🗜️ 2D High-Cycle Fatigue🥱 by Different Energy Formulations

🚀 Fatigue Mechanism in Phase Field Fracture

To account for fatigue effects, the phase-field governing equation is modified by introducing a fatigue degradation function \( f(\bar\alpha) \):

\[ \dot d = -L\left(\frac{\partial \Psi_{\text{loc}}}{\partial d} - \kappa(\bar\alpha)\Delta d \right) \]

The local energy is defined as: \[ \Psi_{\text{loc}} = \omega(d)\Psi_{\text{activate}} + f(\bar\alpha)\Psi_{\text{fracture}} \]

🔑 The fatigue degradation enters both the gradient and fracture energy terms, \[ \kappa = \frac{2lf(\bar\alpha)G_{c(0)}}{c_0}, \quad \text{and} \quad f(\bar\alpha)\Psi_{\text{fracture}} = \frac{f(\bar\alpha)G_{c(0)}}{c_0 l} \alpha(d) \] This shows that applying \( f(\bar\alpha) \) is equivalent to uniformly scaling the fracture toughness \( G_c \) throughout the formulation, i.e., \(G_c = f(\bar\alpha) G_{c(0)}\).

⚙️ Fatigue Degradation Function \( f(\bar\alpha) \)

\[ f(\bar\alpha) = \begin{cases} 1, & \bar\alpha \leq \alpha_{\text{critical}} \\ \left( \frac{2 \alpha_{\text{critical}}}{\bar\alpha + \alpha_{\text{critical}}} \right)^2, & \bar\alpha > \alpha_{\text{critical}} \end{cases} \]

where \( \alpha_{\text{critical}} = 62.5 \, \text{MPa} \). This function gradually reduces fracture toughness after the accumulated fatigue energy \( \bar\alpha \) exceeds the threshold, mimicking the progressive weakening observed in real materials.

⚙️ Fatigue Accumulation \( \bar\alpha \)

The fatigue history variable \( \bar\alpha \) is updated using the Incremental Cycle-based Linear Accumulation (ICLA) rule: \[ \bar\alpha_t = \bar\alpha_{t-1} + \Psi^f_t \times \Delta N_t \] where \( \Psi^f_t \) is the instantaneous fatigue energy and \( \Delta N_t = N_t - N_{t-1} \) is the increment in cycle count.

⚙️ Instantaneous Fatigue Energy \( \Psi^f_t \)

Three formulations are available to compute \( \Psi^f_t \):

1. Mean Load Approach (default):

\[ \Psi_t = 2E\epsilon_{\text{max}}^2 \left( \frac{1+R}{2} \right)^2 \left( \frac{1+R}{2} \right)^n \]

where \( \epsilon_{\text{max}} \) is the maximum principal strain. For this benchmark, \( R = 0.5 \), \( n = 0.5 \).

2. Elastic Energy Approach:

\[ \Psi_t = 0.5 \boldsymbol{\sigma} : \boldsymbol{\epsilon} \]

3. Spectral Activation Approach:

\[ \Psi_t = 0.5 \boldsymbol{\sigma}^+ : \boldsymbol{\epsilon} \]

where \( \boldsymbol{\sigma}^+ \) is the tensile-activated part of the stress tensor obtained via spectral decomposition.

🚀 Multi-Physics System and Boundary Conditions

📋 Benchmark Configuration

✅ This is a metallic material with \(E = 210\,\text{GPa}\), \(\nu = 0.3\). Therefore, the AT2 model is adopted, i.e., \(\alpha(d) = d^2\) and \(c_0 = 2\).

✅ The simulation includes 38,000 cycles to evaluate fatigue-induced degradation.

✅ Stress spectral decomposition is used to split the elastic energy into activated and inactivated components.

⚠️ Reminder: \(\bar\alpha\) denotes accumulated fatigue energy, while \(\alpha(d)\) is the crack geometric function.

📊 Simulation Results (Click figure to see animation)

Performance Comparison

✅ The macro-mechanical behaviors are approximately the same.
✅ The mean load approach offers greater flexibility by adjusting load ratio \(R\) and exponential coefficient \(n\).
✅ The reason why \(f(\bar\alpha)\) doesn’t spread to the left is because the pre-existing crack absorbs most of the deformation there, leading to minimal fatigue degradation.

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