Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

Analytical solution


* calculation of stresses and deformations. For a loading under simple tension, a uniaxial stress state is obtained that is homogeneous at all points:

:math:`\sigma =\left[\begin{array}{ccc}\sigma & 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]` and :math:`\varepsilon =\left[\begin{array}{ccc}\varepsilon & 0& 0\\ 0& \gamma & 0\\ 0& 0& \gamma \end{array}\right]`

So the equivalent quantities are :math:`\{\begin{array}{}{\sigma }_{\mathrm{VMIS}}=\mid \sigma \mid ={\sigma }_{\mathrm{TRESCA}}\\ {\sigma }_{\mathrm{VMIS}-\mathrm{SG}}=\sigma \end{array}`

and :math:`\{\begin{array}{}{\varepsilon }_{\mathrm{INVA}-2}=\frac{2}{3}\mid \varepsilon -\gamma \mid \\ {\varepsilon }_{\mathrm{INVA}-\mathrm{2SG}}=\frac{2}{3}\mid \varepsilon -\gamma \mid \ast \mathrm{sign}\left[\frac{\varepsilon +2\gamma }{3}\right]\end{array}`


* then manual calculation of the cycles using the RAINFLOW method, as well as the loading amplitudes (:math:`\frac{\Delta \sigma }{2}` or :math:`\frac{\Delta \varepsilon }{2}`).


.. csv-table::

    "cycles", ":math:`\Delta \sigma /2` "," :math:`\Delta {\varepsilon }_{\mathrm{INVA}-2}/2`"
    "1
    2
    3
    4", 1".
    0.5
    1.
    3.5", "0.8667
    0.433315
    0.8667
    3.03335"



* finally transfer these values to the Wöhler or Manson-Coffin curves to estimate the unit damage at each cycle :math:`i`, i.e. :math:`{\mathrm{Du}}_{i}=\frac{1}{{N}_{i}}` (:math:`{N}_{i}` being the number of cycles to break for a given amplitude), as well as the cumulative damage :math:`D=\underset{i}{\Sigma }D{u}_{i}` (linear accumulation rule of MINER).

**Note:**

*We will use as equivalent stress* :math:`{\sigma }_{\mathrm{VMIS}-\mathrm{SG}}` *and as equivalent deformation* :math:`{\varepsilon }_{\mathit{INVA}\mathrm{-}\mathrm{2SG}}\mathrm{=}\frac{2}{3}\mathrm{\mid }\varepsilon \mathrm{-}\gamma \mathrm{\mid }\mathrm{\times }\mathit{sign}\left[\frac{\varepsilon +2\gamma }{3}\right]` *.*


Benchmark results
----------------------


* Given the values of the load parameters used, we simply obtain at the end of loading (increment 8) :math:`\sigma =-3.` :math:`\varepsilon =-3.` :math:`\gamma =0.9` :math:`{\varepsilon }_{\mathit{INVA}\mathrm{-}2}\mathrm{=}2.6`.


* For the calculation of the damage, we obtain:

:math:`{D}_{\mathit{Wöhler}}\mathrm{=}\mathrm{4,8133}\mathrm{.}{10}^{\mathrm{-}3}\mathrm{=}\underset{i\mathrm{=}1}{\overset{4}{\Sigma }}D{u}_{i}`

:math:`{D}_{\mathit{Manson}}\mathrm{=}\mathrm{4,67}\mathrm{.}{10}^{\mathrm{-}3}\mathrm{=}\underset{i\mathrm{=}1}{\overset{4}{\Sigma }}D{u}_{i}`

Bibliographical references
---------------------------


1. DOWNING and SOCIE, 1982. "Simple Rainflow Counting Algorithms." Int. J. Fatigue, January 1982 (p. 31).