\[ \sigma_{ij}(r, \theta) = \frac{\color{red}{K_I}}{\sqrt{2\pi r}} f_{ij}^{I}(\theta) + \frac{\color{red}{K_{II}}}{\sqrt{2\pi r}} f_{ij}^{II}(\theta) + \frac{\color{red}{K_{III}}}{\sqrt{2\pi r}} f_{ij}^{III}(\theta) + \mathcal{O}(1), \quad \color{gray}{\sigma_{ij}(r, \theta) \underset{r \to 0}{\to} \infty.} \]

Stress Intensity Factors (SIFs)

Denoted \(K_m\), they :

• represent the crack tip sollicitation,
• depend on geometry and load.

Crack propagation criteria

Propagation criteria rely on the SIFs \(K_m\), e.g.,

\(\quad \displaystyle \color{red}{K_I} \leq K_{Ic}\), or

\(\quad \displaystyle G = \frac{\color{red}{K_{I}}^2 + \color{red}{K_{II}}^2}{E'} + \frac{\color{red}{K_{III}}^2}{2 \mu} \leq G_c\).

How to determine the SIFs ?

Different methods are employed :

• analytical (mostly for simple case, or (semi-)infinite domains),
• numerical (e.g., FEM simulations),
• experimental (e.g., photo-elasticity, DIC.).

\(\to\) Known solutions have been compiled in handbooks!

Fracture mechanic design handbooks
(Tada et al., 2000)

Methodology in handbook

1. Dimensional analysis \[ K_I = \bar{\sigma} \sqrt{W} \color{red}{\hat{K}_I}(\color{green}{A}), \quad \bar{\sigma} = \frac{P}{W B}, \quad A = \frac{a}{W}. \]
2. Data gathering (not if analytical method)
   • Observations : \((\color{green}{A}, \color{red}{\hat{K}_I})\)
   • Experimental/Numerical methods
3. Fit polynomial solution to data \[ \small \color{red}{\hat{K}_I}(\color{green}{A}) = \frac{(2+A)(0.76 + 4.8A - 11.58A^4 + 11.43 A^3 - 4.08 A^4)}{(1-A)^{3/2}} \]

Limitations

Dense polynomials, limited to few parameters, etc.

Parameters and Observations

1. Dimensional analysis

\[ K_I = \sigma \sqrt{a} \color{red}{\hat{K}_I}\left(\color{green}{\frac{a}{b}}\right) , \quad \sigma = \frac{P}{b}. \]

2. Data gathering

• Observations (elastic simulations) : \(\left(\color{green}{\frac{a}{b}}, \color{red}{\hat{K}_I}\right)\)

3. Sparse regression

\[\color{green}{\mathbf{p}}= \left[ \color{green}{\frac{a}{b}}, \exp\left(\color{green}{a/b}\right), \frac{1}{\sqrt{1-\color{green}{a/b}}} \right]\]

• Hyperparameters :
  • Vary \(\beta\) \(\to\) find good accuracy-simplicity tradeoff
  • \(d=2\), increase until satisfied.

Code output

=== Model 1
Assessment :
    - Hyperparameter beta    : 3.83e+02
    - Number of coefficients : 1
    - R2 score               : 0.000
LaTeX expression:
\hat{K}_{I}(a/b) =
    +45.6

=== Model 2
Assessment :
    - Hyperparameter beta    : 1.02e+01
    - Number of coefficients : 2
    - R2 score               : 0.967
LaTeX expression:
\hat{K}_{I}(a/b) =
    -21.1
    +17.9 \frac{1}{\sqrt{1-a/b}}^2

=== Model 3
Assessment :
    - Hyperparameter beta    : 5.11e-01
    - Number of coefficients : 3
    - R2 score               : 0.997
LaTeX expression:
\hat{K}_{I}(a/b) =
    +5.43
    -14.4 \exp(a/b)^2
    +22.9 \frac{1}{\sqrt{1-a/b}}^2

Code output

=== Model 1
Assessment :
    - Hyperparameter beta    : 1.79e+03
    - Number of coefficients : 1
    - R2 score               : 0.000
LaTeX expression:
\hat{K}_{I}(a/b) =
    +45.6

=== Model 2
Assessment :
    - Hyperparameter beta    : 3.43e+00
    - Number of coefficients : 2
    - R2 score               : 0.999
LaTeX expression:
\hat{K}_{I}(a/b) =
    +3.75
    +4.11 \frac{1}{\sqrt{1-a/b}}^3

=== Model 3
Assessment :
    - Hyperparameter beta    : 6.13e-01
    - Number of coefficients : 3
    - R2 score               : 1.000
LaTeX expression:
\hat{K}_{I}(a/b) =
    +5.32
    -0.383 \exp(a/b)^3
    +4.19 \frac{1}{\sqrt{1-a/b}}^3

Experimental setup

By G. Eliyahu-Yakir and P. Reis (EPFL)

Illustration

7-parameter problem

• Geometry: \(\color{green}{W, L, a, L_c}\).
• 3D volumic loads:
  • centrifugal : \(\boldsymbol{f}_\Omega(r, \theta) \propto \color{green}{\Omega}\)
    • along \(\boldsymbol{e}_r\) and \(\boldsymbol{e}_\theta\),
  • Euler : \(\boldsymbol{f}_\dot{\Omega}(r, \theta) \propto \color{green}{\dot{\Omega}}\)
    • along \(\boldsymbol{e}_2 = \cos(\color{green}{\alpha}) \boldsymbol{e}_\theta - \sin(\color{green}{\alpha}) \boldsymbol{e}_z\).

Data generation

Elastic FEM simulations

Summary of the methodology

1. Physics : Dimensional analysis + Superposition
   • Isolate independent SIFs
   • Find the proper dimensionless parameters
2. Generation of the data
   • Sampling of the parametric space (LHS)
   • Elastic simulations
3. Machine learning : Sparse regression
   • Large polynomial + L1-regularization
   • Injection of non-linear terms as model input
4. Result : Sparse analytical approximation of the SIFs

Benefits in fracture mechanics

A problem of fracture can now be fully analytical facilitating

• uncertainty quantification,
• design (parametric optimization of structure),
• stability analysis,
• etc.