2. Benchmark solution#
2.1. Elastic solution#
In elasticity, for an infinite plate, comprising a hole of diameter \(a\), subjected to a loading \(P\) according to \(y\) to infinity, the analytical solution in plane stresses and polar coordinates \((r,\theta )\) is [bib1]:
\({\sigma }_{\mathit{rr}}\mathrm{=}\frac{P}{2}\left[(1\mathrm{-}{(\frac{a}{r})}^{2})\mathrm{-}(1\mathrm{-}4{(\frac{a}{r})}^{2}+3{(\frac{a}{r})}^{4})\mathrm{cos}2\theta \right]\)
\({\sigma }_{\theta \theta }\mathrm{=}\frac{P}{2}\left[(1+{(\frac{a}{r})}^{2})+(1+3{(\frac{a}{r})}^{4})\mathrm{cos}2\theta \right]\)
\({\sigma }_{r\theta }\mathrm{=}\frac{P}{2}\left[(1+2{(\frac{a}{r})}^{2}\mathrm{-}3{(\frac{a}{r})}^{4})\mathrm{sin}2\theta \right]\)
In particular, at the edge of the hole (\(r\mathrm{=}a\)): \({\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}=P[(1+2\mathrm{cos}2\mathrm{\theta })]\)
And along the \(x\) axis: \({\sigma }_{\theta \theta }\mathrm{=}{\sigma }_{\mathit{yy}}\mathrm{=}\frac{P}{2}\left[(1+{(\frac{a}{r})}^{2})+(1+3{(\frac{a}{r})}^{4})\right]\)
Numerically, for \(P\mathrm{=}1\mathit{MPa}\), and for an infinite plate, we have:
Point |
Component |
Calculation |
\(\mathrm{MPa}\) |
\(A\) |
|
|
—1 |
\(B\) |
|
|
3 |
For a plate of finite dimension, the [bib1] charts make it possible to obtain the stress concentration coefficient, and we find that for a traction of \(1\mathrm{MPa}\), maximum \(\mathit{SIYY}\) is equal to approximately \(3.03\mathit{MPa}\) at point \(B\).
2.2. Bibliographical references#
Boundary analysis of cracked structures and strength criteria. F. VOLDOIRE: Note EDF/DER /HI/74/95/26 1995
Stress concentration factors. R.E. PETERSON Ed. J. WILEY p150