Effect of crack angle and concrete strength on the dynamic fracture behavior of rock-based layered material containing a pre-existing crack

Abstract

Dynamic fracture behavior of rock-based layered material containing a pre-existing crack was investigated, which is crucial to evaluate the stability of rock-based systems. Dynamic tests on semi-circular bend (SCB) samples were conducted with a split-Hopkinson Pressure Bar (SHPB) system and the cracking process was collected using digital image correlation (DIC). An MTS Insight 30 electromechanical test equipment was adopted to perform static fracture experiment for comparison. Different typical fracture modes were classified: interfacial fracture, combined mode of tensile–shear fracture, and combined mode of compression–shear fracture. Regardless of static load or dynamic load, the fracture mode is controlled by crack dip angle, and the peak load is proportional to crack dip angle. The increase in concrete strength makes a strengthening impact on the bearing capacity of SCB sample, which can prevent the propagation of secondary cracks in the concrete. Both static and dynamic fracture toughness are significantly affected by dip angle. The difference between static and dynamic fracture toughness of mixed-mode reduces initially and subsequently increases with an increase in dip angle. Furthermore, comparing with static fracture toughness, dynamic fracture toughness substantially depends stronger on dip angle. The dip angle and concrete strength have an obvious effect on the fracture resistance.

1 Introduction

Rock-based layered structures are frequently utilized in engineering construction projects, including tunneling, civil engineering, and mining. Examples include surrounding rock-shotcrete/rock-lined structures, concrete gravity dams, and cement paste backfill (CPB) structures [1,2,3,4,5,6]. Barrett et al. [1] proposed four deterministic design models for the shotcrete surrounding rock structure. Based on these models, studies show that adhesion is crucial for the support ability of shotcrete, but after the peak, flexural strength and toughness play a controlling role. Johnston et al. [2] investigated the interface shear behavior between concrete and rock in bored piles and found that the lateral shear performance was determined by the shear properties of the rough interface between concrete and rock. Ulfberg et al. [3] investigated the effect of concrete material parameters on the ultimate bearing capacity of a scale model dam based on scale model tests. The results indicate that the material parameters of concrete play a vital role in the load-bearing capacity of the dam when there is a significant geometric change in the concrete–rock dam foundation interface. Fishman et al. [4] explained the fracture mechanism of concrete structures on rock foundations under compression–shear conditions through experimental analysis. Alneasan et al. [5] studied inclination interface crack between two dissimilar rock layers, revealing that the extension path of interface crack is determined by the elastic stiffness ratio of two rock layers and crack dip angle. Huang et al. [6] conducted an experimental study for CPB-rock structures and found that different binder types and contents have significant effects on mode I fracture toughness of interface. Interfaces and bedrock flaws at various scales may exist in rock-based structures. The interface between the two materials is a weak point in rock-based layered system, and flaws in the rock could cause a hidden threat to the stability of layered structure, which will lead to the weakening of the disaster resistance of the overall structure. Furthermore, the layered material is highly prone to disasters due to the influence of human activities and geological environment variables [7,8,9,10,11,12,13]. For example, Sujatha et al. [7] pointed out that the interface will aggravate the crack extension under load, and based on the fracture mechanics method, a friction model considering the interaction between interface was proposed. Fishman et al. [8] discussed the damage mechanism of rock–concrete structures and proposed a practical method for studying the ultimate turning resistance of structures. According to a modified interface cracking model, the stress intensity factor in different materials was obtained [9, 10]. Zhou et al. [11, 12] found that fractures and joints have an essential influence on the overall fracture characteristics of surrounding rock, and elucidated their relationship with the fracture behavior of material. Meng et al. [13] experimentally investigated the influences of bedding plane and fracture angle on the cracking behavior of layered materials in mixed mode, and revealed the fracture mechanism of layered material. Thus, a comprehensive understanding of the fracture propagation behavior and fracture characteristics for rock-based bi-material systems with cracks of varying dip angles is critical to assessing the overall stability.

Research scholars on the fracture characteristics of rock–concrete have conducted plenty of work in terms of experiments, numerical simulations, and theoretical analysis, where three-point bending (TPB) tests and four-point bending (FPB) methods [14,15,16,17] are usually used. For example, the evolution of fractures in layered rock was investigated through TPB experiments, revealing the mechanism of fracture penetration/deflection of layered planes [14, 15]. Luo et al. [16] presented a method through TPB experiments to quantify the energy at material failure. Marsavina et al. [17] elucidated the mechanism of interface crack extension in bi-material samples via FPB experiments. Charalambides et al. [18] investigated the interface fracture characteristics based on the FPB test method. Suo and Hutchinson [19] presented a sandwich configuration sample and reported cracking behavior for interface crack. Lee and Buyukozturk [20] also analyzed the fracture characteristics of bi-material interface with FPB tests using sandwich samples. Buyukozturk and Hearing [21] reported cracking in the interfacial region of two-phase composites and parameters affecting the fracture behavior of concrete by TPB experiments. Agrawal et al. [22] demonstrated that the interface fracture characteristics of aluminum–vinyl ester bi-material samples were controlled by the mode mixing ratio based on FPB experiments. Yang et al. [23] investigate the fracture characteristics of the rock–concrete using different fracture modal mixtures in terms of TPB and FPB tests. Zhong Hong et al. [24] analyzed the relationship between cracking behavior of bi-material and modal mixing ratio using an FPB method. Chen et al. [25] performed TPB experiments to verify the fracture toughness of materials obtained from the wedge splitting model. Guan et al. [26,27,28] proposed a simple closed-form solution and a simplified two-point method that can be utilized to obtain fracture toughness through a series of TPB and FPB experiments. Dong Wei et al. [29,30,31] conducted TPB and FPB tests on rock–concrete layered samples with various interface roughness, presented a cracking criterion for rock–concrete interface, and calculated related mechanical parameters. These studies are undoubtedly conducive to understanding the fracture characteristics of rock–concrete systems and are crucial in improving the construction quality and safety of rock foundation systems. It is worth noting that the above bending tests are inappropriate for beam members containing weak interfaces, because such beam samples may crack or even break prematurely under their high self-weight. Therefore, an SCB method was further proposed and recommended as a standard approach to structural fracture research [32]. This type of configuration sample has several strengths over other samples, including cost-effectiveness, convenience processing and testing procedures, and universal testing device requirements. Moreover, by introducing cracks at various angles in SCB configuration samples, the cracks can experience a variety of stress states, and then realize modes including tension, shear, and combination mode. Hence, the SCB test approach is frequently utilized to investigate the flaw growth behavior and fracture toughness of laminated medium [33,34,35,36,37,38]. Scholars used SCB samples to detailedly investigate the influence of loading conditions, mechanical characteristics, and bedding structure on the fracture behavior of layered media. The above studies indicated that mechanical behavior is significantly affected by bedding strength and loading direction, in which propagation path and fracture toughness are determined from loading direction [33,34,35]. Nejati et al. [36] proposed an improved SCB test method for layered rocks, which presents the relationship between different anisotropy ratios and orientations and normalized stress intensity factors. The primary purpose of the above studies, however, is to study the effect of various bedding structure parameters on fracture behavior of layered media. Recently, Huang and Fang et al. [6, 37, 38] used SCB samples to analyze the fracture characteristics of CPB–rock structure, which revealed that the fracture behavior of composite structures was determined by cementing materials, environmental variables, and loading methods.

The above studies are conducive to clearly understanding of the mechanical characteristics of composite structures under static loading. For concrete construction projects on practical rock foundations, structure-rock foundation system are more susceptible to dynamic disturbances, which include hydrodynamic loads on concrete hydraulic dam foundations, oscillation of vehicles driving on tunnels and roadbeds and impact loads such as blasting and rock bursts in underground tunnel construction. Investigation on rock/concrete fracture mechanics under dynamic loads has always been a hot topic, and researchers have conducted many experiments and achieved important research results [39,40,41,42,43,44]. Unfortunately, the lack of studies on the mechanical characteristics of cracked combined structures under dynamic conditions limited our understanding of the cracking behavior of this structure type. Recently, Qiu Hao et al. [45, 46] studied dynamic crack extension behavior and failure path of rock-mortar samples under dynamic conditions. Research shows that interface roughness and impact speed obviously influence the crack propagation path and composite fracture toughness of composite structures. Focusing on the influence of interface crack tip position and related parameters, Sundaram et al. [47] analyzed fracture characteristics of layer structures, and explained the crack penetration-branching mechanism. Their results showed that the crack angle plays a significant role in interfacial crack growth and fracture toughness.

These basic studies are significantly beneficial to understand the fracture mechanism of rock-based layered systems, which is important for improving its security and durability. In practical construction projects, the rock foundation may have several natural flaws and form arbitrary angles with the interface. Since the crack–interface interaction, the fracture characteristics of rock-based layered system are particularly complicated and depend on the combined effect of interface properties and its whole behaviors. Although scholars have struggled to investigate the interface cracking of composite structures, they have mainly focused on interface cracking behavior, with a few studies focused on crack–interface interaction; thus, our knowledge of the stability of layered structures is limited. To overcome this issue, the current study performed experiments on rock-based layered SCB samples that adopted the Split Hopkinson Pressure Bar (SHPB) test system, with the high-speed camera device being utilized to monitor the fracturing process. The cracking behavior under various dip angles and concrete strength is analyzed. Furthermore, the stress intensity factors of rock-based layered SCB samples were solved utilizing ABAQUS software, and the effects of varied crack dip angle and concrete strength on its fracture toughness were investigated. Static tests were then performed on rock–concrete samples utilizing an MTS Insight 30 electromechanical control test system for comparison.

2 Experimental methods

2.1 Sample and test procedure

Figure 1 depicts the overall process for preparing rock-based layered SCB samples. The specific preparation method is as follows. Natural sandstone blocks without obvious fissures were made into sandstone plates with dimensions of 300 mm × 110 mm × 50 mm, and artificial grooves with dimensions of 1.8 mm × 1.4 mm × 2.2 mm (depth × width × spacing) were cut on the surface of the slabs for roughening to promote interfacial adhesion. The sandstone slab was then placed in a mold, which is full of 40 mm of thick concrete. Then, it was cured in a standard curing room for 28 days under the same room temperature and humidity, the disc samples with diameter (D) and thickness (B) of 75 mm and 30 mm were machined. In this process, dimensional errors of each disc were carefully restricted within ± 0.5 mm, and flatness of both end surfaces was strictly controlled within ± 0.1 mm. Finally, the SCB sample was obtained by cutting the disc in half, and a crack of 2a (a = 16 mm) in length and 1 mm in width (t) was created in SCB sample utilizing a precision cutting device. Six pre-existing cracks (θ = 0°, 15°, 30°, 45°, 60°, and 75°) were selected to analyze the impact of cracks on fracture characteristics of SCB samples in detail (Fig. 1). The span denoted by 2S, between the two bearings measures 37.5 mm (Fig. 2).

Fig. 1

Preparation process for rock-based layered SCB sample

Fig. 2

Schematic diagram of layered sample for SCB test

There were 72 samples used in this study, 18 were subjected to static load testing, and the remaining 54 were subjected to dynamic load testing. The geometry of test samples as well as the average experiment results is presented in Table 1. The samples’ numbers are S-θ, where S stands for the static condition and θ stands for the angle of pre-existing crack to interface. Likewise, for samples marked DC-θ, D is dynamic conditions, C stands for A, B, or C, denoting various concrete strength of SCB samples, and θ denotes the angle of pre-existing crack to interface.

Table 1 Basic parameters and mechanical properties

To effectively use the experiment findings to evaluate the stability and durability of rock-based structures in practice, black sandstone rock masses with no obvious fractures were chosen in the field, and concrete materials were used to conduct laboratory experiments of layered structures. Figure 3 illustrates the sand particle-size gradation curves. Three various mix proportions of concrete were adopted, with mass ratios of cement, sand, and water of 1.0:2.1:0.5, 1.0:1.6:0.4, and 1.0:1.0:0.35, respectively. The material properties of sandstone and concrete are listed in Table 2. Among them, the dynamic tensile strength was obtained by conducting SHPB tests on rock and concrete Brazilian disc samples (with diameter of 50 mm and thickness of 25 mm) prepared in the laboratory.

Fig. 3

Grading curves of sand

Table 2 Material properties

2.2 Testing apparatus and procedure

2.2.1 Test setups

Static SCB tests were conducted with a 30 kN MTS Insight 30 system, as presented in Fig. 4. The sample was supported on a loading plate by two steel round bars (3 mm in diameter). The current tests were conducted adopting displacement control (0.02 mm/min) method. The displacement and load in the vertical direction of the sample were measured by a high-precision (0.001 mm) linear variable differential transformer and an electronic load transducer (load accuracy: ± 0.5%), respectively.

Fig. 4

Static SCB tests. LS low speed

Figure 5a indicates that an SHPB system was employed for the dynamic SCB testing, which included a bullet-shaped striker, incident and transmitted bar, and a buffered bar. Many investigations have proven that mechanical behavior is significantly affected by the friction between bars and sample end face [48,49,50]. As a result, petroleum jelly is necessary to uniformly coat the loading end of the SCB sample to decrease the frictional effect. By opening the gas valve, the conical strikers are launched from the pressure vessel and strike the right side of incident bar, generating a half-sine wave that passes on incidence bar to sample, as shown in Fig. 5b. There are two types of waves that can result from an incident wave reaching a sample: the reflected wave and transmitted wave. The three waves were captured using two strain gauges and recorded with a digital oscilloscope.

Fig. 5

Dynamic SCB tests: a schematic diagram; b loading equipment. HS high-speed, In incident, Tr transmission, Re reflection, SG strain gauge

2.2.2 DIC technique

The fracture processes of layered SCB samples under static conditions were obtained using a low-speed photography system. Figure 4 presents the primary components of the low-speed photography equipment, namely a camera including an image sensor and two LED flashes (ZF-3000). The two LED flashlights were placed in front of the sample for illumination. An LS camera with a resolution of 4096 × 3000 pixels is adopted, and the frame rate was set to 15 fps (frames per second), which can effectively capture the fracture propagation process.

In the dynamic test, the HS camera device was adopted to capture the dynamic fracture process simultaneously. Two LED flashlights were lighted on the surface of samples, and the HS equipment (Vision Research Inc.—Phantom V711) with a resolution setting of 256 × 256 pixels and an inter-frame time of 12.6 µs was used to capture the dynamic cracking and expansion process, as shown in Fig. 5. When the voltage value measured via SG is greater than − 34 mV, and then, oscilloscope releases a transistor–transistor logic pulse to trigger the HS equipment, which automatically records the complete deformation image of the sample.

The DIC method can efficiently record strain field and deformation path data on the surface of material and is extensively used for the investigation of the fracture characteristics of objects [11, 51, 52]. It was employed to collect deformation field information image of rock-based layered material in the current investigation. Then, the DIC post-processing calculations were performed in the MATLAB environment using the Ncorr [53] program. This program was primarily derived from Pan’s reliability-guided DIC approach [54]. The computational procedure is shown below: (1) Set an initial image as a reference and then select target images; (2) introduce the semi-circular region of interest (ROI); (3) calculate the final displacement; (4) obtain the strain field information.

2.2.3 Test scheme

To thoroughly study the impact of crack dip on the cracking behavior of layered structures under dynamic and static conditions, dynamic and static experiments were conducted on SCB samples with six prefabricated crack dips, respectively, as illustrated in Fig. 1. In addition, three different concrete strengths were considered in dynamic testing to assess the effect of bi-material strength ratio on the dynamic fracture characteristics of SCB samples. In the current work, tests were repeated three times for each dip angle and concrete strength to assure the validity of results. The SCB samples were placed on a loading frame with two supports, and the LS photography device was debugged to maintain acquisition synchronization with the MTS loading device (Fig. 4). The SCB samples were then placed between the incident and transmitted bars in the dynamic test, and data were collected using the HS camera equipment in synchronization with the SHPB load device.

2.3 Testing principle

Dynamic balance was thoroughly checked for all dynamic test samples to assure data validity. Figure 6 shows the dynamic balance of a typical SCB sample. In Fig. 6a, the strain gauge monitors the initial pulse signal under dynamic load. Figure 6b indicates that the dynamic pressure P₁ (pressure at the incidence end) and P₂ (pressure at the transmission end) at both ends of sample are basically similar. According to the aforementioned findings, the dynamic balance requirement is satisfied under dynamic loading, indicating that the inertial effect can be ignored, since the force differential that creates is negligible in the sample [55].

Fig. 6

Dynamic force history in SCB tests: a original signal and b dynamic force balance

The dynamic force P₁ and displacement u₁ at the incident bar–sample interface, and dynamic force P₂ and displacement u₂ at the transmitted bar–sample interface are obtained using the following expressions, derived from one-dimensional stress wave theory [56]:

$$P_{1} = EA\left[ {\varepsilon_{i} (t) + \varepsilon_{r} (t)} \right],P_{2} = EA\varepsilon_{t} (t)$$

(1)

$$u_{1} = C_{b} \int_{0}^{t} {\left[ {\varepsilon_{i} (t) - \varepsilon_{r} (t)} \right]} {\text{ d}}t,u_{2} = C_{b} \int_{0}^{t} {\varepsilon_{t} (t)} {\text{ d}}t.$$

(2)

In these equations, A, E, and C_b refer to the bar cross-sectional area, Young’s modulus, and elastic wave velocity, respectively, and ε_i(t), ε_r(t) , and ε_t(t) denotes the incident strain, reflected strain, and transmitted strain, respectively.

After verifying in strict dynamic equilibrium, the average force and relative displacement δ(t) of the sample can be obtained by

$$\overline{P} (t) = \frac{1}{2}\left[ {P_{1} (t) + P_{2} (t)} \right],\delta (t) = u_{1} (t) - u_{2} (t).$$

(3)

3 Test results

3.1 Fracture behavior analysis

3.1.1 Fracture process

The crack dip angle under static and dynamic loading has an important impact on the development process and propagation path of the sample. Given that development procedures under static and dynamic loading possess similarity, this study takes the fracture development process under dynamic loading as an example. Furthermore, high-speed camera and DIC were utilized to analyze the dynamics of vertical strain ε_yy, horizontal displacement u_x and vertical displacement u_y in SCB samples over time. Figure 7a–g presents the characteristic images corresponding to the typical time of crack development process for samples with various crack dip angles, containing original failure images, and evolutions of strain and displacement. In Fig. 7, different-colored bars represent different sizes of ε_yy, u_x, and u_y for layered SCB samples. It is positive for line strain elongation among them (under tensile stress). Positive values for a displacement field are along the positive axis, whereas negative values represent the opposite displacement.

Fig. 7

Fracture process of layered SCB sample under dynamic load conditions

At the start of the loading (0 μs), there are no noticeable strains or displacements in the sample. Figure 7a presents the crack development of SCB sample at θ = 0°. It is observable that ε_yy occurs in the region near the crack tip when interface crack initiates at the pre-existing crack tip (75.6 μs). Then, a secondary crack occurs in the concrete (100.8 μs). Subsequently, a new secondary crack appeared in the rock (176.4 μs), and coalescence formation along the interface led to sample failure.

The wing crack initiates first (88.2 μs) at the pre-crack tip, and ε_yy, u_x, and u_y increase significantly near the crack tip when θ = 15° (Fig. 7b). Then, it develops along the direction of the loading end on the right (113.4 μs), and secondary cracks appear in the concrete. As the time reaches to 151.2 μs, the wing crack coalescence occurs at the loading end on the right side, and the strain field and displacement field change significantly.

The failure modes of SCB samples with θ = 30° and 45° are illustrated in Fig. 7c, d, which are comparable to the sample with θ = 15°. The primary distinction between these two cases refers to the moment of wing crack emergence and coalescence. Notably, the new crack starts from the deviation of crack tip for SCB samples with θ = 60° and 75°, and ε_yy is formed in the area away from the crack tip; following that, it develops toward interface, and eventually propagates to the right load end. Moreover, the development length of wing crack and the distance between crack origination location and crack tip both increase with an increase in dip angle. In addition, pre-existing cracks of SCB samples slowly close with increasing dip angle, indicating that the fracture mode of SCB samples has shifted from combined tensile–shear fracture to combined compression–shear fracture.

3.1.2 Failure pattern

The impacts of concrete strength and crack dip on failure results for SCB samples in the current dynamic test are illustrated in Fig. 8. The fracture patterns of the samples, which contain three strength ratios of SCB samples, are basically similar. However, due to the difference in concrete strength, the location and quantity of secondary cracks are various, and the secondary crack extension length of the concrete is weakened when the concrete strength is relatively large. The increase in concrete strength leads to a reduction in the length of cracks that develops along the interface, which has an impact on the crack development path. In the previous studies [57], it was found that interface strength plays a critical role in the crack propagation path. When the interface strength is low, interface crack generally appear in SCB sample. The length of interface crack gradually reduces with increasing interface strength, and wing cracks develop. It is noteworthy that when the dip angle is larger (60° and 75°), cracks start far away from the tip. Additionally, the expansion direction of wing cracks is controlled by the dip angle.

Fig. 8

Effects of concrete strength. σ_cs concrete strength, σ_rs rock strength

Figure 9 shows the fracture propagation paths of SCB samples with different crack dip angles under dynamic and static loading. Figure 9a and Fig. 9b indicate that when θ = 0°, new cracks appear at the tip of pre-cracked, and then develop along the interface, that is, tensile fracture (interfacial fracture) occurs in the sample. For the samples with θ = 15°, 30° and 45°, a wing crack appears at the crack tip and extends to interface along the load direction, and the SCB sample undergoes a combined tensile–shear fracture. New cracks start from the deviation of crack tip as the dip angle increases further (60° and 75°). Then, it propagates toward interface and extends to right loading end eventually, and the sample undergoes compression–shear failure. Further analyzing the cases θ = 60° and 75°, it is revealed that the deviation between crack initiation position and crack tip increases with increasing dip angle. Notably, the pre-cracked is open at $\theta = 0^\circ ,{ 1}5^\circ {, }30^\circ {\text{ and }}45^\circ$, while the pre-cracked is closed at $\theta = 60^\circ {\text{ and }}75^\circ$. The above observations show that the crack dip angle made an important impact on the fracture modes of layered SCB sample. The tensile fracture mainly emerges along interface when the dip angle is low. As the dip angle increases to a certain value, a combined shear and tension fracture is formed, and the sample undergoes a combined compression–shear fracture along the loading direction as the dip angle ulteriorly increases.

Fig. 9

Effect of dip angle on failure pattern

Based on the results of sample failure, the three fracture modes are classified below (Fig. 9c):

• 1. The fracture that develops along the interface is called interface fracture (tensile fracture).

• 2. The fracture which originates from the tip of pre-existing crack, then extends along the loading direction to the interface, and finally develops to the loading end is called combined fracture (combined tensile–shear).

• 3. The fracture that initiates far away from the crack tip, then develops to the interface, and eventually propagates to the loading end is known as combined fracture (combined compression–shear).

3.2 Deformation response and peak load

Figure 10a clearly presents static load–displacement curves of samples at various crack dip angles. Early in this loading process, the curve presents nonlinear variation, which is due to the pores and microcracks inside the samples. The microcracks in the sample gradually develop with increasing the load, and the curve shows linear elastic change. After reaching the peak, the primary crack penetrates the sample, and the curve decreases rapidly. It can be seen that the peak load increases gradually when crack dip angle varies from 0° to 75°. The findings suggest that the peak load of SCB sample is highly affected by the crack dip angle. It is further verified that both the peak load and displacement are maximum at θ = 75°, indicating that the crack dip angle determines the deformation of SCB samples [58].

Fig. 10

Load–displacement curves of SCB samples

Figure 10b presents dynamic load–displacement curves of SCB samples at various crack dip angles. The cumulative deformation prior to failure is utilized to determine the displacement corresponding to peak load. The incident energy of the sample before the peak load will be converted into its deformation energy in accordance with the principle of energy conservation. The deformation energy is subsequently obtained via multiplying the load and deformation of samples. The incident energy is constant in the current investigation, and the bigger the load capacity, the smaller the deformation; hence, the deformation reduces with the dip angle varying from 0 to 75°.

Figure 11 presents the peak load comparison under static and dynamic conditions. As the crack dip angle increases, the peak load increases. Under static loading, both the peak load and growth rate are obviously lower than those under dynamic loading. The peak load increases slowly for both static and dynamic loading when the dip angle is between 0° and 30°, while it increases dramatically at θ > 30°.

Fig. 11

Comparison of peak loads

Figure 12 indicates the relationship between peak load and crack dip angle of samples at various concrete strengths. The peak load increases gently when the dip angle varies from 0° to 30°, and after 30°, it increases significantly as the dip angle increases. The above results indicate that the peak load is obviously impacted by the dip angle θ. In addition, it is demonstrated that the whole strength of the sample is greatly influenced by the increase in concrete strength. Previous studies [59] indicate that the interface tensile strength is closely related to the cohesion between rock and concrete. An increase in concrete strength for rock–concrete structures leads to fewer pores at the interface and stronger bonding effect, which sequentially affects the mechanical properties (such as the cohesion) of rock–concrete interface. Additionally, there are many hydration products on the interface surface, which can improve the bonding performance of rock–concrete interface, thereby increasing the nominal tensile strength of rock–concrete sample. Figure 12 presents that the peak load of sample increases with increasing concrete strength, indicating that the increase of concrete strength can enhance the bonding force between rock and concrete. In our previous studies [57], it was found that the peak load significantly increases with increasing interface strength. The peak load tends to stabilize when the interface strength increases to a certain value. As interface strength increases, the difference in peak load between various dip angles increases.

Fig. 12

Effects of dip angle on peak load of SCB samples at different concrete strength

4 Establish finite-element model

4.1 Calculation principle

Fracture toughness is frequently applied in evaluating the rock/concrete fracture resistance, which can be determined by the dimensionless stress intensity factor (SIF) and the critical load. For the SCB tests, the mode I and mode II SIFs in the static and dynamic case (samples satisfy the dynamic equilibrium under dynamic loading) can be expressed as follows [48, 49, 60]:

$$K_{{_{\rm I} }}^{{{\text{s}},d}} = \frac{{P_{{}} \sqrt {\pi a} }}{2RB}Y_{\rm I} \left(\frac{a}{R},\frac{S}{R}\right) \,$$

(4)

$$K_{{_{{{\rm I}{\rm I}}} }}^{s,d} = \frac{{P_{{}} \sqrt {\pi a} }}{2RB}Y_{{{\rm I}{\rm I}}} \left(\frac{a}{R},\frac{S}{R}\right){,}$$

(5)

where K_I^s,d and K_II^s,d represent mode I and mode II SIFs under both static and dynamic loading conditions, respectively, P refers to loading force, R and B refer to radius and thickness of SCB sample, respectively, and Y_I and Y_II are the mode I and mode II dimensionless SIFs, both of which can be determined through finite-element approach.

Mixed mode I–II SIF K_eff is generally referred to as the effective SIF, which reflects the fracture characteristics under the mixed-loading mode and is defined below [61]

$$K_{{_{{{\text{eff}}}} }}^{s,d} = \sqrt {(K_{{_{\rm I} }}^{s,d} )^{2} + (K_{{_{{{\rm I}{\rm I}}} }}^{s,d} )^{2} } {.}$$

(6)

In addition, the mixed parameter M^e can be introduced to discuss the mechanical characteristics under various loading conditions, which is described below

$$M^{e} = \frac{2}{\pi }{\text{ tan}}^{ - 1} \left(\frac{{K_{\rm I} }}{{K_{{{\rm I}{\rm I}}} }}\right).$$

(7)

The material is in mode I loading when K_II = 0, M^e = 1, and it is in mode II loading if K_I = 0, M^e = 0. Various mode mixity ratios $\left( {M^{e} = \, 1,{ (0} - {\text{1) and }}0} \right)$ are determined via adjusting the dip angle of pre-existing crack.

The elastic theory of interfacial fracture of dissimilar materials serves as a theoretical foundation for the current study, which uses finite-element software to determine the SIF using the J-integral approach. For the interfacial fracture in the case of plain strain, the correlation between stresses and SIFs of the crack tip at interface of two dissimilar materials can be determined as follows [12]:

$$\sigma_{22} + i\sigma_{12} = \frac{{\left( {K_{I} + iK_{{{\rm I}{\rm I}}} } \right)r^{i\varepsilon } }}{{\sqrt {2\pi r} }},$$

(8)

where $i = \sqrt { - 1}$ and ε represent the oscillation index, and

$$\varepsilon = \frac{1}{2\pi }\ln \left( {\frac{1 - \beta }{{1 + \beta }}} \right),$$

(9)

where β denotes a Dundurs’ parameter

$$\beta = \frac{1}{2}\frac{{G_{A} (1 - 2\nu_{B} ) - G_{B} (1 - 2\nu_{A} )}}{{G_{A} (1 - 2\nu_{B} ) + G_{B} (1 - 2\nu_{A} )}},$$

(10)

where G denotes shear modulus and v refers to Poisson’s ratio. Subscripts A and B denote material A and B, respectively.

The energy release rate and SIFs are related, which is described as the following formula:

$$G = \frac{1}{2}\left( {1 - \beta^{2} } \right)\left[ {\frac{{1 - \nu_{A}^{2} }}{{E_{A} }} + \frac{{1 - \nu_{B}^{2} }}{{E_{B} }}} \right]\left| K \right|^{2}$$

(11)

$$\left| K \right| = \sqrt {K_{\rm I}^{2} + K_{{{\rm I}{\rm I}}}^{2} } .$$

(12)

4.2 Computation and analysis of SIF

Figure 13 presents the numerical model developed by the ABAQUS software for solving SIFs of SCB samples. The geometric size of the numerical model of the sample corresponds to that of test sample. There are 19,986 elements and 60,609 nodes in the mesh, with refinement of the mesh in crack tip region. The J-integral method is used as a calculation principle to determine the SIF of the crack tip. Five contours are distributed from inside to outside in the crack tip region, and the SIF of fifth contour is utilized in this study.

Fig. 13

Calculation model for SIF: a Model I and b Model I–II

The results obtained in this study are compared with those of the isotropic SCB sample proposed by Lim et al. [60]. Mode I stress intensity factors with different a/R are compared in Table 3. The calculation results in the study are generally in accordance with those of Lim et al. [60].

Table 3 Comparison for SIFs

The impacts of a/R, θ, and E_B/E_A on dimensionless SIF (Y_I and Y_II) are depicted in Fig. 14a and b illustrate that for a given crack length ratio, Y_I first decreases and subsequently increases as θ increases, while Y_II increases first and then decreases as θ increases. Additionally, at lower or larger dip angles, the absolute value of Y_I increases significantly with increasing the crack length ratio. When θ > 30°, Y_I is lower than 0 and Y_II changes irregularly, which is due to the additional traction caused by friction along crack face after closure of pre-cracked. As illustrated in Fig. 14c and d, Y_I and Y_II increase as the modulus ratio increases. When θ = 45° approximately, Y_I = 0, the shear stress is largest, that is, pure-mode II fracture occurs. Furthermore, Fig. 14d presents that Y_II reaches its maximum value at a dip angle of about 31°, suggesting that pure-mode II fracture is not based on the maximum of Y_II. Figure 14b and d presents that Y_II of layered material is not 0 under mode I loading, which is different from that of single material, indicating that the material difference of the layered material has a significant influence on Y_I and Y_II at the tip of crack.

Fig. 14

Effects of θ and E_B/E_A on normalized SIFs: a Y_I versus θ, b Y_II versus θ, c Y_I versus E_B/E_A, and d Y_II versus E_B/E_A

5 Fracture toughness analysis and discussion

5.1 Static fracture toughness

The effect of dip angle on static fracture toughness (K^s_Ic, K^s_IIc and K^s_effc) for different concrete strength is shown in Fig. 15. It is observed from Fig. 15a and b that the static fracture toughness varies nonlinearly with increasing crack dip angle, and mode I static fracture toughness K^s_Ic reduces with an increase in dip angle at θ < 45°, but increases with increasing the dip angle at θ ≥ 45°. Moreover, mode II static fracture toughness K^s_IIc shows a contrary trend when θ < 45° and θ ≥ 45°. Notably, Fig. 15a presents that the change rate of K^s_Ic is small for lower dip angles (θ ≤ 15°). Figure 15b shows that the change rate of K^s_IIc is large for lower and higher dip angles ($\theta \le 15^\circ {\text{ and }}\theta \ge 60^\circ$), while if 15° < θ < 60°. The change rate of K^s_IIc shows an inverted U-shaped change. As illustrated in Fig. 15c, the static mixed fracture toughness K^s_effc initially reduces and subsequently increases with an increase in dip angle, while when θ = 45°, it is at its lowest. It is because mode II fracture occurs at θ = 45°. Additionally, K^s_Ic, K^s_IIc, and K^s_effc can be enhanced by increasing concrete strength. This is because the increase in concrete strength leads to the increase of interface cohesion, which results in higher loads required for sample fracture.

Fig. 15

Impacts of dip angle on fracture toughness: a K^s_Ic, b K^s_IIc, and c K^s_effc

5.2 Dynamic fracture toughness

The impact of dip angle on dynamic fracture toughness (K^d_Ic, K^d_IIc, and K^d_effc) for different concrete strength is shown in Fig. 16. As demonstrated in Figs. 15 and 16, static and dynamic fracture toughness varies similarly. Moreover, the study suggests that dynamic fracture toughness improves with an increase in concrete strength.

Fig. 16

Influences of dip angle on dynamic fracture toughness for different concrete strength: a K^d_Ic, b K^d_IIc, and c K^d_effc

Figure 17 depicts the correlation between static and dynamic fracture toughness at different dip angles. As shown in Fig. 17a, K^s_Ic and K^d_Ic vary similarly with dip angle, while K^d_Ic varies more significantly. The overall trend of the difference between K^s_Ic and K^d_Ic reduces initially and subsequently increases with increasing dip angle. If the dip angle is low or high (θ = 0° or θ = 75°), the difference between K^s_Ic and K^d_Ic is larger, which are 3.46 MPa·m^1/2 and 3.44 MPa·m^1/2 separately, that is, K^d_Ic is about six times K^s_Ic. When θ = 45°, the difference between K^s_Ic and K^d_Ic is basically 0, explaining the dominance of mode II fracture. This is because the mode I fracture toughness K^s_Ic and K^d_Ic is 0 at θ = 45°.

Fig. 17

The variations in fracture toughness at various dip angles: a K^s,d_Ic, b K^s,d_IIc, and c K^s,d_effc

Figure 17b presents that the difference between K^s_IIc and K^d_IIc increases first and reduces subsequently with increasing dip angle. The difference between K^s_IIc and K^d_IIc is 0.05 MPa·m^1/2 at θ = 0°, while when the dip angle approaches 40°, it increases to 1.64 MPa·m^1/2. For θ = 90°, the difference between K^s_IIc and K^d_IIc is 0.36 MPa·m^1/2. The results reveal that the difference between K^s_IIc and K^d_IIc is larger at medium dip angles. This is because mode I fracture mainly occurs when θ is small or large.

The comparison of K^s_effc and K^d_effc at various dip angles is indicated in Fig. 17c. The difference between K^s_effc and K^d_effc is 3.45 MPa·m^1/2 at θ = 0°, 1.64 MPa·m^1/2 when θ = 45°, and 3.46 MPa·m^1/2 when θ is further increased to 75°. This is due to the weak interfacial bonding, which causes tensile fracture along the interface at θ = 0° (loading along the interface). Shear fracture appears primarily in samples for θ = 45°, while combined compression–shear failure occurs for θ = 75°. The above results indicate that dip angle plays an important role in determining the disparities between static and dynamic fracture toughness, and the latter of which is more reliant on dip angle than the former.

5.3 Effects of various factors on fracture resistance

The relationship between K^s,d_Ic and K^s,d_IIc is illustrated in Fig. 18. It is observed in Fig. 18a and b that the variations of K^s,d_Ic and K^s,d_IIc are similar, both of which are semi-ellipse, and the curves of K^s,d_Ic and K^s,d_IIc gradually expand outward with increasing concrete strength, indicating that the increase in concrete strength has a strengthening impact on crack resistance of sample.

Fig. 18

Correlation between K^s,d_Ic and K^s,d_IIc: a K^s_Ic versus K^s_IIc, b K^d_Ic versus K^d_IIc, and c K^s,d_IIc/K^s,d_Ic versus θ

The static and dynamic modal mixing ratio K^s,d_IIc/K^s,d_Ic characterizes the correlation between tensile stress and shear stress [62]. All combinations of K^s,d_Ic and K^s,d_IIc form an envelope, reflecting the fracture characteristics of samples under various compositions of tensile and shear stresses. It can be concluded that the higher the K^s,d_IIc/K^s,d_Ic, the higher the ratio of K^s,d_IIc, indicating that the shear fracture is more prone to appear. Figure 18c indicates the impact of various concrete strengths and θ on the modal mixing ratio K^s,d_IIc/K^s,d_Ic. The results show that K^s,d_IIc/K^s,d_Ic is the largest when θ = 45°. This is because the sample mainly undergoes slip failure (mode II fracture) at θ = 45°. The proportion of K^s,d_IIc/K^s,d_Ic is larger at 30° < θ < 60°, and it increases first and decreases subsequently with an increase in θ, showing that the mode II fracture toughness is greater in the medium dip angle range, that is, shear failure mainly occurs.

The fracture toughness ratio (K_effc/K_Ic) characterizes the fracture resistance and reflects the cracking behavior of the sample. For the purpose of investigating the fracture resistance of rock–concrete composites with different concrete strengths, the mixing parameters M^e of different crack dip angles are determined based on Eqs. (4), (5) and (7). The material undergoes pure-mode I failure when M^e = 1, and it undergoes pure-mode П failure for M^e = 0. If 0 < M^e < 1, mixed-mode I-П fracture occurs. The relationship between K_effc/K_Ic and M^e of layered SCB samples for different concrete strength is presented in Fig. 19. This study indicates that as the modal mixing parameter M^e decreases and moves from mode I to mode II, and K_effc/K_Ic is significantly weakened. This is because of the sample fracture from mode I to mode II, resulting in a decrease in the proportion of mode I fracture toughness and an increase in mode II fracture toughness. The sample is prone to slip failure, and the presence of interface and crack also weakens its bearing capacity. The investigation carried out by Lim et al. [60] reveals similar results. Furthermore, Fig. 19 shows that an increase in concrete strength has a strengthening effect on K_effc/K_Ic, indicating that the fracture resistance of layered structures is improved via increasing the concrete strength.

Fig. 19

Influence of concrete strength on fracture resistance (K_effc/K_Ic)

6 Summary and conclusions

The fracture characteristics of rock-based layered materials were carefully investigated via static and dynamic SCB experiments. The impacts of various crack dip angles and concrete strength on fracture mode and fracture toughness were discussed. The main conclusions were summarized:

• 1. Three fracture modes can be classified under static and dynamic loading: interfacial/tensile fracture, combined tensile–shear fracture, and combined compression–shear fracture. The crack dip angle θ plays a critical role in the formation of fracture modes. When θ = 0°, interface fracture mainly occurs in the layered SCB samples, and the fracture mode changes to combined tensile–shear fracture when θ = 15°, 30°, and 45°, and combined compression–shear fracture occurs when θ further increases to 60° and 75°.

• 2. Under dynamic loading, the strength of concrete is highly connected to secondary cracks, and the decrease in concrete strength will induce an increase in the number of secondary cracks. The crack dip angle plays a controlling role in the fracture expansion path of layered SCB samples regardless of static or dynamic loading. With increasing dip angle, the angle of wing crack with respect to the interface initially increases and subsequently reduces. When θ > 45°, the pre-existing crack closes, and new cracks are not opening anymore from the crack tip. Additionally, the crack dip angle and concrete strength have a strengthening effect on the peak load.

• 3. Y_I and Y_II present significant dependence on dip angle and modulus ratio. Under mode I loading, Y_II of layered material is not 0, indicating that the material difference of layered material has an essential effect on Y_I and Y_II of the composite crack tip. If θ is about 45°, pure-mode II fracture occurs. The K^s,d_Ic and K^s,d_IIc turn around at θ = 45° and 40°, respectively, and the K^s_effc and K^d_effc increase as θ increases. The θ plays a vital role in controlling the disparity between static and dynamic fracture toughness, and the latter of which depends more strongly on the dip angle θ than the former. Further analysis indicated that the static and dynamic modal mixing ratio K^s,d_IIc/K^s,d_Ic turns around at a dip angle of 45°, and it increases initially and subsequently decreases with an increase in dip angle θ. As the modal mixing parameter M^e decreases and moves from mode I to mode II, and K_effc/K_Ic is obviously weakened. Furthermore, the increase in concrete strength results in a strengthening impact on the cracking behavior of rock-based layered structures.