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:

(2.1)#\[{\sigma }_{\mathit{rr}}\mathrm{=}\frac{P}{2}\mathrm{.}\left[(1\mathrm{-}{(\frac{a}{r})}^{2})\mathrm{-}(1\mathrm{-}4\mathrm{.}{(\frac{a}{r})}^{2}+3\mathrm{.}{(\frac{a}{r})}^{4})\mathrm{.}\mathrm{cos}(2\theta )\right]\]
(2.2)#\[{\sigma }_{\theta \theta }\mathrm{=}\frac{P}{2}\mathrm{.}\left[(1+{(\frac{a}{r})}^{2})+(1+3\mathrm{.}{(\frac{a}{r})}^{4})\mathrm{.}\mathrm{cos}(2\theta )\right]\]
(2.3)#\[{\sigma }_{r\theta }\mathrm{=}\frac{P}{2}\mathrm{.}\left[(1+2\mathrm{.}{(\frac{a}{r})}^{2}\mathrm{-}3\mathrm{.}{(\frac{a}{r})}^{4})\mathrm{.}\mathrm{sin}(2\theta )\right]\]

In particular, at the edge of the hole (\(r=a\)), we have:

(2.4)#\[{\sigma }_{\theta \theta }\mathrm{=}p\mathrm{.}\left[1+2\mathrm{.}\mathrm{cos}(2\theta )\right]\]

And along axis \(x\):

(2.5)#\[{\sigma }_{\theta \theta }={\sigma }_{\mathrm{yy}}=\frac{P}{2}\mathrm{.}\left[(1+{(\frac{a}{r})}^{2})+(1+3\mathrm{.}{(\frac{a}{r})}^{4})\right]\]

Numerically, for \(P=1\mathit{MPa}\), and for an infinite plate, we have

Point

Component

Calculation

\(\mathrm{MPa}\)

\(A\)

\(\mathit{SIXX}\)

\({\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}(r=a,\mathrm{\theta }=\mathrm{\pi }/2)\)

—1

\(B\)

\(\mathit{SIYY}\)

\({\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}(r=a,\mathrm{\theta }=0)\)

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 \(\mathrm{SIGYY}\) is equal to approximately \(3.03\mathit{MPa}\) at point \(B\).

2.2. Elastoplastic solution (limit load)#

In elastoplasticity, by a static approach using plane stresses, it is possible to obtain an upper bound of the limit load for a band of finite width \(\mathrm{2L}\) and of infinite length, including a hole of width \(\mathrm{2a}\) and subject to a constraint imposed at infinity \(p\):

(2.6)#\[{p}_{\text{lim}}^{\text{-}}\mathrm{=}\frac{{\sigma }_{y}\mathrm{.}(L\mathrm{-}a)}{L}\]

Here we get the upper bound of the limit load: \({p}_{\text{lim}}^{\text{-}}\mathrm{=}0.9\mathrm{\times }270\mathrm{=}243\mathit{MPa}\). (We take \({\sigma }_{y}=270\mathrm{MPa}\) here, because the load limit is the same between a perfect elastoplastic material and a material whose tensile curve has a horizontal asymptote at \(270\mathrm{MPa}\)). In this test (in particular modeling B), we would like to find, through an elastoplastic calculation, an approximation of this limit load, knowing that analytical methods make it possible to know an upper bound. So we will take the value \({p}_{\text{lim}}^{\text{-}}\) as a reference.

2.3. Bibliographical references#

  1. Boundary analysis of cracked structures and strength criteria. F. VOLDOIRE: Note EDF/DER /HI/74/95/26 1995

  2. « Stress concentration factors », Peterson R.E., Wiley, 1974.