PSI - Issue 75

4

Author name / Structural Integrity Procedia (2025)

Laurent Gornet et al. / Procedia Structural Integrity 75 (2025) 129–139

132

4. Wöhler (S-N) curve : analytical and modeling Neural Network modellings Python is widely used in scientific computing and machine learning due to its clear syntax and the richness of its ecosystem. Core libraries such as “ NumPy ” (for numerical computations), “ SciPy ” (for optimization routines), “ Matplotlib ” (for data visualization), and “ Pandas ” (for data handling) provide a solid foundation for analyzing experimental fatigue data. In the context of modeling “ Wöhler curves (S-N curves) ” , which describe the relationship between stress amplitude and fatigue life in materials, these libraries are crucial for preparing and exploring the data. Building on this ecosystem, “ TensorFlow ” an open-source deep learning framework developed by Google enables the construction of neural network models that can learn complex, nonlinear relationships directly from experimental observations. “ Keras ” , the high-level API within TensorFlow, allows us to define, train, and evaluate neural networks with ease. For example, one can develop models that approximate S-N curves with physical constraints such as asymptotes or slope continuity, using physics-informed neural networks (PINNs) or hybrid approaches. These tools are particularly well suited for fatigue modelling, as they allow combining prior knowledge from material mechanics with data-driven learning, resulting in predictive models that are both accurate and interpretable. 4.1. Wöhler (S-N) curve: analytical modelling Numerous models have been proposed in the literature to describe Wöhler (S-N) curves for material. For the TR50/R367-2 carbon fiber Epoxy matrix laminated composite material, the existence of a fatigue limit is assumed. Consequently, the selected model must account for an asymptotic behavior, typically characterized by the parameter ₀ . This approach is used for woven carbon thermoplastic materials by Muller et al. 2021. This requirement excludes classical Wöhler and Basquin models, which do not incorporate such a feature. The focus is placed on reconstructing the S-N fatigue curves of carbon/epoxy laminates using advanced neural network-based approaches and a Strohmeyer’s model (1) :

1

b N

( ) log b A

( )

(

) 0

log

log N a b S S = + − With

a

=

S S

= +

(1)

0

A

We propose a modified version of the Strohmeyer’s model (Equation 2, Gornet- fatigue’s model). The equation models the stress-life (S-N) curve used in fatigue analysis, describing how the stress S decreases as the number of cycles N increases, approaching a fatigue limit.

Sat S S c N − +

S

S

=

+

0

(2)

(

)

0

1

b d

Where : S is the stress amplitude at N cycles. Sat S is the saturation stress, i.e., the stress level at very low cycles (almost static strength). 0 S is the fatigue limit, the asymptotic minimum stress the material can withstand indefinitely. c, b, d parameters are positive parameters controlling the shape and curvature of the curve. c controls the rate at which the stress decreases with cycles. b controls the nonlinearity with respect to N . Parameter d adds additional flexibility to the curve’s shape, allowing a smoother or sharper transition . The S-N Fatigue tests identification for the [+45/-45/90/0] S laminate is presented on figure 2. The Python code is presented in appendix A.