Logarithmic Strains
============================

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-dimensionalright 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 in-plane principal streches :math:`{\gamma}_i = {\gamma}_i(\bm{X},t), i = 1,2` are found by solving the eigenvalue problem for the right Cauchy-Green deformation tensor:

.. math::
	({{\gamma}_i}^2 \bm{1} - \bm{C}) \cdot \bm{n}_i = \bm{0}

where :math:`\bm{n}_i`, the eigenvectors, gives the principal directions of the right Cauchy-Green deformation tensor.

.. note::
	Because of the two-dimensional nature of the DIC measurements, the third principal direction :math:`\bm{n}_3` is assumed to be normal to the surface of the specimen, i.e. the shear strains through the thickness of the specimen are assumed negligible.

The in-plane **principal logarithmic strains** are then calculated from the principal stretches:

.. math::
	{\epsilon}_i = ln({\gamma}_i), i = 1,2

where the rotation of the principal strains is given by the eigenvectors :math:`\bm{n}_i`. 

.. note::
	In *eCorr* it is also possible to obtain the third principal strain component :math:`{\epsilon}_3`, i.e. through the thickness. However, this is only valid where we can assume negligible elastic strains and plastic incompressibility. 
	Then, the third component is estimated as follows: :math:`{\epsilon}_3 = -({\epsilon}_1 + {\epsilon}_2)`.

	Further, an effective strain measure :math:`{\epsilon}_{eff}` based on the von Mises norm is available. This is defined as
	:math:`{\epsilon}_{eff} = \sqrt{\frac{2}{3}({{\epsilon}_1}^2 + {{\epsilon}_2}^2 + {{\epsilon}_3}^2)} = \sqrt{\frac{4}{3}({{\epsilon}_1}^2 + {{\epsilon}_1}{{\epsilon}_2} + {{\epsilon}_2}^2)}`

A two-dimensional strain matrix is established using the calculated principal strains:

.. math::
	\bm{\epsilon}_{princ} = \begin{bmatrix} {\epsilon}_1 & 0 \\ 0 & {\epsilon}_2 \end{bmatrix}

Now, this principal strain matrix :math:`\bm{\epsilon}_{princ}` is associated with the principal directions :math:`\bm{n}_i`. 
To obtain the strain matrix for a specific direction, the matrix may be rotated using a two-dimensional rotation matrix :math:`\bm{R}`:

.. math::
	\bm{\epsilon}_{rot} = \bm{R}^T \bm{\epsilon}_{princ} \bm{R}

Thus, by using the rotation provided by the principal directions :math:`\bm{n}_i`, the strain matrix can be rotated back to the coordinate domain, giving the **logarithmic coordinate strains**:

.. math::
	\bm{\epsilon}_{coord} = \begin{bmatrix} {\epsilon}_{xx} & {\epsilon}_{xy} \\ {\epsilon}_{yx} & {\epsilon}_{yy} \end{bmatrix} = \bm{R}^T \bm{\epsilon}_{princ} \bm{R}

.. note::
	When time-series of strains are plotted or exported in *eCorr*, the strains are always calculated in the center of an element.