Creep behavior of sandstone under the coupling action of stress and pore water pressure using three-dimensional digital image correlation

Abstract

Understanding the damage evolution and time-dependent property of rock creep is of great significance for predicting geohazards and evaluating the long-term stability of geotechnical structures. In this study, a three-dimensional digital image correlation system was adopted to investigate the creep behavior of sandstone under the coupling action of stress and pore water pressure. The apparent strain fields, deformation characteristics of the localization zone, and micromorphology of the fracture surface were analyzed. The results demonstrated that when the applied deviatoric stress level was above σ_ci (crack initial stress) or σ_cd (crack damage stress), the increase in pore water pressure promoted creep deformation evidently, improved the creep rate significantly and shortened the time-to-failure of the rock obviously. In the radial strain field, the localized development of substantial microcracks on the rock surface was concentrated in the steady-state creep, while the microcracks interconnected to form macroscopic shear cracks that dominated the accelerating creep, and this damage evolution characteristic can be used as a precursor and early warning of rock creep failure. Besides, increasing the pore water pressure also would cause the divergence point of strain curves inside and outside the localization zone to appear earlier at the secondary creep, and produce a wider localization zone at the tertiary creep. The creep fracture surface of the rock was dominated by intergranular microcracks. Increasing the pore water pressure would result in the deterioration of the cemented structure and breakage of the cemented matrix more seriously, thus stimulating the generation of more microcracks.

Keywords

Three-dimensional digital image correlation (3D-DIC)creep strain field evolution strain curves microcrack damage

Introduction

The creep behavior of rocks not only controls the precursory phase of important geological hazards such as earthquake ruptures and volcanic eruptions (Bell et al., 2011; Lyakhovsky and Ben-Zion 2008), but also influences the integrity of underground mines and excavations and the long-term storage of hazardous waster (Brantut et al., 2013; Chen et al., 2023; Jiang et al., 2023; Yang and Edward 2000). In fact, because of the geographical environment in which they are located, the rocks at the foundations of tunnels, dams, and slopes often contain water, and their pores are saturated and potentially subjected to pore water pressure (Chu et al., 2022). It is known that rocks generally exhibit remarkable creep deformation under high load-to-stress ratios in humid environments, which is commonly regarded as a triggering factor for geological and geotechnical engineering disasters, such as rainfall-induced landslides, water inrush in tunnels, and deformation of dam foundations (Huang et al., 2020; Liang et al., 2022; Wu et al., 2005; Zhao et al., 2013). Therefore, the investigation of the creep behavior of brittle rocks under the coupling action of stress and pore water pressure is critical for evaluating the long-term stability of geotechnical structures in a water-rich environment.

Creep is defined as the stress remaining constant while the strain increases with time, and the creep curve is divided into three stages (1) primary or decelerating creep, (2) secondary or steady-state creep, and (3) tertiary or accelerating creep (Brantut et al., 2013). Extensive research has been studied on the creep characteristics of rocks and it has been established that the difference in the load-to-stress ratio significantly influences the time-to-failure and creep strain rate (Chen et al., 2021; Fabre and Pellet 2006; Wu et al., 2022). Fujii et al. (1999) found that the applied deviatoric stress increased from 301 to 302 MPa (only a 4% increase in differential stress), resulting in an approximately six-fold increase in the creep strain rate and less time-to-failure. Zhang et al. (2012) discovered that the steady creep rate increased nonlinearly with deviatoric stress in an exponential form, and confirmed that the creep effect on rocks was more evident at high stress levels. Brantut et al. (2013) regarded that the increase of applied deviatoric stress would induce the increase of the stress intensity factor at the microcrack tips inside the rock, resulting in a large increase in the subcritical crack growth rate.

Furthermore, previous research has suggested that water accelerates the creep behavior of rocks, which mainly manifests in increasing the creep deformation rate and shortening the time-to-failure (Hashiba et al., 2018). Kranz et al. (1982) found that the time-to-failure of westerly granite under saturated conditions was reduced by about three orders of magnitude compared to ambient humidity conditions. Grgic and Amitrano (2009) observed that the addition of water to porous samples during creep tests induced a significant increase in both the acoustic emission activity and dilatant inelastic volumetric strains. Xu et al. (2018) found that the minimum creep strain rate increased from 1 × 10⁻⁸ to 2∼3 × 10⁻⁸ s⁻¹ and the time-to-failure decreased from 19 to 12 h upon increasing the pore pressure from 5 to 20 MPa. Some researchers highlighted that pore fluid would evidently decrease the specific fracture energy of rock, thereby reducing their fracture toughness, and fluid-rock reactions that occur at the crack tips would accelerate crack growth (Baud et al., 2000; Brantut et al., 2013).

In order to deeply investigate the creep behavior of rock, researchers had adopted several different experimental techniques to carry out experiments, such as holographic interferometry, acoustic emission (AE), nuclear magnetic resonance (NMR), and computed tomography (CT). Kurita et al. (1983) adopted holographic interferometry to observe the surface deformation of granite in creep experiments under a confining pressure of 50 MPa, and highlighted that surface deformation was fairly uniformly distributed at primary and secondary creep, and the surface strain became clearly localized at approximately the onset of tertiary creep. The location of microcracks inside the rock during creep monitored by the AE technique revealed that the AE hypocenters were distributed in several small clusters at decelerating and steady-state creep and became localized along the nascent rupture plane at accelerating creep (Lei et al., 2000, 2003). Ren (2002) performed a CT real-time test of the creep damage propagation of soft rock and found that the crack expanded stably and accelerated in the secondary and tertiary creep stages, respectively. Zhou et al. (2021) adopted NMR technology to detect the micropore characteristics of sandstone before and after the creep test and found that there was an exponential relationship between the porosity increment and loading ratio. The application of the above techniques has revealed the rock damage evolution in creep tests from different aspects. However, due to the invisible limitation of the triaxial pressure chamber, the initiation, propagation, and connection of rock surface microcracks under triaxial creep loading conditions have rarely been studied in previous research. Moreover, the time-dependent characteristic of rock surface microcrack extension is the key to evaluating and predicting rock failure. Therefore, it is necessary to combine the visual triaxial pressure chamber with the three-dimensional digital image correlation (3 D-DIC) method to explore the creep behavior of rock under complex loading conditions.

In this study, creep tests of sandstone under the coupling action of pore water pressure and stress were performed using a transparent triaxial compression servo-control testing system and the 3 D-DIC system. During the creep loading period, the microcrack damage initiation, propagation, and connection were determined by evaluating the strain field on the rock surface. Virtual extensometers were set up at different locations of the strain field to obtain the strain-time curves (Munoz et al., 2016). Furthermore, the divergence of strain evolution inside and outside the localization zone and the width of the localization zone were analyzed. Finally, the micromorphologies of the fracture surfaces of the creep failure samples were scanned and analyzed using an electron microscope.

Test method

Rock and experimental set-up

The sandstone samples used in the experiments were retrieved from Jingkou Town, Chongqing. The diameter and height of the sample were 25 mm and 50 mm (Figure 1(a)), respectively, and the preparation process strictly complied with the requirements of the International Society of Rock Mechanics (ISRM) (Aydan et al., 2013). The original porosity T2 diagram of the sandstone was obtained using NMR tests. The porosities of the three samples were 12.34%, 12.24%, and 12.39%, respectively (Figure 1(b)). There are two peaks in the porosity distribution curve, mainly focusing on the development of small and medium pore sizes (Yan et al., 2015). X-ray diffraction analysis revealed that the main minerals in the sandstone samples were quartz (∼42.5%), albite (∼33.7%), potassium feldspar (∼9.10%), muscovite (∼7.4%), and a small amount of chlorite (∼7.3%). The grain structure was revealed by a thin section under a polarized light microscope (Figures 1c-1d). Most of the particle sizes were distributed in the 0.05 – 0.10 mm range. The particles are mainly in close-line contact, and the cement between the particles is mainly chlorite. The samples had a density of 2.03 g/cm³ and their mechanical properties are listed in Table 1.

Table 1.

Mechanical properties of sandstone under confining pressure of 3 MPa.

Peak strength (saturated) (MPa)	61.52
Young’s modulus (saturated) (GPa)	9.82
Poisson’s ratio (saturated)	0.37
Peak strength (1 MPa pore water pressure) (MPa)	59.41
Young’s modulus (1 MPa pore water pressure) (GPa)	9.32
Poisson’s ratio (1 MPa pore water pressure)	0.38

Figure 2 shows a schematic of the transparent triaxial compressive servo-control testing system and the 3 D-DIC system. The servo tester shown in Figure 2(b) has a loading capacity of 500 kN, a displacement range of 10 mm, and a frame stiffness of 5 GN/m. The longitudinal displacement and axial force were measured by a linear variable differential transformer (LVDT) at the top of the sample and a strain-gauge load cell (SGLC) at the bottom of the sample, respectively. The triaxial pressure chamber was designed to be transparent and visible, and a transparent heat shrink tubing (thickness 0.1 mm) was wrapped over the sample to isolate the confining oil, thus the rock surface deformation and damage evolution could be captured and analyzed using a 3 D-DIC system.

To capture the deformation and damage evolution of the rock surface from multiple angles, six complementary metal–oxide–semiconductor cameras were used in the 3 D-DIC system, which was divided into three groups of 3 D image acquisition units (L1, L2, and L3) and placed evenly around the pressure chamber (Figure 2(c)). Three groups of light-emitting diode (LED) lights provided a stable light source environment around the cameras. These three groups of LEDs and cameras could be controlled separately or simultaneously using the 3 D-DIC system software via the control box. In addition, for the one-to-one synchronous correspondence of instantaneous image acquisition and force loading signal, the force signals of SGLC were collected by the data acquisition channel of the 3 D-DIC system in real-time. Prior to the creep tests, the cameras of the three acquisition units need to be calibrated through the customized triangular prism standard target to establish a unified spatial coordinate system. More importantly, in order to calibrate and eliminate errors caused by the chamber and confining oil, a small cubic custom calibration plate was placed in the transparent chamber filled with oil to calibrate each of the three acquisition units (Tang et al., 2022). The 3 D-DIC system could calculate the displacement and strain of point, line, and plane elements on the rock surface with a precision of 0.01 pixel.

Principle of 3 D-DIC

3D-DIC is a non-contact measurement method based on speckle image analysis to obtain the displacement and deformation information of the sample surface. The principle of the 3 D-DIC method is shown in Figure 3, in which binocular stereo vision and digital image matching are key technologies for realizing deformation tracking and calculation (Sutton et al., 2009). Figure 3(a) shows the principle model of binocular stereo vision technology. The corresponding image points of object point P in the left and right cameras are P₁ and P₂ respectively, and the unique image point P in three-dimensional space is determined by the intersection of projective ray O₁P₁ and O₂P₂ (Chen et al., 2020).

Figure 3(b) shows the process of digital image correlation matching, which involves 2n pictures in n states during the deformation process. Firstly, the analysis area is delineated in the image, and the seed points are selected in the left camera analysis area and matched with the right camera seed points to determine the connection between the left and right cameras for the same object point information. Then the analysis area in the first image (undeformed) is used as a reference to complete the matching of the analysis area in the subsequent images (deformed). After the matching is complete, the displacement and strain field of the sample surface are obtained based on the correlation algorithm. The strain components are calculated by Taylor’s expansion as follows (Sutton et al., 2009):

ε_{x x} = \frac{\partial u}{\partial x} + \frac{1}{2} ({(\frac{\partial u}{\partial x})}^{2} + {(\frac{\partial v}{\partial x})}^{2})

(1)

ε_{y y} = \frac{\partial v}{\partial y} + \frac{1}{2} ({(\frac{\partial u}{\partial y})}^{2} + {(\frac{\partial v}{\partial y})}^{2})

(2)

ε_{x y} = \frac{1}{2} (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}) + \frac{1}{2} (\frac{\partial u \partial u}{\partial x \partial y} + \frac{\partial v \partial v}{\partial x \partial y})

(3)

Where ε_xx and ε_yy are the strain components in the x and y direction, respectively. And ε_xy is the shear strain.

Stress thresholds of rock

It is known that the failure process of brittle rocks under compression can be divided into five distinct stages: (1) crack closure, (2) linear elastic deformation, (3) crack initiation and stable crack growth, (4) unstable crack growth, and (5) failure and post-peak behavior. The stress thresholds for dividing the two adjacent stages are σ_cc (crack closer stress), σ_ci (crack initial stress), σ_cd (crack damage stress), and σ_c (peak stress), respectively (Hoek and Martin 2014; Lu et al., 2022; Martin and Chandler 1994). Stress thresholds are important not only for estimating the progressive failure of rocks in engineering projects but also for evaluating the long-term stability of underground structures (Turichshev and Hadjigeorgiou 2016). For instance, the long-term strength can be estimated using σ_ci and σ_cd as the lower and upper limits, respectively (Diederichs et al., 2004). In fact, the stress thresholds are closely related to the “degree of initial microcrack damage” inside the rock; a sample with a high level of initial microcrack damage is expected to creep deformation faster than a sample with a lower level of initial microcrack damage.

In the present study, Martin’s volumetric method was adopted to obtain the stress thresholds through triaxial compression tests. As shown in Figure 4, by calculating the fracture volume and volume strain curve of saturated sandstone, it is obtained that the stress thresholds σ_ci and σ_cd are 34.45 MPa (56% of the peak strength) and 52.29 MPa (85% of the peak strength), respectively. Similarly, the stress thresholds σ_ci and σ_cd of sandstone under 1 MPa pore water pressure are 31.48 MPa (53% of the peak strength) and 48.12 MPa (81% of the peak strength), respectively. Therefore, the stress levels in the creep test were selected as 50%, 80%, and 95% of the sample peak strength, which are close to the stress thresholds σ_ci or σ_cd and are located in the elastic, stable crack growth, and unstable crack growth stages, respectively. The creep test scheme is shown in Table 2.

Table 2.

Test scheme of creep.

Sample no.	Confining pressure /MPa	Pore water value /MPa	Stress value /MPa	Stress level	Test time
JS-1	3.0	0.0	30.8	50% of peak stress	1 × 10⁵ /s or creep failure
JS-2			49.2	80% of peak stress
JS-3			58.4	95% of peak stress
JSW-1		1.0	29.7	50% of peak stress
JSW-2			47.5	80% of peak stress
JSW-3			56.4	95% of peak stress

Experiment procedure

According to the creep test scheme designed in Table 2, 50%, 80%, and 95% of the peak strength of sandstone are selected as the stress level at the start of creep, the confining pressure was set at 3 MPa, and the pore water pressure was selected at 0 MPa (saturated) and 1 MPa. The specific creep test steps are as follows.

A random speckle field with white background and black spots was made on the surface of the processed sample. After the paint was dried, apply negative pressure (-1 MPa) to saturate the sample for 48 h in a vacuum vessel filled with water. Then, took out the sample and wiped off the surface water for testing. Wrapped the transparent heat-shrinkable tube on the surface of the sample, then blew the tube evenly with a heat gun and sealed it with glue. Assembled the transparent chamber and placed it on the test bench.

The experimental load application process is shown in Figure 5. First, an axial prestress of 3 MPa was applied to the sample, followed by a confining pressure of 3 MPa. Subsequently, a pore water pressure value of 1 MPa was applied to the top of the saturated sample. To ensure the uniformity and constancy of the internal pore water pressure of the sample before the creep test, the stress state and pore water pressure were maintained until water dripped from the water outlet channel at the bottom of the chamber, after which the water outlet channel was closed, and the sample remained in an undrained state during the test. Subsequently, the axial deviatoric stress was applied at a rate of 0.5 MPa/s to a predetermined stress value (50%, 80%, and 95% of the sample peak strength). Thereafter, the applied deviatoric stress, confining pressure, and pore water pressure were maintained constant until the end of the test.

To avoid the influence of the previous stage load, a creep test was performed using a single-step axial-stress-controlled testing procedure. Besides, considering the continuous and stable working time of the 3 D-DIC system, the creep test time was set at 1 × 10⁵ s (27.8 hours). If the sample failed during this period, the test was stopped immediately.

Analysis method of rock surface strain

Through the 3 D-DIC method, the deformation of the rock surface over a wide range of areas can be monitored, and the rock deformation can be obtained by arranging virtual extensometers inside the analysis area (Tang et al., 2019). As shown in Figure 6, the axial height, radial arc length, and arc angle of the rock surface analysis area obtained by the three groups of observation units (L1, L2, L3) were approximately 38 mm, 23 mm, and 110°, respectively. The strain at different positions and directions can be obtained by arranging virtual strain extensometers in different directions at different positions in the strain field (A1–A3 and R1–R3 in Figures 6(a) to (c)). In addition, to study the difference in strain evolution inside and outside the localized zone during rock creep failure, virtual strain extensometers E1–E6 were arranged, as shown in Figure 6(d). For simplicity, we analyzed the strain–time curves (obtained by strain extensometers E1–E6) only within the observation plane that completely covered the strain-localization zone. Furthermore, by extracting the average strain values of the grid nodes in the analysis area (white points in Figure 6(d)), we can analyze the evolution of various strains inside the strain-localized zone, such as axial, radial, and shear strains, which will be presented and illustrated in Figure 17.

Figure 7 shows the axial strain–time curves obtained by the LVDT (test machine) and virtual extensometers (3 D-DIC system). Figure 7(a) shows that compared with the LVDT strain, the strain measured by the 3 D-DIC system is the actual deformations of the sample and is free from bedding error (Munoz et al., 2016). In addition, it can be seen from Figure 7(b) that the deformation measured by the LVDT coincides with the deformation measured by extensometer L3-A3, indicating the reliability and stability of the 3 D-DIC system used to measure rock deformation in the creep test. The rock strain curves obtained from different extensometers (L1-A1, L2-A2, and L3-A3) diverged in the secondary creep, which was mainly caused by the localized deformation of the rock surface during the creep process. This phenomenon will be illustrated and thoroughly examined in the subsection dedicated to strain localization characteristics.

Experimental results

Strain–time curves

Figure 8 shows the evolution of the axial (ε₁) and radial (ε₃) strains with time under different stress levels and pore water pressures, which were obtained by calculating the average values of strain extensometers A1 (R1), A2 (R2), and A3 (R3). It can be observed from the figure that the strain–time curves exhibit clear initial attenuation and steady flow under a 50% stress level, and the creep deformation is small. For the sample that failed under 80% or 95% stress levels, the strain–time curves exhibited three distinct phases: decelerating, steady-state, and accelerating creep stages (Xie et al., 2023). The reason for this phenomenon is that the creep deformation of the brittle rock reflects the motion of defects in the crystalline structure within the constituent minerals and stable microfracturing (Brantut et al., 2013; Zhang et al., 2012). When the sample was loaded to a lower stress level, the original pores and cracks inside the rock gradually closed under loading, the closing effect gradually weakened with time, and no new microcracks were generated, resulting in the gradual stability of the sample deformation. When the applied deviatoric stress level increased, the internal damage to the rock intensified at the initial stage of creep. The longer the creep time, the more serious the internal microcrack damage, and the clearer the response of the rock strain–time curve, indicating that there is large axial compressive deformation and lateral dilatancy deformation inside the rock.

At the 50% stress level, the increase in pore water pressure has a slight effect on rock creep deformation. Under the condition of saturated water and pore water pressure of 1 MPa, and after the creep lasts for 10⁵ s, the changes in axial strains are 1.37 × 10⁻⁴ and 1.04 × 10⁻⁴, respectively, and the radial strain variations are -1.62 × 10⁻⁴ and -2.13 × 10⁻⁴, respectively. Conversely, under 80% and 95% stress levels, the pore water pressure increases to 1 MPa, resulting in a significant reduction in time-to-failure. This is because, at the 50% stress level, the primary microcracks inside the sample closed after loading; therefore, the pore water cannot act on the microcrack tip, which plays a weak role in promoting the propagation of microcrack damage. However, when the applied stress level increased to 80% or 95%, the sample was in the microcrack stable development stage or unstable development stage of microcracks (Figure 4), respectively. The pore water can act on the tip of cracked microcracks to stimulate and accelerate their growth and development, resulting in a significant shortening of the creep failure time.

Strain rate–time curves

Figure 9 shows the temporal evolution of the axial and radial strain rates under different stress levels and pore water pressures during creep in a semi-logarithmic chart. When the applied deviatoric stress level was 50%, the strain rates first decreased rapidly with time, followed by the steady-state stage of the creep rate, and the creep rate gradually decreased to zero, that is, there was no deformation inside the rock, and it remained stable. For samples that failed under 80% and 95% stress levels, the strain rates decreased inversely to time first, then the strain rate changed gradually to reach its minimum, and turning tended to increase rapidly toward final failure. This process is generally classified into three regions (Okubo et al., 1991).

Furthermore, under stress levels of 80% and 95%, the strain rate increased with increasing pore water pressure, and the degree of rate increase was more pronounced in the radial strain rate (Figure 9(b)). In particular, the radial strain rate was much larger than the axial strain rate in the three stages of creep when the applied deviatoric stress is above σ_ci, such as when the minimum axial and radial strain rates were 1.24 × 10^−6/s and 5.09 × 10⁻⁶/s at 95% stress level under pore water pressure of 1 MPa, indicating that the radial expansion may dominate the creep time-dependent deformation until failure.

Strain field pattern

The failure process of brittle rock is closely related to microcrack sprouting and propagating, which can be well identified and discussed by the strain field varying with time during loading (Song et al., 2016). As shown in Figures 10 –13, the radial (Figures 10–11), axial (Figure 12), and shear (Figure 13) strain fields of the rock surface at different creep times were selected. The value in the strain field was calculated with extension as positive and compression as negative, and the value of the chromatographic column was the same for the same strain field. The rightmost picture shows the spatial distribution of the strain field in the combined results of L1, L2, and L3.

Temporal evolution of radial strain field

Figure 10 shows the evolution of the radial strain field with time for saturated sandstone under different stress levels. At 50% (Figure 10(a)) and 80% (Figure 10(b)) stress levels, the radial strain field remained homogeneous after creep for 10⁵ s. At 95% stress level, local strain concentrations appeared immediately at separate locations on the L1, L2, and L3 observation planes at t = 0 s (Figure 10(c-1)). In the second stage, such as t = 500 s and t = 1670 s, the location and range of the larger strain concentration on the rock surface increased, which reflects the local initiation and development of microcracks on the rock surface (Chen et al., 2020; Song et al., 2016). Entering the accelerated creep stage, at t = 2341 s (Figure 10(c-4)), these larger strain concentration locations began to connect with each other, that is, the microcracks on the rock surface tended to connect and penetrate. This phenomenon requires special attention, as it is an important external sign that the rock enters the accelerated creep stage from steady-state creep. At t = 2376.76 s, a localized zone was formed through the sample, and the rock was subjected to shear failure. The location of the strain-localized zone was highly consistent with the location of the macroshear crack in the rock.

As shown in Figure 11, the radial strain fields evolved with time under different stress levels at a pore water pressure value of 1 MPa. At 50% stress level, the rock surface deformed uniformly after 27.8 h of creep (Figure 11(a)). When the applied stress level is 80% (Figure 11(b)), the deformation of the rock surface is uniform during the deceleration stage. At t = 10010 s in the secondary creep, a larger strain concentration area is obliquely distributed on the L3 observation surface, indicating the initiation of microcrack damage. As time passed, the deformation in the large strain concentration area further increased and the distribution range became wider, indicating that the microcracks on the rock surface developed synchronously at multiple locations and began to connect with each other (t = 12010 s). Subsequently, at t = 12873.9 s, the microcracks on the rock surface penetrated to form macroshear cracks. When the applied stress level was 95%, at the initial time t = 0 s, there were large strain-concentration areas at the lower left corner of the L1 observation plane. During the steady-state creep stage, these large strain concentration areas began to gradually expand and develop from the lower left to the upper right, which also indicated the development and expansion path of microcracks on the rock surface. Finally, at t = 634.95 s, a strain localization zone penetrating the sample was formed, indicating the formation of macroshear instability cracks in the rock.

From the above analysis, it can be concluded that the microcrack damage on the rock surface originates in the deceleration creep stage (95% stress level) or steady-state creep stage (80% stress level), which depends on the initial applied deviatoric stress level. Subsequently, the initiation and localized development of a large number of microcracks in the rock surface were generated in the steady-state creep stage. Finally, the microcracks interconnected to form macroscopic instability cracks, which dominated the accelerating creep stage. In addition, at the 95% stress level, the time from initiation to penetration of microcracks on the rock surface was reduced by 1741.81 s under a pore water pressure value of 1 MPa compared with the saturated condition, indicating that the increase in pore water pressure can accelerate the generation and expansion of microcracks inside the rock. Similarly, under a pore water pressure value of 1 MPa, when the applied stress level was increased from 80% to 95%, the development and penetration time of microcracks on the rock surface decreased by 12202.95 s.

Temporal evolution of axial strain field

The axial strain field varying with time during creep tests is shown in Figure 12. When loading to the predetermined stress value, obvious horizontal layered strain compression bands distribute on the rock surface (t = 0 s in Figures 12(b) and (c)). This is mainly because Jingkou sandstone is a sedimentary rock, and the core-taking direction is perpendicular to the bedding. There is a certain degree of inhomogeneous interlayer difference inside; thus, the compressive deformation of the bedding position was more evident than that of other positions during the loading process, resulting in a nonuniform axial strain field with a greater interlayer compressive deformation. Turning to the secondary creep stage, the above compression bands further expanded and developed in the horizontal direction, but they were not connected to each other, as shown at t = 2376.43 s in Figure 12(a-5) and at t = 12860 s in Figure 12(b-5). When a macrocrack was formed by the penetration of microcracks on the rock surface, an inclined through-localization zone was formed simultaneously, and the strain in the localized zone was in a compressed state. At t = 2376.76 s in Figure 12(a-6) and at t = 634.95 s in Figure 12(c-6), the red area with large compressive strain only distributed in the localized zone and its adjacent positions, indicating that axial strain outside the localized band was unloaded and recovered after the formation of the macrocrack.

Comparing the axial strain field at the final failure time in Figures 12(a) to (c), it can be seen that the strain in some parts of the obliquely distributed compression localization zone is positive (blocky blue area), and a large number of blue strip areas with a layered distribution appear in the rock block above the macrocrack, indicating that the strain at this position changes from the compression state before rock failure to the tensile state after failure. The main reason is that after the formation of macrocracks inside the rock, the inelastic strain unload occurred outside the localized zone, and the pore water pressure produced a certain hydraulic fracturing effect between the crack and bedding. In addition, the nonuniformity of friction sliding of the upper rock block along the fracture surface causes tensile deformation at the rock bedding position.

Temporal evolution of shear strain field

The evolution of the shear strain field with time presented in Figure 13 is important for understanding the creep behavior of rock. It can be seen that the strain localization development was not obvious in the primary and secondary creep stages, but was mainly concentrated in the tertiary creep when the microcracks interconnected with each other (t = 2341 s in Figure 13(a) and t = 620 s in Figure 13(c)). These phenomena were rarely observed in previous studies, and it was indicated that the development time of the local larger deformation in the shear strain field is more delayed compared to the radial strain fields. Therefore, it can be speculated that the initiation, development, and penetration of microcracks in a radial strain field may play a leading role in the process of rock creep failure. In addition, according to the previous study of AE signals during triaxial creep of rock, it was concluded that tensile cracks were the main failure mode in the steady-state creep stage, while in the acceleration stage, both tensile crack and shear crack increased significantly, but the shear crack increased more (Wang 2018). Therefore, it can be inferred that the development of deformation localization in different strain field was closely related to the types of microcracks generation. In other words, the localization development in the radial strain field was dominated by tensile cracks and concentrated in the steady-state creep and accelerating creep stages, while the localization development in the shear strain field was dominated by shear cracks and mainly concentrated in the accelerating creep stage.

Strain localization characteristic

Strain–time curves inside and outside strain localization zone

By arranging the virtual extensometers E1–E6 using the approach described in the test method subsection (Figure 6(d)), the strain–time curves inside and outside the localized zone were obtained, as shown in Figure 14. Overall, the strain–time curves of the inside and outside localized zones exhibited good consistency in the decelerating creep stage, indicating that the difference in strain inside and outside the localized zone was not significant at this stage. In the steady-state creep stage, the strain–time curves of virtual extensometers E2 (E4) and E3 (E6) gradually diverged, implying that the strain in the localized zone gradually developed dominantly. In the acceleration stage, the curves of E2 (E4) inside the localization zone exhibited a sharp increase, whereas the curves of E3 (E6) outside the localization zone exhibited inelastic unloading. This phenomenon demonstrates that deformation in the accelerated creep stage mainly originates from the formation of macroscopic shear cracks inside the localized zone. It is worth noting that an increase in applied stress level or pore water pressure would cause the divergence point of strain curves to appear earlier.

In particular, the divergence point of the radial strain curves inside and outside the localized zone occurred earlier than that of the axial strain curves. For instance, as shown in Figure 13(c), the divergence point of axial strain inside and outside the localization zone occurred at t = 349 s, whereas the difference in radial strain inside and outside the localization zone began at t = 150 s. This phenomenon can be confirmed by the evolution of deformation localization in the radial and axial strain fields (Figures 10 to 12), that is, the microcrack damage develops more prominently in the radial strain field. In addition, the curve of E1 (E4) could not reflect the difference between the internal and external strains of the localized zone, but reflected the comprehensive strain relationship of the shear failure zone and the undamaged region.

Width of strain localization zone

As the localization of the apparent strain supposedly reflects the evolution of microcrack damage, the width of the localization zone can reflect the damage range and degree of the rock in the creep test. As shown by the strain field patterns in Figures 10 to 11, the oblique localization zone takes the macroscopic shear crack as the center, and a certain width of band is formed around it, which is characterized by a higher strain than other positions. Therefore, an observation surface with a completely developed localization zone was selected, and the center of the localization zone was taken as the axis (red dotted line in Figure 15). Parallel section lines were arranged at equal intervals (0.5 mm) on both sides of the axis. The vertical distance of any section line from the center of the shear crack is d_i, and the width of the strain-localization zone is calculated using the following formula:

D_{l z} = 2 d_{i} (ε_{i} - \bar{ε} \geq 0)

(4)

where ε_i and ε denote the average values of all strain elements (grid nodes) on the section line and observation surface, respectively. If the difference between the average strain of the section line d_i from the crack center and the average strain of the complete observation plane is zero or positive, the section line is located inside the strain localization zone; otherwise, it lies outside the strain localization region. Thus, the width D_lz of the localization zone on the rock surface can be obtained.

Figure 16 shows the distribution of the strain difference calculated using the above formula with the macroscopic shear crack as the center. It can be observed from the figure that the strain difference at the primary creep stage is close to zero, indicating the uniformity of the rock’s apparent strain. With an increase in time, the difference in strain values near the crack center gradually increases (larger than zero), indicating that the dominant development of deformation is formed first near the crack center. As shown in Figure 16(c), a macroscopic crack was formed at t = 656.12 s, and the strain difference distribution was bell-shaped, indicating higher strains near the center of the crack than at further distances.

Moreover, it can be found that at the 95% stress level, when the pore water pressure increases from 0 to 1 MPa, the width of the rock strain localization zone increases by 0.6 mm, indicating that the increase in pore water pressure intensifies the extent of damage during rock failure, and more microcracks develop around the macroscopic fracture surface. However, under a pore water pressure value of 1 MPa, when the deviatoric stress level increased from 80% to 95%, the width of the localization zone decreases by 1.26 mm. This is mainly because the time-to-failure of the rock is longer when the applied stress level is lower, implying that the coupling action time of stress and pore water pressure is longer. Therefore, the more sufficient the physical dissolution, chemical erosion, and mechanical action between the water and rock. The looser the rock structure, the more fully developed the microcracks near the fracture surface, resulting in an increase in the width of the localized zone. This can also be confirmed by the microscopic topographical scanning of the rock fracture surface in Figure 18.

Temporal evolution of radial, axial, and shear strains in localization zone

According to the width of the rock surface strain localization zone determined by the proposed method in the previous subsection (Figure 15), the average values of the absolute radial (ε_x), axial (ε_y), and shear (ε_xy) strains of grid nodes (Figure 6(d)) within the localized zone are obtained (approximately 500 grid nodes, accounting for approximately 19.5% of the entire observation surface) to obtain the strain–time curves shown in Figure 17.

As shown in Figure 17(a), it can be seen that in the primary and secondary creep stages of saturated sandstone, the average radial strain increases in the localized zone were 7.61 × 10⁻⁴ and 2.21 × 10⁻³, the average axial strain increases were 3.43 × 10⁻⁴ and 7.66 × 10⁻⁴, and the variation of shear strain were 7.20 × 10⁻⁵ and 3.61 × 10⁻⁴, respectively. This phenomenon illustrated that the strain within the localized zone mainly increased in the radial strain during primary and secondary creep. A similar phenomenon is observed in Figures 17(b) and (c), which is also supported by the comparison and illustration of the corresponding strain field evolution (Figures 10 to 13).

In addition, it can be seen from the figures that the turning point of the rapid increase in the radial strain was significantly ahead of the axial and shear strains. As shown in Figure 17(a), the rapid increase point of rock radial strain was at t = 2000 s, and the turning point of axial and shear strains was at t = 2190 s. Combined with the strain field evolution depicted in Figures 10 to 13, it can be inferred that the development of microcrack damage inside the localized zone in the secondary creep mainly caused a clear increase in rock radial strain, and the interconnection between microcracks in the tertiary creep caused a rapid increase in radial strain first. Therefore, the evolution of the rock surface radial strain field and radial strain inside the localization zone can be used as precursors and early warnings for rock creep failure.

Scanning electron microscope of fracture surface

To investigate the coupling effect of stress and pore water on rock creep failure characteristics, the fracture surfaces of the failure samples were scanned using a scanning electron microscope (SEM) as shown in Figure 18. Three different positions on each sample shear fracture surface were selected for observation. From the original morphology of sandstone in Figure 18(a), it can be seen that the sandstone mineral particles were closely arranged in linear contact, the surface was smooth, and there was no clear crack damage development. The scaly cementitious material chlorite tightly wraps the surface of mineral particles, and the cementation between the mineral particles has good integrity and plays a strong role in connecting the mineral particles.

As shown in Figures 18(b) to (d), intergranular cracks mainly developed on the fracture surface of sandstone, the cementation matrix was severely broken, the cementation structure was damaged, and the mineral particles were arranged loosely. Moreover, with the increase in pore water pressure, numerous microscopic cracks were connected and collected on the fracture surface of the sample (Figures 18(c) to (d)), and the fracture of the cemented matrix was more serious. These phenomena indicate that the pore water pressure increase can stimulate and promote the development of more abundant microcracks inside the rock and aggravate the damage and deterioration of the internal structure of the rock. This also confirms, from a microscopic perspective, that an increase in pore water pressure could shorten the rock time-to-failure and increase the width of the strain localization zone.

Discussion

Through the analysis of the apparent strain field (Figures 10 to 13) and the micromorphology of the fracture surface (Figure 18), it can be determined that the creep behavior of brittle rocks such as sandstone is controlled by the initiation, propagation, and interconnection of microcracks. Furthermore, the effect of the pore water pressure on the creep characteristics of brittle rock depends on the microcrack state at the initiation of the creep test. The sample was in the elastic stage when the deviatoric stress level was 50% of the peak strength. The closure of the original microcracks and the deformation coordination between mineral particles make the rock deformation uniform as a whole (Zhang et al., 2012). Because the applied stress was lower than the microcrack initiation stress level, even if the pore water pressure increased, the influence on the creep deformation of the rock was weak, and the phenomenon of time-dependent failure may not appear.

When the applied deviatoric stress level is 80% of the peak strength (exceeding σ_ci), the rock exhibits steady-state creep and uniform surface deformation under saturated condition (Figure 10(b)). However, with an increase in pore water pressure, microcracks generate and expand stably to form shear cracks on the rock surface in a short time (Figure 11(b)). Several possible mechanisms can be summarized to explain the phenomenon, when under the coupling action of stress and pore water pressure, there are weakening effects such as physical dissolution and chemical erosion of water at the tip of the microcrack, such as hydrolysis and fragmentation of cemented mineral chlorite in the micromorphology (Lin et al., 2005). Moreover, when water infiltrates the surface of newly cracked microcracks, its surface energy can be reduced, thereby reducing barriers to the development and propagation of microcracks (Zhou et al., 2018). In addition, there is an additional tension at the tip of the microcrack owing to the action of pore water pressure, which is conducive to the cracking of microcracks and may lead to the expansion of tensile cracks or sliding friction of closed cracks. Therefore, when the pore water pressure increases, the microcrack development and propagation inside the rock is faster, which leads to creep deformation failure.

When the applied differential stress level exceeds σ_cd, such as 95% of the peak strength, the development of microcracks inside the rock was in an unstable state under the load. The lateral deformation of the rock begins to dominate, which can be confirmed in Figures 8 and 17. When the pore water pressure increases, the combined degradation effects of physics, chemistry, and mechanics occur at the microcrack tip, which can promote the growth of microcracks inside the rock significantly, expand, and connect to form shear instability cracks within a shorter time. This can be clearly verified from the evolution of strain localization on the rock surface and the micromorphology of the fracture surface, which shows that under the action of pore water pressure, the cumulative microcrack damage development inside the rock is more serious, and the unstable propagation time of the shear instability crack is advanced, resulting in a shorter creep failure time.

Conclusion

In this study, we performed creep tests on sandstone under the coupled action of stress and pore water pressure. Using 3 D-DIC technology and visualization of confining pressure chambers, strain (strain rate)–time curve evolution, the full-field strain pattern, and strain localization characteristics were analyzed. The conclusions are as follows:

When the applied stress level is at the elastic stage of rock, the increase in pore water pressure has a slight effect on its time-dependent deformation. When the applied deviatoric stress level is at the stage of stable crack propagation or unstable crack propagation, the increase in pore water pressure promotes the creep deformation and creep rate obviously, thus shortening the time-to-failure of the rock.

In the radial strain field, the stable propagation and local accumulation of massive microcrack damage were concentrated in the secondary creep. The mutual penetration between microcracks forms unstable shear cracks that dominate the tertiary creep. The strain localization characteristic of the radial strain field can be regarded as a precursor and early warning for rock creep failure.

The difference in strain inside and outside the localization zone of the rock occurs at the secondary creep and becomes increasingly significant with time. As the pore water pressure increased, the divergence point of strain-time curves inside and outside the localization zone appeared earlier.

A method for determining the width of the rock strain localization zone is proposed based on the DIC technique, and it is determined that the width of the localization zone is in the range of 4.09–5.95 mm. And it is found that the increase of pore water pressure will produce a wider localization zone of rock at tertiary creep.

Under the coupled action of pore water pressure and stress, the development of sandstone creep failure cracks is dominated by intergranular cracks. Increasing the pore water pressure promotes the dissolution of cemented minerals, intensifies the destruction of the cemented structure, and stimulates more abundant microcrack damage generation.

Footnotes

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research presented in this manuscript was supported by the National Natural Science Foundation of China (Grant Nos. 52304099, 51974041, 52192625 and 52174082), the Program for Guangdong Introducing Innovative and Entrepreneurial Teams (2019ZT08G315), the Shenzhen Science and Technology Program (Grant No. RCYX20221008092903013), the Open Fund from Key Laboratory of Deep Earth Science and Engineering (DESE 202108), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021A1515011731), and the Postdoctoral Research Foundation of China (2023M731491).

ORCID iD

Minghui Li

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