Infinitesimal Strains
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In a 2D-DIC analysis, a two-dimensional displacement field :math:`\bm{u}=\bm{u}(\bm{X},t)` is measured. Here :math:`\bm{X}` denotes a position in the reference coordinate system. 
:math:`t` denotes the time, usually associated to the image ID in a sequence of images.

For any position :math:`\bm{X}` and any time :math:`t` the two-dimensional displacement gradient may be calculated as:

.. math::
	\bm{H} = \frac{\bm{\partial u}}{\bm{\partial X}} = \begin{bmatrix} H_{11} & H_{12} \\ H_{21} & H_{22} \end{bmatrix}

The infinitesimal strains are basically defined as the components of the displacement gradient matrix.

.. math::
	{\epsilon}_{11} = H_{11}

.. math::
	{\epsilon}_{12} = {\epsilon}_{21} = \frac{1}{2}(H_{12} + H_{21})
	
.. math::
	{\epsilon}_{22} = H_{22}