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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The fatigue degradation function is required to have a value of 1 until the threshold fatigue damage is achieved, following which it asymptotically reduces to 0. The fatigue degradation function is chosen as the follows f ( α ) = 1 if α ≤ α T 2 α α + α T otherwise (10) where, α T is a material parameter that behaves as the threshold fatigue damage. Fatigue degradation is not observed until this level of damage is exceeded. This phase of damage represents a period of plastic instability. The damage variable is given by α = t 0 g d ψ + 0 where only positive increments in the elastic energy are counted in the damage incrementation. This ensures that we only accumulate fatigue damage during loading. A further requirement is included to the damage accumulation to be able to model a very localized plastic e ff ect. This is a fatigue damage elastic energy threshold H f t which serves as an energy analog to K op . In essence, fatigue damage is accumulated while ψ exceeds a certain threshold value. The purpose of this addition is to simulate crack closure e ff ects in the tensile load region. We choose this threshold according the deformation history of the material, where it is defined as a function of H given by the following H = max τ ∈ [0 , t ] ψ 0 ( τ ) (11) which replaces ψ 0 in the strong form to enforce the irreversibility condition. In the literature, there are several ap proaches to finding K op , such as the following given by Schijve (2004) K op K max = 0 . 45 + 0 . 22 R + 0 . 21 R 2 + 0 . 12 R 3 (12) where R is the stress ratio given by σ min /σ max where these are the min and max load stresses. The formulation here is restricted to the zero-based load case where R = 0, leading to a simple linear relation between the opening and max stress intensity factors. As stress intensity factors are proportional to the stress, the square of the function is considered to arrive at the following linear relation for the threshold function H f t = 0 . 2025 H . (13) The evolution equation for the fatigue damage variable is now defined as the follows ˙ α = g ˙ ψ 0 if g ψ 0 ≥ H f t and ˙ ψ 0 ≥ 0 0 otherwise (14) It will be demonstrated in the following sections that the addition of this threshold is able to simulate overload e ff ects in the tensile region. This is because overloads greatly inflate H f t ahead of the crack tip. This restricts the damage range, leading to slow fatigue degradation and hence slower crack growth. Other relations for H f t may be proposed to model more realistic behavior, subject to the following constraint ∂ H f t ∂ H ≤ 1 . (15) which ensures that at higher amplitudes, fatigue damage accumulates faster, leading to faster propagation rates with higher amplitudes. The fatigue phase field model is included in ABAQUS through a UMAT employing a HETVAL subroutine, using coupled temperature-displacement elements, where temperature serves as a stand-in for the phase field parameter k ∇ 2 T = − r . (16) The similarity of the ∇ 2 T term to the ∇ 2 φ term is exploited in the strong form, allowing us to rearrange the strong form and to introduce a heat source term, with conductance k set to 1 r = − ( φ l 2 0 − 2(1 − φ ) ψ 0 G c l 0 ) . (17)