Rotationless Strains
============================

.. note::
	The rotationless strains must be seen as **experimental**. The strains are acumulated, i.e. the strain values are a sum of all previous strain increments. 
	In DIC there will always be a degree of noise in the measured displacement field. Thus, an acumulated strain value will also accumulate the noise components from all preceeding images.

.. hint::
	It may be beneficial to carry out some degree of filtering of the raw DIC displacement field before trying to use the rotationless strain.

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

First, a polar decomposition is applied to the deformation gradient, splitting the deformation gradient in a rotation matrix :math:`\bm{R}` and a strecth matrix :math:`\bm{U}`:

.. math::
	\bm{F} = \bm{RU}

The velocity gradient :math:`\bm{L}` is defined as

.. math::
	\bm{L} = \bm{\dot{F}} \bm{F}^{-1} = \frac{\bm{F}(\bm{X},t_i) - \bm{F}(\bm{X},t_{i-1})}{dt} \bm{F}_i^{-1}


where :math:`\bm{\dot{F}}` is the time derivative of the deformation gradient, :math:`t_i` indicates the current time step and  :math:`dt=t_i - t_{i-1}` is the duration of the last timestep.

The rate of deformation tensor :math:`\bm{D}` is further defined as:

.. math::
	\bm{D} = \frac{1}{2}(\bm{L} + \bm{L}^T)

And then the rotationless deformation tensor :math:`\hat{\bm{D}}` is found as:

.. math::
	\bm{\hat{D}} = \bm{R}^T \bm{D} \bm{R}

The rotationless strain increments is given by the components of the rotationless deformation tensor:

.. math::
	{\hat{\epsilon}}_{xx} = \hat{D}_{11}

.. math::
	{\hat{\epsilon}}_{xy} = \hat{D}_{12}

.. math::
	{\hat{\epsilon}}_{yx} = \hat{D}_{21}

.. math::
	{\hat{\epsilon}}_{yy} = \hat{D}_{22}

The rotationless strain at a time :math:`t_i` is calculated as a sum of all strain increments from the start of the test :math:`t=t_0` until the current time :math:`t=t_i`:

.. math::
	{\epsilon}_{xx}^{rot} = \sum_{t = t_0}^{t_i}{{\hat{\epsilon}_{xx}}}

.. math::
	{\epsilon}_{xy}^{rot} = \sum_{t = t_0}^{t_i}{{\hat{\epsilon}_{xy}}}

.. math::
	{\epsilon}_{yx}^{rot} = \sum_{t = t_0}^{t_i}{{\hat{\epsilon}_{yx}}}

.. math::
	{\epsilon}_{yy}^{rot} = \sum_{t = t_0}^{t_i}{{\hat{\epsilon}_{yy}}}