Green Strains¶

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

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

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

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

$\bm{C} = \bm{F^T}\bm{F}$

The Green strain matrix is defined as:

$\bm{E} = \frac{1}{2}(\bm{C} - \bm{1})$

At component level this gives:

$E_{11} = \frac{1}{2}(C_{11} - 1)$

$E_{12} = E_{21} = \frac{1}{2}C_{12} = \frac{1}{2}C_{21}$

$E_{22} = \frac{1}{2}(C_{22} - 1)$