Phase-field modeling of fracture in heterogeneous materials: jump conditions, convergence and crack propagation

The investigations presented in this contribution build on a fully diffuse framework for crack propagation simulations in heterogeneous materials. The crack is described following the phase-field approach. Suppose, a one-dimensional rod \(x\in (-\infty ,\infty )\) of infinite length is cracked at \(x=0\). The sharp crack location can be fixed using the Dirac distribution, cf. Figure 1a. In contrast to a sharp crack, the phase-field approach introduces an additional scalar fieldwith a characteristic length \({\ell _\text {c}}\), cf. Figure 1b, where \(c=1\) and \(c=0\) resemble intact and fully broken material, respectively. Consequently, a three-dimensional generalization for the corresponding surface energy of the crack [3, 27]can be derived, which is a measure for the energy required to form the corresponding crack surface. The fracture toughness \({\mathcal {G}}_\text {c}\) stems from the energetic cracking criterion by Griffith [6]. In other words, the crack surface has been regularized [5]. In case of a homogeneous fracture toughness, Eq. (1) is obtained by rewriting Eq. (2) for the one-dimensional case and subsequent minimization of the functional, subject to the boundary conditions \(c(0)=0\) and \(c'(x\rightarrow \pm \infty )=0\). For a heterogeneous fracture toughness, phase-field profiles different from Eq. (1) are obtained, cf. [25] for more detailed considerations. In Fig. 2a on the left, a phase-field crack \(\varGamma ^{\ell _\text {c}}\) is depicted, which possibly interacts with the interface between the two subdomains \(\varOmega _1\) and \(\varOmega _2\). Next, the diffuse representation of these subdomains is outlined.

In the present work, two linear elastic domains \(\varOmega _i\) with \(i=1,2\) are considered. A static order parameter \(p\in [0,1]\) describes for every material point whether it belongs to the one \(\varOmega _1\) or other \(\varOmega _2\) subdomain, and introduces a smooth transition in the vicinity of the former sharp interface \(\varGamma ^\text {i}\), cf. Figure 2a. The transition width is controlled by the interface length scale \({\ell _\text {i}}\). The order parameter does not change in time and is a priori known, e.g., derived from a direct imaging technique. Finally, the motion is described by the displacement \(\varvec{u}\in \mathbb {R}^3\). The elastic energy density of each domain corresponds towhereis a quadratic degradation function to account for previously introduced phase-field cracks. Various other degradation functions are proposed in [28, 29], including multiple functions for isotropic and anisotropic contributions [30] and compared for example in [31, 32]. The residual stiffness \(\eta \ll 1\) prevents numerical problems for fully degraded material \(c=0\). Furthermore, only the positive part of the strain energy density is degraded to account for the tension compression asymmetry of cracks. Note, that positive does not refer to the sign of the strain energy density but to a one-dimensional analogy, where only positive stresses are degraded. A correct and physically meaningful, yet simple and efficient, splitting of the strain energy has been subject of discussion of several publications, cf. for example [27, 28, 33‐38]. In case of large deformation kinematics, similar approaches can be found [39‐43]. In this contribution, the so-called tensile split [27, Section 3.3]with the Lamé constants \(\overset{i}{\lambda },\overset{i}{\mu }\) and Macaulay brackets \(\langle \bullet \rangle _\pm =\tfrac{1}{2}(\bullet \pm \vert \bullet \vert )\) is adopted. For further information on the strains \(\overset{i}{\varvec{\varepsilon }}_\pm \), please refer to the cited section. It is noted that the diffuse interface model is not restricted to the tensile split, but can be applied to any decomposition of type (3).

The diffuse interface region with \(p\in (0,1)\) in the vicinity of the formerly sharp interface \(\varGamma ^\text {i}\) is referred to as \(\varGamma ^{\ell _\text {i}}\). The order parameter p, which recovers the values \(p=0\) for \(\varOmega _1\) and \(p=1\) for \(\varOmega _2\) far away from the interface, cf. Figure 2a, varies according towhich corresponds to a three-dimensional generalization of the one-dimensional solution of the Modica–Mortola functional [44], cf. Figure 2b. The signed distance \(d(\varvec{x})\) measures the shortest distance from any point in \(\varOmega _1\cup \varOmega _2\) to the interface mid-surface in the direction of the interface normal \(\varvec{n}^{\text {i}}=\nabla p / \vert \nabla p \vert \), cf. Figure 2a on the right. In this contribution, the signed distance \(d(\varvec{x})\), and thus \(p(\varvec{x})\) are a priori known. Within the diffuse region, the elastic energy density is defined aswhich also holds for the individual phases, because one or the other contribution vanishes for \(p=0\) or \(p=1\). The approach of interpolating the individual constituents is a widespread approach, cf. for example [18‐21, 24]. An alternative representationcan be obtained by inserting Eq. (3) into (7). The individual dependencies on \(\overset{i}{\varvec{\varepsilon }}\) are omitted above for the sake of readability. Analogously to Eq. (7), an interpolationof the individual strains is assumed in the diffuse region. Together with the strain jump , the independent variables for the strain energy density are now changed from \(\lbrace \overset{1}{\varvec{\varepsilon }},\overset{2}{\varvec{\varepsilon }} \rbrace \) to with the relationswhich is more convenient in view of the presented model. The strain is obtained from the displacement \(\varvec{u}\) bywhereas different homogenization assumptions for are discussed in Sect. 2.4. The next section combines the energy contributions from the phase-field crack and the linear elastic deformation, and derives the coupled field problem.

Starting from the findings and definitions of the previous section, the internal energy densityis presented, which is used in the following derivations of the governing differential equations. It contains bulk material and crack phase-field contributions, \(\psi ^\text {el}\) and \(\psi ^\text {c}\), respectively. The global incremental potentialis defined according to [45, Equation (90)]. The index n refers to the previous, converged time increment, whereas quantities without the index are related to the current \((n+1)^\text {th}\) increment. The time increment is denoted by \(\tau =t_{n+1}-t_n\). It is noted that a first order Euler backward time integration scheme is employed for the rate \(\dot{c}\) of the crack phase-field, which enters the dissipation potential \(\varPhi (\dot{c})=\frac{\eta _\text {f}}{2}\dot{c}^2\). Here, \(\eta _{\text {f}}\ge 0\) is a kinetic parameter, often referred to as viscosity, which is of purely numerical nature in this contribution. Its choice is discussed in Sect. 3. The incorporation of \(\dot{c}\) can be understood as a viscous regularization, which makes it possible to use a monolithic solution approach. Various other approaches and their implications can be found in literature, see e.g., [2, 46]. The vector \(\bar{\varvec{t}}\) denotes given external tractions on Neumann boundaries \(\partial \varOmega _{\varvec{t}}\). The strain jump, which enters \(\psi \), is determined within a local minimization procedure, cf. Section 2.4, and is therefore known which is indicated by the star . Hence, \(\varvec{u}\) and c are obtained by the global minimization principlewith admissible setsThe first constraint in \({\mathcal {C}}\) is only relevant for the initial crack along \(\varGamma ^\text {c}_0\). Afterwards, the second condition, also known as fracture-like irreversibility constraint [3, 47], suffices, where \(c_\text {th}\ll 1\) is a small, positive threshold value. Previous investigations [25] confirmed, that setting \(c_\text {th}=0.03\) does not impact the crack path, while ensuring irreversibility. A discussion on different irreversibility constraints can be found in [47]. Subsequently, the Euler–Lagrange equationscan be derived, subject to Neumann boundary conditionswhere \(\varvec{n}^{\text {b}}\) denotes the outward normal vector along the corresponding boundary. Equation (4) has been used in Eq. (19). The thermodynamically consistent relation for the Cauchy stress tensor is defined byIt is noted that the chain rule has been used together with Eqs. (10) and (11). The next section is dedicated to the determination of the strain jump which enters the incremental potential (14). The validity of Eq. (22) is demonstrated in “Appendix A”.

In this section, two different approaches from the homogenization theory are examined to determine the unknown strain jump , which enters the incremental potential (14) as . In particular, the Voigt–Taylor approach and a partial rank-I relaxation are introduced. In contrast to existing works [20], the combination of an additive strain energy decomposition and the above approaches is presented. This work incorporates the tensile split by Miehe [27] but is not limited to it.

Voigt–Taylor approach  In classical homogenization theory, the Voigt–Taylor approach assumes equal strains in every constituent, i.e.,From that, the strain jump follows, which is inserted in the incremental potential above. In other words, the Voigt–Taylor approach is automatically implemented in any diffuse modeling framework similar to this one, if the strain jump is not treated by some means or other.

Different from the assumption of equal strains in the constituents, the Reuss–Sachs approach assumes equal stresses. It is, however, not investigated in this contribution, because there are various other publications doing so [21, 22, 24]. The two approaches yield upper and lower energetic barriers for the energy, respectively, cf. [21, Equation (55)] or [24, Equation (62)].

Partial rank-I relaxation  The previous approaches do not include any information about the interface but only interpolate between the bulk energies, cf. Equation (7). In contrast, the partial rank-I relaxation correctly incorporates all mechanical jump conditions by means of the strain jump, which can be expressed by the Hadamard condition [48, Equation (2.2.9)]without loss of generality,¹ and the stress jump \(\llbracket \varvec{\sigma } \rrbracket ^\top \cdot \varvec{n}^{\text {i}}={\varvec{0}}\). The former is stated here for a symmetric tensor \(\varvec{\varepsilon }\). The vector \(\varvec{a}\) is the strain jump amplitude and \(\varvec{n}^{\text {i}}= \nabla p /\vert \nabla p \vert \) the interface normal. The latter is fulfilled as follows: In order to calculate \(\varvec{a}\), a pointwise minimization procedure [21, Equation (50)]with the associated necessary conditionis pursued, which is a relaxation of with respect to \(\varvec{a}\) [24]. The stresses are calculated according to Eq. (22). For the cases \(p=0\) or \(p=1\), which indicate bulk material away from the diffuse interface region, the condition is fulfilled and the jump \(\llbracket \varvec{\varepsilon }\rrbracket ={\varvec{0}}\) either way. For \(p\in (0,1)\), the expression in the curly braces has to be zero, which is similar to the aforementioned stress jump condition at sharp interfaces. Here, a pointwise enforcement throughout the whole diffuse interface region is required. In absence of a crack, \(g(1)=1\), the analytical solution for linear elastic material according to [24, Equation (42)] is recovered because the additive strain energy decomposition is unnecessary. The same is valid for \(g(c)<1\) if there is no strain energy decomposition at all, cf. [20], because g(c) does not exhibit any jump across the interface. In this contribution, the key novelty lies in the incorporation of any (variational) additive decomposition of the strain energy into a degraded and persistent part. The nonlinearity of the spectral decomposition for the tensile split requires a local Newton‐Raphson scheme to solve for \(\varvec{a}\) as soon as a crack emerges within the diffuse interface region. In the context of an elasto-plastic multiphase-field model, Schneider et al. [19] had to solve a very similar Eq. (26) because of the nonlinearity due to plasticity. The local Newton–Raphson scheme is presented in “Appendix B”.

The weak form including consistent linearization for the global Newton–Raphson scheme can be found in “Appendix C”. The spatial discretization is carried out using locally refined Truncated Hierarchical non-uniform rational B-splines [49] (NURBS) with quadratic shape functions in Sect. 3.1 and linear shape functions in Sect. 3.2. An adaptive refinement strategy is pursued, which allows for efficient computations with a high resolution of the steep gradient in regions where the crack develops and propagates. Additionally, the mesh is prerefined along the interface \(\varGamma ^{\ell _\text {i}}\) to resolve the diffuse transition for p sufficiently. Spatial convergence with respect to the crack length scale \({\ell _\text {c}}\) is ensured by choosing the finest element level in a way, that the characteristic element size is at least three times smaller than \({\ell _\text {c}}\), cf. Figure 3. All simulations are carried out using a MATLAB-based in-house finite element code.

For verification of the implementation, the convergence of the presented modeling approach with respect to the sharp interface solution is tested. For this purpose, a quarter of a circular bi-material disc is subjected to a radial displacement \(u_\text {R}=r_2/1000\), cf. Figure 3. In the study, the simulated domain is reduced to a square. The geometry parameters are given as \(r_1={3}~\mathrm{mm}\), \(r_2={15}~\mathrm{mm}\) and \(a={8}~\mathrm{mm}\). Plane strain is assumed. The signed distance for Eq. (6) readsSymmetry boundary conditions are applied along \(\partial \varOmega _\text {left}\) and \(\partial \varOmega _\text {bottom}\). Along \(\partial \varOmega _\text {top}\) and \(\partial \varOmega _\text {right}\), a traction \(\bar{\varvec{t}}\) is applied, cf. Eq. (73) in “Appendix D”. Two error norms, the well known energy normand total error in energyare defined, where the index \((\bullet )_\text {ex}\) stands for the exact solution and \(\vert \bullet \vert \) stands for the absolute value. The subscript ‘loc’ for the energy norm \(e_\text {loc}\) refers to the fact, that the error is calculated locally before integration. For the case without a crack, the analytical solution, cf. “Appendix D”, is used for \((\bullet )_\text {ex}\), and the denominator in \(e_\text {tot}\) is replaced by the analytical expression (69). For the cracked case, an overkill solution with a sharp interface serves as ‘exact’ solution. The overkill solution is calculated on a much finer mesh. The evaluation of the energy norm \(e_\text {loc}\) is not possible because of differing integration point locations.

Investigation without a crack  At first, the model is tested for the linear-elastic case in the absence of a crack. The material parameters are set to \(\nu _1=\nu _2=\nu =0.3\), \(E_2={100}~{\mathrm{GPa}}\), \({\ell _\text {c}}={50}~\upmu {\mathrm{m}}\) and a very high value \({\mathcal {G}}_\text {c}=10^{5}~\hbox {kN mm}^{-1}\) to prevent any crack formation or propagation. The viscosity \(\eta _{\text {f}}\) is set to zero. Two cases \(E_1/E_2=\lbrace 2,20\rbrace \) allow to study the influence of the intensity of the elastic dissimilarity. The coarsest element level of the mesh is depicted in Fig. 3 on the bottom left. An \(h{\ell _\text {i}}\)-refinement study is undertaken, where the interface length scale and the mesh size are reduced simultaneously, i.e., more refinement levels are added along the arc with radius \(r_1\). The ratio \({\ell _\text {i}}/h\approx 2.7\) is kept constant.

The results of the convergence study are depicted in Fig. 4. In general, it can be seen, that the partial rank-I scheme exhibits the same or better convergence rates and error levels than the Voigt–Taylor approach with respect to the analytical sharp interface solution. Furthermore, the obtained results are consistent with literature, compare Fig. 4a to [17, Figure 8b], or Fig. 4b to [24, Figure 5b]. Additionally, the Voigt–Taylor approach reacts more sensitive to a more intense elastic dissimilarity which can be seen when comparing the left and right plots in Fig. 4a and b. This is also reported in [18].

Investigation with a crack across the interface  Next, a phase-field crack along \(\varGamma ^\text {c}_0\) with \(e={0.5}~\mathrm{mm}\) is introduced as shown in Fig. 3, with the corresponding finite element mesh on the bottom right. The mesh is prerefined to the finest level along the crack to rule out any inaccuracies due to approximation errors of the phase-field. It serves as coarsest refinement level and is gradually refined along the interface for the \(h{\ell _\text {i}}\)-convergence study. All parameters and the boundary conditions are chosen similar to the previous investigation, except that only results for \(E_1/E_2=2\) are considered for the sake of brevity. It is noted that the traction \(\bar{\varvec{t}}\) does not reproduce the analytical solution as before, nor does it lead to a circumferentially constant radial displacement \(u_\text {R}\), which is why an overkill solution is used as reference. For the overkill solution, the NURBS mesh has to account for the physical \(C^0\)-continuity of the displacement along the arc \(r_1\). Additionally, the crack boundary condition along \(\varGamma ^\text {c}_0\) has to be set. This is why the finite element mesh is different from the mesh of the previous convergence study. Convergence of the overkill solution has been verified.

The results for three different values of the residual stiffness \(\eta \), cf. Eq. (4), are depicted in Fig. 5. All plots are identical, i.e., the residual stiffness does not influence the convergence, which is expected. However, for crack propagation simulations, a zero value most probably leads to numerical problems. Furthermore, the obtained results are more or less identical to the intact case, cf. Fig. 5 to the left plot in Fig. 4b: Convergence rate and error level are matching.

The two convergence studies suggest the successful verification of the implemented method and validation by means of literature results. The major question whether any impacts on propagating cracks can be observed is answered in the next section.

Besides the improved convergence of the presented model with respect to the sharp interface solution it is of great interest whether any impacts on crack propagation at diffuse interfaces are recognized. Hansen-Dörr et al. [25] thoroughly investigated many different setups, where a crack approaches a possibly inclined interface and, depending on the ratio of the elastic moduli on both sides of the interface and the ratio of the bulk material and interface fracture toughnesses, decides to branch or to go straight into the adjacent bulk material. A comparison to analytical results from He and Hutchinson [26] and to a sharp interface reference revealed inaccuracies of the Voigt–Taylor approach, which was used therein. This section demonstrates, that the presented model extension to a partial rank-I relaxation yields the same critical deflection ratios as the sharp interface reference, while the Voigt–Taylor gives different results.

The setup for this study, which is similar to [25], is depicted in Fig. 6. The corresponding NURBS mesh is shown in Fig. 7a. A square domain with an initial phase-field crack \(\varGamma ^{\ell _\text {c}}_0\) is subjected to a horizontal displacement load. For this purpose, the concept of the so-called surfing boundary condition [50] is exploited, which proved to be very suitable for quantification purposes [50, 51]. The key idea of this approach is to introduce a point with the time dependent position \(\bar{\varvec{x}}\), which is moving along the y-axisA displacement of hyperbolic tangent-like shape is applied on the side edges \(\partial \varOmega _{u, \text {s}}\),with respect to the moving point, assuming \(u_\text {ref} = {7.5}~{\upmu \mathrm{m}}\), \(d={0.5}~\mathrm{mm}\), \(v={0.3}~\hbox { mm s}^{-1}\) and \({\bar{y}}_0={-4}~{\mathrm{mm}}\). Similar parameters proved to be suitable in previous studies [25]. Homogeneous Neumann boundary conditions are considered along the top and bottom edges. A horizontal interface, implied by the grey hatched bar in Fig. 6, divides the domain in two halves. The fracture toughness of the interface is different from the bulk material. This is accounted for by locally reducing the fracture toughness according to a Gaussian-like distributioncf. Figure 6 blue plot in the middle. Far away from the interface, the bulk material fracture toughness \({\mathcal {G}}_\text {c}^\text {b}={2.7}\hbox {N mm}^{-1}\) is recovered. Along the interface, a potential crack is supposed to propagate for a critical energy release rate equal to \({\mathcal {G}}_\text {c}^\text {i}\). However, the crack experiences a bulk material influence due to the interaction of both length scales \({\ell _\text {i}}\) and \({\ell _\text {c}}\) [25], and tends to propagate at critical energy release rates, which are higher compared to the minimum of the function in Eq. (32). Hence, a compensated interface fracture toughness \(\hat{{\mathcal {G}}}_\text {c}^\text {i}\) is specified in a way, that a crack propagates along the interface midline at the true interface fracture toughness \({\mathcal {G}}_\text {c}^\text {i}\), i.e., \(\hat{{\mathcal {G}}}_\text {c}^\text {i}<{\mathcal {G}}_\text {c}^\text {i}\). For more details on the compensation procedure, the reader is referred to [25, 52, 53]. The ratio of the elastic moduli with \(E_1={210}{\hbox { GPa}}\) varies, while the Poisson ratio \(\nu =0.3\) is kept constant, i.e., the interpolation in Eq. (7) directly translates into a hyperbolic tangent function for \(E_1\) and \(E_2\), cf. orange line in Fig. 6 on the right. For the sharp interface reference solution, the elastic moduli vary according to the purple step function. The elastic dissimilarity is described with the first Dundurs’ parameter [54]Following the study of Hansen-Dörr et al. [25], the length scale parameters are set to \({\ell _\text {i}}={18.75}~{\upmu \mathrm{m}}\) and \({\ell _\text {c}}={15}~{\mathrm{mm}}\), while the viscosity and the residual stiffness take the values \(\eta _{\text {f}}={10^{-5}}~\hbox {kN s} \upmu \mathrm{m}^{-2}\) and \(\eta =10^{-5}\), respectively.

A parameter study with \(\alpha =\lbrace -0.5,-0.25,0.25,0.5 \rbrace \) and different ratios \({\mathcal {G}}_\text {c}^\text {b}/{\mathcal {G}}_\text {c}^\text {i}\) is conducted, and the results for the sharp interface and the diffuse interface description using the Voigt–Taylor approach or partial rank-I relaxation are compared with respect to the crack behavior at the interface: Fixing \(\alpha \), the tendency of the crack to branch is higher for larger ratios \({\mathcal {G}}_\text {c}^\text {b}/{\mathcal {G}}_\text {c}^\text {i}\). The NURBS mesh is adaptively refined according to the crack path. All elements, where the phase-field fell below a value of \(c=0.5\), are marked for refinement [49]. For the sake of brevity, only the results where the two homogenization approaches yielded different crack phenomena at the interface are presented.

Figure 7b–d present the crack patterns shortly after the crack met the interface, with \(\alpha =-0.5\) and \({\mathcal {G}}_\text {c}^\text {b}/{\mathcal {G}}_\text {c}^\text {i}=8\), i.e., \({\mathcal {G}}_\text {c}^\text {b}/\hat{{\mathcal {G}}}_\text {c}^\text {i}=23\). For better interpretation, the elements with \(c<0.1\) are made invisible. It can be seen, that the sharp interface solution and the diffuse interface approach with a partial rank-I relaxation predict a branching of the crack. On the contrary, the Voigt–Taylor approach yields a straight crack with no such tendency. Even setting \({\mathcal {G}}_\text {c}^\text {b}/{\mathcal {G}}_\text {c}^\text {i}=10\), i.e., \({\mathcal {G}}_\text {c}^\text {b}/\hat{{\mathcal {G}}}_\text {c}^\text {i}=30.3\), did not change this fact. This is in line with the presumption of Hansen-Dörr et al. [25], who also observed such deviations. For further loading, all cracks started to propagate further in y-direction, while the two crack branches for the sharp interface and the diffuse partial rank-I relaxation arrested. For \(\alpha =-0.25\) and \({\mathcal {G}}_\text {c}^\text {b}/{\mathcal {G}}_\text {c}^\text {i}=5.5\), i.e., \({\mathcal {G}}_\text {c}^\text {b}/\hat{{\mathcal {G}}}_\text {c}^\text {i}=12\), a similar behavior is observed. For \(\alpha =\lbrace 0.25,0.5 \rbrace \), the three approaches always yield the same outcome. On the one hand this could imply an increased sensitivity of the crack behavior towards the homogenization scheme for negative \(\alpha \) values. On the other hand, only discrete values of \({\mathcal {G}}_\text {c}^\text {b}/{\mathcal {G}}_\text {c}^\text {i}\) are investigated and it is possible, that only those were picked from the infinite amount of possible ratios, where the results are identical. This is, why future model investigations will include a deeper and also three-dimensional analysis of the phenomena reported above. To conclude with, it was shown, that the present model extension meaningfully influences the crack behavior and yields closer results to the sharp interface solution.