Abstract
We address brittle fracture in anisotropic materials featuring two-fold and four-fold symmetric fracture toughness. For these two classes, we develop two variational phase-field models based on the family of regularizations proposed by Focardi (2001), for which Γ-convergence results hold. Since both models are of second order, as opposed to the previously available fourth-order models for four-fold symmetric fracture toughness, they do not require basis functions of C1-continuity nor mixed variational principles for finite element discretization. For the four-fold symmetric formulation we show that the standard quadratic degradation function is unsuitable and devise a procedure to derive a suitable one. The performance of the new models is assessed via several numerical examples that simulate anisotropic fracture under anti-plane shear loading. For both formulations at fixed displacements (i.e. within an alternate minimization procedure), we also provide some existence and uniqueness results for the phase-field solution.
Highlights
[17], was recently extended to materials with anisotropic fracture toughness
We address brittle fracture in anisotropic materials featuring two-fold and four-fold symmetric fracture toughness and develop two variational phase-field models based on the family of regularizations proposed by Focardi [16], who proved their Γ-convergence
The main novelty lies in the model for four-fold symmetric anisotropy: as opposed to all previously available models for the same type of anisotropy, our proposed model is of second order, it does not require higher-continuity basis functions nor mixed variational principles for finite element discretization
Summary
One of the examples is sketched, where it is assumed that there exists a plane of anisotropy A ⊂ R2 within the body Ω ⊂ R3 such that for any point x ∈ A and with the polar coordinate system within A associated to x, Gc is viewed as a function of the polar angle θ ∈ [0, 2π), that is, Gc = Gc(θ) This is in contrast to the simple isotropic case when Gc ≡ const. In the following, when considering the cases of isotropic and anisotropic fracture toughness we refer to them as ’isotropic fracture’ and ’anisotropic fracture’, respectively In the former case, we assume Gc(θ) ≡ const such that, owing to (1) where γ(θ) ≡ 1, it is Gc(θ) = G0. The term ’anisotropic fracture’ is accompanied by the specification ’with k-fold symmetric Gc’
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
