A Pre-peak Elastoplastic Damage Model of Gosford Sandstone Based on Acoustic Emission and Ultrasonic Wave Measurement

The rock material tested here is Gosford sandstone collected from the Gosford Quarry, New South Wales, Australia. Gosford sandstone is widely distributed around the Sydney region and used as civil construction material. The mechanical property of Gosford sandstone was extensively investigated in previous studies (Roshan et al. 2016, 2017; Masoumi et al. 2016; Lv et al. 2019; Keneti et al. 2021). Due to its homogeneity and isotropy, Gosford sandstone can be regarded as an ideal material for rock testing. The major mineral in Gosford sandstone is 86% quartz, 7% illite, 6% kaolinite and 1% anatase (Roshan et al. 2016; Masoumi et al. 2017), with poorly cemented medium grains (0.2–0.3 mm). The typical porosity of Gosford sandstone is 5% and its density is around 2226.5 kg/m³ (Ord et al. 1991). In this research, rock samples used in triaxial loading experiments were carefully examined to ensure no pre-existing cracks. The samples were core drilled into cylinders with 50 mm in diameter and 100 mm in height. The ISRM suggested procedure was followed during the preparation of the sandstone samples (Fairhurst, 1997).

Six Gosford sandstone samples were prepared to be tested under triaxial compressive loading. All triaxial tests were conducted under a servo-controlled hydraulic universal testing frame with a loading capacity of 3600 kN (Li et al. 2016a, b), as shown in Fig. 1. The frame enables to select either force-controlled or displacement-controlled loading cycles. In this work, the axial displacement was selected as the feedback signal to control the whole loading process.

The loading cell is the HTRX-140XL rock triaxial cell supplied by Geotechnical Consulting and Testing Systems (GCTS Testing Systems) in the US, with the maximum confining pressure of 140 MPa. The largest sample that can be placed in the loading cell is 76 mm in diameter and 152 mm in height. A pressure intensifier with a maximum capacity of 120 MPa was utilized to provide confining pressure.

The axial and circumferential strain were measured by two Linear Variable Differential Transformers (LVDTs) and their displacement measurement range is 25 mm, which is sufficient to measure the rock deformation in the designed triaxial tests. All the LVDTs were calibrated by the supplier GCTS to ensure the accuracy of displacement measurement.

High-frequency sinusoid ultrasonic waves (1000 kHz) can be motivated from the loading platen for P wave velocity measurement by the GCTS ULT-200 system (input voltage: 12 V), as shown in Fig. 1. Ultrasonic velocity measurement was conducted every five seconds during the test, ensuring continuous monitoring of rock damage development. To reduce the influence of possible voids generated between the loading platen and the rock sample, honey was used as the coupling material between them, as suggested by GCTS.

To detect the internal behaviour of test material, especially the microcrack generation and propagation, AE monitoring was adopted in this study. During the whole loading process, an eight-channel AE data acquisition system provided by MISTRA was also applied to record and process AE data, as shown in Fig. 1. Eight AE sensors (AE-HTRX-50 from GCTS, frequency range: 0–500 kHz) were adhered onto the thermal shrinkage outside the rock sample using hydrothermal glue to collect waveform singles generated by rock failure. The layout of the sensor array can be seen in Fig. 1. The arrival time of AE hits was automatically captured and then Geiger’s method was used to back-calculate the location of AE events.

In this study, six triaxial tests were conducted with a variety of confining pressure. The confining stresses were set as 5 MPa, 10 MPa, 15 MPa, 20 MPa, 30 MPa and 40 MPa for all six samples. At the beginning of the test, confinement was applied to the rock sample with a constant loading rate at around 0.1 MPa/s. Then the servo-control valve motivated the top platen to add axial load on the rock specimen until the peak stress was reached and stress-softening behaviour was observed.

Figure 2 presents the pre-peak stress–strain curves of six sandstone samples under a variety of confinements (5 MPa, 10 MPa, 15 MPa, 20 MPa, 30 MPa and 40 MPa). \({\varepsilon }_{1}\), \({\varepsilon }_{3}\), and \({\varepsilon }_{V}\) indicate axial strain, lateral strain and volumetric strain, respectively. The overall stress–strain curve can be divided into two or three different stages. The 5 MPa, 10 MPa, 15 MPa and 20 MPa scenarios start from the crack closure stage (\({\sigma }_{c}\)), where a concave non-linear stress–strain relation was observed when axial loading was just applied (Zhang and Tang 2020). Then it is followed by the elastic stage, where stress increased linearly with recoverable strain changes. The curves presented here are in the deviatoric stress space meaning that the application of hydrostatic pressure is not included. In the 30 MPa and 40 MPa scenarios, the confinement is high enough to close microcracks before the application of axial load, and hence the crack closure stage is not identified in those two scenarios. By the end of the elastic stage, the stress strain curve (both lateral and volumetric strain) became non-linear again. The stress at this point was defined as the crack initiation stress (\({\sigma }_{ci}\)), where the stable crack growth began inside the rock with the presentation of associated unrecoverable plastic strain (Nicksiar and Martin 2012; Zhao et al. 2015). Two methods were applied to identify the crack initiation stress (\({\sigma }_{ci}\)). The first method is the lateral strain method, where crack initiation stress (\({\sigma }_{ci}\)) is defined as the point, where the lateral stress–strain curve is not linear. The second method is the volumetric strain method, where crack initiation stress (\({\sigma }_{ci}\)) is the end of linear correlation between the axial strain and volumetric strain (Brace et al. 1966; Martin and Chandler 1994; Zhao et al. 2015). The crack initiation stress (\({\sigma }_{ci}\)) identified by these two methods shows great consistency. After that, the unstable crack growth stress was reached (which is named the crack damage stress, \({\sigma }_{cd}\)), and at this point the volumetric strain started to reverse (from compassion to dilation) (Vásárhelyi and Bobet 2000; Zhang et al. 2021b). This stage was noted as the crack damage stage, which was characterised by a drop in ultrasonic wave velocity, intensified plastic deformation, the boost of AE events, and damage initiation (will be discussed in Sects. 3.2 and 3.3). Then, the final failure occurred when the peak stress was reached (\({\sigma }_{f}\)).

An interesting observation is that the reverse of volumetric strain did not appear in Fig. 2e, f (30 MPa and 40 MPa confinement, respectively). That is due to the fact that the high hydrostatic stress confines the lateral expansion of rock mass prior to reaching the peak stress. The microcracks are compacted by the high confinement and once sliding is initiated, the asperities would be sheared off so that dilation tendency is restricted (Patton 1966). Therefore, the volumetric expansion was not observed in the pre-peak stage for samples under high confinements, which is also illustrated in the previous sandstone loading tests (Bésuelle et al. 2000; Klein et al. 2001).

The crack initiation stress (\({\sigma }_{ci}\)) indicates that rock has already entered the plastic stage. In other words, the crack initiation stress is actually on the initial plastic yielding surface. Any additional stress applied beyond that point will lead to irreversible plastic strain. In addition, the yielding surface (also referred to as the loading surface) expands as the increase of material loading until the failure stress is exceeded, namely the material hardening process. The peak (failure) stress (\({\sigma }_{f}\)) is on the failure surface, which is the ultimate yielding surface after which strain-softening replaces hardening. Therefore, the plastic yielding surface is an essential part to control the plastic flow and should be determined carefully. To ensure that the plastic yielding surface is selected properly, the evolution of crack initiation stress (\({\sigma }_{ci}\)) and peak stress (\({\sigma }_{f}\)) are shown in Fig. 3, which are represented by the corresponding first and second stress invariant. The Drucker–Prager criterion is proved to be sufficient to describe the yielding surface in this research, as formulated in the following equation:where \({f}_{p}\) is the plastic yielding surface, \({\tilde{I }}_{1}\) and \({\tilde{J }}_{2}\) are the first invariant of the effective stress tensor and second invariant of the deviatoric effective stress tensor. The definition of effective stress will be systematically introduced in Sect. 3.3. \(\alpha\) is a factor considering the effect of hydrostatic pressure on the plasticity. k is the D–P parameter to measure plastic hardening, and \({k}_{0}\) and \({k}_{f}\) are two constants indicating the hardening degree for the initial plastic yielding surface and final failure surface, respectively. From the test results, we observed that the hydrostatic pressure variable \(\alpha\) can be considered as a constant (− 0.2341) for the whole plastic evolution range from \({\stackrel{\sim }{\sigma }}_{ci}\) to \({\stackrel{\sim }{\sigma }}_{f}\). \(k\) is increasing with the progressive development of plasticity (from \({k}_{0}=\) 19.96 to \({k}_{f}\)=37.10), which can be considered as a factor related to material hardening (expansion of the yielding surface).

The location of AE events can provide a direct measurement of the microcrack initiation and propagation inside rock materials. The AE results should be strongly related to the damage and plasticity evolution of rock specimens. Herein, we use the sample with 10 MPa confinement as an example to explain the relationship between AE and non-linear rock deformation. Figure 4 shows the spatial and temporal (in colour) distribution of AE events and their corresponding energy (pseudo energy, which is defined as the integration of signal voltage to time) in different loading stages, including the elastic, stable crack growth and unstable crack growth stages. In the elastic stage, only a few AE events were detected, and the corresponding energy of AE events was low (below \(200\) \({v}^{2}\times \mathrm{\mu s}\)). Once entered the stable crack growth stage (from \({\sigma }_{ci}\) to \({\sigma }_{cd}\)), although the number of AE events only slightly increased, the energy of AE events was much higher than that in the elastic stage (up to \({10}^{3}{v}^{2}\times \mathrm{\mu s}\)). The spatial distribution of AE events in these two stages was random, without a clear trend of coalescence. As for the unstable crack growth stage, both the number and energy of AE events increased significantly and reached their maximum. The number of AE events initiated per strain change also surged when the axial loading approached the crack damage stress. A clear cluster of AE events can be observed indicating crack coalescence, and the event cluster aligned with the final failure plane of rock samples. The AE results suggested a phased damage evolution of rock samples under triaxial loading, which can assist the characterisation of rock damage initiation.

By assuming rock damage and the associated property change are isotropic, a scalar damage parameter was applied in this research. Although the accuracy of isotropic damage assumption is less compared with the two- or four-dimensional damage tensor (Krajcinovic 1996; Murakami and Kamiya 1997), the scalar damage parameter brings unparalleled simplicity in engineering practices and thus it is well accepted by a number of previous studies (Balieu and Kringos 2015; Chen et al. 2015; Miao et al. 2021). According to the equivalent strain theory proposed by Kachanov (1958, 1986, 1992, 1993), damage can be expressed as the reduction of the cross-sectional area of rock material due to microcrack generation. Considering a cylindrical rock sample with a cross-sectional area of \(A\), due to the generation and opening of microcracks and defects, the cross-sectional area of the solid skeleton (\(\tilde{A }\)) in the sample to sustain load would be less than \(A\). Assuming the area of those defects is \({A}_{\omega }\), the effective cross-sectional area can be written as the following equation:

The scalar damage variable \(\omega\) can be defined as the percentage of cross-sectional area reduction:

The direct measurement of damage is challenging since the internal microcrack evolution is difficult to be measured over the triaxial loading experiments. Therefore, to obtain damage evolution, Lematitre (1984, 2012) and Lemaitre and Chaboche (1994) proposed the hypothesis of strain equivalence, which states the constitutive equation of the damaged material can be simply represented by replacing the stress with effective stress (stress exerted on rock grains, excluding cracks and voids), as shown in Eq. (4) and Fig. 5:where \(\stackrel{\sim }{\sigma }\) is effective stress and \(\tilde{E }\) is the effective Young’s modulus. Both can be used to describe the property of damaged material. To measure damage, the deterioration of the elastic modulus (effective Young’s modulus) is normally adopted, namely:where \({C}_{ijkl}\left(\omega \right)\) and \({C}_{ijkl}^{0}\) are the fourth-order material stiffness matrix for damaged and undamaged cases. In addition, the P wave velocity can be expressed by the Young’s modulus:where \({V}_{p}\) is P wave velocity. \(\nu\) is Poisson’s ratio, which can be considered as a constant during loading due to the isotropic damage assumption. \(\rho\) is the density of rock material. Thus, the P wave velocity can become an effective parameter to identify damage evolution inside a rock specimen and its simplicity has already attracted field-scale applications (Zhang et al. 2016).

However, it is not recommended to back calculate Young’s modulus directly from the measured P wave velocity using Eq. (6) as a result of potential wave diffusion (Carcione and Picotti 2006). The P wave velocity measurement with low frequency suffers from attenuation in rock micro-grain structure, and therefore, high-frequency P wave with normalization was applied in this research. The application of P wave damage identification on concrete shows that P wave frequency ranging from 0.1 to 1 MHz can be a reasonable approximation (Berthaud 1988; Lemaitre and Desmorat 2005). Hence, the normalized P wave velocity was used to quantify the damage inside a rock sample, as shown in the following equation (Lemaitre and Desmorat 2005):where \({V}_{p(\mathrm{max})}\) is the maximum P wave velocity and \({V}_{p}\) is the P wave velocity measured over triaxial loading.

Figure 6 shows the P wave velocity and volumetric strain evolution of rock samples (since the total rock mass is conserved in the whole test, the density change can be easily calculated by volumetric strain) under different confinement pressures (5 MPa, 10 MPa, 15 MPa, 20 MPa, 30 MPa and 40 MPa). The reduction of P wave velocity can be observed in all six scenarios at around the moment when the axial stress passing \({\sigma }_{cd}\), indicating the generation of microcracking damage in the rock specimen.

Note that we did not observe volumetric strain expansion in the pre-peak stage for 30 MPa and 40 MPa confinement cases and their corresponding P wave velocity drop is less than that in the 10 MPa and 20 MPa confinement cases (same as previous studies Bésuelle et al. 2000; Klein et al. 2001)). This is because the dilation generated by the shearing of microcracks in low confining stress can reduce the P-wave velocity and rock density, whereas high confinement leads to microcrack shearing off.

On the other hand, for the 5 MPa scenario, despite its low confinement, not much damage was recorded in the pre-peak stage due to the strong brittleness of the rock sample, which resulted in the rapid failure of the rock sample with insignificant P wave velocity drop. With the increase of confinement, rock material would typically experience the brittle–ductile transition, yielding more pre-peak damage. Hence, the damage generated in the 10 MPa case is higher than the 5 MPa one.

It is interesting to find that the damage variable \(\omega\) is closely correlated with the volumetric and lateral strain expansion (dilation), as shown in Fig. 6. As soon as the transition from volumetric compression to dilation occurs, the P wave velocity drops, indicating microcrack-induced damage starts to affect rock mechanical properties (Fig. 6a–d). This phenomenon is also noticed by Ortiz (1985), who defined dilation (\({\varepsilon }^{+}\)) as the indicator of damage for rock material using the following equation:where \(H({\varepsilon }_{k})\) is the Heaviside step function for the \({k\mathrm{th}}\) principal strain and \({V}^{k}\) is the direction vector of the \({k\mathrm{th}}\) principal strain direction. The start of damage is normally accompanied by the initiation of \({\varepsilon }^{+}\) (Shao et al. 2005). Therefore, we proposed that the normalized P wave velocity drop and volumetric strain reversal can be considered as effective parameters to determine the damage evolution of rock materials. The P wave velocity-based damage variable calculated from Eq. (7) would be used as the input parameter for the following elastoplastic damage model.

Based on the analysis of lab test results, this section aims to develop an elastoplastic damage constitutive model under the framework of both the plastic and damage theory for the pre-peak behaviour of rock samples. In the high confinement cases, rock damage was restricted but notable plastic yielding was still generated (see Fig. 2). Therefore, in this model, damage was believed to be affected by the confinement and uncoupled with plastic deformation. A non-associated plastic flow model was adopted that the plastic strain can be induced by both hydrostatic and deviatoric stress in rock samples (considering the effect of confinement). An isotropic P wave velocity-based damage parameter was used to simulate the reduction of rock properties due to microcracking. We also assumed that the whole loading process was isothermal, and the strain increments were infinitesimal. Classically, the total strain can be separated into elastic and plastic components, as shown in the following equation:\(\mathrm{where} {\varepsilon }_{ij}^{e}\) here refers to the elastic strain and \({\varepsilon }_{ij}^{p}\) indicates the plastic strain. In addition, from a thermodynamics point of view, the reversible energy stored in the rock sample can be expressed by the Helmholtz free energy. A pragmatic approach is to assume that the plastic and damage components of the Helmholtz free energy can be written as two separated parts. This method was also adopted in Rousselier’s theory (1981), as explained in Eq. (10) and Fig. 7:where \({\gamma }_{p}\) is the plastic hardening variable and \(\omega\) is the damage variable. Though a few studies proposed that the plastic and damage evolution are fully coupled (Salari et al. 2004; Chen et al. 2010; Wang and Xu 2020), based on the observation in Sect. 3.3, it is suggested that damage and plasticity are not so closely related. As discussed in Fig. 6, less damage is observed under high confinement, while plastic deformation still develops. In addition, previous researchers found that the damage initiation point is prior to or inferior to the plastic initiation point in the loading process (Diederichs et al. 2004; Wang et al. 2016). These results support the argument that the damage evolution and plastic flow can be independent. In addition, in Fig. 7, considering an ideal plastic material with both plasticity (unrecoverable strain) and damage (deterioration in elastic modulus) presence, the Helmholtz free energy is separated into two parts \({\varphi }_{p}\) and \({\varphi }_{\omega }\). In addition, considering the complicated rock engineering environment, the separation of damage and plastic component can largely simplify the model and make it easy for engineering applications. Here, the only difference between the plastic and damage components of Helmholtz free energy is that effective stress (\({\stackrel{\sim }{\sigma }}_{ij}\)) is used in calculating plastic Helmholtz energy.

Under the scope of continuum mechanics, the second law of thermodynamics (the Clausius–Duhem inequality) is used to ensure the material is thermodynamically allowable. The Clausius–Duhem inequality indicates the rate of dissipated energy should be positive, as defined in the following equation:\(\mathrm{where}\, d\varphi\) is the total derivative of the Helmholtz free energy, consisting of its partial differential of three independent variables, total strain (\({\varepsilon }_{ij}\)), plastic hardening (\({\gamma }_{p}\)) and damage (\(\omega\)).

Substituting Eq. (10) into Eq. (11):

For any strain tensors, the inequality must be satisfied. Therefore, the coefficient before the total strain should be zero and the inequality can be rewritten as Eqs. (13), (14) and (15). Noticing that the damage and plastic strain are unrecoverable from the whole loading process, and therefore, the dispassion from the whole test should be non-negative. Although the P wave measured damage slightly violates the second law proposed here, it is still in an acceptable range and the overall damage evolution trend is consistent with the theoretical equation (see later in Fig. 11):

For simplicity, the Helmholtz free energy can be written as

Therefore, substituting Eq. (16) into Eq. (13):

Since one of the major failure modes of rock material is the shear failure, the associated plastic flow may not be applicable to describe the plastic flow of rock shear failure. This is because the direction of plastic strain generated is no longer normal to the plastic yielding surface and the hydrostatic stress can also induce plastic strain. A detailed explanation can be found in Fig. 8.

Therefore, rather than the associated flow rule, the non-associated plastic flow rule was applied in this study, which implies the plastic yielding surface and plastic potential function is no longer coincide with each other. For simplicity, a plastic yielding surface and plastic potential functions were proposed in Eqs. (18) and (19), respectively:where \({f}_{p}\) is the plastic yielding surface (loading surface) and \({g}_{p}\) is the plastic potential function. \({\alpha }_{p}\) is another plastic hardening variable, which is a conjugated force according to the plastic hardening variable \({\gamma }_{p}\), as shown in Eq. (20). \({\alpha }_{p}\) controls the hardening (expansion) of the yielding surface:

The plastic flow rule is, therefore, can be expressed aswhere \({\lambda }_{p}\) is the plastic multiplier, controlling the magnitude of plastic increment, and \(\frac{\partial {g}_{p}({\stackrel{\sim }{\sigma }}_{ij},{\alpha }_{p})}{\partial {\stackrel{\sim }{\sigma }}_{ij}}\) is the gradient of plastic yield potential, which shows the direction of plastic increment. Due to the infinitesimal strain assumption, the consistency law is adopted showing that the plastic increment is small enough. Before and after the plastic increment strain is generated, the stress state is still on the plastic loading surface:

Substituting Eq. (17) into Eq. (22), we can get that (Shao et al. 2006; Grassl and Jirásek 2006):

Substituting Eq. (21) into Eq. (23), the evolution of plastic multiplier can be achieved:

Similar to the plastic definition, the damage yielding surface and damage potential can be expressed aswhere \({f}_{\omega }\) is the damage yielding surface (loading surface) and \({g}_{\omega }\) is the damage potential. \(\omega\) is the damage variable, which can be written aswhere \({\lambda }_{\omega }\) is the damage multiplier. Since the damage defined here is a scalar without direction, the gradient of damage potential \(({g}_{\omega })\) can be ignored. The damage evolution also obeys the damage consistency law, proposed as

Therefore, the evolution of damage multiplier can be expressed as

To evaluate the plastic strain and plastic multiplier (\({\lambda }_{p}\)) of a rock sample in its plastic deformation stage, the corresponding plastic yielding surface, plastic potential, and hardening variables are determined in this section. The rock sample is assumed to be isotropic, whose modulus matrix only has two independent variables.

There are two relations that need to be fit by experiment results, i.e., the plastic hardening Eq. (35) and plastic potential Eq. (37). To obtain the best fitting parameters for the elastoplastic constitutive model, MATLAB curve fitting toolbox was used to fit these equations. The fitting method is the well-accepted nonlinear least square method. The optimization algorithm is the Trust-Region algorithm, where a region is defined around the best solution. Then the Trust-Region algorithm uses a defined model (e.g., quadratic functions in this research) to approach the target function, ensuring the best convergence of fitting results. Then, those two relations can be fitted accordingly.

Using the parameters summarised in Table 1, the mechanical response of rock material under triaxial loading can be obtained. The axial and circumferential loading behaviour are presented in Figs. 9 and 10, respectively. Comparing the theoretical model with the lab test results, a good consistency can be observed at all six different confinements. Since the post-peak stage can be highly affected by the randomness of rock fracture initiation, where a stress drop and lateral expansion can suddenly occur violating the continuity hypothesis in the continuum mechanics, the theoretical model was only applied to the pre-peak stage. In addition, the failure plane generated in the post-failure stage shows strong anisotropy and heterogeneity, and therefore, the scalar damage variable is no more applicable in that case. Further study is still required to implement the elastoplastic damage model in the post-peak stage.

In previous sections, the plastic evolution law is discussed in detail. The damage evolution is similar to the plastic counterpart, but with its own hardening parameter and yielding surface. To explain the damage evolution, from the thermodynamic point of view, the damage energy release rate \(Y\) (conjugate force of damage variable) can be expressed as

Since the scalar damage variable is assumed, the damage potential surface can be ignored. The damage identification can be separated into two subsequence questions, namely, how does damage initiate and how does it propagate. Inspired by previous studies, we proposed the damage initiation energy release rate is proportional to the confinement, as shown in Fig. 11a. From Fig. 6, damage accelerates as the sample approaches its peak stress. Hence, an exponential function was often used to describe the damage yielding surface and its hardening variable, which is also adopted by a few studies (Rinaldi et al. 2007; Cao et al. 2010; Chen et al. 2015). The damage yielding surface is defined aswhere \({a}_{\omega }\) and \({b}_{\omega }\) are fitting parameters to control the initiation of damage and the damage evolution rate. Apparently, the whole damage process was strongly related to the confinement of specimens in these tests, as illustrated in Fig. 6. Hence, \({\sigma }_{3}\) was used to reflect the effect of confinement on the damage yielding surface. Note that for the 5 MPa confinement scenario, owing to great brittleness, the damage evolution is not sufficient (less than 2%), and therefore, we do not interpret this group of data. The fitting parameters based on test results are listed in Table 2. The damage evolution can be modelled by these parameters, as shown in Fig. 11.

A high agreement can be observed between the model results and the test damage results calculated based on Eq. (7). Since all the data analysed here are for the pre-peak stage, this observation shows that damage can be induced even before the structural failure of a rock specimen. Interestingly, we found that damage can be restricted by high confinement (Fig. 11), where the ultimate damage drops from 0.16 at 20 MPa to only around 0.1 and 0.12 at 30 MPa and 40 MPa confinement.