2. Benchmark solution

2.1. Calculation method

2.1.1. A, C, E, G modeling

The stress intensity factor \({K}_{I}\) is tested in relation to the analytical value without contact.

For a specimen in bending and a right crack with an abscissa \([-a\phantom{\rule{2em}{0ex}},\phantom{\rule{2em}{0ex}}a]\), the analytical value of \({K}_{I}\) without closure of the crack is determined by the Bui formula [7]:

\({K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}\underset{\mathrm{-}a}{\overset{a}{\mathrm{\int }}}\mathrm{-}t\sqrt{\frac{a\mathrm{-}t}{t+a}}\mathit{dt}\)

where \(a\) refers to the half-length of the crack.

From the integral above, we calculate the following simplified expression for \({K}_{I}\):

\({K}_{I}\mathrm{=}\frac{{a}^{\frac{3}{2}}\sqrt{\pi }}{2}\)

Indeed, in the integral, several changes of variables are made:

\(t\to \mathit{at}\) we have \({K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{a}^{2}\underset{\mathrm{-}1}{\overset{1}{\mathrm{\int }}}\mathrm{-}t\sqrt{\frac{1\mathrm{-}t}{t+1}}\mathit{dt}\)

\(t\to t\mathrm{-}1\) we have \({K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{a}^{2}\underset{0}{\overset{2}{\mathrm{\int }}}\mathrm{-}(t\mathrm{-}1)\sqrt{\frac{2\mathrm{-}t}{t}}\mathit{dt}\)

\(t\to \mathrm{2t}\) we have \({K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{\mathrm{2a}}^{2}\underset{0}{\overset{1}{\mathrm{\int }}}\mathrm{-}(\mathrm{2t}\mathrm{-}1)\sqrt{\frac{1\mathrm{-}t}{t}}\mathit{dt}\)

\(t\to {t}^{2}\) we have \({K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{\mathrm{2a}}^{2}\underset{0}{\overset{1}{\mathrm{\int }}}\mathrm{-}2({\mathrm{2t}}^{2}\mathrm{-}1)\sqrt{1\mathrm{-}{t}^{2}}\mathit{dt}\)

\(t\to \mathrm{sin}t\) we have \({K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{\mathrm{2a}}^{2}\underset{0}{\overset{\frac{\pi }{2}}{\mathrm{\int }}}\mathrm{2cos}\mathrm{2t}{\mathrm{cos}}^{2}t\mathit{dt}\)

After product development by \(\mathrm{cos}\):

\({K}_{I}\mathrm{=}\frac{1}{\sqrt{\pi a}}{\mathrm{2a}}^{2}\underset{0}{\overset{\frac{\pi }{2}}{\mathrm{\int }}}(\frac{1}{2}+\mathrm{cos}\mathrm{2t}+\frac{1}{2}\mathrm{cos}\mathrm{4t})\mathit{dt}\)

Hence the final expression (after integration):

\({K}_{I}\mathrm{=}\frac{{a}^{\frac{3}{2}}\sqrt{\pi }}{2}\)

The half-length being unitary (\(a=1\)), we therefore test the stress intensity factor:

\({K}_{I}=\frac{\sqrt{\mathrm{\pi }}}{2}\approx \mathrm{0,88629}\)

By construction, there is no mode II. So we test: \({K}_{\mathit{II}}=0\)

On the other hand, the rate of return of analytical energy is equal to:

\(G=\frac{1-{\mathrm{\nu }}^{2}}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\approx \mathrm{7,853982}E-7\)

2.1.2. B, D, F modeling

For a specimen in bending and a right crack with abscissa \([-a\phantom{\rule{2em}{0ex}},\phantom{\rule{2em}{0ex}}a]\), the analytical value of \({K}_{I}\) with crack closure is determined by the Bui formula [7]:

\({K}_{I}=\sqrt{\frac{2}{\pi (a-c)}}\underset{c}{\overset{a}{\int }}t\sqrt{\frac{t-c}{a-t}}\mathit{dt}\)

where \(c\) is the abscissa of the point where the crack starts to open (see [])

and \(a\) refers to the half-length of the crack.

We show analytically [7] that the abscissa \(c\mathrm{=}\mathrm{-}\frac{a}{3}\)

So, \({K}_{I}=\sqrt{\frac{3}{2\pi a}}\underset{-a/3}{\overset{a}{\int }}t\sqrt{\frac{t+a/3}{a-t}}\mathit{dt}\)

From the integral above, we calculate the following simplified expression for \({K}_{I}\):

\({K}_{I}={(\frac{2}{3}a)}^{\frac{3}{2}}\sqrt{(\pi )}\)

Indeed, in the integral, several changes of variables are made:

\(t\to \mathit{at}\) we have \({K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\underset{-1/3}{\overset{1}{\int }}t\sqrt{\frac{t+1/3}{1-t}}\mathit{dt}\)

\(t\to t-\frac{1}{3}\) we have \({K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\underset{0}{\overset{4/3}{\int }}(t-\frac{1}{3})\sqrt{\frac{t}{4/3-t}}\mathit{dt}\)

\(t\to \frac{4}{3}t\) we have \({K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\frac{4}{9}\underset{0}{\overset{1}{\int }}(\mathrm{4t}-1)\sqrt{\frac{t}{1-t}}\mathit{dt}\)

\(t\to {t}^{2}\) we have \({K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\frac{4}{9}\underset{0}{\overset{1}{\int }}({\mathrm{4t}}^{2}-1)\frac{{\mathrm{2t}}^{2}}{\sqrt{1-{t}^{2}}}\mathit{dt}\)

\(t\to \mathrm{sin}t\) we have \({K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\frac{4}{9}\underset{0}{\overset{\pi /2}{\int }}({\mathrm{4sin}}^{2}t-1)({\mathrm{2sin}}^{2}t)\mathit{dt}\)

After developing the powers of \(\mathrm{sin}t\):

\({K}_{I}=\sqrt{\frac{3}{2\pi a}}{a}^{2}\frac{4}{9}\underset{0}{\overset{\pi /2}{\int }}(2-\mathrm{3cos}\mathrm{2t}+\mathrm{cos}\mathrm{4t})\mathit{dt}\)

Hence the final expression (after integration):

\({K}_{I}\mathrm{=}{(\frac{2}{3}a)}^{\frac{3}{2}}\sqrt{(\pi )}\)

The half-length being unitary (\(a=1\)), we therefore test the stress intensity factor:

\({K}_{I}={(\frac{2}{3})}^{\frac{3}{2}}\sqrt{(\mathrm{\pi })}\approx \mathrm{0,9648017}\)

By construction, there is no mode II. So we test: \({K}_{\mathit{II}}=0\)

On the other hand, the rate of return of analytical energy is equal to:

\(G=\frac{1-{\mathrm{\nu }}^{2}}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\approx \mathrm{9,308423}E-7\)

2.2. Reference quantities and results

We’re testing \({K}_{I}\), \({K}_{\mathit{II}}\), and \(G\).

2.2.1. A, C, E, F, G models

\({K}_{I}=\frac{\sqrt{\mathrm{\pi }}}{2}\approx \mathrm{0,88629}\)

\({K}_{\mathit{II}}\mathrm{=}0\)

\(G\mathrm{=}\frac{1\mathrm{-}{\nu }^{2}}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\mathrm{\approx }\mathrm{7,853982}E\mathrm{-}7\)

2.2.2. B and D modeling

\({K}_{I}={(\frac{2}{3})}^{\frac{3}{2}}\sqrt{(\mathrm{\pi })}\approx \mathrm{0,9648017}\)

\({K}_{\mathit{II}}\mathrm{=}0\)

\(G=\frac{1-{\mathrm{\nu }}^{2}}{E}({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\approx \mathrm{9,308423}E-7\)

2.3. Uncertainty about the solution

Weak, semi-analytical solution.

2.4. Bibliographical references

1. GENIAUT S., MASSIN P.: eXtended Finite Element Method, Code_Aster Reference Manual, [R7.02.12]

2. GENIAUT S.: XFEM approach for contact cracking of industrial structures, PhD thesis GeM Laboratory, 2006.

3. H.D. Bui: Mechanics of brittle fracture, Ed. Masson, 1978.

4. THRESHER R.W., SMITH F.W.: The Partially Closed Griffith Crack, International Journal of Fracture, vol. 9, no. 1, pp. 33-41, 1973

5. Rice, J.R. (1968), « A path independent integral and the approximate analysis of strain concentration by notches and cracks, » Journal of Applied Mechanics 35: 379—386