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The realistic approximation of structural behavior in a post fracture state by the phase-field method requires information about the spatial orientation of the crack surface at the material point level. For the directional phase-field split, this orientation is specified by the crack orientation vector, that is defined perpendicular to the crack surface. An alternative approach to the determination of the orientation based on standard fracture mechanical arguments, i.e. in alignment with the direction of the largest principle tensile strain or stress, is investigated by considering the amount of dissipated strain energy density during crack evolution. In contrast to the application of gradient methods, the analytical approach enables the determination of all local maxima of strain energy density dissipation and, in consequence, the identification of the global maximum, that is assumed to govern the orientation of an evolving crack. Furthermore, the evaluation of the local maxima provides a novel aspect in the discussion of the phenomenon of crack branching. As the directional split differentiates into crack driving contributions of tension and shear stresses on the crack surface, a consistent relation to Mode I and Mode II fracture is available and a mode dependent fracture toughness can be considered. Consequently, the realistic simulation of rock-like fracture is demonstrated. In addition, a numerical investigation of \(\Gamma \)-convergence for an AT-2 type crack surface density is presented in a two-dimensional setup. For the directional split, also the issues internal locking as well as lateral phase-field evolution are addressed.
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AT-2 is a common choice in the computational mechanics community, see e.g. Borden et al. (2012), Hofacker et al. (2009) and Kuhn and Müller (2010), due to the straight forward implementation and onset of phase-field evolution at external loading. Nevertheless, the additional numerical effort to constrain the phase-field evolution in AT-1 is rewarded by an initial elastic phase before the phase-field evolution, which is a better approximation for brittle fracture than the softening behavior of AT-2 prior to the crack, see Alessi et al. (2018) and Tanné et al. (2018) for a discussion. Furthermore, higher order functionals can be used for the approximation of materials with anisotropic fracture toughness and an enhanced approximation of \(\Gamma _l\) at the expense of higher order gradients, see e.g. Borden et al. (2014).
According to Griffith, the formation of crack surface requires energy, that is dissipated from the strain energy stored in the deformation of the structure. In the initial considerations, a tensile force on the atomic structure is assumed. As soon as the distance between the atoms exceeds a critical value, the atomic bond is released and a separation of the initially continuous body, i.e. a crack, is obtained. In the framework of continuum mechanics, the counterpart to force and separation in the sense of Griffith are stresses and strains, respectively. Furthermore, the energy associated with forces acting along the separation distance has its local continuum mechanical counterpart in the strain energy density at the material point. Hence, in a similar way as \(\gamma _l\) is the representation of the regularized crack surface \(\Gamma _l\) at the material point level, the strain energy density \(\psi _\varepsilon \) is the local basis to obtain the phase-field analogue of the structural level quantity energy release rate \(\mathcal G\). Assuming linear elasticity, the strain energy density
The key feature of the stress based directional decomposition is the information about the orientation of the crack surface. This information is provided in a unique way by the crack orientation vector \(\mathbf r\) perpendicular to the crack surface, see Fig. 4. In combination with vectors \(\mathbf s\) and \(\mathbf t\), both perpendicular to \(\mathbf r\) and to each other, a Cartesian crack coordinate system (CCS) is established. The decomposition of a linear elastic ground stress tensor based on Eq. (4) with respect to the CCS is visualized in Fig. 4. In order to approximate the crack kinematics visualized in Fig. 3, the crack driving stress tensor
includes the shear components on the crack surface as well as the stress components perpendicular to the crack surface with tensile magnitude by \(\langle \bullet \rangle _\pm =(\bullet \pm \vert \bullet \vert )/2\). Furthermore, the continuum description of the crack kinematics originating from a discrete crack behavior requires an additional correction term to account for Poisson effects during crack opening, see Steinke and Kaliske (2018). It can be shown, that Eq. (6) is identical to the recent development of a representative crack element theory in case of isotropic linear elasticity at small strains, see Storm et al. (2020).
containing the remaining components of the total stress tensor as well as the correction term with negative sign. This ensures, that linear elastic behavior is recovered in case of no phase-field crack is present. In reverse manner, the crack driving and persistent strain energy densities are obtained from the stress and strain tensors by
In this work, the combination of both approaches, that is already published in Steinke and Kaliske (2018), is applied in order to benefit from the elegant simplicity of a formulation with a damage-like history while obtaining a proper profile for fully evolved cracks. A critical value for the phase-field \(p_\text c\) is defined. Based on this value, a history variable \(\mathcal H(t)\) is introduced, that depends on the current solution time \(t_n\) as well as on a previous step of the simulation at time \(t_n-1\). The history variable reads
A major ingredient of the directional split is a realistic definition of the crack orientation vector \(\mathbf r\). The explicit definition of the crack orientation forms further the basis of the generalization of the directional split in the framework of representative crack elements, see e.g. (Storm and Kaliske 2021; Storm et al. 2020, 2021; Yin et al. 2021). The definition \(\mathbf r=\nabla p/\vert \nabla p\vert \) proposed in Strobl and Seelig (2015) seems an intuitive approach. It results in a realistic orientation in the special case of the through crack discussed by Strobl and Seelig. However, this definition fails both at the crack tip as well as within fully degraded elements. At the crack tip, the gradient of the phase-field does not represent the actual orientation of the crack, see Fig. 5. While this is the critical region, where crack propagation is expected, the phase-field evolution is affected by a wrong decomposition of the stresses. Furthermore, the stress free boundary of the crack surface requires an approximation by at least one row of connected fully degraded elements, see Steinke et al. (2016). However, in a fully degraded element, i.e. all nodes having \(p=1\), the gradient of the phase-field is not specified properly. While this region is critical for the proper approximation of the crack kinematics, again the gradient definition fails to deliver a realistic approximation of the crack.
In an isotropic linear elastic material, the principal directions of strains and stresses are identical. A similar conclusion is obtained for the strains and the effective stress tensor for special cases, e.g. the spectral and volumetric deviatoric split, see Steinke (2021) for a detailed investigation of the spectral split on this issue. However, an analogous derivation is hardly possible for the directional split. The reason is the composition of the stress tensors in Eqs. (6) and (7), which depend on the crack orientation vector \(\mathbf r\), as well as the dependence on the specific value of the degradation function. In consequence, analytical conclusions on the principal directions of the effective stress tensor of the directional split, i.e. whether or not they are identical to the principal directions of the strain tensor, are not available. Rather, the principal directions of the strains are evaluated and employed.
where \(\mathbf n_1\) is the direction of the largest principal strain, i.e. \(\hat\varepsilon _1>\hat\varepsilon _2\) holds. In combination with the irreversibility approach, the change of the crack orientation is formulated in an explicit scheme by
The fracture toughness \(G_\text c\) is a measure for the material inherent energetic barrier against crack evolution. In combination with the crack surface density \(\gamma _l\), the resistance against crack propagation is quantitatively specified at the material point level. In the phase-field method, the crack driving component of the local strain energy density is the energetic counterpart to that resistance. Crack evolution is a result on the global level, that is based on the local formulation and evaluation of driving force and resistance by Eq. (17). The specific amount of crack driving strain energy density depends on the state of strain. Furthermore, it strongly depends on the orientation of the crack surface. In the previous section, an intuitive alignment of the crack orientation vector, i.e. along the direction of the largest principal strain, is postulated. It can be shown, that in the majority of possible strain states, this approach yields the maximum driving force for isotropic elastic material due to maximization of the crack driving strain energy density with respect to the crack orientation vector. However, certain states of strain exhibit the maximum in the crack driving strain energy density for a crack orientation, that is at a specific angle to the principle direction. Also recent experimental observations in Rozen-Levy et al. (2020) support the hypothesis that the principal of maximum energy dissipation also applies to the propagation of cracks.
In this section, an analytical approach to obtain the maximum crack driving component of the strain energy density is proposed and evaluated for a linear elastic material in a 2D setup. At the basis of known eigenvectors \(\mathbf n_1\) and \(\mathbf n_2\) of the 2D strain tensor \(\varvec\varepsilon \), a principal direction coordinate system (PCS) is established. The representation of both eigenvectors in the RCS and the PCS reads