, effects of the breaking load in Brazilian disc concrete specimens each containing a single hole (i.e., ring type rock-like specimens) of different size is studied first. The effect of distribution of the initial lateral stress on the crack propagation process of these specimens is investigated. The stress distribution and crack propagation analyses are accomplished both experimentally and numerically and the results are compared and discussed. Then, some pre-holed Brazilian discs and rectangular specimens are tested under compression and the experimental results are compared with the numerically simulated results obtained by using the higher order displacement discontinuity method (a version of the indirect boundary element method).

It is concluded that the crack propagation mechanism in the brittle solids such as rocks and concrete may occur due to the cracks coalescence phenomenon in the bridge area, which is mainly caused by propagation of tensile radial cracks emanating from the surface of the central holes. There is an attempt to simulate all the experimental tests by the proposed indirect boundary element method, and it has been observed that the corresponding numerical results are in good agreement with the experimental results. Comparing the experimental and numerical models illustrates that the tensile cracks and the cracks propagation paths are mainly produced by the coalescence phenomenon of the pre-existing multi-holes in PHDS (prepared from a special type of fine grained concrete).

However, based on the present analyses, it may be also concluded that the cracks may be initiated radially near the surface of the holes, the stress concentration can be released and finally, stresses in the specimen can redistributed to attain a new equilibrium condition. Then, the final breaking of the pre-holed disc specimens may be due to the propagation of radially induced tensile cracks initiated from the surface of the central hole and propagating toward the direction of diametrical loading and/or perpendicular to it (for the case of specimens with larger holes). In the case of disc specimens with multiple holes, the cracks propagation and cracks coalescence may occur simultaneously in the breaking process of PHDS under diametrical compressive loading. In the case of rectangular specimens with a different arrangement of the holes, it is shown that the crack propagation process depends strongly on the location of cylindrical holes.