Reference problem
=====================


Modeling A
--------------

The analysis consists in determining the criterion of CROSSLAND and the criterion of DANG VAN - PAPADOPOULOS at a point in a structure subjected to periodic radial multiaxial loading.

**Criterion of CROSSLAND:**

:math:`{\tau }_{a}+a{P}_{\mathit{max}}\mathrm{-}b\mathrm{\le }0` where:

:math:`\begin{array}{cc}{\tau }_{a}\mathrm{=}\frac{1}{2}\underset{0\mathrm{\le }{t}_{0}\mathrm{\le }T}{\mathit{Max}}\underset{0\mathrm{\le }{t}_{1}\mathrm{\le }T}{\mathit{Max}}∥\tilde{s}({t}_{1})\mathrm{-}\tilde{s}({t}_{0})∥\mathrm{=}& \frac{1}{2}\underset{0\mathrm{\le }{t}_{0}\mathrm{\le }T}{\mathit{Max}}\underset{0\mathrm{\le }{t}_{1}\mathrm{\le }T}{\mathit{Max}}\sqrt{\frac{1}{2}({\tilde{s}}_{11}^{2}+{\tilde{s}}_{22}^{2}+{\tilde{s}}_{33}^{2}+2{\tilde{s}}_{12}^{2}+2{\tilde{s}}_{13}^{2}+2{\tilde{s}}_{23}^{2})}\\ \text{amplitude de scission}& \text{avec}\tilde{s}\text{déviateur du tenseur des contraintes}\sigma \end{array}`

:math:`{P}_{\mathit{max}}\mathrm{=}\underset{0\mathrm{\le }t\mathrm{\le }T}{\mathit{Max}}(\frac{1}{3}\mathit{trace}(\sigma ))\mathrm{=}` maximum hydrostatic pressure

:math:`a\mathrm{=}({\tau }_{0}\mathrm{-}\frac{{d}_{0}}{\sqrt{3}})\mathrm{/}\frac{{d}_{0}}{3}` and :math:`b\mathrm{=}{\tau }_{0}`

with :math:`{\tau }_{0}` = endurance limit in pure alternating shear

:math:`{d}_{0}` = endurance limit in pure alternating traction-compression

The criterion is: :math:`\mathrm{Rcrit}={\tau }_{a}+a{P}_{\mathrm{max}}-b`

If :math:`\mathrm{Rcrit}` is negative or zero, there is no damage. If :math:`\mathrm{Rcrit}` is positive, there is likely to be damage.

**Criteria of DANG VAN - PAPADOPOULOS:**

:math:`{K}^{\text{*}}+a{P}_{\mathit{max}}\mathrm{-}b\mathrm{\le }0`

where :math:`{K}^{\text{*}}\mathrm{=}R\mathrm{/}\sqrt{2}` where :math:`R` radius of the smallest sphere circumscribed to the loading path in the space of the stress deviators :math:`\tilde{s}`.

:math:`R\mathrm{=}\underset{0\mathrm{\le }t\mathrm{\le }T}{\mathit{Max}}\sqrt{(\tilde{s}(t)\mathrm{-}{C}^{\text{*}})\mathrm{:}(\tilde{s}(t)\mathrm{-}{C}^{\text{*}})}` where :math:`{C}^{\text{*}}` is the center of the hypersphere

:math:`{C}^{\text{*}}\mathrm{=}\underset{C\epsilon K}{\mathit{Min}}\underset{0\mathrm{\le }t\mathrm{\le }T}{\mathit{Max}}\sqrt{(\tilde{s}(t)\mathrm{-}C)\mathrm{:}(\tilde{s}(t)\mathrm{-}C)}`

:math:`{P}_{\mathit{max}}\mathrm{=}\underset{0\mathrm{\le }t\mathrm{\le }T}{\mathit{Max}}(\frac{1}{3}\mathit{trace}(\sigma ))\mathrm{=}` maximum hydrostatic pressure

:math:`a\mathrm{=}({\tau }_{0}\mathrm{-}\frac{{d}_{0}}{\sqrt{3}})\mathrm{/}(\frac{{d}_{0}}{3})` and :math:`b\mathrm{=}{\tau }_{0}`

with :math:`{\tau }_{0}` = endurance limit in pure alternating shear

:math:`{d}_{0}` = endurance limit in pure alternating traction-compression

The criterion is: :math:`{R}_{\mathit{crit}}\mathrm{=}{K}^{\text{*}}+a{P}_{\mathit{max}}\mathrm{-}b`

If :math:`\mathit{Rcrit}` is negative or zero, there is no damage. If :math:`\mathrm{Rcrit}` is positive, there is likely to be damage.


Material properties
~~~~~~~~~~~~~~~~~~~~~~~~~

:math:`{\tau }_{0}` = endurance limit in pure alternating shear = :math:`352.\mathit{MPa}`

:math:`{d}_{0}` = endurance limit in pure alternating traction-compression = :math:`540.97\mathit{MPa}`


Loading history
~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    ":math:`t` ", "1. ", "2. ", "3."
    ":math:`{\sigma }_{\mathrm{xx}}(t)` ", "411. ", "0. ", "—411."
    ":math:`{\sigma }_{\mathrm{xy}}(t)` ", "205. ", "0. ", "—205."
    ":math:`{\sigma }_{\mathit{yy}}(t)\mathrm{=}{\sigma }_{\mathit{zz}}(t)\mathrm{=}{\sigma }_{\mathit{xz}}(t)\mathrm{=}{\sigma }_{\mathit{yz}}(t)` ", "0. ", "0. ", "0."


Charging is considered to be periodic.


B modeling
---------------

The material properties and the loading history are identical and obtained from test cases SSLV135A.


C modeling
---------------

The material properties and the loading history are identical and obtained from test cases SSLV135E.


D modeling
---------------

The material properties and the loading history are identical and obtained from test case SSLV135F.


Modeling E
---------------

The material properties and the loading history are identical and obtained from test cases SSLV135B.


F modeling
---------------

     The material properties and the loading history are identical and obtained from test cases SSLV135G


G modeling
---------------

     Material properties and loading history are identical and obtained from test cases SSLV135H and SSLV135A