PSI - Issue 42

Sarim Waseem et al. / Procedia Structural Integrity 42 (2022) 1692–1699 Waseem et al. / Structural Integrity Procedia 00 (2019) 000–000

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Gri ffi th’s theory was constrained by its requirements for a pre-existing crack and well-defined crack path, which Francfort and Marigo (1998) managed to overcome with the introduction of the variational form of Gri ffi th’s energy balance E = E s ( Γ ) + E e ( Γ , u ) (2) where E is the total energy of the system, E s is the crack surface energy and E e is the bulk energy. The key assumption here being that the system evolves in a way to minimize the global energy of the system. The following approximation may be made for the crack surface energy: E s = G c d Γ ≈ G c γ d Ω (3) where Γ ⊂ Ω represents the crack surface in the domain and is subject to the irreversibility condition ˙ Γ ≥ 0 which ensures that the crack surface cannot shrink and may only grow. γ ( φ, ∇ φ ) is the crack density function (see Miehe et al. (2010)), describing the presence of cracks over the domain which is dependent on the phase field parameter and additionally incorporates non-local gradient e ff ects. The deriva tion of this follows from the one-dimensional crack distribution and incorporates a length scale which controls the width of the di ff use crack. While l 0 was initially introduced as a numerical parameter to regularize sharp cracks, it is now widely accepted that l 0 should be considered as a material parameter. For the bulk energy, ψ e is a function of both strain and the phase field parameter. The following simplification is used E e = ψ e ( φ, ε ) d Ω = g ( φ ) ψ 0 ( ε ) d Ω . (5) Here, ψ 0 is the undamaged elastic energy for linear elasticity. Meanwhile, g is called the degradation function and it is solely a function of the phase field parameter. The degradation function used in our formulation is given as follows g ( φ ) = (1 − φ ) 2 . (6) There are alternatives to this choice of degradation function. For example, cubic degradation functions have been found to more e ff ectively capture the linear elastic phase of the material (see e.g. Borden et al. (2016)). The energy functional is now given by: E ( φ, ε ) = g ( φ ) ψ 0 ( ε ) d Ω + G c γ d Ω (7) Both the displacement field and the phase field are solutions to the minimization problem of this functional. The following equations then define the strong form of the problem ∇ σ = 0 G c ( φ l 0 − l 0 ∇ 2 φ ) − 2(1 − φ ) ψ 0 ( ε ) = 0 (8) where, σ is the degraded stress given by g ( φ ) σ 0 where σ 0 is the non-degraded stress given by C : ε . The addition of fatigue e ff ects is done through the simple introduction of a fatigue degradation function. Essentially, fatigue is captured through the degradation of the fracture toughness γ = φ 2 2 l 0 + l 0 2 |∇ φ | 2 (4)

φ l 0 −

2 φ ) − 2(1 − φ ) ψ

f ( α ) G c (

l 0 ∇

0 ( ε ) = 0 .

(9)

Here, α is the fatigue damage variable, representing the total damage accumulation through the history of the loading. α = 0 represents the case where no fatigue damage has been taken.