2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Boundary element method, with quadratic elements [bib1].
The calculation of \({K}_{I}\) and \({K}_{\mathrm{II}}\) is carried out by a contour integral (integral M [bib2]) in which the stresses and displacements calculated in the part are involved, as well as the stresses and displacements deduced from asymptotic solutions defined analytically, in which \({K}_{I}\) and \({K}_{\mathrm{II}}\) are alternately zero.
For comparison, the calculation of \(K\) is also performed by the virtual extension method.
2.2. Benchmark results#
The results of the reference solution are shown in the table below, for the various values of the angle and for the two ends of the crack, with
.
Method |
Left side |
Right side |
|||||||
\(\theta =15°\) |
\(\theta =30°\) |
\(\theta =45°\) |
\(\theta =60°\) |
\(\theta =15°\) |
\(\theta =30°\) |
\(\theta =45°\) |
\(\theta =60°\) |
||
integral |
\({F}_{I}\) |
1.0115 |
0.7868 |
0.5211 |
0.2770 |
1,1266 |
0.9910 |
0.7646 |
0.4919 |
\(M\) |
\({F}_{\mathrm{II}}\) |
0.4434 |
0.6244 |
0.6723 |
0.5804 |
0.0862 |
0.2961 |
0.4056 |
0.4057 |
extension |
\({F}_{I}\) |
1,0110 |
0.7864 |
0.5210 |
0.2769 |
1,1260 |
0.9904 |
0.7643 |
0.4919 |
Virtual |
\({F}_{\mathrm{II}}\) |
0.4429 |
0.6240 |
0.6720 |
0.5801 |
0.0865 |
0.2960 |
0.4055 |
0.4056 |
The relationship between the global energy return rate \(G\) and \({K}_{j}\) is written as follows [bib3]:
\(G=\beta ({K}_{I}^{2}+{K}_{\mathrm{II}}^{2})\)
with:
\(\beta \mathrm{=}\frac{1}{16C{h}^{2}(\alpha \pi )}(\frac{1+{\kappa }_{1}}{{\mu }_{1}}+\frac{1+{\kappa }_{2}}{{\mu }_{2}})\) and \(\begin{array}{c}{\kappa }_{i}\mathrm{=}\frac{3\mathrm{-}{\nu }_{i}}{1+{\nu }_{i}}\\ {\mu }_{i}\mathrm{=}\frac{{E}_{i}}{2(1+{\nu }_{i})}\\ \alpha \mathrm{=}\frac{1}{2\pi }\mathrm{ln}\left[(\frac{{\kappa }_{1}}{{\mu }_{1}}+\frac{1}{{\mu }_{2}}){(\frac{{\kappa }_{2}}{{\mu }_{2}}+\frac{1}{{\mu }_{1}})}^{\mathrm{-}1}\right]\end{array}\)
2.3. Uncertainty about the solution#
Estimated at less than 0.1%. It should be noted that the difference between the contour integrals method and the virtual extension method is generally less than 0.05%.
2.4. Bibliographical references#
Stress intensity factor analysis of interface crack using boundary element method. Application of contour-integral method. N. MIYAZAKI, T. IKEDA, T. SODA, and T. MUNAKATA. Engng.Fract.Mechs., 45, no. 5, 599-610, 1993.
An analysis of interface cracks between dissimilar isotropic materials using conservation integrals in elasticity. J.F. YAU and T.C. CHANG. Engng.Fract.Mechs., 20, 423-432, 1984.
The strength of adhesive joints using the theory of cracks. B.M. MALYSHEV and R.L. SALGANIK. Int.J.Fract.Mech., 1, 114-128, 1965.