A three-field phase-field model for mixed-mode fracture in rock based on experimental determination of the mode II fracture toughness

The phase-field approach to fracture is based on the variational formulation by Francfort [25] and its subsequent regularization by Bourdin [16, 17]. Here, the discrete crack is approximated using a scalar variable s, the so-called phase-field, which smears the crack over a regularization width \(l_0\). The phase-field parameter attains a value of zero on the crack and is one if the material is undamaged. Crack propagation is considered as a minimization problem of the associated functionalHere, \(G_c\) is the critical fracture energy, g(s) is the degradation function which models the loss of stiffness in the material due to damage, w(s) is the energy dissipation function, and \(c_w\) is a scaling parameter. Formulation (1) suffers from interpenetration of crack surfaces and non-physical crack patterns in compression. Thus, commonly, an additive split of the elastic strain energy density \(\Psi (\varvec{\varepsilon }) = \Psi ^+(\varvec{\varepsilon }) + \Psi ^-(\varvec{\varepsilon })\) is used. Based on the spectral split proposed in [39], a hybrid formulation was introduced in [3]. Here, the split is only accounted for in the phase-field equation, which results in a linear elastic problem. This reduces the computational effort while providing comparable results [3]. Following variational theory, the Euler–Lagrange equations of the functional equation [Eq. (1)] can be derived, which yields the following coupled system of equations for the hybrid formulation: Here, the AT-2 model with \(w(s)=1-s^2\) and \(c_w = 1/2\) [52] is used. A quadratic degradation function \(g(s) = (1-\eta )\,s^2 + \eta\) is chosen, where \(\eta\) is a small numerical parameter which prevents full degradation of the material. The coupled system (2) is subject to the boundary conditionsThe history variable \({\mathcal {H}}\), as introduced by Miehe [39], ensures irreversibility of the phase-field and facilitates the use of a staggered solution scheme. It is defined aswhere \(\langle \,\cdot \, \rangle\) denotes the Macaulay brackets with \({\langle \, \cdot \, \rangle _+ = \frac{1}{2}(x - |\,x\,|)}\) and \(\varvec{\varepsilon }_+\) is the positive part of the strain tensor resulting from a spectral split. In the coupled system (2), the ratio \({\mathcal {H}}/G_c\) drives the evolution of the phase-field. Although the critical fracture energies for mode I and mode II fracture can vary considerably, this is not accounted for in the formulation above. To overcome this limitation, Zhang [59] proposed a mixed-mode modification of (2), where two critical fracture energies \(G_{c_I}\) and \(G_{c_{II}}\) are introduced. The driving force is replaced with a weighted average of mode I and mode II driving forces weighted by their respective critical fracture energies. The phase-field equation (2a) is modified according towhere the mode I and mode II driving forces \({\mathcal {H}}_{I}\) and \({\mathcal {H}}_{II}\) are defined asThe mixed-mode formulation by Zhang [59] uses a single length-scale parameter for both tensile and shear fracture, and thus, is not able to account for different tensile and shear strength of the material. To overcome this limitation, we extend the formulation above to a three-field problem, which will be introduced in the next section.

Based on the formulation by Zhang [59], we propose a three-field phase-field formulation which introduces different scalar variables for tensile and shear failure. The scalar variables \(s_I,\,s_{II} \in [0,1]\) represent mode I and mode II fracture, respectively, with \(s_I=0\) on a tensile crack and \(s_{II}=0\) on a shear crack, as depicted in Fig. 1. The driving force is split up following 8. Here, it is assumed that the tensile phase-field \(s_I\) is driven by \({\mathcal {H}}_{I}\), while the shear field \(s_{II}\) is driven by \({\mathcal {H}}_{II}\). Adopting a phase-field evolution according to (2b) for each of the damage variables the three-field problem is obtained as Here, a degradation function \(g(s_I,\,s_{II})\) is defined aswhich accounts for the damage of mode I and mode II cracks. Different length-scale parameters \(l_{0,I}\) and \(l_{0,II}\) for shear and tensile fracture, respectively, are introduced. In the case of pure mode I or pure mode II failure, the formulation falls back to the original mixed-mode formulation (7). It should be noted that the split of the elastic strain energy density in tensile and shear components as proposed by [59] suffers from an overestimation of the force response under pure mode I loading [59]. Alternatively, different approaches based on the split by Amor [4] or directional-dependent splits based on local crack coordinates as proposed by Strobl [50] or Steinke [48] can be integrated in the proposed three-field formulation. An overview of existing splitting methods can be found in [23].

The Finite Cell Method (FCM) is based on an implicit representation of the geometry. Instead of generating a boundary conforming mesh, the actual geometry is recovered during integration with the help of an indicator function. As depicted in Fig. 2, the physical domain \(\Omega _{phy}\) is embedded into a larger domain of simple shape \(\Omega _{\cup } = \Omega _{phy} \cup \Omega _{fict}\) which can easily be meshed. To account for the actual geometry, an indicator function \(\alpha ({{\varvec{x}}})\) is defined which takes a value close to zero in the surrounding, the so-called fictitious domain, and is equal to one in the physical regionHere, \({\tilde{\alpha }}\) is a small numerical parameter greater than but unequal to zero to ensure stability [44]. The weak form is multiplied by \(\alpha ({{\varvec{x}}})\) eliminating contributions from the fictitious domain. Advanced integration schemes such as quad- and octree-subdivision approaches are needed for a sufficiently accurate integration of the cells cut by the domain boundary [21]. For further elaboration on the FCM and its combination with multi-level hp-adaptive refinement, the reader is referred to [21] and [58].

Let the trial spaces for the displacement solution \(\varvec{{\mathcal {S}}}_u\), the mode I phase-field solution \({\mathcal {S}}_{s_{I}}\), and the mode II phase-field solution \({\mathcal {S}}_{s_{II}}\) be defined aswhere \(H^1\) refers to the Sobolev space of degree one. Furthermore, let the spaces for the test functions be defined asThe weak formulation of the coupled three-field problem for the FCM states:

Find \({\varvec{u}} \in \varvec{{\mathcal {S}}}_u\), \(s_I \in {\mathcal {S}}_{s_{I}}\) and \(s_{II} \in {\mathcal {S}}_{s_{II}}\), such that In (19a), the penalty method is used to apply Dirichlet boundary conditions for the elastic problem in a weak sense, where \(\beta\) is the penalty parameter.

The system (19) is discretized in a finite-element setting using integrated Legendre polynomials as basis functions for the finite test and trial spaces as explained in [42]. In each displacement step of the quasi-static simulation, a staggered solution scheme is used to solve the discretized equations. First, Eq. (19b) is solved for the tensile field, then, Eq. (19c) is solved for the shear field, and finally, Eq. (19a) is solved for the displacements \({\varvec{u}}\), which are used to update the history variables \({\mathcal {H}}_{I}\) and \({\mathcal {H}}_{II}\). In the next staggered step, the phase-field equations are solved using the updated history variables and the staggered iterative scheme continues until convergence. As a stopping criterion for the staggered procedure, the residua of the three solution fields are compared against a certain threshold. The iterations are terminated after staggered step i ifwhere \({\mathcal {R}}_{{\varvec{u}}}, {\mathcal {R}}_{s_I}\), and \({\mathcal {R}}_{s_{II}}\) denote the residual of the elastic, mode I phase-field, or mode II phase-field problem, respectively. In contrast to the classic two-field mixed-mode approaches and similar to Fei [24], the history variables \({\mathcal {H}}_{I}\) and \({\mathcal {H}}_{II}\) are updated depending on the dominating crack mode. If \({\mathcal {H}}_{I} \ge {\mathcal {H}}_{II}\) or \(s_{I}({{\varvec{x}}}) < 0.5\), we assume that a mode I is present and only perform an update \({\mathcal {H}}_{I}\). Accordingly, if \({\mathcal {H}}_{I} < {\mathcal {H}}_{II}\) or \(s_{II}({{\varvec{x}}}) < 0.5\), we assume that a mode II crack is present and update \({\mathcal {H}}_{II}\).

In addition to the simple shear crack between the two notches as observed in Bahrami [7], two additional crack patterns are obtained during the DNBD tests on SPK and PFD. As shown in Fig. 4, two mixed-mode patterns are observed. Type A features two tensile wing cracks which initiate at the top and bottom notch, and emerge symmetrically. Further displacement contributes to the formation of a shear band connecting the notches, which leads to abrupt rupture of the specimen. Type B features similar tensile cracks. However, small deviations in the geometry lead to shear cracks emerging from the tips of the tensile cracks connecting to the notches. Type C is the pure shear crack observed by Bahrami [7]. In Fig. 4, right, the number of crack types observed for SPK and PFD are listed. Crack Type B is observed once for each type of rock. This crack pattern does not feature a shear crack connecting the two notches and consequently is excluded from the computation of the mode II fracture toughness. On the contrary, the values obtained from experiments yielding a Type A pattern give the true mode II fracture toughness despite the development of the tensile cracks. This is based on the assumption that an arrested tensile crack does not significantly influence the stress field along the crack plane. This yields nine admissible experiments for SPK and ten admissible experiments for PFD, which are used to compute the mode II fracture toughness in Sect. 3.2.2.

The mode II fracture toughness is defined as the critical mode II stress intensity factor at failure. Following Bahrami [7], the mode II fracture toughness can be computed based on the failure load F, the geometrical dimensions of the specimen, and the mode II stress intensity factor \(K_\text {II}^{*}\) as follows:where t is the thickness of the disk, a is the notch length, and R is the radius of the disk. Bahrami [7] performed finite-element analyses to determine the normalized mode I and mode II stress intensity factors for different DNBD geometries. The relevant values for the selected geometry are summarized in Table 1. Following Eq. 21, the mode II fracture toughness is calculated and the average of all admissible experiments is taken. The experimental results including the observed failure loads and the computed mode II fracture toughness for both types of rock are summarized in Table 2. The average failure load of all admissible experiments for SPK is 16.74 kN. For PFD, we observe lower failure loads which yield an average load of 14.3 kN. A mode II fracture toughness of 4.79 is calculated for SPK, while a lower \(K_\text {II}\) of 3.99 is obtained for PFD. The mode I fracture toughness of SPK is taken from [47]. For PFD, it is calculated based on an empirical relationship stated in [57] and the material parameters determined in [49]. As stated in Table 2, this yields a \(K_\text {I}/K_\text {II}\) ratio of 2.78 for PFD, and a ratio of 4.74 for SPK.

As explained in Sect. 3.2.1, three different crack patterns could be observed in the DNBD tests: the mixed tensile-shear crack (A), the circular mixed tensile-shear crack (B), and the pure shear crack (C). To assess the possibilities of the proposed model, geometrical and material parameters are varied to see if all crack types can be reproduced. For crack types A and C, a notch width of 1.1 mm is used, which corresponds to the geometry prepared by a water jet cutter. For Type B, a notch width of 2.2 mm is used and the notches are shifted perpendicular to their connecting line using an offset \(\delta\) of size 1.0 mm. The latter corresponds to the geometry prepared using a hand saw.

The results of the phase-field simulations for both types of rock, SPK and PFD, are shown in Fig. 6. Here, for visualization purposes, a combined scale with a value of \(s=1\) on a tensile crack, and a value of \(s=-1\) on a shear crack is chosen. As can be seen, the proposed model is able to correctly capture all three crack types. The mixed tensile-shear crack (A) features two tensile wing cracks emerging from the tips of the notches followed by a shear crack connecting the notch tips. This crack pattern can be reproduced for both SPK and PFD using the geometric parameters of the specimens prepared by a water jet cutter. Due to the higher \(K_I/K_{II}\) ratio of SPK, the tensile wing cracks propagate further before shear failure occurs compared to PFD. The circular mixed tensile-shear crack (B) exhibits similar tensile wing cracks; however, shear cracks develop between the tips of the tensile cracks and the respective, closest notch tips. Similar to Type A, it can be obtained using both SPK and PFD material parameters. Interestingly, the numerical results confirm that Type B occurs due to the imperfectly captured geometry when cutting the notches by hand saw. Due to the misaligned notches, the shear cracks start to initiate from the tips of the tensile wing cracks and evolve towards the notch tips, instead of the shear band forming in the center of the specimen as in the case of crack Type A. Here, the proposed model gives valuable insights concerning the formation and type of cracks, while experimental determination would require advanced experimental techniques and expensive equipment. The pure shear crack (C) shows a shear band connecting the two notches. In contrast to Type A, no or only minor tensile wing cracks have formed before shear failure occurs. Whereas crack types A and B are triggered by geometric differences in the specimen, the pure shear crack is obtained numerically by decreasing the \(K_I/K_{II}\) ratio and occurs for both geometries. For SPK, a \(K_I/K_{II}\) ratio of 2.23 is chosen, while for PFD, a ratio of 2.12 yields a pure shear crack. Whereas the numerical result for PFD represents a pure shear crack, the result obtained for SPK shows small tensile cracks which initiate before shear failure occurs.

In summary, the proposed model can reproduce all three experimentally observed crack patterns. A detailed study on the influence of both material and geometrical parameters on the resulting crack pattern, including stochastic analysis, might generate new valuable insights and is part of future research.

In this section, a detailed analysis of the mixed tensile-shear crack (Type A) is presented. To this end, only experiments yielding this specific crack pattern are considered. This corresponds to 5 SPK and 5 PFD experiments of the geometry prepared using a water jet cutter, which yielded mostly Type A crack patterns. In Fig. 7, left, experimental and numerical crack patterns are compared for both types of rock. On the left, five different phases of crack propagation are evaluated for the experiments based on the high-speed camera recordings and compared with the simulation results. Here, the first column shows the displacement in the x-direction computed with DIC, while the second column shows the corresponding crack path. The computed phase-field crack paths are depicted in the third column (SPK) and the fourth column (PFD). In phase \(\textcircled {1}\), a wing crack starts to initiate at the notch where the displacement is applied. This behavior can be observed both in the experiments and in the numerical simulation. However, in the phase-field analysis, the second tensile crack initiates much earlier and not only after the first tensile crack has almost fully developed. This difference in behavior likely stems from the different boundary conditions applied in experiments and simulations. In the experiment, the disk can compress and sink into the soft wood, resulting in a change of the contact area over time. This behavior is not captured by the boundary conditions set for the numerical simulation. In phase \(\textcircled {2}\), the propagation of the tensile cracks continues with increasing displacement. The growth of the tensile wing cracks steadily decelerates as soon as the wing cracks propagate up to the height of the opposite notch tip (phase \(\textcircled {3}\)). Next, a shear band starts to develop in the center of the disk. In contrast to the tensile cracks, which initiate locally and grow from the crack tip with increasing load, the shear failure follows a different pattern. As can be seen, the shear crack initiates at the center of the disk and the associated damage covers a wider area. The damage increases gradually along the connection line between the notches (phase \(\textcircled {4}\)) until failure occurs abruptly in phase \(\textcircled {5}\).

In summary, the numerical behavior, including the localization of the shear band in the center of the specimen, agrees very well with the experimental observations. The fully developed crack patterns for SPK and PFD are shown in Fig. 7, right. A direct comparison of the crack patterns of the two types of rock shows that the tensile wing cracks propagate further in the case of SPK. Due to the higher \(K_I/K_{II}\) ratio, the initiation of shear cracks occurs later, which enables the wing cracks to propagate beyond the notch tip of the opposite notch. This difference is also visible in the experimental crack paths. The computed load–displacement curves obtained with the three-field model are shown in Fig. 8 for SPK, top, and PFD, bottom. The experimentally measured failure loads for the chosen geometry are indicated by dotted lines, and the averaged failure load as well as the standard deviation are presented for each type of rock. Due to the different boundary conditions in the experiment, no direct comparison of experimental and simulated load–displacement curves is presented here. The plastic deformations of the wooden jaws, which are used to transfer the load to the specimen, cannot be captured by the numerical model. Consequently, absolute values of the computed failure loads are compared against the averaged experimental failure load. For both types of rock, the crack phases \(\textcircled {1}\) - \(\textcircled {5}\) are marked. First, the force increases linearly until the tensile wing cracks start to initiate (phase \(\textcircled {1}\)). This results in a sudden drop in force which occurs at 9.1 kN in the case of SPK and at 15.7 kN in the case of PFD. Due to the different length scales for tensile cracks (\(l_{{0,\text {I}}_{\,\text {SPK}}}=0.259\) and \(l_{{0,\text {I}}_{\,\text {PFD}}}=0.916\)), the loss in force is higher for PFD. The tensile wing cracks are not yet fully developed, when the force starts increasing again. The slope is almost linear until \(\textcircled {3}\), when the shear band starts to develop. Failure occurs at the maximum bearable force of \(F_\text {max, SPK} = 19.63\) kN and \(F_\text {max, PFD} = 17.5\) kN, respectively. At this point (\(\textcircled {4}\)), the shear damage is already clearly visible. Once the shear band has fully developed (\(\textcircled {5}\)), the force drops to zero. As can be seen, the computed failure loads are in very good agreement with the experimental values. For both types of rock, the computed failure loads lie within the range of values observed experimentally. The failure loads show a relative deviation of \(4.4\%\) for SPK and of \(1.2\%\) for PFD from the respective averaged experimental failure loads. For a comparison with a standard two-field approach, the same setup is computed using the mixed-mode approach by Zhang [59]. Here, the length-scale parameter is calibrated using the tensile strength of the material, i.e., \(l_{0_{\text {SPK}}} = l_{{0,\text {I}}_{\,\text {SPK}}}\) and \(l_{0_{\text {PFD}}} = l_{{0,\text {I}}_{\,\text {PFD}}}\). The respective force–displacement curves are shown in Fig. 8. As expected, the standard mixed-mode approach is able to reproduce the tensile wing cracks and the subsequent shear failure. For both types of rock, the tensile wing cracks initiate earlier. This is due to the fact that there is no clear distinction between mode I and mode II cracks in the classic mixed-mode approach. Both modes in terms of the weighted average of their crack driving forces contribute to the damage that forms the wing cracks. In the case of SPK, shear failure occurs at a force of 29.1 kN. This is to be expected, as mode I cracks are now governed by \(l_{{0}_{\,\text {SPK}}}=0.259\) instead of \(l_{{0,\text {II}}_{\,\text {SPK}}}=0.722\). Following Eq. 25, a smaller length scale corresponds to a higher material strength and thus a higher failure load. A reverse effect is observed for PFD, where we obtain a lower failure load of 13.8 kN compared to the three-field approach. It is clear that, with a relative error in failure loads of \(26.8\%\) in the case of PFD and \(54.8\%\) in the case of SPK, the standard mixed-mode approach fails to reproduce the experimental observations.

The geometry and boundary conditions for the numerical simulation are depicted in Fig. 9, right, and the simulation parameters are listed in Table 5. The boundary conditions are set as follows: y-displacements are fixed on the top surface, while a positive displacement is applied in y-direction. Steps of size 0.02 mm are applied until a total displacement of 0.2 mm, at which the step-size is decreased to 0.005 mm. The geometry is represented using FCM. To this end, the core sample is embedded into a Cartesian mesh with \(44 \times 92 \times 44\) elements and integration is performed using an octree subdivision approach with partitioning depth \(d = 3\). This results in a total number of \(2\,168\,105\) DOFs for the three-field system. Due to the limited access to the material of the rare drill core, the parameters for this dolostone could not be determined experimentally. Consequently, it is assumed that the MSC-4 dolostone behaves similar to the PFD and material parameters are taken from Table 4. However, the mode II fracture toughness needs to be chosen differently, as the value determined for PFD results in premature cracking and a crack pattern which does not relate to the one observed experimentally. Based on a parameter study, we choose \(G_{c_\text {II}} = 5 \cdot 10^{-3}\) kN/mm, for which we obtain good agreement in both the failure load as well as the observed crack pattern. The higher value of \(G_{c_\text {II}}\) can be attributed to the three-dimensional setting, in which not only mode I and mode II, but also mode III cracks occur. However, the extension of the proposed two-field problem to account for mode III cracks is the subject of future research and is beyond the scope of this work. The simulation is performed on the SuperMUC-NG cluster at the Leibniz Supercomputing Centre using 24 nodes with 48 cores and 96 GB memory per node. The hybrid MPI and OpenMP parallelization of the three-field phase-field problem is based on the framework presented in [34, 35].

In Fig. 10, the computed fracture pattern is shown for different displacement steps. The tensile cracks and shear cracks are visualized as iso-volumes of their respective phase-field based on a choice of \(s_I \le 0.03\) for the tensile cracks shown in blue, and \(s_{\text {II}} \le 0.03\) for the shear cracks shown in red. At a displacement of \(u_{y_1}=-0.22\) mm, cracks initiate around the large central pore. Due to the lower \(G_c\) value for mode I fracture, most cracks are of tensile nature. Only a few shear cracks emerge from the left and right sides of the central pore directed towards the smaller pores. A tensile crack starts to initiate on an internal pore, which is highlighted with a black arrow. With further displacement, the crack propagates upwards, connecting the large, central pore with the longitudinal pore on the back of the drill core (step \(u_{y_2}\)). Additionally, a shear crack occurs, which initiates on the right side of the central pore and propagates towards the specimen’s back, where it connects to the bottom end of the longitudinal pore. At displacement step \(u_{y_3}\), the vertical tensile crack has propagated further towards the top plate of the drill core, while shear damage accumulates at the back of the central pore (step \(u_{y_3}\)). At a displacement of \(u_{y_4}=-0.268\) mm, a large shear crack has emerged from the back of the central pore leading to complete failure of the sample.

For a detailed comparison of the final crack pattern, the experimental cracks are highlighted in Fig. 11 and contrasted with the numerical result. A common feature is the vertical tensile crack which initiates on top of the internal pore and propagates upwards. In contrast to the experiment, where a nearly straight vertical crack is observed, the computed crack tends to lean towards the outer surface of the specimen. This behavior is related to the boundary conditions. First, as can be seen in the experimental setup (cf. Figure 9), the top and bottom surfaces of the drill core are not completely parallel. This results in a real displacement which differs from the pure in-axis displacement applied in the simulation. Moreover, as a consequence of the phase-field formulation, the crack is not able to penetrate the Dirichlet boundary. Instead, it isrepelled from the top and bottom surface of the core sample. Therefore, the part on the right side of the core sample that falls off in Fig. 9, \(\textcircled {3}\), is smaller in the simulation than in the experiment and shaped differently. The vertical cracks connecting the larger pores to the bottom side of the specimen cannot be reproduced in the simulation. Similar to the experiment, shear and combined tensile-shear cracks connecting the different pores are visible outside the specimen. Since the fracture pattern could only be recorded from one side, it is difficult to judge if the final failure occurred due to a shear crack. However, the visible experimental crack pattern is captured remarkably well, and the numerical simulation generates interesting insights, including the fact that the vertical tensile crack initiates on top of the internal pore, as highlighted in Fig. 9, step \(u_{y_1}\). In Fig. 12, the experimental load–displacement curve is shown and compared with the computed result. The experimental curve clearly shows a brittle fracture behavior predicting failure of the sample at a force of 147.0 kN and a displacement of 0.24 mm. The computed curve closely follows the slope of the experimental curve until a displacement of 0.224 mm when the maximum load of 142.31 kN is reached. At this point, the tensile crack starts to develop which results in a drop in force. Once the tensile crack has stabilized, the curve slowly starts to ascend again. Shear damage accumulates which results in complete failure at a displacement of 0.256 mm. The deviation of the computed failure load corresponds to \(3.2\, \%\) of the force measured experimentally.