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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 $X_{\alpha}(\alpha=1,2,3)$ moves into a position with the spatial (or EULER) coordinates xi(i=1,2,3). Thus, the motion is described by the function $\xb={\chib}(\Xb,t)$. Using a less exact notation we can write $\xb=\xb(\Xb,t)$. The deformation gradient is defined as

\begin{eqnarray*}\Fb=F_{i\alpha}=x_{i,\alpha}=\frac{\partial x_i}{\partial X_{\alpha}}= \frac{\partial\xb}{\partial\Xb} \end{eqnarray*}

1.2 Stretching

EULER's stretching tensor $\Db$ is obtained as the symmetric part of the velocity gradient $\Lb=\mbox{grad}\,\vbb=v_{i,j}=\dot{x}_{i,j}$. 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*}

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*}

Wolfgang Fellin
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