Restrictions in phase field modeling of brittle fracture

Abstract

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...

Analysis of Hertzian indentation fracture using a phase field approach

“…Thus, such approaches generally violate crack boundary conditions and lead to a spurious phase field evolution [10]. Thus, the deviation of the approximated surface energy from the theoretical value becomes significant unless the crack is many times longer than ℓ, as illustrated in [12]. While the energy consumption of a smeared crack is approximated quite well for a small, but non-vanishing internal length ℓ the crack tip consumes additional energy which is not accounted for in the method [11].…”

“…traction-free, it is feasible here to omit any split in the calculation of stress (1) and consider a full stiffness degradation (3) 1 . Referring to the discussion in [12], the degradation function g = (4 − a s ) S 3 + (a s − 3) S 4 of fourth-order with a s = 1 × 10 −4 is an optimal compromise to reduce the internal length sufficiently, but also guarantees that the phase field approaches the broken state. Since brittle materials tend to fail normal to the maximum principal stress, the phase field S should be driven by positive principal stresses rather than strains.…”

“…Since the physically motivated separate choice of stiffness degradation (3) 1 and crack driving energy (3) 2 would cause unsymmetric entries in the tangent matrix of the fully coupled set of equations, we use an alternate minimization procedure where displacements u and phase field S are computed separately in an alternating manner as proposed by [7]. For any finite discretization the mechanical field inevitably influences the phase field solution at a crack and thus leads to an overestimation of the approximated surface energy [12,14]. Numerical studies are carried out on a quadratic domain of size 100a×100a for different indenter radii a = 0.5 .…”

“…While the energy consumption of a smeared crack is approximated quite well for a small, but non-vanishing internal length ℓ the crack tip consumes additional energy which is not accounted for in the method [11]. Thus, the deviation of the approximated surface energy from the theoretical value becomes significant unless the crack is many times longer than ℓ, as illustrated in [12]. Obviously, this requirement cannot be fulfilled during crack formation in absence of a pre-crack.…”

“…2.1, the internal length ℓ depends on the choice of the degradation function g(S) and has to be sufficiently small compared to z 0 . Referring to the discussion in [12], the degradation function g = (4 − a s ) S 3 + (a s − 3) S 4 of fourth-order with a s = 1 × 10 −4 is an optimal compromise to reduce the internal length sufficiently, but also guarantees that the phase field approaches the broken state. For fixed E and G c the internal length thus varies monotonically as a function of the tensile strength as worked out in [13] with values in the range…”