2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The reference solution is an analytical solution from SIH [bib1] and [bib2].

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Note that the angle \(\alpha\) here designates the parametric angle of the point \(M\) (angle with respect to the axis \(\mathit{Ox}\) projected of \(M\) onto the circle of radius \(b\)) and not the polar coordinate of this point.

Here: \(a\mathrm{=}25\mathit{mm}\) and \(b\mathrm{=}6\mathit{mm}\), so \(k\mathrm{=}\mathrm{0,9707728}\)

The values of the elliptical integral \(E(k)\) are tabulated in [bib3], according to \(\mathit{asin}(k)\) which is equal here to \(\mathrm{76,11}°\). We then find: \(E(k)\mathrm{=}\mathrm{1,0672}\).

Hence the stress intensity factor in \(\mathit{MPa.}\sqrt{}\mathit{mm}\): \({K}_{I}(\mathrm{\alpha })=4.0680{\left[{\mathrm{sin}}^{2}\mathrm{\alpha }+\frac{{b}^{2}}{{a}^{2}}{\mathrm{cos}}^{2}\mathrm{\alpha }\right]}^{1/4}\)

Then, using Irwin’s formula (plane deformation):

\(g(\mathrm{\alpha })=\frac{1-{\mathrm{\nu }}^{2}}{E}{K}_{I}{(\mathrm{\alpha })}^{2}\)

2.2. Bibliography#

  1. G.C. SIH: Mathematical Theories of Brittle Fracture - FRACTURE, VOLII‑academic Press - 1968

  2. M.K. KASSIN and G.C. SIH: Three-dimensional stress distribution around an elliptical crack under arbitrary loadings J. Appl. Mech., 88, 601-611, 1966.

    1. TADA, P. PARIS, G. IRWIN: The Stress Analysis of Cracks Handbook - Third Edition - ASM International - 2000