PSI - Issue 13

Zhengkun Liu et al. / Procedia Structural Integrity 13 (2018) 787–792

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2 Z. Liu et al. / Structural Integrity Procedia 00 (2018) 000–000 influences of the angle of the direction of anisotropy for crack path in polycrystalline materials are depicted. At last, conclusions and an outlook for the next steps are given.

2. Phase-field model of anisotropic fracture

In the variational principle for brittle fracture, the total potential energy functional E can be stated as E ( u , Γ ) = Ω ψ e dV + Γ G c d Γ . (1) Here, the first term is the elastic energy and the second term is the fracture energy. ψ e is the elastic energy density with the linearized strain tensor ε and the fourth-order elastic sti ff ness tensor with cubic symmetries C . In Eq. 1, the critical energy release rate G c denotes the critical energy required to create a unit area of new crack. Due to fracture, the elastic strain energy density takes the form as ψ e ( ε, s ) = ( s 2 + η ) 1 2 ε : ( C : ε ) , (2) where the small positive dimensionless parameter 0 < η � 1 is used to ensure a numerically well-conditioned system for a fully-broken state ( s = 0). The Voigt notation of the anisotropic fourth-order elastic sti ff ness tensor can be expressed in matrix form as C = PC 0 P T , (3) in which P is a matrix which transposes the principal sti ff ness matrix C 0 to the oriented sti ff ness matrix C in cartesian axes. The transformation matrix P can be formulated as P =    cos 2 ( α ) sin 2 ( α ) 2 cos( α ) sin( α ) sin 2 ( α ) cos 2 ( α ) − 2 cos( α ) sin( α ) − cos( α ) sin( α ) cos( α ) sin( α ) cos 2 ( α ) − sin 2 ( α )    , (4) where α denotes the material orientation. In the principal sti ff ness matrix C 0 , only three independent material param eters are used, namely C 0 =    C 11 C 12 0 C 12 C 11 0 0 0 C 44    . (5) In the phase-field model for anisotropic brittle fracture, a second order structural tensor ω , being invariant with respect to rotations, as an additional material parameter for the directional dependency of the fracture resistance was intro duced by Clayton and Knap (2016). The coe ffi cients of the second order structural tensor ω in the two-dimensional case reads ω = 1 + β sin 2 ( θ ) − β cos( θ ) sin( θ ) − β cos( θ ) sin( θ ) 1 + β cos 2 ( θ ) , (7) with the direction of anisotropy θ (the angle of unit vector normal to the preferential cleavage plane). In Eq. 6, if the di ff use parameter κ tends towards zero, the phase-field approximation of the fracture energy density ψ f rac is exact. The governing equations of phase-field model for the brittle fracture in anisotropic materials can be expressed as div σ = 0 , ˙ s M = − 2 s ψ e + G c 1 − s 2 κ + 2 κ ∇ s ∙ ( ω ∇ s ) , (8) with respect to the mobility factor M which should be chosen large enough. The fracture energy can be approximated by Γ G c d Γ = V G c 1 4 κ (1 − s ) 2 + κ ∇ s ∙ ( ω ∇ s ) ψ f rac dV . (6)