Simulation of Shear and Tensile Fractures Using Ductile Phase Field Modelling with the Calibration of P Wave Velocity Measurement and Moment Tensor Inversion

Following a general form of the entire Helmholtz free energy, we attempt to determine the elastic, plastic and damage components of Eq. (1), by considering both tensile phase field damage variable and the shear phase field damage variable. According to previous studies, the form of elastic Helmholtz free energy for two damage variables could be complex, based on the sound theoretical and empirical considerations (Cordebois and Sidoroff 1982; Chiarelli et al. 2003; Lemaitre and Desmorat 2005a).

However, there are currently no one-size-fits-all solutions for the form of elastic Helmholtz free energy with two independent phase field damage variables. The most common approach is to perform a volumetric-deviatoric decomposition on the elastic Helmholtz free energy such that the tensile damage deterioration of elastic Helmholtz free energy affects only the volumetric part, and on the other hand, the deviatoric elastic Helmholtz free energy is influenced by the shear damage. However, this hypothesis seems to deny the relationship between shear microcracks and the volumetric expansion of microcracks, whereas the contributions of shear cracks to volumetric expansion were highlighted in previous studies (Guéguen and Palciauskas 1994; Chandler 2013). Therefore, the shear phase field damage variable is considered in the second term of Eq. (1), as mentioned in a previous study (Iannucci and Ankersen 2006).

Based on the aforementioned studies, we have proposed a new elastic Helmholtz free energy for the rock materials that incorporates both tensile and shear phase field damage variables. The proposed equation is given by:where \({d}_{t}\) and \({d}_{c}\) are the tensile and shear phase field damage variables, respectively. The determination of those two variables is based on ultrasonic wave velocity measurement and acoustic emission monitoring, which will be introduced later in Sect. 4. \(\lambda\) and \(\mu\) are the Lamé constants of rock samples. The application of Lamé constants indicates an isotropy assumption is adopted for rock material in this study. \(tr\left[{\varepsilon }_{e}\right]\) is the trace of elastic strain tensor and \(tr\left[{\left({\varepsilon }_{e}\right)}^{2}\right]\) is the trace regarding the square of the elastic strain tensor.

In addition, the function describing how the phase field damage variables couple with the total Helmholtz free energy is referred to as the degradation function. Generally, multiple forms of degradation functions can be applied, such as exponential, quadratic or high order polynomial. In this study, a quadratic form of degradation function is selected (see \({\left(1-{d}_{t}\right)}^{2}\) and \({\left(1-{d}_{c}\right)}^{2}\) in Eq. (2)). There are two main reasons for using quadratic degradation functions in this study. First, the quadratic form of the degradation function has been adopted by most previous studies, demonstrating its robustness in simulating energy degradation in phase field damage models (Santillán et al. 2017; Zhou et al. 2018). Second, the quadratic degradation functions also have clear physical meaning from a traditional damage perspective (Ju 1989; Li et al. 2023b), indicating the application of energy equivalence hypothesis in many studies of traditional damage mechanics (Darabi et al. 2012).

A well-accepted plastic Helmholtz free energy does not exist in current research, as plastic flow and the generation of plastic strain are rather complex, especially for rock materials. The specific plastic characteristic of rock materials, including dilation, non-associated plastic flow and hardening relationships require making subtle hypotheses based on phenomenological observations.

The plastic Helmholtz free energy is related to the energy stored by the plastic strain and is closely related to the rock heterogeneity, such as the random distribution of microcracks and the different properties of mineral composition in rock materials (Li et al. 2023b). Prior research have sought to incorporate plastic Helmholtz free energy within the framework of phase field damage models by leveraging the Ramberg–Osgood law, a formulation that effectively elucidates crack propagation phenomena in metallic materials (Alessi et al. 2018; Farrahi et al. 2020). However, plasticity in rock materials is far beyond the capacity of the Ramberg–Osgood model, such that the previous studies may not handle the plasticity incremental theory, which is widely used in rock materials.

Hence, in this study, we use a phenomenological relationship to deduce the plastic Helmholtz free energy, as one of only a few reasonable approaches to expand the phase field damage model into ductility crack propagation. This is also a pragmatic way to ensure the thermodynamic framework will not be violated in this work. The determination of plastic Helmholtz free energy is predicated upon the conjugate relationship between the hardening variable and the hardening parameter, as suggested in Eq. (3)where \(\alpha\) is the hardening parameter applied in this study and \(k\) is the hardening variable. Similar to the stress–strain correlation, the interrelation between the hardening parameter and hardening variable is likewise expressed through the differential of the comprehensive Helmholtz free energy. The definition of the hardening variable (\(k\)) is to measure the expansion and shrinkage of the yielding surface, with the unit of stress. On the other hand, the description of the hardening parameter (\(\alpha\)) is based on the cumulative plastic strain, such as the volumetric plastic strain or the second invariant of plastic strain, depending on the assumptions made in plastic flow.

A plastic Helmholtz free energy coming from a previous study is applied, and its expression is (Abu Al-Rub and Voyiadjis 2003; Voyiadjis et al. 2008):where \(\mathcalligra{h}\left({d}_{t}, {d}_{c}\right)\) is a degradation function based on two phase field damage variables (\({d}_{t}\) and \({d}_{c}\)). \(b\) is a fitting parameter related to the speed of hardening. The parameter \(b\) is based on laboratory observations, which will be introduced in the later sections. \(Q\) is another parameter, which is related to the plastic flow of this study.

Then, by substituting Eq. (4) into Eq. (3), the hardening law is then obtained:

We notice that the term (\(1-{e}^{-b\alpha }\)) is a normalised term to simulate the evolution of plastic hardening variables with hardening parameters. Hence, the parameter \(Q\) in Eq. (5) must have a unit of stress (Pa), to honour the definition of the plastic hardening variable.

Finally, we turn attention to the plastic degradation function (\(\mathcalligra{h}\left({d}_{t}, {d}_{c}\right)\)) to describe the deterioration of the plastic Helmholtz free energy. In this study, we only consider the effect of shear damage on the plastic Helmholtz free energy. Therefore, the degradation function can be written as: \(\mathcalligra{h}\left({d}_{t}, {d}_{c}\right)\)=\(\mathcalligra{h}\left({d}_{c}\right)\)=\({\left(1-{d}_{c}\right)}^{2}\). The exclusion of tensile damage’s impact on plastic evolution can be attributed to two underlying rationales. First, a previous study (Wu et al. 2006) only considered the intercorrelation of shear damage and plastic Helmholtz free energy. They found generation of shear cracks is due to the slippage between the mineral grains and rock particles in brittle rocks. On the other hand, tensile cracks tend to form without significant plastic strain under tensile stress, leading to catastrophic failure once the crack is generated. Hence, shear cracks are more responsible for the plastic behaviour of rock samples. Second, the post-peak stage in the rock triaxial tests is ignored in the model calibration. Rock internal damage and plastic strain develop rapidly before the ultimate failure of rock samples, as the eventual shear failure plane is formed (Goodman 1989; Jaeger et al. 2009). Consequently, the influence of tensile phase field damage is disregarded within the formulation of the plastic Helmholtz free energy.

The energy required to form a microcrack is path independent, since the formation of each unit of microcrack consumes energy. In fracture mechanics, this energy consumed per unit of microcrack is described by the critical fracture energy per unit of area, which is calculated as a path integral of cracking energy along the microcrack.

However, we note that the significant deviation between the formation of tensile and shear cracks inside rock samples is that the rock resistance to shear cracks is much higher than that to tensile cracks. This observation indicates the necessity of separating the mode I and mode II surface energy according to the rupture across the microcracks. Hence, the surface energies induced by tensile and shear cracks are written as:where \({\mathcal{G}}_{c,I}\) and \({\mathcal{G}}_{c,II}\) represent the critical fracture energy per unit of area for mode I and mode II failure, respectively. \({\Gamma }^{t}\) and \({\Gamma }^{c}\) denote the tensile and shear cracks generated in the RVE system. The surface crack Helmholtz free energy is also decomposed into its tensile component \({\varphi }_{f}^{\left({d}_{t}\right)}\) and shear component \({\varphi }_{f}^{\left({d}_{c}\right)}\). Although the critical fracture energy per unit of area may depend on certain parameters, such as confinement, length scale parameter and temperature, we treat them as constants according to previous studies (Borden et al. 2016; You et al. 2021). The determination of these parameters can be achieved through the Semi-Circular Bend (SCB) test, which will be introduced in Sect. 5.2.

To formulate the path integral in Eqs. (7) and (6), “phase field damage variables” are proposed to transfer the path integral of microcracks into measurable parameters. The path integral in Eqs. (7) and (6) only represents “sharp cracks” without either aperture or FPZ. The phase field method is used to smear both tensile and shear cracks into diffusive phase field damage variables within the length scale parameters (\({{\ell}}^{t}\) and \({{\ell}}^{c}\), respectively), as shown in Eqs. (9) and (8) (Miehe et al. 2010):where \({\mathcal{r}}^{c}\) and \({\mathcal{r}}^{t}\) are the shear and tensile surface crack densities. Equations (7) and (6) are then transformed from path dependent integrals to volume integrals of the RVE system (\(\Omega\)). The more detailed form of the parameters \({\mathcal{r}}^{c}\) and \({\mathcal{r}}^{t}\) can be written as (Ambrosio and Tortorelli 1990; You et al. 2021):

Then by combining the elastic, plastic and surface energy parts of Helmholtz free energy that were introduced in Sects., 2.1, 2.2 and 2.3 respectively, the eventual form of the overall Helmholtz free energy is specified as^*:

Remark (^*): The rock material used in this study is Gosford sandstone. The plasticity of Gosford sandstone is obvious, as proved by our previous studies (Li et al. 2022, 2023b). For high strength rock materials such as granite or marble, where the plasticity is ignorable due to its brittleness, modifications may be necessary, especially in the plastic description of Helmholtz free energy. A pragmatical approach is to delete the plastic part of Helmholtz free energy. Hence, the free energy of high strength rocks can be written as (\({\varphi }^{(d)}={\left(1-{d}_{t}\right)}^{2}{\left(1-{d}_{c}\right)}^{2}\lambda {\left(tr\left[{\varepsilon }_{e}\right]\right)}^{2}+{\left(1-{d}_{c}\right)}^{2}\mu \left(tr\left[{\left({\varepsilon }_{e}\right)}^{2}\right]\right)+{\mathcal{G}}_{c,II}\frac{1}{2}\left(\frac{{\left({d}_{c}\right)}^{2}}{{{\ell}}^{c}}+{{{\ell}}^{c}\left|\nabla {d}_{c}\right|}^{2}\right)+{\mathcal{G}}_{c,I}\frac{1}{2}\left(\frac{{\left({d}_{t}\right)}^{2}}{{{\ell}}^{t}}+{{\ell}}^{t}{\left|\nabla {d}_{t}\right|}^{2}\right)\)).

Since the internal damage in rock materials is not visible and cannot be directly measured, traditional damage variables are generally treated as internal variables. Traditional damage variables are related to the presence of microcracks inside the rock samples, as the development of these microcracks deteriorates the rock’s mechanical properties. One effective approach for determining traditional damage variables is to use a combination of measurements of ultrasonic wave velocity (the change of rock modulus) and acoustic emission events (precise location and profile of every microcrack obtained from analysing the focal mechanism and source parameters). A relationship between the variation of modulus and traditional damage variables is established in this study. This approach is inspired by a few previous studies that have used double scalar damage variables (Tang et al. 2002; Lu et al. 2022). The effective Lamé constants, shown as Eqs. (13) and (14), incorporate traditional damage variables in this study:where \(\widetilde{\lambda }\) and \(\widetilde{\mu }\) are the effective Lamé constants. \({\omega }_{t}\) is the tensile traditional damage variable and \({\omega }_{c}\) is the shear traditional damage variable. Note that the definition of effective Lamé constants in Eqs. (13) and (14) also indicates that there is no coupling effect between \({\omega }_{t}\) and \({\omega }_{c}\). The presence of microcrack-induced voids within the rock specimen results in the attenuation of ultrasonic wave propagation. Furthermore, the ultrasonic wave velocity can also be mathematically represented by the modulus of the rock sample. The ultrasonic wave velocity defined by the Lamé constants is:where \({v}_{p}^{0}\) is the ultrasonic wave velocity for the intact sample and \(\rho\) is the density of the rock sample. Then the ultrasonic wave velocity of the damaged rock sample is obtained by substituting Eqs. (13) and (14) into Eq. (15):where \({v}_{p}^{\omega }\) is the ultrasonic wave velocity for the damaged rock sample expressed by traditional damage variables. Even though the density may change in the laboratory triaxial tests performed in this study, the influence of density is not involved to avoid the potential coupling between plasticity and damage, as has been explained in previous studies (Lemaitre and Desmorat 2005b; Li et al. 2022). Hence, the variance of ultrasonic wave velocity is purely due to the change of effective Lamé constants.

The increase of rock traditional tensile and shear damage variables is accompanied by a decrease in ultrasonic wave velocity, as the wave propagation is restricted by microcracks (see Eq. (16)). Since the calibration in this study is based on the pre-peak stage of the complete stress–strain curve, the traditional tensile and shear damage variables applied in this study are small (less than 10%, see Fig. 3). Hence, full differential on Eq. (16) is performed to analyse how the increment of both traditional tensile and shear damage variables contributes to the ultrasonic wave velocity:

However, since there are two unknown variables \(d{\omega }_{t}\) and \(d{\omega }_{c}\), it is not possible to directly calculate the traditional damage variables (\({\omega }_{t}\) and \({\omega }_{c}\)) from Eq. (17) alone. To complement Eq. (17), acoustic emission analysis and moment tensor inversion techniques are employed to comprehend the characteristics and distribution of microcracks. For example, the ratio between the increment of traditional tensile and shear damage (\(d{\omega }_{t}/d{\omega }_{c}\)) can be used to calculate traditional tensile and shear damage variables \({\omega }_{t}\) and \({\omega }_{c}\) from Eq. (17).

From the moment tensor inversion, the profile of microcracks is linked to the traditional damage variables in micromechanics, stating the traditional damage variable is an accumulative effect by each individual microcrack (Pensée et al. 2002; Nemat-Nasser and Hori 2013). A well-acceptable equation (\({D}_{ij}=\sum_{{\text{k}}=1}^{n }\frac{{v}_{k}}{V}\left({\overrightarrow{n}}_{k}\otimes {\overrightarrow{n}}_{k}\right)\)) indicates that the anisotropic damage variable can be determined based on the volume and rupture plane of microcracks (Shao et al. 1999, 2005; Shao and Rudnicki 2000). This equation is modified to ensure that the traditional damage variable and the phase field damage variable discussed in this study share a similar isotropic assumption \(({\overrightarrow{n}}_{k}\otimes {\overrightarrow{n}}_{k}=\left[\begin{array}{ccc}1& & \\ & 1& \\ & & 1\end{array}\right])\). Similar modifications have been observed in a few pioneering studies via laboratory observations and constitutive modelling (Feng and Yu 2010; Zhang et al. 2022). Consequently, the traditional damage variables can be determined by leveraging the volume quantification of microcracks.where \(V\) is the total volume of rock RVE. \({V}^{c}\) and \({V}^{t}\) are the volume of shear and tensile cracks, respectively.

Acoustic sensors are attached to the rock samples to identify the tensile and shear cracks inside the rock sample and to determine the detailed profile of microcracks, such as the volume and the rupture direction of microcracks. If the rupture of microcracks is assumed to be an inhomogeneous point source in the infinite displacement field, the microcrack rupture can be expressed by Green’s function:where \({u}_{i}\) is the displacement in the \({i}_{th}\) direction of the source-induced displacement field at the position \({{\varvec{x}}}_{{\varvec{f}}}\) and time \(t\). \({{\varvec{x}}}_{0}\) and \({t}_{0}\) are the location and occurrence time of the point rupture source. Note that the bold notifications indicate a vector variable, and therefore \({{({\varvec{x}}}_{0})}_{k}\) is the \({k}_{th}\) direction of the source event. \({M}_{jk}\) is the moment tensor used to describe the rupture of the seismic event source. \({G}_{ij}\) is Green’s function, describing the overall displacement field generated by the rupture of the point source event.

If the linear elastic assumption is taken in the microcrack rupture, the moment tensor can be written as:where \(n\) and \(v\) are the rupture plane direction of microcracks and the direction of rupture respectively. \({u}_{d}\) is the displacement induced by the microcrack rupture and \(s\) is the surface area of microcracks. Then the trace components of the moment tensor (\(M\)) can then be written as:where \(dV\) is the volume change owing to the rupture of microcracks. Equation (22) builds the relationship between the trace of the moment tensor and the volume change of microcracks. \(dV\) could be \(d{V}^{c}\) and \(d{V}^{t}\) according to the rupture across the microcracks.

Also, the dot product between the direction of rupture and the direction of the microcracking plane (\(n\cdot v\)) can be expressed by eigenvectors of the moment tensor:

Another concern is to distinguish between tensile and shear microcracks based on the moment tensor analysis. It should be noted that microcracks may not be simply classified into tensile or shear cracks, and there could be mixed-mode cracks. However, in this study, microcracks were forced to be classified as either tensile or shear cracks based on the dominant rupture mechanism. In this study, \(R\) variable was used to illustrate the ‘polarity’ of each microcrack from moment tensors:where \({m}_{i}^{*}\) is the eigenvalues of the deviatoric part of the moment tensor. If \(R\) variable ranges from 30 to 100%, then the microcracks are considered as a tensile failure. Shear microcracks show the \(R\) variable between -30% and 30%. Finally, microcracks belonging to compression failure (\(R\)<− 30%) are not accounted for in this study as healing mechanics are normally ignored in damage constitutive models.

Then the ratio between traditional tensile and shear damage increments (\(d{\omega }_{t}/d{\omega }_{c}\)) can be determined by applying Eqs. (18), (19), (22) and (24) to inverse the traditional damage variables along with Eq. (17).

Akin to the thermodynamic framework of phase field damage variables mentioned in Sect. 2, the traditional damage variables are also derived from a hypothesis of energy equivalence. The most common form of Helmholtz free energy regarding traditional damage variables is a double-term equation, only containing elastic and plastic components (Salari et al. 2004; Shao et al. 2006), whilst the damage effect is manifested as the degradation function in front of the elastic and plastic Helmholtz free energy. Hence, we only consider a two-component Helmholtz free energy for traditional damage variables (Chen et al. 2015; Wang and Xu 2020).

The complete Helmholtz free energy for the traditional damage variables is denoted as \({\varphi }^{(\omega )}\), whilst \({\varphi }_{e}^{\left(\omega \right)}\) and \({\varphi }_{p}^{\left(\omega \right)}\) represent the elastic and plastic Helmholtz free energy. The form of elastic Helmholtz free energy for traditional damage variables is also various, dependent on the specific material being studied, especially when two independent damage variables are engaged. First, decomposition is employed on elastic Helmholtz free energy to discern the effect of traditional tensile and shear damage variables separately. A previous study employed traditional damage variables within the framework of strain-equivalence hypothesis, whereby volumetric and deviatoric decompositions of the Helmholtz free energy were conducted (Desmorat et al. 2007). In addition, the coupling effect between traditional tensile and shear damage variables on the Helmholtz free energy also needs further consideration, such as in Matzenmiller et al. (1995) and Alfarah et al. (2017), a multiplier term \((1-{\omega }_{t})(1-{\omega }_{c})\) is introduced to describe the degradation on the entire Helmholtz free energy by traditional damage variables.

Inspired by the abovementioned investigations, the form of elastic Helmholtz free energy for the traditional damage variables is:

The degradation function in front of volumetric components of Helmholtz free energy is \({\left(1-{\omega }_{t}\right)}^{2}{\left(1-{\omega }_{c}\right)}^{2}\). The degradation function is related to the square of traditional tensile and shear damage variables, incorporating the application of energy equivalence hypothesis (Darabi et al. 2012).

The component of plastic Helmholtz free energy for the traditional damage variables is:

The degradation function \(\mathcalligra{h}\left({\omega }_{t}, {\omega }_{c}\right)\) for traditional damage variables is consistent with the elastic Helmholtz free energy in previous studies (Shao et al. 2006; Chen et al. 2015). Herein, the significant contribution of shear cracks on the plastic flow encourages the form of degradation function as \(\mathcalligra{h}\left({\omega }_{t}, {\omega }_{c}\right)={(1-{\omega }_{c})}^{2}\).

Phase field damage variables identified in this study are independent fields besides elastic and plastic strains, which are evolved to minimise the overall Helmholtz free energy defined in Eq. (12). Rock samples in the phase field damage model follow the functional of overall Helmholtz free energy.where \({d}_{t}({\varvec{x}})\) and \({d}_{c}({\varvec{x}})\) are the phase field tensile and shear damage variables, written as a function of position \({\varvec{x}}\) in the rock sample*. This is similar to the “test functions” in COMSOL software to resolve partial differential equations in FEM models. The parameter \(I({d}_{t},{d}_{c})\) is the function of phase field damage variables (\({d}_{t}\) and \({d}_{c}\)). \(\nabla {d}_{t}\) and \(\nabla {d}_{c}\) are the spatial derivative of phase field damage variables. Equation (28) identified the overall energy of the rock system in FEM (e.g. rock sample in this study). Hence, the minimised value of Eq. (28) is the most stable state for the entire system. Based on the Euler–Lagrange equation (Hamilton’s Principle), the minimised value for \(I\left({d}_{t},{d}_{c}\right)\) occurred when:

Remark (*): Writing \({d}_{t}({\varvec{x}})\) and \({d}_{c}({\varvec{x}})\) as functions of position \({\varvec{x}}\) may somehow violate Eq. (1), where the \({d}_{t}\) and \({d}_{c}\) are independent variables. However, Sect. 2 introduces the theoretical framework of the phase field damage model whereas this section focuses on the numerical interpretation of the model. The study area in Sect. 2 is a homogeneous RVE system whereas \({\varvec{x}}\) here represents any points in the FEM. The variational method diffuses all internal variables into functions of position (\({\varvec{x}}\)) and uses test functions to find the minimised “virtual work” according to the internal variables.

Then substituting Eq. (12) into Eqs. (29) and (30), we get the phase field damage evolution law as follows:

Equations (31) and (32) introduce the evolution law of tensile and shear phase field damage variables, respectively. Note that \(\Delta\) is the Laplacian operator and its expression is \(\Delta =\nabla \cdot \nabla\) (Alessi et al. 2018).

The formation of microcracks is an irreversible process that involves energy consumption and dissipation. To avoid rock healing, an easy method is to introduce another historical field of rock energy in the FEM system. The historical field is the maximum energy in history and allows for a one-to-one correlation between the phase field damage variables and the maximum energy recorded. The new evolution law is written as:where \({\mathcal{H}}^{t}\) and \({\mathcal{H}}^{c}\) are the historical fields for tensile and shear energies. They are defined as:

The maximum energy fields \({\mathcal{H}}^{t}\) and \({\mathcal{H}}^{c}\) for tensile and shear microcracks independently range from \({\mathcal{H}}^{t}, {\mathcal{H}}^{c}\in \left[0\right., \left.+\infty \right]\).

The evolution of traditional damage variables is explicit, related to their conjugate forces, as the differential of the total Helmholtz free energy with respect to the traditional damage variables. Then the conjugate forces according to the traditional damage variables are written as:where \({Y}_{{\omega }_{t}}\) and \({Y}_{{\omega }_{c}}\) are the thermodynamic conjugate forces according to the traditional tensile and shear damage variables. The damage loading envelope is generally taken as a simple form with its conjugate forces:where \({f}_{{\omega }_{t}}\) and \({f}_{{\omega }_{c}}\) are the yielding and loading envelopes for the traditional tensile and shear damage variables, respectively. \({r}_{{\omega }_{t}}\left({\omega }_{t}\right)\) and \({r}_{{\omega }_{c}}\left({\omega }_{c}\right)\) are the tensile and shear energy release thresholds at given traditional tensile and shear damage variables. The selection of \({r}_{{\omega }_{t}}\left({\omega }_{t}\right)\) and \({r}_{{\omega }_{c}}\left({\omega }_{c}\right)\), serving as the damage hardening process, controls the evolution of traditional damage variables.

The evolution of traditional damage variables can be described using a plastic incremental theory. In the case of isotropic damage, the increment of traditional damage variables is equivalent to the plastic multipliers:where \(d{\lambda }_{{\omega }_{t}}\) and \(d{\lambda }_{{\omega }_{c}}\) are the damage multiplier for the traditional tensile and shear damage variables. The determination of those variables stands on the damage consistency theory, indicating that during the incremental loading path, the traditional damage variables and their conjugate forces continuously locate on the damage loading surface. That is to say, the full differential of the damage loading surface is zero:

Then the damage multipliers (\(d{\lambda }_{{\omega }_{t}}\) and \(d{\lambda }_{{\omega }_{c}}\)) can be back-calculated from Eqs. (41) and (42).

In the authors’ previous study, the correlation between the single-phase field damage variable and the traditional damage variable was conducted (Li et al. 2023b). In this section, we extend our previous research from a single-phase field damage model to a double-phase field damage model that includes both tensile and shear phase field damage variables. Evolution law of traditional damage variables is compared with that of phase field damage variables, and their tensile components are:

The term \({l}^{t}\Delta {d}_{t}\) measures the diffusive effect of microcracks inside the rock sample. However, in our calibration procedure, the ultrasonic wave velocity measurements are based on the ultrasonic nodes at both ends of the cylinder sample, which only provides a homogenous representation of rock damage. Therefore, the smeared field (\(\Delta {d}_{t}\) in the phase field damage model) can be ignored to match the traditional damage variable, as suggested by a previous study (You et al. 2021) (\({l}^{t}\Delta {d}_{t}\approx 0\)).

The form of \(\frac{\partial \left({\mathcal{H}}^{t}\right)}{\partial {d}_{t}}\) and \(d{Y}_{{\omega }_{t}}\) are similar, namely the differential of elastic Helmholtz free energy to the traditional or phase field damage variable, since the elastic Helmholtz free energy selected in this study is similar (see Eqs. (2) and (26)). Then the last term of the tensile phase field and traditional tensile damage variables are \(\frac{{d}_{t}}{{l}^{t}}\) and \(\frac{\partial {r}_{{\omega }_{t}}}{\partial {\omega }_{t}}d{\omega }_{t}\), respectively. For the tensile phase field damage variable, a linear trend is observed (\(\frac{\partial \left({\mathcal{H}}^{t}\right)}{\partial {d}_{t}}\sim {d}_{t}\)) if Eq. (2) is substituted into Eq. (45). This indicates that a linear damage evolution law is a valid fitting relationship to satisfy both equations in (45), as mentioned in previous studies (Shao et al. 2006). Hence, we have found that the tensile phase field damage variable and the traditional tensile damage variable may coincide, leading to a possible bridge to use the parameters in the tensile traditional damage variable to express the evolution law of the tensile phase field damage variable:

For the shear components of both traditional and phase field damage variables, there exists a similar relationship as shown in Eq. (46):

The conjugate force of traditional damage variables \({Y}_{{\omega }_{t}}\) and \({Y}_{{\omega }_{c}}\) can be measured from the complete stress–strain curves in rock triaxial tests. Hence, the length scale parameters \({l}^{t}\) and \({l}^{c}\) is able to be calibrated.

To calibrate the phase field damage and traditional damage models, a series of triaxial tests were performed in this study. The experiments utilise the multiphysics high-pressure high-temperature rock testing system, which was developed within the mining geomechanics laboratory at UNSW. Four triaxial tests were performed, with varying confinements of 10 MPa, 15 MPa, 20 MPa and 30 MPa. We also performed a uniaxial loading test to calculate the uniaxial compressive strength (UCS: \({f}_{c}\), 48.2 MPa), which is an important parameter in plastic modelling. The triaxial cell used in this study is HTRX-140XL, manufactured by GCTS in the US, with a maximum confinement stress limited to 140 MPa. A high-stiffness loading frame, under the governance of a servo-control system, imparts axial loading onto the rock specimens. Two loading platens are embedded with ultrasonic transducers to measure the ultrasonic wave velocity passing through a rock sample. The ultrasonic node at the top of the rock sample continuously emits P waves to pass through the rock sample. These waves are then received by the bottom node. Then the P wave velocity can be calculated using the length of the rock sample and the time taken from the top to the bottom node.

Eight acoustic sensors are attached around the rock sample, receiving the seismic waveforms from the microcracking process. The raw waveforms are first processed by the AEwin software from MISTRA. Then the triggering waveforms are transferred into Insite software by ITASCA for further processing. The entire setup of the laboratory equipment is illustrated in Fig. 1.

Rock samples used in this study were core-drilled from the same block of Gosford sandstone to minimise heterogeneity. The Gosford sandstone is medium-grained material, with the grain size of 0.2–0.3 mm, as reported by Zhao et al. (2014). The cementation of Gosford sandstone is not particularly strong, and quartz constitutes only 70–80% of its composition. The remaining composition includes impurities such as clay and feldspar. The surface of each sample was carefully inspected to ensure the rock samples were intact. Cylinder samples of the Gosford sandstone had a diameter of 50 mm and a height of 100 mm. Furthermore, the surface of the Gosford sandstone sample was ground to guarantee the flatness of the sample within a tolerance of \(\pm 0.01\) mm.

Before conducting the loading test, a 1 kN axial pre-load was applied to the rock sample to ensure that the platen adhered closely to the rock samples. Subsequently, hydrostatic pressure was applied using hydraulic oil to levels of 10 MPa, 15 MPa, 20 MPa and 30 MPa through a hand pump. The confinement provided by the hand pump was monitored during the triaxial test, ensuring variation remained no more than 2%. Following the application of confinement, an axial load was applied using a high-stiffness loading frame (with a stiffness greater than 5 MN/mm), which was regulated by the displacement servo-control system. The triaxial test was conducted at a loading speed of \(3\times {10}^{-6}\) m/s to ensure that the triaxial loading was quasi-static. The data acquisition system, used to measure force and defamation, had a frequency of 10 Hz, surpassing the 1 Hz requirement set by the ISRM standard. The entire loading test was conducted in strict accordance with the ISRM standard (Fairhurst and Hudson 1999).

In fracture mechanics, parameters \({\mathcal{G}}_{c,I}\) and \({\mathcal{G}}_{c,II}\) are known to regulate the energy consumed to form per unit length of microcracks. The determination of these parameters has been extensively studied in the literature (Kuruppu and Chong 2012; Kuruppu et al. 2014; Backers and Stephansson 2015). In our previous study, we determined \({\mathcal{G}}_{c,I}\) to validate the single-phase field damage model (Li et al. 2023b). In this section, we focus on the determination of mode II critical fracture energy per unit length using the semi-circular bend specimen.

The Semi-Circular Bend test is the ISRM-suggested method to determine rock toughness (\({K}_{IIc}\)), based on a previous study (Ayatollahi and Aliha 2006). We take a semi-circular rock sample with a diameter of 95 mm. A notch is cut at the centre of the rock sample with a length of 23.75 mm, half of the radius of the rock sample. The notch is 45° inclined to ensure the crack propagates from the tip of the rock sample in the mixed mode. The width of the rock sample is also 47.5 mm. The setup of the rock sample can be observed in Fig. 2.

According to the ISRM standards, the width of the notch should be cut via a blade thinner than 1.5 + 0.2 mm (Kuruppu et al. 2014). The blade we used here for cutting is 1.5 mm and the width of the notch is within 1.7 mm, as measured by both a straightedge and a calliper as shown in Fig. 2b.

Then the toughness of mode II fracture (\({K}_{IIc}\)) can be determined according to the maximum force that the semi-circular sample can sustain in the SCB test. The equation applied to calculate the \({K}_{IIc}\) is (Ayatollahi and Aliha 2006):where \({Y}_{II}\) is named the mode II geometry factor as a function of sample geometry. The determination of \({Y}_{II}\) requires crack length (\({\alpha }_{l}\)), sample diameter (\(D\)), and the distance between two support rods in the SBC test (\(S\)). The value of the parameter \({Y}_{II}\) in this study is taken from a previous study (Ayatollahi and Aliha 2004, 2006).

Another concern is the relationship between the toughness of mode II fracture (\({K}_{IIc}\)) and the (\({\mathcal{G}}_{c,II}\)) critical energy required to form a unit length of microcracks. The relationship between those two parameters from elastic mechanics is summarised as^*:

Then, the parameter \({\mathcal{G}}_{c,II}\) can be back-calculated according to Eqs. (48) and (49), as an important parameter of the entire double-phase field damage model.

Remark (*): The relationships between toughness in mode I (\({K}_{Ic}\)) and mode II (\({K}_{IIc}\)) fractures, and their critical energy required to form a unit length of microcracks for mode I (\({\mathcal{G}}_{c,I}\)) and mode II (\({\mathcal{G}}_{c,II}\)) are based on the linear elastic solution of an existing fracture in an infinite field. This assumption of elastic and isotropic behaviour implies that the compression modulus of the rock material is equal to its tension modulus. In previous studies, the transition from \({K}_{Ic}\) to \({\mathcal{G}}_{c,I}\) (Zhang et al. 2003; Anderson 2017; Aliha et al. 2023) and from \({K}_{IIc}\) to \({\mathcal{G}}_{c,II}\) (Shen et al. 2014; Chen and Wong 2018; Li et al. 2018) does not distinguish between compression and tension modulus, but instead uses ‘elastic modulus’. In accordance with these pioneering studies, this study also employs the elastic modulus (\(E\)) in Eq. (49).

Given the absence of distinction in the plastic Helmholtz free energy between traditional damage variables and phase field damage variables, the plastic law remains consistent for both scenarios. In this study, the yielding surface employed is given by (Menetrey and Willam 1995; Papanikolaou and Kappos 2007):where \(\xi\) is related to the first invariant of the stress tensor and \(\rho\) is a measurement of the second invariant of the stress tensor, \(\theta\) is the Lode’s angle (Bao et al. 2013). Since only triaxial tests are performed in this study (\({\sigma }_{2}={\sigma }_{3}\)), the variable \(r\left(\theta ,e\right)\) is constantly equal to one in the yielding envelope. \(k\) is the hardening variable, representing the expansion of entire yielding surface, whilst \(\alpha\) is the hardening parameter, representing the plastic flow generated in the rock sample. \(m\) is the internal rock friction. The expression of variable \(m\) is (Cervenka et al. 1998; Červenka and Papanikolaou 2008):where \({f}_{c}\) is the UCS of rock samples, \({f}_{t}\) is the uniaxial tensile strength of rock samples. \(e\) is the out-of-roundness of rock material.

Under triaxial conditions, the yielding surface proposed in Eq. (50) is degraded into a Hoek–Brown yielding envelope. A few pioneering studies applied the yielding surface of Eq. (50) in rock and concrete materials and found that the versatile yielding envelope has a significant capacity to formulate rock plastic response (Pietruszczak et al. 1988; Grassl et al. 2002).

The form of plastic hardening law is given by Eq. (5), which relates the hardening variable and the hardening parameter. The hardening variable is chosen as the volumetric plastic strain (\({\varepsilon }_{v}^{p}\)), which has a more solid physical interpretation compared to a higher order polynomial (Papanikolaou and Kappos 2007; Bao et al. 2013).where \({k}_{0}\) is the initial hardening variable, at the initial yielding surface. When the hardening parameter (\(\alpha\)) is zero, the hardening variable approaches its initial value, \({k}_{0}\). Conversely, when \(\alpha\) approaches infinity, the hardening variable increases to \({f}_{c}\).

The evolution law of plasticity is usually non-associated, indicating the direction of plastic increment may not coincide with the gradient of the yielding surface. In this study, as laboratory experiments only contain triaxial tests, the plastic potential does not necessitate Lode’s angle effect. The plastic potential is then expressed as:where \(g\) is the plastic potential in this study, \(A\) and \(C\) are two fitting parameters.

The laboratory results from the triaxial test are illustrated in Fig. 3. We noticed that the axial stress–strain relationship starts with a non-linear trend at the very beginning of the stress–strain curves. Then the linear relationship between stress and strain replaces the initial non-linear trend as typical elastic behaviour in this stage. The ultrasonic wave velocity increases at this stage, owing to the closure of microcracks inside the rock samples. Indeed, the damage variable is not presented in this stage since the drop of ultrasonic wave velocity is not observed (the healing mechanism is not accounted for (Lemaitre and Desmorat 2005b; Li et al. 2022)). On the other hand, when the non-linear axial stress–strain relationship is illustrated, both rock-induced damage and plastic deformation start to control the rock’s mechanical response. An increasing trend of both tensile and shear components of traditional damage variables can be observed.

Another interesting finding is that in the 10 MPa, 15 MPa and 20 MPa cases in Fig. 3a–c, respectively, the deviation between the traditional tensile and shear damage variables is significant. More tensile damage components are generated than shear damage components. However, for the 30 MPa scenario in Fig. 3d, the difference between tensile damage and shear damage reduces, where the tensile damage and shear damage develop simultaneously. This observation represents a typical pressure sensitivity of rock samples, indicating that the mechanical response of rock samples changes with the confinements applied. With the increasing confinements, the transition from the tensile-dominated failure mode to tensile–shear mixed failure mode represents an increasing trend of shear components in rock samples, as also reported by previous studies (Jiang et al. 2016; Huang et al. 2021).

The elastic modulus of rock samples is estimated from the triaxial laboratory test results (see Fig. 3), where the slope of linear increasing stress–strain curves illustrates the axial modulus of rock samples. In this study, the Gosford rock sample is considered isotropic (Sufian and Russell 2013; Yin and Zhao 2014) and only two independent variables are sufficient to describe the elasticity of the rock sample (\(E\) and \(\nu\)). We also observed that the modulus of the specimen is related to confinement, as a common observation in previous studies (Asef and Reddish 2002; Li et al. 2023a), where rock samples become stiffer after the application of confinement. In Fig. 3, Young’s modulus \(E\) for confinements of 10 MPa, 15 MPa, 20 MPa and 30 MPa is 3.47 GPa, 3.75 GPa, 3.78 GPa and 3.90 GPa, respectively.

The plasticity parameter is based on the yielding and ultimate failure stress in the triaxial tests. The confinements, yielding stress and failure stress are recorded to fit the yielding and ultimate failure surface in Eq. (50). Then the plastic parameters can be summarised as: \(m=\) 19 and \({k}_{0}=0.56{f}_{c}\). By substituting \(m\) and \({k}_{0}\) into Eq. (50), the plastic hardening variable can be calculated at given confinements and axial stress. The hardening parameter, as the volumetric plastic strain obtained from the stress–strain curves, is used to fit the hardening law, with the result of \(b=300\). The plastic potential is determined according to the fitting of plastic increment theory, and the parameters are summarised as: \(A=\) 0.0181 and \(C=\)-0.0247.

The progression of damage within the proposed model requires the initial determination of the conjugate force according to the traditional damage variables. Next, the (\({\mathcal{G}}_{c,II}\)) critical fracture energy per unit length of shear cracks is determined from Sect. 5.2 (44.17 J/m²) and the (\({\mathcal{G}}_{c,I}\)) critical fracture energy per unit length of tensile cracks is taken from a previous study (131 J/m²) (Li et al. 2022). Then the length scale parameter for the tensile and shear cracks can be calculated as: \({l}^{t}=2.7\times {10}^{-6}\) m and \({l}^{c}=4\times {10}^{-7}\) m.

Then the model calibration is illustrated in Figs. 4 and 5, where the laboratory observations, the traditional damage model and the phase field damage model all demonstrate commendable consistency.

Note that the fittings shown in Figs. 4 and 5 are results from COMSOL software. In these simulations, an axisymmetric sample was constructed to simulate the triaxial tests of cylinder rock samples. Two platens were applied to the top and bottom of the rock sample. The grids used for both the platens and rock sample were set to ‘Finer’ grids to accurately capture elasticity, plasticity and damage evolution. One hundred sixty-eight ‘Finer-grid’ elements are automatically generated in COMSOL with the dimension of 4.2 mm × 0.8 mm. In the COMSOL simulation, three internal variables (\({d}_{t}\), \({d}_{c}\) and \(\alpha\)) were solutions from three independent physical fields. The non-linear method applied was ‘constant (Newton)’ to ensure better convergence. For each distinct field, the calculation was terminated once the tolerance (relative error of the physical fields, as 0.1% in this study) was met.

Figure 6a to d shows COMSOL simulation results for the 15 MPa confinement experiment. Figure 6a indicates the setup of triaxial test with rock sample and loading platens. Due to the axial symmetry of the triaxial test, rotating a 2D plane around the symmetrical axis can accurately simulate the mechanical behaviour of the rock sample. The distribution of \(1-{(1-{d}_{t})}^{2}\) \({(1-{d}_{c})}^{2}\) is illustrated in Fig. 6b with damage mainly concentrated at the centre and the corner of rock samples. The distribution of \({d}_{c}\) and \({d}_{t}\) is also illustrated in Fig. 6c and d. Shear phase field damage variables are mainly observed at the centre of the rock sample. The tensile damage concentrates on the diagonal line of rock sample, indicating the proactive role that tensile cracks play in the formation of the ultimate failure of rock samples. However, in Fig. 6, crack initiation and coalescence in rock sample are not observed, since the rock sample in this model is purposely calibrated as continuous and intact, without the influence of microcracks.

Then the damage distribution of numerical results is compared with the laboratory observations. Acoustic events monitored in the laboratory test are shown in Fig. 6f, as well as the eventual failure plane depicted from Fig. 6e. Acoustic events are found to concentrate along the ultimate failure plane, as microcracking damage inside rock sample. We then focus on the focal mechanism of acoustic events within 10 mm to the ultimate failure plane (see Fig. 6g), presenting the formation of ultimate failure plane. We found that in the centre of rock sample, more shear failures are observed than the tensile failure, which is also proved via Fig. 6c to d that \({d}_{c}\) is larger than \(2.5\times {10}^{-2}\) and \({d}_{t}\) is between \(1\times {10}^{-2}\) to \(1.5\times {10}^{-2}\). Also, at two ends of failure plane, tensile failure is more common than their shear counterparts, this is also captured by the double-phase field damage model, that \({d}_{c}\) is ranged from \(1\times {10}^{-2}\) to \(1.5\times {10}^{-2}\) and \({d}_{t}\) is larger than \(2.5\times {10}^{-2}\). Hence, the double-phase field damage variable is comparable to the actual failure generated in rock triaxial test.

In this study, the validation of the proposed model is performed in COMSOL software. In COMSOL, multiple pre-installed physical fields are embedded into the FEM system. The “test function” is applied to obtain a close approximation of the partial differential equations. The test functions are first satisfied by the proposed boundary conditions, iterated to ensure the error between the partial differential equations and the FEM result is small enough. The weak form of the control equations is summarised as:where \({R}_{u}\), \({R}_{{d}_{c}}\) and \({R}_{{d}_{t}}\) are the residuals of displacement, shear phase field damage variable and tensile phase field damage variable. These values need to be minimised with respect to the location-dependent test function \({T}_{u}\), \({T}_{{d}_{c}}\) and \({T}_{{d}_{t}}\). \(\Omega\) indicates the volume of the RVE system and \(\partial {\Omega }_{t}\) is the boundary element with Neumann boundary conditions \(\sigma \cdot n=\overline{t }\) (e.g. force boundary conditions).

The simulation result of the proposed elastoplastic double-phase field damage model is illustrated in Fig. 7. A 45° inclined notch is set at the centre of the rock sample, with a width of 50 mm and a height of 100 mm, which sustains a constant loading rate (0.01 mm/s) at the top of the rock sample. To accurately capture the crack propagation from the notch, a fine-grid zone that covers the notch area is defined with refined grid sizes. The other part of the rock sample has relatively coarse grids. The solving setups and fields are the same as the COMSOL model in Sect. 5.5. To illustrate the crack propagation under both tensile and shear phase field damage conditions, an equivalence phase field damage variable considering both tensile and shear damage is summarised as \(1-{(1-{d}_{t}^{ })}^{2}{(1-{d}_{c}^{ })}^{2}\).

The evidence of microcrack generation can be observed when a 2 mm axial displacement is exerted onto the loading platen. The phase field damage zone is mainly concentrated on the tips of the pre-existing notch, which can be explained by the stress concentration theory. According to this theory, the end of microcracks is more likely to aggregate the stress distribution. The actual shape of microcracks is observed when the axial platen advancement reaches 4 mm, where lateral and vertical cracks initiate around the tip of microcracks.

The sequence of microcracks generated in the numerical model may not exactly match the laboratory observations, particularly the generation of vertical wing cracks after the lateral cracks. One possible reason for this discrepancy is that the threshold for crack generation is not explicitly defined in the current framework of the phase field damage model but is set to zero in a previous study (Miehe et al. 2010). To address this issue and improve the accuracy of the model, a reconstruction of the thermodynamic framework for the double-phase field damage model may be necessary, which could be a topic for future research.

The microcracking mechanism of rock can be separated into tensile and shear modes according to the definition of the phase field damage model. In the shear phase field damage model, \({\varphi }_{p}^{\left(d\right)}\) represents the plastic contribution to the shear fractures, whilst \({\varphi }_{{f}^{ }}^{\left({d}_{c}\right)}\) manages the shear failure due to microcrack propagation. Therefore, the microcracking process may involve different mechanisms that are hard to be directly identified in laboratory tests. In this section, we aim to determine the mechanism of crack propagation, leveraging the proposed double-phase field damage model.

The simulation of crack propagation via the phase field damage model also necessitates cross-checking with well-established laboratory tests. This is essential both to verify the proposed model and to highlight its limitations. In this context, pioneering research from the last few decades was reviewed (Feng et al. 2019; Li et al. 2019a, b; Lin et al. 2019; Wang et al. 2020), demonstrating that crack initiation from notched rock samples can be categorised into wing cracks and secondary cracks, as reported by Lajtai (1974); Nemat‐Nasser and Horii (1982); Horii and Nemat‐Nasser (1985); and Bobet and Einstein (1998). Further distinction amongst secondary cracks reveals coplanar and oblique cracks, as identified by Park and Bobet (2009) and Cao et al. (2019).

Figure 8a shows the concentration of shear damage at the initial stage of loading, highlighting the significant role that plastic strain plays in crack propagation. This observation is similar to those made using high-speed and digital cameras (Ju et al. 2018) and Digital Image Correlation (DIC) (Yang and Jing 2011; Zhou et al. 2018; Liu et al. 2021). Figure 8b illustrates how the pathway of vertical cracks forms in our numerical results. As axial loading increases beyond 3.4 mm, vertical (wing) cracks begin to propagate upwards. Notably, the formation of wing cracks includes a shear component, suggesting that shearing behaviour contributes to the formation of these cracks. This is supported by recent studies indicating the coexistence of mixed-mode I–II cracks in wing crack formation (Liu et al. 2022). Figure 8c further indicates that plasticity affects the propagation of shear cracks from the notched sample, with a highly plastic zone observed from the tip of the pre-notch to the top of the rock sample (similar to ‘S’ shape fracture in (Dong et al. 2020)).

The results of this study are compared with laboratory observations in Fig. 8d (Liu et al. 2022). The existence of wing cracks, coplanar secondary cracks and oblique secondary cracks are observed in this numerical simulation. However, we also note that the formation of wing cracks lags behind that of oblique secondary cracks, which contrasts with laboratory findings (Wong and Einstein 2006). Nevertheless, the general shape and pathway of wing cracks align with laboratory observations, curving parallel to the axial loading direction. The direction of coplanar secondary crack propagation also matches laboratory results, with a similar inclination angle to the pre-made notch. In addition, oblique cracks form at a significant inclination angle, as shown in Fig. 8d (Zhao et al. 2018; Wang et al. 2019). Overall, the proposed model captures key characteristics of crack propagation from a notched rock sample, although further improvements are required based on existing laboratory investigations. The difference between laboratory observations and proposed phase field damage model can be owing to the instability of the numerical model, where the convergence in FEM cannot be reached when the fractures are massively developed. A more detailed matching between phase field damage model and laboratory results will be an interesting direction for future research.

Compared with a previous phase field study where the plastic component of Helmholtz free energy is not considered (Jarrahi et al. 2021), our promising results prove the existence of interaction between crack propagation and plasticity (Schmidt 1980). Therefore, the significance of plastic Helmholtz free energy (Sect. 2.2) should be emphasised in simulating rock failure behaviour. In summary, the proposed double-phase field damage model successfully reveals the microcracking mechanism in the crack propagation process, including the influence of plastic strain and the percolation of tensile and shear microcracks inside rock samples.