2. Benchmark solution

2.1. Benchmark results

2.1.1. Calculation of stress intensity factors

The reference solution for an opening crack of depth a in a plate of thickness \(W\) stressed in pure \(I\) mode (force \(\sigma\) along the axis \(y\)) is as follows:

\({K}_{I}\mathrm{=}Y\sigma \sqrt{a}\), with \(Y\mathrm{=}1.99\mathrm{-}0.41\frac{a}{W}+18.7{(\frac{a}{W})}^{2}\mathrm{-}38.48{(\frac{a}{W})}^{3}+53.85{(\frac{a}{W})}^{4}\)

Numerical application: for a crack \(15\mathit{mm}\) deep (\(W\mathrm{=}72\mathit{mm}\)) and the load \({f}^{0}\), we get \({K}_{I}\mathrm{=}\mathrm{2,874}\mathit{MPa.}\sqrt{\mathit{mm}}\).

2.1.2. Calculation of the stresses around the crack bottom

The analytical solution for the stress on a circle with radius \(r\) around the crack bottom, obtained for a crack in an infinite environment, is as follows:

\({\sigma }_{\theta \theta }\mathrm{=}\frac{{K}_{I}}{4\sqrt{2\pi r}}(3\mathrm{cos}\frac{\theta }{2}+\mathrm{cos}\frac{3\theta }{2})\)

Digital application: for loading \({f}^{0}\) and for radius \(r\mathrm{=}{R}_{\mathit{AMORC}}\mathrm{=}\mathrm{0,046}\mathit{mm}\), we get:

\(\theta \mathrm{=}0\): \({\sigma }_{\theta \theta }\mathrm{=}53.5\mathit{MPa}\); \(\theta \mathrm{=}\pi \mathrm{/}2\): \({\sigma }_{\theta \theta }\mathrm{=}18.9\mathit{MPa}\)

2.1.3. Calculation of the priming factor

The priming factor associated with the geometric singularity in \(P\) is calculated analytically using the procedure prescribed in RCC -M [3]. All the moments provided are considered to be local extremes and are therefore retained in the calculation.

We note \({s}^{0}\) the value of \({\sigma }_{\theta \theta }\) taken at a radius \(r\) from the bottom of the crack, at an angle \(\theta\) given for loading \({f}^{0}\). We can then build the table of the amplitudes of variation of \({\sigma }_{\theta \theta }\) for all the possible combinations:

Transitional 1		Transitional 2			math: 10s° |:math: 110s° |:math: 50s°
\({\mathrm{Nb}}_{\mathrm{occur}}=i\)		\({\mathrm{Nb}}_{\mathrm{occur}}=j\)
\(t=0\)	\(t=1\)	\(t=0\)	\(t=1\)	\(t=2\)
Transitional 1	\({\mathit{Nb}}_{\mathit{occur}}\mathrm{=}i\)	\(t=0\)		\(100s°\)
		\(t=1\)			\(90s°\)	\(10s°\)	\(50s°\)
Transitional 2	\({\mathrm{Nb}}_{\mathrm{occur}}=j\)	\(t=0\)				\(100s°\)	\(40s°\)
		\(t=1\)					\(60s°\)
		\(t=2\)

Taking into account the load ratio \(R\) of each combination makes it possible to calculate the amplitude of variation of the effective stresses \(\Delta {\sigma }_{\mathit{eff}}\):

Transitional 1		Transitional 2			math: 10s° |:math: 110s° |:math: 50s°
\({\mathrm{Nb}}_{\mathrm{occur}}=i\)		\({\mathrm{Nb}}_{\mathrm{occur}}=j\)
\(t=0\)	\(t=1\)	\(t=0\)	\(t=1\)	\(t=2\)
Transitional 1	\({\mathrm{Nb}}_{\mathrm{occur}}=i\)	\(t=0\)		\(100s°\)
		\(t=1\)			\(\mathrm{95,6}s°\)	\(\mathrm{22,6}s°\)	\(\mathrm{72,1}s°\)
Transitional 2	\({\mathrm{Nb}}_{\mathrm{occur}}=j\)	\(t=0\)				\(\mathrm{105,9}s°\)	\(\mathrm{45,6}s°\)
		\(t=1\)					\(\mathrm{83,6}s°\)
		\(t=2\)

The calculation of the priming factor \(\mathrm{FA}\) is carried out according to an iterative process:

1. identification of the transient number of non-zero occurrences leading to the maximum of

   

;

2. calculation of the associated elementary priming factor, using the law of fatigue;

3. updating the number of occurrences of the processed combination.

Two cases are studied in succession with the numbers of occurrences of the following two transients: \(i=j=1\) and \(i=\mathrm{1 };j=2\). We denote \({N}_{\mathrm{kl}-\mathrm{mn}}\) the number of cycles admissible for the combination of the \(l\) -th time step of the -th time step, the \(k\) -th transitory and the \(n\) -th time step of the \(m\) -th transient (calculated from

of this combination and of the law of fatigue).

1st case:

2nd case:

Numerical application: for modeling A of this test case, we assume that \(s°=1\). So we have:

\({\mathrm{FA}}_{1}=\mathrm{2,005}{.10}^{-\mathrm{10 }}\); F \({A}_{2}=\mathrm{2,710}{.10}^{-10}\)

For B modeling, the value of \(s°\) depends on the angle

, and can be calculated using the formula in paragraph 2.1.2.

:

, \({\mathrm{FA}}_{1}=\mathrm{0,607}\); \({\mathrm{FA}}_{2}=\mathrm{0,819}\)

:

, \({\mathrm{FA}}_{1}=\mathrm{2,01}{.10}^{-3}\); \({\mathrm{FA}}_{2}=\mathrm{2,72}{.10}^{-3}\)

2.2. Bibliographical references

1. W.F. BROWN, J.E. STRAWLEY: « Plane Strain Crack Toughness Testing of High Strength Material », American Society of Testing and Materials, STP 410

2. J.B. LEBLOND: « Mechanics of fragile and ductile fracture », Lavoisier, Paris, 2003.

3. RCC -M: « Mechanics of fragile and ductile fracture », Lavoisier, Paris, 2003.