Reference solution
=====================


Calculation method
-----------------

According to the general theory of the homogenization of composite materials [:ref:`bib1 <bib1>`], the effective Young's modules :math:`{E}_{\mathit{xx}}^{\mathit{eff}}` and :math:`{E}_{\mathit{yy}}^{\mathit{eff}}` in the directions :math:`X` and :math:`Y` of a mixture having the form given above, are given by the following formulas:

:math:`\frac{1}{{E}_{\mathit{xx}}^{\mathit{eff}}}\mathrm{=}\frac{{f}_{1}}{{E}_{1}}+\frac{{f}_{2}}{{E}_{2}}`

:math:`{E}_{\mathit{yy}}^{\mathit{eff}}\mathrm{=}{f}_{1}{E}_{1}+{f}_{2}{E}_{2}`

:math:`{f}_{1}` and :math:`{f}_{2}` are the volume fractions of each material, in our case:

:math:`{f}_{1}\mathrm{=}{f}_{2}\mathrm{=}0.5`


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


1. M. BORNET, T. BRETHEAU and P. GILORMINI: Homogenization in the mechanics of materials (T1). Hermes Science Publications - 2001.