Benchmark solution
====


Elastic solution
----

In elasticity, for an **infinite plate,** comprising a hole of diameter :math:`a`, subjected to a loading :math:`P` according to :math:`y` to infinity, the analytical solution in plane stresses and polar coordinates :math:`(r,\theta )` is [:ref:`bib1 <bib1>`]:

:math:`{\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]`

:math:`{\sigma }_{\theta \theta }\mathrm{=}\frac{P}{2}\left[(1+{(\frac{a}{r})}^{2})+(1+3{(\frac{a}{r})}^{4})\mathrm{cos}2\theta \right]`

:math:`{\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 (:math:`r\mathrm{=}a`): :math:`{\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}=P[(1+2\mathrm{cos}2\mathrm{\theta })]`

And along the :math:`x` axis: :math:`{\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 :math:`P\mathrm{=}1\mathit{MPa}`, and for an **infinite** plate, we have:


.. csv-table::

    "Point", "Component", "Calculation", ":math:`\mathrm{MPa}`"
    ":math:`A` "," :math:`\mathit{SIXX}` "," :math:`{\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}(r=a,\mathrm{\theta }=\mathrm{\pi }/2)` ", "—1"
    ":math:`B` "," :math:`\mathit{SIYY}` "," :math:`{\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}(r=a,\mathrm{\theta }=0)` ", "3"


For a plate of **finite** dimension, the [:ref:`bib1 <bib1>`] charts make it possible to obtain the stress concentration coefficient, and we find that for a traction of :math:`1\mathrm{MPa}`, maximum :math:`\mathit{SIYY}` is equal to approximately :math:`3.03\mathit{MPa}` at point :math:`B`.


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. R.E. PETERSON Ed. J. WILEY p150