In the phase field approach for fracture an additional scalar field is introduced in order to describe the state of the material between intact and fully broken. So far, for the loading dependent degradation of stiffness (damage) either the volumetricdeviatoric split of strain [1,2] or the spectral decomposition [3,4] is used. In contrast to such an isotropic degradation of stiffness, the fully broken state represents a crack with a particular orientation. Both aforementioned approaches do not take the crack orientation into account. This may lead to the violation of the crack boundary conditions. In order to satisfy these conditions the phase field approach is modified here by taking the orientation of the crack into account. A key feature of the phase field approach compared to other numerical methods is the smooth approximation of the crack, which avoids numerical difficulties. The scalar phase field S(x, t) represents the state of material between S = 1 for undamaged material and S = 0 for fully broken material. Evolution of cracks is based on the energetic criterion for brittle fracture proposed by Griffith. The crack surface energy is approximated by a regularized crack surface density γ ℓwhich reproduces in the limit ℓ → 0 a sharp crack. Formulating the Lagrangian and employing Hamilton's principle leads to the coupled Euler-Lagrange equations for the momentum balance and the phase field evolution [4] ρü − div σ = 0 andFor a body Ω the corresponding boundary conditions are u =ū on ∂Ω u , σ ·n =t on ∂Ω σ and ∇S ·n = 0 on ∂Ω with the unit normal vector n. The constitutive behavior is included in (2) determining the stress σ(S) = ∂ψ el /∂ε ε ε and ψ deg (S) = ∂ψ el /∂S. The latter term includes only those parts of the elastic energy density being responsible to cause further damage, for more details see [1] or [3]. The constitutive equations depend on the scalar phase field, which is able to describe isotropic damage. However, in order to satisfy the boundary conditions on a crack the crack orientation has to be taken into account. Directional decomposition close to cracksIn brittle materials damage occurs essentially in case of tension. In addition, the state of fully degraded material (S ≪ 1) should reproduce a crack. This means that the positive normal stress perpendicular to the crack surface should vanish as well as the shear stresses along the frictionless crack surface. The orientation of a crack surface is represented by its normal, which can be computed in the underlying approach from the gradient of the phase field: n s = −∇S/|∇S|. In the sequel, all strains and stresses in the vicinity of a crack are computed with respect to an orthonormal coordinate system with the base vectors n s , t 1 and t 2 . The requirement of a non-positive normal stress on the crack surface leads to an inequality condition. For isotropic linear elastic material this restriction can be written in terms of strain including the material parameters and the normal strains in tangential directionsIn order to enforce this condit...
The formation and further evolution of cracks caused by the compression of a stiff indenter onto the surface of an initially defect‐free brittle solid is a fascinating problem of fracture mechanics. Its prediction, however, is still a challenging task since crack nucleation is caused by a rather weak stress concentration in the contact near‐field. The present contribution focuses on phase field simulation of indentation fracture, including crack formation at some a priori unknown location outside of the contact region and the subsequent formation of a cone crack. While the phase field method, at first glance, appears to be a promising tool to simulate the current problem we elaborate critical issues and discuss essential modifications. Finally, the indentation fracture process is simulated showing the effect of varying indenter radii on crack initiation and the influence of Poisson's ratio on the angle of cone crack propagation in good agreement with experimental findings and other theoretical studies.
A key feature of phase field approaches to fracture is the relatively simple structure of the method. However, current approaches inevitably exhibit some limitations which do not seem to be obvious and are therefore neglected in many investigations. The assumption of isotropic stiffness degradation by a scalar phase field parameter does not capture the anisotropy introduced by a crack and severely restricts the scope of application. Widely used tension-compression splits violate the specific conditions at cracks like traction free crack surfaces and, in addition, introduce mesh orientation dependency. Accurate approximation of the surface (fracture) energy is another important requirement which is discussed for a non-vanishing internal length ℓ.1 Current phase field approaches to brittle fractureIn the phase field approach for fracture, a crack is approximated in a smooth manner and the surface energy of a discrete crack is replaced by a surface energy density (1) 2 defined in the whole domain. The hereby introduced scalar phase field S describes the current state of material between S = 1 (undamaged) and S = 0 (broken) while an internal length parameter ℓ controls the width of the transition zone between these states. The method can be based on a variational principle with the corresponding Lagrangian given in (1) Evaluation of Hamilton's principle yields the governing field equations and boundary conditions. A common choice for the local surface potential in (1) 2 is w 0 = (1 − S) 2 with c w = 1, see e.g [1]. Tension-compression asymmetric responseIn order to account for the asymmetric tension-compression response of a crack, usually the elastic energy density in (1) 1 is decomposed into active and passive portions ψ act and ψ pas , respectively. However, the common volumetric-deviatoric or spectral splits, which are isotropic and do not account for the crack orientation, do generally violate crack boundary conditions and thus lead to an unphysical behavior [2,3]. Besides that, all kinds of tension-compression splits introduce a mesh orientation dependency which results from the only partial (split-dependent) degradation of stiffness components in conjunction with the limited kinematics of finite elements. As a)ū ,F S ≈ 0 b) 0 0.5 1 1.5 0 0.002 0.004 norm. displacementū norm. forceFFig. 1: Through-cracked block subjected to simple shear. a) Initial crack modeled by imposing S ≈ 0 on at least one continuous row of finite elements. b) Non-vanishing reaction force for mesh not aligned with crack orientation.an illustrative example, we consider a block under simple shear (Fig. 1a) with a horizontal throughcrack modeled by imposing S ≈ 0 on at least one continuous row of elements. In case of the FE mesh being aligned with the crack direction, the volumetricdeviatoric split correctly reproduces the behavior of the through-cracked block with almost no reaction force (Fig. 1b). By contrast, using the same split, a mesh which is not aligned with the crack yields a stiff elastic response as shown by the load-displacement...
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