PSI - Issue 52

Tong-Rui Liu et al. / Procedia Structural Integrity 52 (2024) 740–751 T.-R. Liu, F. Aldakheel, M. H. Aliabadi / Structural Integrity Procedia 00 (2023) 000–000

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Figure 1: Sketch of solid domain Ω with sharp crack Γ s (left) and its phase field regularized representation Γ d (right)

2. Mathematical foundation

2.1. Phase field model of brittle fracture

As shown in Fig.1, Consider Ω ⊂ R n dim ( n dim = 1 , 2) as a solid domain, with its external boundary denoted by ∂ Ω ⊂ R n dim − 1 , the outwards normal unit vector to the external boundary is denoted as n . The kinetic and fracture responses at a point x ∈ Ω and time t ∈ T = [0 , T ] can be described by admissible displacement u ( x , t ) and fracture phase field d ( x , t ): u : Ω ×T → R n dim ( x , t ) → u ( x , t ) and d : Ω ×T → [0 , 1] ( x , t ) → d ( x , t ) with ˙ d ≥ 0 (1) where d ( x , t ) = 0 and d ( x , t ) = 1 denote the intact and fully damaged state of the material, respectively. ˙ □ = d □ / dt represents the time derivative of □ . The deformation is measured by the strain field ε ( x , t ) : Ω ×T → [ R n dim × n dim ] sym under infinitesimal strain theory, such as: The symbol ∇ sym ( · ) denotes the symmetric gradient with respect to spatial coordinates. For the phase field problem, as shown in Fig. 1(right), the sharp crack topology Γ s is regularized by a di ff usive phase field d ( x ) : Γ d → [0 , 1] over a localized band Γ d ⊆ Ω which is unknown a priori, and its exterior domain Ω \ Γ d keeping intact. By introducing the crack surface density function γ ( d , ∇ d ), the sharp crack is regularized by phase field approximation in a purely geometric context as: Γ d ψ c ( d , ∇ d )d V = Γ d G c γ ( d , ∇ d )d V phase field regularized crack ≈ Γ s G c d A sharp crack (3) ε = ∇ sym u = sym[ ∇ u ] : = 1 2 ∇ u + ∇ u T . (2)