Tungsten-Twin: Defect-Aware Fatigue Simulator

Introduction

This mini-project is based on models for structural integrity and the lifetime of EB-PBF tungsten. It builds a digital twin: synthetic defect populations → Kitagawa/Murakami logic → short-crack Paris growth (MPa√m units) → Monte-Carlo life scatter. Outputs include a designer Kitagawa chart, S–N scatter with percentiles, and what-if studies (surface finishing, pore shape, clustering).

What to look for

Kitagawa designer chart: allowable Δσ vs. defect √area.

S–N (fixed Δσ): life scatter + median/5–95% markers.

What-ifs:

Surface finishing (remove top 100–300 μm) → life ↑, scatter ↓.

Elongated pores (low AR) → life ↓.

Clustering (size mixture) → more short-life outliers.

Sensitivity: ΔK_th, Paris (C, m), nominal Δσ.

Scope

Defect → fatigue modelling with thresholded short-crack life and Monte-Carlo scatter. No physical testing. FE-ready: you can later replace the SCF surrogate with FE hot-spots (Δσ) or ΔK/J fields.

Method (Lite path)

• Synthetic pores (size, aspect ratio, surface/subsurface, depth)
• Stress concentration proxy, Kitagawa–Takahashi limit
• Short-crack growth (Paris-like) from a0 to a_crit
• Monte Carlo across pore populations

Results

Designer chart (Kitagawa):

Baseline S–N with percentiles:

What-ifs (unit-corrected):

Surface finishing (remove top 100 µm)	Pore shape (round vs. elongated)	Clustering (isolated vs. size-mixture)
Removes near-surface pores → ↑ median life, ↓ scatter.	Elongated (low AR) pores raise Kt → ↓ life, wider scatter.	Clusters increase chance of a critical defect → more short-life outliers.

What-if comparisons (Δσ_nom fixed; Monte-Carlo across defect populations).

Key Methods & Equations

Kitagawa–Takahashi (allowable stress vs. defect size)

Let the equivalent crack length be $a \approx \frac{\text{area}}{\pi}$ $$\quad\text{with}\quad \text{area} = (\sqrt{\text{area}})^2 $$

For small defects (threshold control), $$\Delta\sigma_{\text{allow}} \approx\ \frac{\Delta K_{\text{th}}}{\sqrt{\pi,a}} ;$$

Transition to a long-crack/strength limit is smoothly blended to form the designer curve.

---

Initial flaw from pore (Murakami-style mapping)

Map a pore to an initial crack size using $a_0 \propto \sqrt{\text{area}}$ $$\quad\text{(larger proportionality for surface pores)}$$

---

Paris short-crack growth (MPa$\sqrt{\text{m}}$ units)

$$ \frac{da}{dN} =\C,(\Delta K)^m,\qquad \Delta K ;=; Y,\Delta\sigma_{\text{MPa}},\sqrt{\pi,a} ;$$

With a threshold check, $\Delta K < \Delta K_{\text{th}}$ $$\Rightarrow$$ $$N_f \to \infty $$

Integrate from (a_0) to (a_c) to obtain the life (N_f).

---

Monte-Carlo workflow

Randomise pore $$(\sqrt{\text{area}})$$, aspect ratio, surface proximity, and depth; compute $$(N_f)$$ per specimen to get scatter, medians, and percentile bands.

Future Scope

FE drop-in: Replace SCF with FEniCSx/CalculiX Δσ hot-spots or ΔK/J fields (pipeline already accepts them).

Data-driven pores: Swap synthetic pore stats with XCT-derived distributions.

References & Background

1. Murakami, Y. Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions. 2nd ed., Elsevier, 2019.
   Why cited: Canonical √area approach and surface vs. subsurface treatment—used to map pores to initial crack size (a_0).

2. Kitagawa, H.; Takahashi, S. “Applicability of Fracture Mechanics to Very Small Cracks or Cracks in the Early Stage.” In Proceedings of the 2nd International Conference on Mechanical Behavior of Materials, 1976.
   Why cited: Foundation of the Kitagawa–Takahashi diagram you use for allowable stress vs. defect size.

3. El Haddad, M. H.; Topper, T. H.; Smith, K. N. “Prediction of Nonpropagating Cracks.” Engineering Fracture Mechanics, 1980.
   Why cited: Classic short-crack/Kitagawa–El-Haddad bridging concept that underpins your threshold-to-long-crack blending.

4. Sanaei, N.; Fatemi, A. “Defects in Additive Manufactured Metals and Their Effect on Fatigue Performance: A State-of-the-Art Review.” Progress in Materials Science, 2021.
   Why cited: Comprehensive AM-fatigue review—supports your assumptions on defect types, surface finishing (100–300 µm removal), and scatter drivers.