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


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

Boundary element method, with quadratic elements [:ref:`bib1 <bib1>`].

The calculation of :math:`{K}_{I}` and :math:`{K}_{\mathrm{II}}` is carried out by a contour integral (integral M [:ref:`bib2 <bib2>`]) in which the stresses and displacements calculated in the part are involved, as well as the stresses and displacements deduced from asymptotic solutions defined analytically, in which :math:`{K}_{I}` and :math:`{K}_{\mathrm{II}}` are alternately zero.

For comparison, the calculation of :math:`K` is also performed by the virtual extension method.


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

The results of the reference solution are shown in the table below, for the various values of the angle and for the two ends of the crack, with

.. image:: images/Object_1.svg
    :width: 172
    :height: 49


.. _RefImage_Object_1.svg:

.


+---------+-------------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+
|Method   |                         |                   |Left side                              |                   |                   |Right side                             |                   |
+---------+-------------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+
|         |                         |:math:`\theta =15°`|:math:`\theta =30°`|:math:`\theta =45°`|:math:`\theta =60°`|:math:`\theta =15°`|:math:`\theta =30°`|:math:`\theta =45°`|:math:`\theta =60°`|
+---------+-------------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+
|integral |:math:`{F}_{I}`          |1.0115             |0.7868             |0.5211             |0.2770             |1,1266             |0.9910             |0.7646             |0.4919             |
+---------+-------------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+
|:math:`M`|:math:`{F}_{\mathrm{II}}`|0.4434             |0.6244             |0.6723             |0.5804             |0.0862             |0.2961             |0.4056             |0.4057             |
+---------+-------------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+
|extension|:math:`{F}_{I}`          |1,0110             |0.7864             |0.5210             |0.2769             |1,1260             |0.9904             |0.7643             |0.4919             |
+---------+-------------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+
|Virtual  |:math:`{F}_{\mathrm{II}}`|0.4429             |0.6240             |0.6720             |0.5801             |0.0865             |0.2960             |0.4055             |0.4056             |
+---------+-------------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+-------------------+

The relationship between the global energy return rate :math:`G` and :math:`{K}_{j}` is written as follows [:ref:`bib3 <bib3>`]:

:math:`G=\beta ({K}_{I}^{2}+{K}_{\mathrm{II}}^{2})`

with:

:math:`\beta \mathrm{=}\frac{1}{16C{h}^{2}(\alpha \pi )}(\frac{1+{\kappa }_{1}}{{\mu }_{1}}+\frac{1+{\kappa }_{2}}{{\mu }_{2}})` and :math:`\begin{array}{c}{\kappa }_{i}\mathrm{=}\frac{3\mathrm{-}{\nu }_{i}}{1+{\nu }_{i}}\\ {\mu }_{i}\mathrm{=}\frac{{E}_{i}}{2(1+{\nu }_{i})}\\ \alpha \mathrm{=}\frac{1}{2\pi }\mathrm{ln}\left[(\frac{{\kappa }_{1}}{{\mu }_{1}}+\frac{1}{{\mu }_{2}}){(\frac{{\kappa }_{2}}{{\mu }_{2}}+\frac{1}{{\mu }_{1}})}^{\mathrm{-}1}\right]\end{array}`


Uncertainty about the solution
---------------------------

Estimated at less than 0.1%. It should be noted that the difference between the contour integrals method and the virtual extension method is generally less than 0.05%.


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


1. Stress intensity factor analysis of interface crack using boundary element method. Application of contour-integral method. N. MIYAZAKI, T. IKEDA, T. SODA, and T. MUNAKATA. Engng.Fract.Mechs., 45, no. 5, 599-610, 1993.


2. An analysis of interface cracks between dissimilar isotropic materials using conservation integrals in elasticity. J.F. YAU and T.C. CHANG. Engng.Fract.Mechs., 20, 423-432, 1984.


3. The strength of adhesive joints using the theory of cracks. B.M. MALYSHEV and R.L. SALGANIK. Int.J.Fract.Mech., 1, 114-128, 1965.