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


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

Finite element calculation with code ABAQUS. Contact nonlinearity is modeled using unidirectional GAP elements. The stress intensity factor is calculated from the integral :math:`J`.


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


.. image:: images/1000000000000AD00000077EE61C05FA97298E78.png
    :width: 5.2575in
    :height: 3.4827in


.. _RefImage_1000000000000AD00000077EE61C05FA97298E78.png:

Dimensional stress intensity factor as a function of the mechanical load factor,

in the case of loading no. 3


+-------+----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------+
|Rating:|                                                                                  |linear dimensionless intensity factor (obtained by a linear combination of the effects of autofrettage and mechanical loading, in broken lines)|
+       + .. image:: images/Object_2.svg                                                   +                                                                                                                                               +
|       |    :width: 114                                                                   |                                                                                                                                               |
+       +    :height: 25                                                                   +                                                                                                                                               +
|       |                                                                                  |                                                                                                                                               |
+       +                                                                                  +                                                                                                                                               +
|       |                                                                                  |                                                                                                                                               |
+-------+----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------+
|       |                                                                                  |                                                                                                                                               |
+-------+----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------+
|       |                                                                                  |nonlinear dimensionless intensity factor (obtained by taking into account the non-linearity of contact, in solid lines).                       |
+       + .. image:: images/Object_3.svg                                                   +                                                                                                                                               +
|       |    :width: 114                                                                   |                                                                                                                                               |
+       +    :height: 25                                                                   +                                                                                                                                               +
|       |                                                                                  |                                                                                                                                               |
+       +                                                                                  +                                                                                                                                               +
|       |                                                                                  |                                                                                                                                               |
+-------+----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------+
|       |                                                                                  |                                                                                                                                               |
+-------+----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------+
|       |                                                                                  |                                                                                                                                               |
+       + .. image:: images/Object_4.svg                                                   +                                                                                                                                               +
|       |    :width: 114                                                                   |                                                                                                                                               |
+       +    :height: 25                                                                   +                                                                                                                                               +
|       |                                                                                  |                                                                                                                                               |
+       +                                                                                  +                                                                                                                                               +
|       |                                                                                  |                                                                                                                                               |
+-------+----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------+


**Empirical formula for the stress intensity factor under external radial tension**


.. image:: images/Object_5.svg
    :width: 114
    :height: 25


.. _RefImage_Object_5.svg:

**Empirical formula for the stress intensity factor in autofretting in full section**


.. image:: images/Object_6.svg
    :width: 114
    :height: 25


.. _RefImage_Object_6.svg:


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


1. H.M. SHU, J. PETIT, and G. BEZINE: Stress intensity factors for radial cracks in thick walled cylinders. I. Symmetrical cracks II. Combination of autofrettage and internal pressure. Engng.Fract.Mechs., 49, no. 4, 611-629, 1994.