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


Geometry
---------


.. image:: images/100004A400001F6B00001D8FC4FC058CBFAE53E1.svg
    :width: 307
    :height: 292


.. _RefImage_100004A400001F6B00001D8FC4FC058CBFAE53E1.svg:

Straight section of a thick tube with a

internal radial crack


.. csv-table::

    "Shelf report", ":math:`b={r}_{2}/{r}_{1}=2({r}_{1}=1\mathrm{mm},{r}_{2}=2\mathrm{mm})`"
    "", ""
    "Crack depth", ":math:`a/({r}_{2}-{r}_{1})=\mathrm{0,05}`"



Material properties
----------------------

The material is standard isotropic linear thermoelastic.


.. csv-table::

    "Young's module", ":math:`E=1000\mathrm{MPa}`"
    "Poisson's Ratio", ":math:`\nu =\mathrm{0,3}`"
    "Linear expansion coefficient", ":math:`{\alpha }_{T}=1E-6`"
    "Elastic limit", ":math:`{\sigma }_{0}=1\mathrm{MPa}` (used to define the initial stress field created by the auto-fretting process under the hypothesis of a previous Von Mises elastoplastic type behavior)"



Boundary conditions and loads
-------------------------------------

Boundary conditions (for half a room in region :math:`y\ge 0`)

Block :math:`\mathrm{UY}=0` on segment :math:`\mathrm{AB}` and on ligament :math:`\mathrm{DE}` (symmetry).

Linear relationship :math:`\mathrm{UX}(A)+\mathrm{UX}(E)=0` (to block horizontal translation)

**Loading**


+--------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Loading no. 1:|radial traction :math:`{\sigma }_{\mathrm{rr}}({r}_{2})={\sigma }_{0}` on the external face; this mechanical loading produces the same :math:`{K}_{I}` as an internal pressure acting simultaneously on the internal radius :math:`{r}_{1}` and on the lips of the crack, without taking into account the non-linearity of contact.                                                                                                                                                                                                                                          |
+--------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+--------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Loading #2:   |thermal loading equivalent to autofrettage defined as follows:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+              +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+              + .. image:: images/Object_1.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|              |    :width: 307                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+              +    :height: 292                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+              +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+              +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+--------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|              |In these formulas, :math:`\rho` designates the maximum radius of the zone that has undergone autofrettage, :math:`{T}_{1}` the temperature at radius :math:`{r}_{1}` and :math:`{T}_{\rho }` the temperature at radius :math:`r=\rho` in the thick, non-cracked tube. In the application referred to here, :math:`\rho ={r}_{2}` is taken, which corresponds to the autofretting of the entire section of the thick tube, and contact nonlinearity is not taken into account. A negative :math:`K` is expected for positive temperatures (compression of the uncracked tube).|
+--------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+--------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Loading #3:   |linear combination loading no. 2 :math:`\text{+}` :math:`\alpha` * loading no. 1, :math:`\alpha` (:math:`\ne \alpha T`!) designating the mechanical load factor; contact nonlinearity is taken into account here, which implies an incremental application of the mechanical load.                                                                                                                                                                                                                                                                                           |
+--------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+