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Subsections
1. Definitions
Details on this item can be obtained from [3].
1.1 Deformation
A motion consists of translation, rotation and deformation. A material point with the material (or initial or LAGRANGE) coordinates
moves into a position with the spatial (or EULER) coordinates
xi(i=1,2,3). Thus, the motion is described by the function
. Using a less exact notation we can write
. The deformation gradient is defined as

1.2 Stretching
EULER's stretching tensor is obtained as the symmetric part of the velocity gradient
. Thus we have
![\begin{eqnarray*}\Db=\frac{1}{2}[\grad \vbb + (\grad \vbb)^T] = D_{ij}=\frac{1... ...,i}) =\frac{1}{2}(\dot{x}_{i,j}+\dot{x}_{j,i})=\dot{x}_{(i,j)}. \end{eqnarray*}](https://www2.uibk.ac.at/images/584x45/geotechnik/res/hypo_versions/img7.gif)
CAUCHY's spin tensor is obtained as the antimetric part of the velocity gradient:
![\begin{eqnarray*}\Wb=\frac{1}{2}[\grad \vbb - (\grad \vbb)^T] =W_{ij}=\frac{1}... ...{1}{2}\left(\dot{x}_{i,j}-\dot{x}_{j,i}\right)= \dot{x}_{[i,j]}. \end{eqnarray*}](https://www2.uibk.ac.at/images/593x45/geotechnik/res/hypo_versions/img8.gif)
Wolfgang Fellin
1999-10-01