Green 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 deformation gradient may be calculated as:

.. math::
	\boldsymbol{F} = \frac{\bm{\partial x}}{\bm{\partial X}} = \bm{1} + \frac{\bm{\partial u}}{\bm{\partial X}}

The two-dimensional right Cauchy-Green deformation tensor :math:`\bm{C}=\bm{C}(\bm{X},t)` is then calculated as:

.. math::
	\bm{C} = \bm{F^T}\bm{F}

The Green strain matrix is defined as:

.. math::
	\bm{E} = \frac{1}{2}(\bm{C} - \bm{1})

At component level this gives:

.. math::
	E_{11} = \frac{1}{2}(C_{11} - 1)

.. math::
	E_{12} = E_{21} = \frac{1}{2}C_{12} = \frac{1}{2}C_{21}

.. math::
	E_{22} = \frac{1}{2}(C_{22} - 1)