Effect of cavity defect on the triaxial mechanical properties of high-performance concrete

Abstract

The stress concentration of pipe structure or cavity defect has a great effect on the mechanical properties of the high-performance concrete (HPC) members in deep underground locations. However, the behaviour of HPC with cavities under triaxial compression is not understood, especially when pressurized liquid flows into the fractures from the cavity. This study aims to investigate the effect of the cavity and the confining pressure on the failure mechanisms, strengths, and deformation properties of HPC with a new experimental scheme. In this experiment, the pressurized liquid can only contact the surface of the sample in the cavity, while the other surfaces are isolated from the pressurized liquid. To further explore the effect of the cavity, the same experiments are also conducted on sealed and unsealed intact samples without a cavity. The failure modes and stress-strain curves of all types of the samples are presented. Under various confining pressures, all the samples with a cavity suffer shear failure, and there are always secondary tensile fractures initiating from the cavity sidewall. Additionally, it can be determined from the failure modes and the stress-strain curves that the shear fractures result from the sidewall failure. Based on the different effects of the cavity on the lateral deformations in different directions, the initiation of the sidewall fracture is well predicted. The experimental results show that both the increase of the confining pressure and the decrease of the cavity size are conducive to the initiation of sidewall fracture. Moreover, the cavity weakens the strength of the sample, and this study gives a modified Power-law criterion in which the cavity size is added as an impact factor to predict the strength of the sample.

Graphical abstract

Keywords

high-performance concrete / cavity / conventional triaxial compression / pressurized liquid / modified power-law criterion

1 Introduction

Due to the high strength, self-consolidating workability, durability, and ductility, high-performance concrete (HPC) has been widely used in dams, offshore oil production platforms, and some other engineering applications. In some concrete members in a state of three-dimensional compression, pipes are necessary, such as diversion cavity and sand discharge cavity at the bottom of dams. Additionally, cavities exist as a common type of defect in concrete. Under triaxial compression, the stress concentration of the provision of such cavities can result in the initiation of the failure, and then the pressurized liquid flows into the fractures through the cavity leading to loss of strength, stiffness, or stability. It is very important to know the effect of the cavities on the mechanical properties of HPC under triaxial compression.

Previous experimental studies on the mechanical properties of concrete with cavities were usually carried out under uniaxial compression [1–5]. The results showed that the provision of the cavity could weaken the strength and ductility of the concrete. Research on the effect of cavities on the mechanical properties of HPC under uniaxial compression is rare. But research on the mechanical properties of the intact HPC samples under triaxial compression has been plentiful [6–13]; many failure criteria have been presented to predict the effect of the confining pressure on the strength of HPC. However, experimental research on the effect of cavities on the triaxial mechanical properties of HPC has not been found in existing literature.

The effect of the cavities on triaxial mechanical properties have been studied for rocks in existing literature [14–26], and three types of fractures have been found to originate around a cavity: primary fracture, remote fracture, and sidewall fracture (often called compressive fracture, spalling, and slabbing, respectively, in some literature) [23–28]. The primary fractures originate from the tensile stress concentration area. The remote fractures begin under combined tension and compression. The sidewall fractures originate from the compressive stress concentration area. However, not all of these three types of fractures can occur in various compression conditions. The cavity size has a great effect on the failure mode of the sample under triaxial compression. When the confining pressure is large and the cavity size is small, only the sidewall fractures usually originate from the cavity [15,16]. The sidewall fracture is the most complex of the three types of fractures because of its starting points in the region of high stress gradient and pure compression.

Zhou et al. [29,30] presented a phase field model for simulating quasi-static and dynamic fracture in rock, and the phase field model was proven to model the fracture initiation, propagation coalescence, and branching in rock. In addition, the modified phase field model has been used to model compressive-shear fracture [31]. However, the failure mechanisms around the cavity in HPC under triaxial compression are not clear. The material properties and mechanical characteristics of rock and concrete are quite different, so research on the triaxial mechanical properties of HPC with cavities is urgently needed.

As a new type of HPC, reactive powder concrete (RPC) has the properties of homogeneous, high-strength, and low permeability [32]. A reduction in the size of coarse aggregates and a decrease in the water-to-cement ratio contribute to reducing the permeability of RPC, and the addition of silica fume also helps decrease the porosity as its particles fill empty spaces between the aggregates and cement paste [33–35]. Tam et al. [36] and Roux et al. [37] reported that the permeability coefficient of RPC was typically lower than that of normal strength concrete by one or two orders of magnitude. Li and Huang [38] have stated that the RPC is impermeable. The low-permeability of RPC reduces the interference factors in the research. Therefore, RPC becomes an ideal material for studying the influence of the pressurized liquid on the failure mechanism of HPC.

This article provides an experimental method for researchers studying rock-like materials, and provides reliable mechanical properties of HPC with a cavity to help designers and structural code workers.

2 Materials and experimental procedures

2.1 Sample preparation

RPC is a new type of HPC with many good mechanical properties [32]. Based on the low-permeability [33–35], RPC was chosen to study the triaxial mechanical properties of HPC with a cavity, and all the samples were made of the same mortar mixture. The materials used for the samples were as follows: 1) ordinary Portland cement P.O.52.5R (an average diameter of 14.5 μm, a density of 3120 kg/m³, and a specific surface area of 3850 m²/kg); 2) silica fume (a diameter of 0.1–0.3 μm, a density of 2214 kg/m³, and a specific surface area of 23310 m²/kg); 3) quartz sand (a diameter of 0.16–0.63 mm and a stacking density of 1750 kg/m³) as fine aggregates; 4) quartz powder (an average diameter of 10 μm and a stacking density of 240 kg/m³); 5) 3301 MH polycarboxylate superplasticizer as a water reducer; 6) defoamer to reduce air bubbles in the mortar; 7) tap water. The material proportions are given in Tab.1. All the samples were of the same size: 175 mm × 70.7 mm × 41 mm. For the samples with a cavity, there was a circular cavity in the center of the sample (Fig.1(a)). The cavity were in four different diameters: 2, 5, 7, and 10 mm. Corresponding to the diameter of the cavity, the samples are referred to as SPH2, SPH5, SPH7, and SPH10, respectively. To distinguish the samples with a cavity from the intact samples without a cavity, the samples with a cavity are referred to as SPH. In addition, the sealed intact samples without a cavity are referred to as SPI, and the unsealed intact samples without a cavity are referred to as UPI (Tab.2).

The production process of the sample was divided into the following steps. Firstly, a smooth iron bar 41mm long was fixed in the center of the steel mold (but this procedure was omitted for UPI and SPI). The iron bars came in four different diameters, namely, 2, 5, 7, and 10 mm. Secondly, the mortar was stirred evenly and then was poured into the steel molds. Thirdly, the poured molds were placed into a standard curing room and demolded 24 h later. Fourthly, the iron bar was removed. Fifthly, to speed up the reaction of the materials, the demolded samples were cured in a steam curing box with a temperature of 90 °C for 120 h. Finally, the samples were stored in the standard curing room again, and all the samples were tested within one month after they were removed from the steam curing box.

2.2 Testing methods

Before assembling the sample, three pieces of PTFE 0.1 mm thick were placed between the sample and the indenter, and some lithium-based lubricant was added between the PTFE films to fully reduce the friction. For SPH, some hot melt adhesive was smeared on four sides of the surfaces of the sample, and then a layer of heat-shrinkable tube was used to cover the sample before the hot melt adhesive solidified. Attention was paid to preventing bubbles from forming during the shrinkage of the heat-shrinkable tube. After the hot melt adhesive solidified, the heat-shrinkable tube and the sample could be firmly bonded together, so that the sample was isolated from contacting the pressurized oil. Additionally, a pinhole was left in the heat-shrinkable tube at the cavity to allow the pressurized oil to flow in, and some dense fiber was placed in the cavity to prevent the crushed concrete from flowing into the oil outside the heat-shrinkable tube. The extensometer for measuring the lateral strain (ε₂) of the wide side (l₂) was placed at the same height as the cavity, while the extensometer for measuring the lateral strain (ε₃) of the narrow side (l₃) was placed within 5 mm above the cavity (ε₁, ε₂, and ε₃ were the strains corresponding to l₁, l₂, and l₃, see Fig.1(a)). Two LVDT sensors were fixed symmetrically to the lateral side of the indenters to measure the axial strain.

SPI was assembled in the same way as SPH, but there was no hot melt adhesive between the heat-shrinkable tube and the sample, and there was no pinhole in the heat-shrinkable tube. UPI was assembled in the same way as SPI, but there were four square openings in the heat-shrinkable tube to allow the pressurized oil to flow in, and the extensometers could also be clamped on the sample directly through the openings. To prevent the crushed concrete from flowing into the pressurized oil, some dense fiber was also placed in the openings of the heat-shrinkable tube on UPI. All the samples were assembled as shown in Fig.1(b).

This test was carried out on an XTR-01 computer-controlled triaxial testing machine. The triaxial cell of the machine had a self-balancing function. The confining pressure and the pressure of the pressurized liquid were all provided by the pressurized liquid, and the pressure of the pressurized liquid in the experiment was equal to the confining pressure. The size of the sample and the force diagram are shown in Fig.1(a). A total of five confining pressure levels were used in this study, namely, 5, 10, 20, 40, and 70 MPa. In addition, the strain produced by the confining pressure is ignored in this paper, because of the deformation of the heat-shrinkable sleeve and the hot melt adhesive. The experimental procedure was as follows.

1) Calibration of the LVDTs and the extensometers with a standard micrometer caliper; the error per millimeter was less than 2%.

2) Assembly of the sample as shown in Fig.1(b).

3) Insertion of the sample into the triaxial cell of the testing machine.

4) After closing the triaxial cell, filling of the triaxial cell with oil and increase of the confining pressure to the required value.

5) Application of the axial load to the sample at a displacement increasing rate of 5 × 10^–6 m/s. If there was some residual stress, the loading stopped when the axial strain reached 10%. If the sample lost all its bearing capacity, the loading ceased immediately.

6) Removal of the sample and photography to show the failure patterns of the sample. Before photographing SPH, the hot melt adhesive on the sample was removed.

3 Results and discussion

3.1 Comparison of the failure patterns

Before the test, five unsealed samples were exposed in the pressurized oil (70 MPa) for 2 h. After being removed, the samples were cleaned and cut into several parts. It could be seen that the thickness of the oil layer in samples was less than 1 mm, while the interiors of the samples were still dry. Therefore, the samples were regarded as impermeable in this study.

As shown in Fig.2, shear fracture dominated the failure mode of SPI when the confining pressure was between 10 and 40 MPa, while several short vertical fractures were distributed in the shear zone when the confining pressure was 5 MPa. Under confining pressure of 70 MPa, there was an irregular shear fracture and several vertical fractures in the sample, and the sample was easily crushed by hand after the experiment.

It can be seen from Fig.2, that the vertical tensile fracture dominated the failure mode of UPI under each confining pressure. The failure patterns of UPI under triaxial compression were similar to the failure pattern of the intact sample under uniaxial compression (Fig.3), but the number of the vertical fractures in UPI under triaxial compression was far less than that in the intact sample under uniaxial compression. Additionally, the crushed ends of the split parts (UPI) occurred at the same end under the confining pressure of 5 and 10 MPa, and the crushed ends of the split parts occurred at different ends under the confining pressure of 20, 40, and 70 MPa.

Fig.2 shows that all the samples of SPH suffered shear failure, but there were always vertical fractures originated from the cavity wall. It should also be noted, for all the visible vertical fractures originated from the cavity wall, that the vertical fractures did not originate from the roof or floor of the cavity, but from the sidewall of the cavity. As described in the previous Ref. [24] on rocks, these vertical tensile fractures originating from the sidewall can be referred to as the secondary tensile fractures. As shown in Fig.2, the secondary tensile fractures originated from the sidewall of the cavity of various sizes under each confining pressure.

As shown in Fig.2, there were also some vertical fractures generated from the shear fracture, but far away from the cavity, in several samples of SPH. These fractures must have originated during the propagation of the shear fractures, and this is great evidence that the shear fractures started from the sidewall failure because, for SPH, there was no residual stress in most of the samples. In other words, these vertical fractures generated from the shear fractures could not initiate after the shear fractures extended to the boundary of the sample. Moreover, if the pressurized liquid did not flow into these vertical fractures generated from the shear fractures but far away from the cavity, these fractures would be arrested by the influence of the confining pressure.

3.2 Comparison of the failure mechanisms

According to the failure modes of the sample and the results of the previous Refs. [13,23–28,39–41], the failure mechanisms of all types of the samples were analyzed as follows.

Fig.4 shows the schematic diagram of the failure mechanism of SPI under triaxial compression. The samples themselves had some defects, arising during the manufacturing process, such as pores or micro-cracks. According to the results of the previous Refs. [13,39–41], as the axial stress increases, the larger defects in the sample nucleate cracks firstly, and the cracks nucleated earlier grow along the direction of the maximum principal stress. However, the cracks nucleated earlier will stop extending because of the confinement of the confining pressure, and then the smaller defects nucleate new cracks as the axial stress continues increasing. When the shear stress ( {\displaystyle \frac{{\sigma }_1-{\sigma }_3}{2}}) reaches the ultimate shear stress of the sample, the defects in the shear band produce cracks spontaneously, leading to shear failure.

Fig.5 shows the schematic diagram of the failure mechanism of UPI under triaxial compression. The samples themselves had some defects that arose during the manufacturing process, and the pit on the surface could be in direct contact with the pressurized liquid. After the cracks nucleated from the pre-existing defects, the pressurized liquid could flow into them and weaken the confinement effect of the confining pressure. The cracks without the confinement of the confining pressure became easier to propagate. As the axial stress continued to increase, one crack would extend to the end of the sample firstly. The sample was split into two parts. The increase in slenderness ratio, the deviation between the direction of the tensile fracture and the direction the axial load, or the distribution of the cracks nucleated from the pre-existing defects caused one end of the split sample to be crushed. It should be noted that under the confining pressures of 5 and 10 MPa, the crushed ends of the two split parts were in the same end of the sample, as shown in Fig.2. In this case, the sequence between the crushing of the end and the initiation of the tensile fracture could not be determined. Under the confining pressure of 20, 40, and 70 MPa, the crushed ends of the two split parts were in different ends of the sample. Therefore, it can be determined that the end crushing must have occurred after the initiation of the tensile fracture under the confining pressure of 20, 40, and 70 MPa.

Fig.6 shows the schematic diagram of the failure mechanism of SPH under triaxial compression. The sample has had a prefabricated cavity and some other pre-existing defects, and the pressurized liquid could only contact the surface of the cavity directly. As the axial stress increased, the stress concentration at the cavity caused sidewall failure. The sidewall failure took the form of the initiation of the first secondary tensile fracture, and the pressurized liquid could then flow into the secondary tensile fracture through the cavity. The effect of confinement of the confining pressure on the secondary tensile fracture was weakened. In addition, the pre-existing defects could also nucleate cracks. The origination sequence of the sidewall failure and the crack initiation from the pre-existing defects depended on the confining pressure and the cavity size. The fractures without pressurized liquid could be arrested by the confining pressure, and then the smaller defects could nucleate cracks. As the axial stress continued to increase, shear fractures propagated from the sidewall failure, and the secondary tensile fractures continued to extend. As the axial load continued to increase, the shear fractures continued to extend, and more vertical tensile fractures initiated from the shear fractures. If the pressurized liquid flowed through the entire shear fractures, the sample lost stability as soon as the shear fractures extended to the boundary of the sample. However, if the confining pressure was large enough, the shear fractures could be squeezed and closed, preventing the pressurized liquid from flowing into the entire shear fracture.

3.3 Relationship between the axial stress and the strains

Fig.7 demonstrates the axial deviatoric stress (q) and the lateral strain (ε₂) corresponding to the same axial strain (ε₁) under each confining pressure. It should be noted that, in order to make the graph tidier, not all the vertical coordinates of ε₂ are listed in Fig.7. In fact, all the ε₂–ε₁ curves start at zero in the plots.

Under each confining pressure, there was always some residual stress in SPI as shown in Fig.7, while there was always no residual stress in UPI as shown in Fig.8. Additionally, there was no residual stress in all the samples of SPH when the confining pressure was between 5 and 20 MPa, while several samples of SPH had residual stress under the confining pressures of 40 and 70 MPa. For SPI, the confining pressure led the sample to remain stable after shear failure. For UPI, the crushing failure at the end of the sample occurred easily, resulting in no residual stress. For SPH, whether there was residual stress remaining depended on whether the pressurized liquid flowed through the entire shear fracture. If the pressurized liquid flowed through the entire shear fracture, the confining pressure could not confine the sample, and the sample easily lost stability, leading to no residual stress. If there was some residual stress in SPH, it could be indicated that the pressurized liquid did not flow through the entire shear fracture. It could be inferred, under the confining pressure of 40 and 70 MPa, that the shear fractures could be squeezed and closed, preventing the pressurized liquid from flowing through the entire shear fracture.

It should be noted that there is always an obvious knee point in each ε₂–ε₁ curve of SPH, while there is no obvious knee point in each ε₂–ε₁ curve of SPI. The knee points in the ε₂–ε₁ curves mean the beginning of the unstable propagation of the cracks, and the discontinuous segments in the ε₂–ε₁ curves mean the end of the shear failure. It can be seen that the axial stress of SPH reached its peak value after the knee point but before the discontinuous segment in the curves (Fig.7). In other words, the axial stress of SPH reached its peak value during the process of the propagation of the shear fractures. Thus, it can be inferred that the fracture propagation reduced the bearing capacity of SPH.

As shown in Fig.8, it was found, for UPI, that ε₂ increased almost linearly with the increase of ε₁ before the axial stress reached its peak value, and ε₂ increased sharply at peak stress. This indicates that the vertical tensile fracture propagation in UPI was an instantaneous process. However, according to the ε₂–ε₁ curves of SPH in Fig.7, it can be seen that the increase of the lateral deformation in SPH was gradual. This was related to the propagation of the shear fractures in SPH. Moreover, it should be noted that the pressurized liquid could also flow into the secondary tensile fractures which originated from the cavity sidewall, but the propagation of the secondary tensile fractures in SPH was a gradual process, which was not like the propagation of the vertical tensile cracks in UPI. This is because the sample outside the shear fractures and the secondary tensile fractures in SPH was always confined by the confining pressure, while the vertical tensile cracks that could not be confined by the confining pressure were distributed throughout UPI.

3.4 The axial peak stresses and the corresponding axial strains

The triaxial compressive strengths of SPI, UPI, and SPH are given in Tab.3, and the average axial peak stresses are plotted in Fig.9(a). The axial strains at peak stress are plotted in Fig.9(b). As shown in Fig.9, it was found that both {\sigma }_{\text{1}}^{\text{u}} and {\epsilon }_{\text{1}}^{\text{u}} of UPI were always between those of SPH2 and SPH10. This was mainly caused by the different failure mechanisms and different sealing conditions between UPI and SPH. The strength of UPI depended on the initiation of the fracture nucleated from the pre-existing defects of the sample, while the strength of SPH reached its peak value during the propagation of the shear fracture. According to the results shown in Fig.8, the propagation of the vertical tensile fractures in UPI was an instantaneous process, and UPI lost its bearing capacity completely once the vertical tensile fractures began propagating. Therefore, the axial peak stress of UPI can be regarded as the axial stress that can lead the fractures to be nucleated from the pre-existing defects. By comparing the strength of UPI and SPH, it can be inferred whether the pre-existing defects in SPH nucleate cracks. If the strength of SPH is larger than that of UPI, there must be cracks nucleated from the pre-existing defects of SPH before the axial stress reaches its peak value. If the strength of SPH is less than that of UPI, the pre-existing defects far away from the cavity cannot nucleate cracks, while the pre-existing defects around the cavity may nucleate cracks because of the stress concentration.

Fig.9 shows that both {\sigma }_{\text{1}}^{\text{u}} and {\epsilon }_{\text{1}}^{\text{u}} increased as the confining pressure increased for the same type of sample. In addition, the cavity size had an obvious effect on the strength and the axial deformation of the HPC samples. Focusing on the sample of SPI and SPH (SPI is one kind of SPH when the cavity size is zero), it was found that both {\sigma }_{\text{1}}^{\text{u}} and {\epsilon }_{\text{1}}^{\text{u}} decreased as the cavity size increased under each confining pressure. Therefore, the cavity weakened the triaxial compressive strength and the corresponding axial deformation.

Based on the above conclusions, it is necessary to give a failure criterion that can predict the triaxial compressive strength of HPC with a cavity. Richart et al. [42] proposed an empirical formula to predict the strength of ordinary concrete expressed as Eq. (1). This formula has played a great role in engineering application. In order to give a simple and effective formula for predicting strength in engineering application, cavity size can be added to the failure criterion as an influence factor.

According to Fig.9, both the cavity size and the confining pressure had a great effect on the triaxial compressive strength of HPC. On the one hand, the cavity size and the confining pressure jointly determined the stress concentration of the cavity. On the other hand, the pressurized liquid in the shear fractures and the secondary tensile fractures weakened the confinement of the confining pressure and then weakened the triaxial strength of the sample. Therefore, in the failure criterion, the cavity size and the confining pressure need to be combined and added (Eq. (2)).

As an improved model, the Power-law failure criterion is more suitable for HPC [9,43–45], and it can be expressed as Eq. (3). In this study, the Power-law criterion can fit the triaxial compressive strength of SPI, as shown in Eq. (5). After the cavity size is added to Eq. (3) as an impact factor, it can be expressed as Eq. (6). In Eqs. (1)–(3) and (6), k₁, k₂, and a are empirical parameters, and f_c is the uniaxial compressive strength of HPC.

When the cavity size is zero, SPI can also be regarded as one kind of SPH. Therefore, {\sigma }_{\text{1}}^{\text{u}} of SPI and SPH are fitted into the same equation, while {\sigma }_{\text{1}}^{\text{u}} of UPI is excluded. Finally, Eq. (7) is obtained by fitting the triaxial compressive strength of SPI and SPH. According to the coefficient of determination, it can be deduced that Eq. (7) can well predict the effect of the cavity size and the confining pressure on the triaxial compressive strength of HPC.

(1)\frac{{\sigma }_1}{f_{\mathrm{c}}}=1+k_1\frac{{\sigma }_3}{f_{\mathrm{c}}},

(2)\frac{{\sigma }_1}{f_{\mathrm{c}}}=1+k_1\frac{{\sigma }_3}{f_{\mathrm{c}}}+k_2\frac{d_{\mathrm{h}}{\sigma }_3}{l_2{f}_{\mathrm{c}}},

(3)\frac{{\sigma }_1}{f_{\mathrm{c}}}=1+k_1{\left(\frac{{\sigma }_3}{f_{\mathrm{c}}}\right)}^a,

(4)f_{\mathrm{c}}=142.1\mathrm{M}\mathrm{P}\mathrm{a},

(5)\frac{{\sigma }_{1-\mathrm{S}\mathrm{P}\mathrm{I}}^{\mathrm{u}}}{f_{\mathrm{c}}}=1+2.74{\left(\frac{{\sigma }_3}{f_{\mathrm{c}}}\right)}^{0.66},R^2=0.9912,

(6)\frac{{\sigma }_1}{f_{\mathrm{c}}}=1+k_1{\left(\frac{{\sigma }_3}{f_{\mathrm{c}}}\right)}^a+k_2\frac{d_{\mathrm{h}}{\sigma }_3}{l_2{f}_{\mathrm{c}}},

(7)\frac{{\sigma }_1^{\mathrm{u}}}{f_{\mathrm{c}}}=1+2.74{\left(\frac{{\sigma }_3}{f_{\mathrm{c}}}\right)}^{0.66}-12.90\frac{d_{\mathrm{h}}{\sigma }_3}{l_2{f}_{\mathrm{c}}},R^2=\mathrm{0.9502.}

Equation (7) cannot explain the effect of the propagation of the shear fractures and the secondary tensile fractures on the triaxial compressive strength of SPH, although it can well fit the triaxial compressive strength of SPH. Therefore, this experimental scheme needs to be further improved to get the effect of the propagation of the shear fractures and the secondary tensile fractures on the triaxial compressive strength.

3.5 The initiation and unstable propagation of the fractures in SPH

q–ε₁ curves, ε_v–ε₁ curves, ε₂–ε₁ curves, and ε₃–ε₁ curves of SPH and SPI are presented in the same graph (Fig.10) under the same confining pressure, and the vertical dashed lines connect the peak point of the volumetric strain (ε_v = ε₁ + ε₂ + ε₃) and the corresponding deviatoric axial stress.

Referring to the failure patterns of SPH, the propagation of the shear fractures and the secondary tensile fractures advanced the increase of ε₂ but had no effect on the increase of ε₃. Therefore, the propagation of the shear fractures and the secondary tensile fractures caused the difference between ε₂ and ε₃ for the same sample. In addition, Fig.10 shows that the ε₂–ε₁ curve and ε₃–ε₁ curve of the same sample of SPI also diverge under the confining pressure of 40 and 70 MPa, while the ε₂–ε₁ curve and ε₃–ε₁ curve of the same sample of SPI almost overlap when the confining pressure is lower than 40 MPa. The difference between ε₂ and ε₃ of SPI is related to the difference between the two sides of the cross-section of the sample, and the surfaces of the shear fractures of SPI were always parallel to l₃. However, Fig.10 shows that the axial strain corresponding to the bifurcation point when the ε₂–ε₁ curve and the ε₃–ε₁ curve of the same sample of SPI diverge is larger than those in SPH, and the difference between ε₂ and ε₃ of SPI is far less than that of SPH, even when the confining pressures are 40 and 70 MPa. Therefore, in SPH, the bifurcation points when the ε₂–ε₁ curve and the ε₃–ε₁ curve diverge can be regarded as the initiation of the sidewall fracture.

In order to better understand the effect of the cavity size and the confining pressure on the initiation of the sidewall fractures in SPH, the crack initiation stresses (σ_ci) are plotted in Fig.11(a). It can be seen from Fig.11(a) that the crack initiation stress (σ_ci) decreased as the cavity size increased under the confining pressure other than 20 MPa, and increased as the confining pressure increased. Therefore, to recap, the increase of the cavity size and the decrease of the confining pressure were all conducive to the sidewall failure.

It should be noted that the trend of the crack initiation stress with the aperture under the confining pressure of 20 MPa was different from the trend under other confining pressures. This may be caused by the inhomogeneity of the defects in the samples.

Additionally, according to the results in the previous literature [46–51], the peak value of the volumetric strain ( {\epsilon }_{\text{v}}^{\text{u}}) refers to the beginning of the unstable propagation of the fractures, and the corresponding axial peak stress is the crack damage threshold (σ_cd). Therefore, it can be considered that {\epsilon }_{\text{v}}^{\text{u}} of SPH corresponds to the beginning of the unstable propagation of the fractures (the shear fractures and the secondary tensile fractures), and the corresponding σ_cd can be obtained. As shown in Fig.10, ε_v of SPI increased with the increase of ε₁ before the axial stress reached its peak value, while ε_v of SPH always reached its peak value before the axial stress reached the peak value. Thus, it can be inferred that the pressurized liquid in the fractures advances the unstable propagation of the fractures. Additionally, both the cavity size and the confining pressure had a great effect on σ_cd, and further analysis of the effect of the cavity size and the confining pressure on σ_cd is plotted in Fig.11(b).

It can be seen from Fig.11(b) that σ_cd of the fractures in SPH decreased as the cavity size increased under each confining pressure, and increased as the confining pressure increased, except for one value of SPH10. Therefore, it is concluded that both the increase of the cavity size and the decrease of the confining pressure are conducive to the fracture propagation in SPH.

4 Conclusions

In order to research the influence of the cavity on the triaxial mechanical properties of HPC surrounded by pressurized liquid, this study explored a new experimental scheme. In this experiment, the samples (SPH) with a cavity of different sizes, the sealed (SPI) and unsealed (UPI) intact samples without a cavity were tested under conventional triaxial compression. For SPH, the pressurized liquid could only be in direct contact the surface of the sample in the cavity, while the other surfaces were sealed. All surfaces of SPI were sealed, while all surfaces of UPI could be in direct contact the pressurized liquid.

Different cavity sizes can lead to different stress concentrations under different confining pressures, and the pressure of the liquid flowing into the fracture can weaken the confinement of the confining pressure. Therefore, under different confining pressures, HPC samples with a cavity of different sizes vary in terms of the strengths, failure modes, and deformation characteristics. The following conclusions can be drawn.

1) Under triaxial compression, the shear fracture dominates the failure mechanism of SPI, and the vertical tensile fracture dominates the failure mechanism of UPI. In addition, the shear fracture dominates the failure mechanism of SPH, while the secondary tensile fractures always originate from the cavity sidewall. Combined with the failure modes and the stress-strain curves, it can be inferred that the shear fractures initiate from the sidewall failure.

2) UPI loses its bearing capacity completely once the tensile fractures begin propagating, while the ultimate shear stress determines the triaxial compression strength of SPI. However, the axial stress of SPH reaches its peak value during the propagation of the shear fractures.

3) Under the confining pressure of 40 and 70 MPa, some residual stresses can be retained in several samples of SPH, while there is always no residual stress under lower confining pressures. This indicates that, if the confining pressure is large enough, the pressurized liquid cannot flow through the entire shear fractures.

4) Under each confining pressure, the triaxial compressive strength of SPH is negatively correlated with the cavity size. This paper gives a modified Power-law criterion to predict the strength of the sample, and the cavity size is added in this criterion as an impact factor.

5) The crack initiation stress and the crack damage threshold of the sidewall fractures in SPH can be obtained with this experimental scheme, and the experimental results show that both the decrease of the confining pressure and the increase of the cavity size are conducive to the initiation and unstable propagation of the fractures.

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