2. Reference solution

2.1. Calculation method used for the reference solution

In plane deformations or in plane stresses, the distribution of displacements is given in this coordinate system \((\mathrm{0,}{x}_{\mathrm{1,}}{x}_{2})\) by:

\(\mathrm{\{}\begin{array}{c}{u}_{1}\mathrm{=}\frac{1+\nu }{E}\sqrt{\frac{r}{2\pi }}({K}_{I}\mathrm{cos}\frac{\theta }{2}(k\mathrm{-}\mathrm{cos}\theta )+{K}_{\mathit{II}}\mathrm{sin}(\frac{\theta }{2})(k\mathrm{-}\mathrm{cos}\theta +2))\\ {u}_{2}\mathrm{=}\frac{1+\nu }{E}\sqrt{\frac{r}{2\pi }}({K}_{I}\mathrm{sin}\frac{\theta }{2}(k\mathrm{-}\mathrm{cos}\theta )\mathrm{-}{K}_{\mathit{II}}\mathrm{cos}(\frac{\theta }{2})(k+\mathrm{cos}\theta \mathrm{-}2))\end{array}\)

with \(k=3-4\nu\) in plane deformations

\(k=\frac{3-\nu }{1+\nu }\) in plane constraints

or in the \((O,X,Y)\) frame by: \(\{\begin{array}{}{u}_{x}=\mathrm{cos}\alpha {u}_{1}-\mathrm{sin}\alpha {u}_{2}\\ {u}_{y}=\mathrm{sin}\alpha {u}_{1}+\mathrm{cos}\alpha {u}_{2}\end{array}\)

On the outline of the plate, we have: \(r=\mathrm{OA}=100\mathrm{mm}\).

We choose to take \({K}_{I}=2.\) and \({K}_{\mathrm{II}}=1.\) and impose the movements on the outline of the circular plate.

2.2. Benchmark results

\({K}_{I}=2.\)
\({K}_{\mathrm{II}}=1.\)
\(G=2.275{10}^{-5}\)	in plane deformations
\(G=2.5{10}^{-5}\)	in plane constraints

2.3. Bibliographical references

1. H.D. BUI Mechanics of Fragile Fracture - Ed. Masson 1978