# Finite element simulation of pressure-loaded phase-field fractures

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## Abstract

A non-standard aspect of phase-field fracture formulations for pressurized cracks is the application of the pressure loading, due to the fact that a direct notion of the fracture surfaces is absent. In this work we study the possibility to apply the pressure loading through a traction boundary condition on a contour of the phase field. Computationally this requires application of a surface-extraction algorithm to obtain a parametrization of the loading boundary. When the phase-field value of the loading contour is chosen adequately, the recovered loading contour resembles that of the sharp fracture problem. The computational scheme used to construct the immersed loading boundary is leveraged to propose a hybrid model. In this hybrid model the solid domain (outside the loading contour) is unaffected by the phase field, while a standard phase-field formulation is used in the fluid domain (inside the loading contour). We present a detailed study and comparison of the \(\varGamma\)-convergence behavior and mesh convergence behavior of both models using a one-dimensional model problem. The extension of these results to multiple dimensions is also considered.

## Keywords

Brittle fracture Phase-field modeling Pressurized cracks Immersed finite element method Finite cell method## 1 Introduction

Over the last decade phase-field models for fracture [1, 2, 3, 4]—which are closely related to traditional gradient-enhanced damage models [5]—have been successfully applied to a wide range of problems, such as dynamic fracturing [6, 7, 8, 9], large deformation fracturing [10, 11], fracturing of electromechanical materials [12, 13], cohesive fracturing [14, 15], fracturing of thermo-elastic solids [16, 17] and many more. The primary advantage of these models is the flexibility with which complex fractures can be simulated, which is a result of the diffuse fracture representation and due to the fact that propagation laws generally follow naturally from energy minimization principles. However, compared to sharp fracture models, phase-field models require high-resolution computational grids in the vicinity of the fractures.

The flexibility of the phase-field framework with respect to the representation of complex fracture patterns is also exploited in the context of fluid-driven fracture propagations (see references below). These models are highly relevant in applications concerning flow in porous media such as hydraulic fracturing. A particularly interesting aspect of the extension of phase-field models to hydraulic fracturing is that the phase-field model not only represents the fracture in a diffuse sense, but also represents the fluid-flow domain.

Various phase-field formulations for fluid-driven fracture propagation have been developed over the last couple of years. In Bourdin et al. [18] a variational formulation for pressure-driven phase-field fracture propagation is considered, in which the work exerted by the pressure inside a crack on the fracture surface is accounted for by a volumetric body force. Crack growth is simulated by quasi-static incrementation of the fracture volume. Alternative formulations for fluid-driven fracture propagation have been developed by enriching a poro-mechanical formulation with a phase-field fracture representation, of which the works by Miehe et al. [19, 20], Mikelić, Wheeler et al. [21, 22], and Wilson et al. [23] are particularly noteworthy. The primary difference between the formulations in these works is the way in which the phase-field influences the flow and action of the fluid on the porous medium.

The above-mentioned models for pressure-loaded phase-field fractures all rely on indirect application of the fracture-surface loading, in the sense that the sharp loading boundary is represented by a phase field. This is a very natural approach in phase-field models, but it also inherently limits the capabilities of the developed models. The absence of a discrete fracture surface in the formulation impedes direct incorporation of a versatile fluid-flow model and fluid-solid interface conditions, and requires a reconstruction of the fracture opening for the fluid-flow model.

In this contribution we study the possibility to directly incorporate the sharp fracture surface pressure-loading in a phase-field fracture representation. The key idea behind this approach is to extract a contour from the phase field and to use it as a standard pressure-loading boundary. We present a detailed analysis for the solution behavior of this model, the behavior of which is intuitively troublesome. In terms of computational techniques, the essential novel ingredient in our approach is the loading-surface extraction routine, which is inspired by the segmentation technique developed in the context of the finite cell method for image-based analysis [24]. To ameliorate the complications associated with the imposition of the mechanical loads on the phase-field fracture, we propose a hybrid diffuse-sharp fracture model, which further exploits the computational capabilities of the finite cell techniques used for the loading-boundary reconstruction. Based on a model problem we present a detailed analysis of both models, which highlights the superior fracture opening approximation behavior of the hybrid model.

In this manuscript, we restrict ourselves to pressure-loaded pre-existing fractures. A detailed understanding of the model—in terms of dependence on the phase-field length scale parameter and discretization parameters—for such stable fractures is relevant in itself, but also a prerequisite for applying it to propagating fractures. To assess the properties of the models as a basis for fracture propagation, we study the behavior of the thermodynamic driving-force field, which is the essential model aspect in the context of fracture propagation. The study presented herein focuses on a constant pressure loading, but the presented results are generally valid for heterogeneous tractions along the fracture surface, thereby enabling enrichment of the model with more sophisticated fracture-loading models.

In Sect. 2 we introduce the sharp pressure-loaded fracture problem, along with its phase-field and hybrid model representations. In Sect. 3 we discuss the employed finite element discretization technique and interface-extraction technique. A detailed analysis of both the phase-field model and the hybrid model is presented for a one-dimensional model problem in Sect. 4. This analysis pertains to the convergence of the models to the sharp model for decreasing length scales, and to the convergence of the finite element solutions under mesh refinement. In Sect. 5 we consider a two-dimensional benchmark problem, for which we demonstrate how the observed behavior in one dimension translates to multiple dimensions. A three-dimensional problem is considered to demonstrate applicability of the models in a volumetric setting. Finally, conclusions are drawn in Sect. 6.

## 2 Pressure-loaded fracture model formulations

### 2.1 Sharp fracture formulation

*p*. Assuming small deformations and deformation gradients, the displacement of the solid under the influence of internal pressurization and in the absence of body forces is governed by:

### 2.2 Phase-field fracture formulation

*d*at \(\varGamma _c\).

*g*(

*d*) is a degradation function with properties \(g(0)=1\), \(g(1)=g^{\prime}(1)=0\), and \(\varvec{\sigma }_e\) is the stress associated with the undamaged, elastic, material. The vanishing derivative of

*g*at \(d=1\) ensures that the fracture driving force vanishes upon completion of damage. The most common choice for the degradation function is \(g(d)=(1-d)^2\), which will also be considered in the remainder of this work. In order to restrict fracture propagation to tensile loading states, the stress degradation function is commonly merely applied to the tensile part of the stress tensor; see e.g. [2] for details.

Imposition of the pressure loading in the phase-field model is a non-standard aspect of the formulation, in the sense that it does not arise naturally as a volumetric term in the smeared problem. Fundamentally, the problem is that the phase-field formulation does provide a smeared representation of the fracture surface area, but that a direct notion of opposing sides of the fracture surface is lost. For that matter there is also no direct notion of a jump in the displacement field.

The absence of a direct notion of fracture opening can be resolved by approximating it in a smeared sense. Following the derivations by Chambolle [25, 26] in Bourdin et al. [18] the fracture volume increase is approximated as \(\int _\varOmega \varvec{u} \cdot \nabla d \,dV\) (conceptually, the jump is obtained by integrating perpendicular to the crack), whereas the fracture jump in References [19, 20, 23] is reconstructed using the stretch of the deformed material. Differences exists between the formulation of Miehe et al. [19, 20] and Wilson et al. [23] in the way the fracture opening influences the fluid flow.

### 2.3 Hybrid fracture formulation

### 2.4 Quantities of interest

## 3 Interface parametrization and Galerkin discretization

The remainder of this work focuses on the finite element approximation of the phase-field model and hybrid model, where the sharp crack model will be used as a reference model. In this section we introduce the employed finite element discretization and the parametrization of the pressure-loading surface and the associated fluid and solid sub-domains.

### 3.1 Galerkin discretization

*h*is a characteristic mesh parameter. As illustrated in Fig. 3 this mesh does in general not conform to the \(\alpha\)-contour. This mesh is used to construct a computational basis for both the phase field and the displacement field:

*k*and

*r*denote the order and regularity of the basis, respectively. The multi-dimensional simulations discussed in Sect. 5 are based on \(C^0\)-continuous linear finite element meshes.

*phase-field model*follows from the system of equations \(\mathbf {K}^{\varvec{u}\varvec{u}}\mathbf {a}^{\varvec{u}}=\mathbf {f}^{\varvec{u}}\), where

*hybrid model*is determined in a two step procedure. In the first step the elasticity problem is solved over the solid domain by \(\mathbf {K}^{\varvec{u}_{\oplus }\varvec{u}_{\oplus }}\mathbf {a}^{\varvec{u}_{\oplus }}=\mathbf {f}^{\varvec{u}_{\oplus }}\), with

### 3.2 Interface parametrization

In order to evaluate the integrals over the fracture surface boundaries in the weak form problems (12) and (14), a parametrization for the \(\alpha\)-contour is constructed. To obtain this parametrization we employ the bi-sectioning-based segmentation scheme proposed in [24] in the context of the isogeometric finite cell analysis of image-based geometric models, and enrich it with functionality to extract the interface between the segmented regions. The merits of this scheme are that it automatically extracts an interface parametrization from a level set function, i.e. the phase field, for \(n_{\mathrm{dim}}=1,2,3\), while having the capability of resolving the smoothness of the \(\alpha\)-contour in the case of higher-order continuous phase-field discretizations.

On one hand this bisection-based tessellation scheme provides a piecewise parametrization of the regions \(\varOmega _{\oplus }\) and \(\varOmega _{\ominus }\), which enables the construction of integration schemes with controllable precision. On the other hand, the tessellation scheme enables the construction of a piecewise parametrization of the interface between these two domains, \(\varGamma _\alpha\). The interface is constructed by collecting all boundary faces (edges in 2-D) in one of the regions that match a boundary face in the other region. For robustness it is crucial that the boundaries of the two regions are tessellated in a consistent way. As for the volumetric regions, this piecewise parametrization of the interface provides us with the possibility to evaluate integrals over this internal boundary. We note that the cells resulting from this bisection-based tessellation scheme may be distorted, but that this is not a fundamental problem from the perspective of integration.

## 4 Analysis of a one-dimensional model problem

*L*, \(\varOmega =[-L,L]\), which is clamped on its outer boundaries and with a fully developed fracture centered at the origin, i.e. \(\varGamma _c = \{ 0 \}\). Deformation of the solid in directions other than the

*x*-direction are confined, and the Hookean stress-strain relation \(\sigma _e = E \varepsilon\) is considered. For the results presented below dimensionless values of parameters are chosen. A length of \(L=5\) is used, the modulus of elasticity is taken as \(E=100\), and the pressure is set to \(p=1\).

### 4.1 \(\varGamma\)-convergence

Prior to studying the finite element approximation behavior we study the dependence of the phase-field solution (32) and hybrid solution (34) on the length-scale parameter \(l_0\). Note that all errors reported below pertain to the difference between the exact solutions presented above and the sharp crack solution (31). The various error norms introduced in Sect. 2.4 have been computed by numerical integration over the domain \(\varOmega {\setminus } \varGamma _c\) on a high-resolution uniform mesh, where one element edge coincides with the sharp crack at \(x=0\). By virtue of the fact that Gauss points are used, this integration procedure automatically removes the singular point \(\varGamma _c=\{ 0 \}\) from the integration domain.

#### 4.1.1 Phase-field model

#### 4.1.2 Hybrid model

#### 4.1.3 Model comparison

The fundamental difference between the phase-field model and hybrid model pertains to the displacement gradients near the loading points. For the phase-field model the displacement gradient is magnified by a factor of \((1-\alpha )^{-2}\) at the exterior of the loading point. This gradient amplification leads to diverging solution behavior in the limit of \(\alpha\) going to one. In contrast, the hybrid model does not suffer from the gradient increase near the loading points, as a consequence of which the solution of the hybrid model converges to the sharp model when the damage threshold goes to one.

Comparison of the \(\varGamma\)-convergence behavior of the phase-field model and hybrid model

Phase-field model (\(u_{\mathrm{phase}}\)) | Hybrid model (\(u_{\mathrm{hybrid}}\)) | |
---|---|---|

\(\left\| \square - u_{\mathrm{sharp}} \right\| _{E(\varOmega {\setminus } \varGamma _{c})}\) | \({\mathcal {O}}\left( l_0^\frac{1}{2}\right)\) | \({\mathcal {O}}\left( (1-\alpha )^{\frac{1}{2}}\right) \cdot {\mathcal {O}}\left( l_0^\frac{1}{2}\right)\) |

\(\left\| \square - u_{\mathrm{sharp}} \right\| _{H^1(\varOmega {\setminus } \varGamma _{c})}\) | \({\mathcal {O}}\left( l_0^\frac{1}{2}\right)\) | \({\mathcal {O}}\left( (1-\alpha )^{\frac{1}{2}}\right) \cdot {\mathcal {O}}\left( l_0^\frac{1}{2}\right)\) |

\(\left\| \square - u_{\mathrm{sharp}} \right\| _{L^2(\varOmega {\setminus } \varGamma _{c})}\) | \({\mathcal {O}}\left( l_0^\frac{3}{2}\right)\) | \({\mathcal {O}}\left( (1-\alpha )^{\frac{3}{2}}\right) \cdot {\mathcal {O}}\left( l_0^\frac{3}{2}\right)\) |

\(| \llbracket \square \rrbracket -\llbracket u_{\mathrm{sharp}} \rrbracket |\) | \({\mathcal {O}}(l_0)\) | \({\mathcal {O}}(1-\alpha )\cdot {\mathcal {O}}(l_0)\) |

\(\left\| \square - u_{\mathrm{sharp}} \right\| _{H^1(\varOmega _\oplus )}\) | \({\mathcal {O}}\left( l_0^\frac{1}{2}\right)\) | 0 |

### 4.2 *h*-convergence

In the remainder of this section we study the convergence behavior of the finite element approximations under uniform mesh refinement for both the phase-field model and the hybrid model. All presented simulations pertain to meshes with an odd number of elements, such that both the loading points, and the center of the fracture are not coinciding with element boundaries. Let us note that in terms of observed solution behavior these results do not essentially differ from results with an even number of uniform elements (i.e., with the fracture coinciding with an element boundary).

The error analysis reported below focuses on the non-standard approximation of the displacement field in the phase-field model and hybrid model. The phase-field is in all cases computed using the finite element system (27), supplemented with a Lagrange multiplier constraint to satisfy the \(d(0)=1\) condition. The influence of the error in the phase field on the error in the energy norm and driving force field is observed to be negligible, as the phase field converges at a faster rate under mesh refinement than the displacement field.

#### 4.2.1 Phase-field model

The finite element approximation of the phase-field model is studied by comparing the discrete solution, \(u^h_{\mathrm{phase}}\), to the phase-field solution, \(u_{\mathrm{phase}}\) (32). For all presented results the error associated with the Neumann boundary condition violation of the phase-field solution (32) was found to be insignificant in comparison to the finite element approximation error. Hence, to examine the effect of the order and smoothness of the approximation space, we consider the solution \(u_{\mathrm{phase}}\) as a reference solution for the finite element approximation \(u^h_{\mathrm{phase}}\). The phase-field discretization error is denoted by \(e_{\mathrm{phase}}^h = u_{\mathrm{phase}} -u_{\mathrm{phase}}^h\).

*The energy norm*The convergence behavior of the error in the energy norm, \(\Vert e_{\mathrm{phase}}^h \Vert _{E(\varOmega {\setminus } \varGamma _c )}\), is shown in Fig. 11a for spline bases of order

*k*and regularity

*r*. The observed rate of convergence of \(\frac{1}{2}\), independent of the order and regularity of the approximation, can be explained by considering the best approximation property of the Galerkin approximation, \(u_{\mathrm{phase}}^h \in {\mathcal {V}}^{u,h} \subset {\mathcal {V}}^{u}\). This property bounds the energy error from above by

*k*(with property \(L_i^k(x_j)=\delta _{ij}\)) constructed over the considered uniform grid with spacing

*h*, yields that on those parts of the domain where the solution is smooth:

*h*going to zero it holds that \(d^h = 1 - |x|/l_0\). From the integrand in (44) it is observed that the divergence of the gradient \(\llbracket u_{\mathrm{phase}}\rrbracket / h\) under

*h*-refinement is compensated for by the vanishing degradation function \(g(d^h)\approx (x / l_0)^2\). It is important to note that although the energy error converges under mesh refinement, it diverges with the length-scale parameter \(l_0\). Decreasing the length scale (to attain \(\varGamma\)-convergence) therefore increases the energy error in the case that the mesh size is not appropriately refined along with it.

*The*\(L^2\)

*-norm and*\(H^1\)

*-norm*The convergence plots for the displacement field in the \(L^2(\varOmega )\)-norm, \(H^1(\varOmega )\)-norm and \(H^1(\varOmega _\oplus )\)-norm over the solid domain are shown in Fig. 12 for polynomial degrees \(k=1,2,3\) and regularities \(r=k-1\) and \(r=0\). The discretization independent observed convergence rate of \({\mathcal {O}}(h^{\frac{1}{2}})\) for the \(L^2(\varOmega )\)-norm as observed in Fig. 12a can be deduced from

*Displacement jump and fracture surface area norm*The mesh convergence behavior of the displacement jump is shown in Fig. 14a, where a convergence rate of 1 is observed independent of the order and regularity of the basis. This rate is expected from the fact that the jump can be expressed as

*h*going to zero, the phase field evidently converges to \(d(u_{\mathrm{phase}})\) with a rate of \({\mathcal {O}}(h)\) in its point value at the sharp crack. It then follows that the corresponding error in the surface area norm converges as

#### 4.2.2 Hybrid model

The finite element solution of the hybrid model, \(u^h_{\mathrm{hybrid}}\), is compared with the exact solution, \(u_{\mathrm{hybrid}}\) (34) for the various error norms introduced in Sect. 2.4. The discretization error of the hybrid model is denoted by \(e^h_{\mathrm{hybrid}} = u^h_{\mathrm{hybrid}} - u_{\mathrm{hybrid}}\). As the hybrid solution follows a standard elasticity problem on the solid domain, and a standard phase-field model on the fluid domain, its convergence behavior evidently derives from the properties of these two models. It is noted that the linear displacement field on the solid domain is represented exactly by the computational basis, as a consequence of which various of the error measures vanish. For generalities sake, below we will also report the standard convergence rates of the solid problem.

*Energy norm*On the solid domain the energy error converges with the standard rate of

*k*, whereas a rate of \(\frac{1}{2}\) was derived above for the fluid domain phase-field model. As a consequence, the rate of convergence of the energy norm is dictated by the phase-field solution, i.e.,

*The* \(L^2\) *-norm and* \(H^1\) *-norm* In general, the \(H^1\)-norm on the solid domain converges with a rate of *k*, and the \(L^2\)-norm with a rate of \(k+1\). As derived above for the phase-field model, on the fluid domain the \(H^1\)-norm diverges with a rate of \(\frac{1}{2}\), whereas the \(L^2\)-norm converges with a rate of \(\frac{1}{2}\). As shown in Fig. 16a, b the fluid domain solution behavior dictates the convergence rates of the hybrid model for both the \(L^2\)-norm and \(H^1\)-norm, i.e., \(\Vert e^h_{\mathrm{hybrid}} \Vert _{L^2(\varOmega )}={\mathcal {O}}(h^{\frac{1}{2}})\) and \(\Vert e^h_{\mathrm{hybrid}} \Vert _{H^1(\varOmega {\setminus } \varGamma _c)}={\mathcal {O}}(h^{-\frac{1}{2}})\)

*Displacement jump and fracture surface area norm*Due to the one-way coupling of the hybrid model, the displacement jump merely depends on the solid solution. Since the energy error for both the primal and the dual problem generally converges with \({\mathcal {O}}(h^k)\), the Babuška–Miller theorem yields

#### 4.2.3 Model comparison

In terms of approximation behavior, the errors are generally dictated by the continuous approximation of the jump over the sharp fracture. Since this approximation in the fluid domain is identical for the phase-field model and hybrid model, in terms of rates of convergence there is no different asymptotic behavior for all error norms that involve the fluid domain. This is reflected in Table 2, where identical rates are observed for the energy error, \(H^1\)-error, \(L^2\)-error, and surface area error. A minor difference between the two models is observed from the \(\alpha\)-threshold dependence. While the phase-field model shows error increases (for example in the energy norm) for increasing values of \(\alpha\) due to the approximation of the steep gradients near the loading points, this dependence is absent for the hybrid model.

Comparison of the *h*-convergence behavior of the phase-field model and hybrid model

Phase-field model (\(u_{\mathrm{phase}}\)) | Hybrid model (\(u_{\mathrm{hybrid}}\)) | |
---|---|---|

\(\left\| \square - \square ^h \right\| _{E(\varOmega {\setminus } \varGamma _{c})}\) | \({\mathcal {O}}\left( h^\frac{1}{2}\right)\) | \({\mathcal {O}}\left( h^\frac{1}{2}\right)\) |

\(\left\| \square - \square ^h \right\| _{H^1(\varOmega {\setminus } \varGamma _{c})}\) | \({\mathcal {O}}\left( h^{-\frac{1}{2}}\right)\) | \({\mathcal {O}}\left( h^{-\frac{1}{2}}\right)\) |

\(\left\| \square - \square ^h \right\| _{L^2(\varOmega {\setminus } \varGamma _{c})}\) | \({\mathcal {O}}\left( h^\frac{1}{2}\right)\) | \({\mathcal {O}}\left( h^\frac{1}{2}\right)\) |

\(| \llbracket \square \rrbracket -\llbracket \square ^h \rrbracket |\) | \({\mathcal {O}}(h)\) | 0 |

\(\left\| \square - \square ^h \right\| _{H^1(\varOmega _\oplus )}\) | \({\mathcal {O}}\left( h^\frac{1}{2}\right)\) | 0 |

\(\left\| \square - \square ^h \right\| _{S}\) | \({\mathcal {O}}\left( h^\frac{1}{2}\right)\) | \({\mathcal {O}}\left( h^\frac{1}{2}\right)\) |

## 5 Multi-dimensional simulations

To demonstrate how the results attained above for the one-dimensional case extend to multiple dimensions, in this section we consider a two-dimensional benchmark problem and perform a detailed study of the mesh convergence and \(\varGamma\)-convergence behavior of the phase-field model and hybrid model. Subsequently we demonstrate the applicability of the models and methods in three dimensions.

### 5.1 A two-dimensional planar crack

We consider the two-dimensional model problem shown in Fig. 17a. The problem consists of a square solid domain \(\varOmega =[-L/2,L/2]\times [-L/2,L/2]\), which is clamped on its outer boundaries and with a fully developed horizontal fracture of length 2*c* centered at the origin, i.e. \(\varGamma _c = [-c,c] \times \{ 0 \}\). The dimensions and parameters are taken from Ref. [18]. The dimension of the domain is taken as \(L=8\) m and the semi-length of the fracture is \(c=0.2\) m. A plane stress Hookean linear elastic law is used with modulus of elasticity \(E=10\) GPa and Poisson ratio \(\nu =0.3\). The pressure loading is taken as \(p=1\) Mpa.

*h*-convergence behavior in Sect. 5.1.1, before considering the \(\varGamma\)-convergence behavior in Sect. 5.1.2.

#### 5.1.1 Mesh convergence

To study the mesh convergence behavior for the two-dimensional problem we take the length-scale parameter as \(l_0=c/4\). All results presented in this section are based on linear finite elements on locally refined triangular meshes as illustrated in Fig. 17b. In a box with a height and width of \(l_0\times (2c+l_0)\) centered at the sharp crack a uniform mesh with mesh parameter *h* is created, where *h* corresponds to the edge lengths of the element edges that coincide with the sharp crack boundary and refinement box. Outside the refinement box the mesh is gradually coarsened as illustrated in Fig. 17b.

*h*and length scale \(l_0\) cannot be selected independently. As a consequence, for decreasing \(l_0\), the mesh insensitivity of the hybrid model compared to the phase-field model is an essential advantage of the former. Another difference in the approximation behavior of the displacement field is seen from the fracture opening evaluated at the loading contour, i.e. Fig. 21b, c. While the hybrid model yields a smooth solution that closely approximates the sharp solution for the considered range of element sizes, the phase-field model shows considerable mesh-dependent fluctuations. These fluctuations are a consequence of the fact that the loading boundary is immersed inside a mesh that continuously interpolates the displacement field, while in the hybrid approach the splitting of the domain effectively makes the loading boundary conforming to the fluid and solid domains.

*h*-convergence behavior the most notable difference between the two models is observed from the fracture opening increase, a quantity of interest that only depends on the solution in the solid domain. The quality of the approximation depends significantly on the mesh size for the phase-field model, since the gradient near the loading boundary must be resolved by the mesh. For this reason, the mesh dependence increases when the \(\alpha\)-threshold approaches 1. In contrast, the volume increase for the hybrid model is virtually independent of the mesh size, since the coarsest meshes considered are already capable of representing the displacement field adequately. These observations are in excellent agreement with our analysis for the one-dimensional model problem; see Sect. 4.2.3.

#### 5.1.2 \(\varGamma\)-convergence

Figure 25 shows the Von Mises stress for the phase-field model and hybrid model. Since both models converge to the sharp model, which has a stress singularity at the tip, significant stress concentrations are observed for both models. Evidently, the phase-field model smears out the stresses ahead of the fracture tip, as a consequence of which the stress concentration only becomes visible for a sufficiently small length-scale parameter \(l_0\). The hybrid model on the other hand represents the elastic problem on the solid domain, and hence the stress concentration is already visible for relatively large \(l_0\). The intensity of the stress concentration increases also in this case as the length-scale parameter decreases, but the dependence is much weaker than for the phase-field model. Similar observations are made for the dependence of the Von Mises stress on the threshold parameter, \(\alpha\), as shown in Fig. 26. Most notably, the stress concentration shows a stronger dependence on the \(\alpha\) parameter for the phase-field model than for the hybrid model.

*h*-convergence behavior, the surface area norm reflects the solution behavior in the fluid domain. Although the fundamental behavior of the displacement field in the fluid domain is the same for the two models, the displacement jump is larger for the phase-field model. As a consequence, also the predicted surface area is larger for the phase-field model.

### 5.2 Three-dimensional interacting penny-shaped cracks

*xy*-plane, and the second penny in a plane that is rotated \(-\,30^\circ\) around the

*y*-axis. The two pennies together constitute the sharp crack boundary \(\varGamma _c\). A Hookean linear elastic law with \(E=10\) GPa and \(\nu =0.3\) is considered, and the internal pressure in the penny-shaped fractures is taken as \(p=1\) Mpa. The displacements on the outer boundaries of the domain are constrained to zero.

In Fig. 32 the computed vertical (*z*-direction) displacement field of the problem is shown for both the phase-field model (Fig. 32a) and hybrid model (Fig. 32b). In agreement with the observations made for the one-dimensional model problem and two-dimensional test case discussed above, the displacement jumps in the phase-field model are magnified due to the large displacement gradients in the phase-field model near the loading boundary. For the particular setting considered here, with \(\alpha =0.8\), an amplification of almost a factor of two is observed in comparison to the hybrid model. As for the other examples, the hybrid model closely resembles the sharp fracture problem. Figure 33 compares the two models in the *xy*-plane for three choices of the threshold parameter \(\alpha\). The increased fracture opening for the phase-field model in comparison to the hybrid model is clearly observed in this figure, where the same color scale is used for both models. In agreement with the results discussed above, the mismatch between the two models decreases as \(\alpha\) decreases. As can be seen from the plotted damage contours, regardless of the selection of \(\alpha\), the discrepancy between the two models vanishes away from the crack, which illustrates that the deviations between the models are essentially localized near the fracture.

## 6 Conclusions

We have studied the possibility of incorporating the loading of pressurized fractures in phase-field fracture formulations as a non-homogeneous Neumann condition over a phase field contour. Computationally, this approach to modeling the fracture loading is enabled by a bisection-based surface tessellation scheme which was originally developed in the context of immersed finite element simulations. The applicability of this tessellation procedure has been demonstrated for two and three dimensional test cases.

In its simplest form the standard phase-field model for fracture is supplemented with an \(\alpha\)-contour integral to incorporate the pressure loading. The behavior of this model is characterized by the presence of an artificial sharp gradient in the solid displacement field near the loading boundary. This gradient—which results in an overestimation of the fracture opening—is a result of the lowered stiffness near the loading boundary when \(\alpha\) is chosen close to 1. Choosing \(\alpha\) sufficiently close to 1 is required, however, as otherwise the geometric information of the sharp fracture boundaries that is encoded in the phase field is not recovered by the extraction operation. For a fixed value of \(0 \ll \alpha < 1\) the sharp problem solution is recovered when the phase field length-scale parameter, \(l_0\), goes to zero.

In terms of mesh dependence, the *h*-convergence rates of various norms is impeded by the discretization effects associated with the non-conforming pressure loading and by the approximation of the displacement jump through a continuous displacement field. Evidently, the mesh size *h* cannot be selected independently of the phase field parameter \(l_0\), since the approximation space should have sufficient resolution in the non-homogeneous part of the phase-field (i.e. near the smeared fractures). For a fixed value of the length-scale parameter, the most notable effect in the phase-field model is the presence of the sharp gradient layer surrounding the fluid domain. This effect does not only result in an \(l_0\)-dependent error of the solid domain deformation, but also creates a significant *h*-dependent error contribution requiring small elements in the solid domain near the pressure loading boundary.

The shortcomings observed for the phase-field model have led us to the development of a hybrid fracture formulation. The idea of this hybrid formulation is to leverage the computational functionality of the tessellation procedure used for the loading boundary extraction to also provide separate sub-domains for the fluid and the solid. On the solid domain the standard (without phase-field degradation) elasticity problem is then solved, thereby avoiding the occurrence of the gradient increases near the loading boundaries. The solution in the solid is then lifted back into the fluid domain by solving the phase-field problem (in the fluid domain). Effectively the result is a two-step solution procedure that mimics the sharp fracture problem in the solid, and the phase-field model in the fluid domain. More specifically, the hybrid model is capable of accurately representing the sharp problem fracture boundary deformation for length-scale parameters, \(l_0\), and mesh sizes, *h*, considerably larger than those needed for the phase-field model to obtain the same accuracy.

In terms of the selection of the \(\alpha\)-threshold parameter, the phase-field model and the hybrid model behave differently. For both models, one, in principle, wants to select \(\alpha\) very close to 1, since doing so would yield a loading condition very similar to that of the sharp model. As indicated above, letting the threshold approach 1 in the phase-field model would result in the creation of an artificial, highly compliant, boundary layer, which, practically, leads to e.g. overestimation of the fracture opening. This erroneous behavior is rigorously resolved by the hybrid formulation, as a result of which the compromising conditions to select \(\alpha\) close to 1 as induced by the phase-field model are significantly relaxed. What remains, however, is that the loading zone (i.e., the fluid domain) must be resolved by the computational grid. More specifically, at least a few elements need to be present across the thickness of the fracture to represent the jump in the displacement over the fracture. Hence, bringing \(\alpha\) closer to 1 while keeping the internal length scale constant means that the minimum mesh size must be reduced accordingly. With respect to the selection of \(\alpha\) we finally note that the considerations herein are based on the stationary setting, and that additional selection aspects might need consideration in a propagating fracture setting.

In our future work we aim at extending the hybrid approach to propagating fractures. The conceptual idea of the envisaged extension is to apply the hybrid model in a staggered solution strategy. Given an initial phase field (a preexisting fracture) and loading condition, the domain is segmented (using the \(\alpha\)-threshold) and the hybrid model is used to compute the displacement field, both in the solid domain and in the fluid domain. From the displacement field the thermodynamic driving force is then computed, which serves as the input to the phase-field problem defined over the entire domain. The updated phase-field, along with an update of the loading condition, then again serves as input to the segmentation process and hybrid model. A necessary condition for this staggered approach to work is that the thermodynamic driving force is approximated correctly by the hybrid model. Therefore, we have already studied the behavior of the thermodynamic driving force in this manuscript. The main conclusion of this study is that the hybrid model is capable of mimicking the driving force similar to that of the phase-field model, while significantly improving the accuracy with which the fracture opening is modeled. A detailed investigation of this propagating setting is a topic of further study, which includes a study of how to select the \(\alpha\) parameter, how to treat nucleation from sharply represented initial fractures, and how to deal with non-monotonous pressure loading.

## Notes

### Acknowledgements

All simulations in this work were performed using the open source software package Nutils (www.nutils.org).

## Funding

This work is part of the Industrial Partnership programme (IPP) ’Computational sciences for energy research’ of the NWO Institutes Organisation (NWO-I). This research programme is co-financed by Shell Global Solutions International B.V.

## Compliance with ethical standards

## Conflict of interest

The authors declare that they have no conflict of interest.

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