Shell knot

The expansion of a shell node leads to a set of nodes lying on a straight line. Therefore, the stretch tensor $ \boldsymbol{U}$ is reduced to the stretch along this line. Let $ \boldsymbol{T_1}$ be a unit vector parallel to the expansion and $ \boldsymbol{T_2}$ and $ \boldsymbol{T_3}$ unit vectors such that $ \boldsymbol{T_2} \cdot \boldsymbol{T_3} =0$ and $ \boldsymbol{T_1} \times \boldsymbol{T_2} = \boldsymbol{T_3}$. Then $ \boldsymbol{U}$ can be written as:

$\displaystyle \boldsymbol{U}=\alpha \boldsymbol{T_1} \otimes \boldsymbol{T_1}+\boldsymbol{T_2} \otimes \boldsymbol{T_2}+\boldsymbol{T_3} \otimes \boldsymbol{T_3}$ (198)

leading to one stretch parameter $ \alpha$. Since the stretch along $ \boldsymbol{T_2}$ and $ \boldsymbol{T_3}$ is immaterial, Equation (198) can also be replaced by

$\displaystyle \boldsymbol{U}=\alpha \boldsymbol{T_1} \otimes \boldsymbol{T_1}+\...
...l{T_2}+\alpha \boldsymbol{T_3} \otimes \boldsymbol{T_3} = \alpha \boldsymbol{I}$ (199)

representing an isotropic expansion. Equation (197) can now be replaced by

$\displaystyle \boldsymbol{\Delta u }=$ $\displaystyle \boldsymbol{\Delta w}+\alpha_0 \left[ \left. \frac {\partial \bol...
...ha \boldsymbol{R} (\boldsymbol{\theta_0}) \cdot (\boldsymbol{p}-\boldsymbol{q})$    
  $\displaystyle + \boldsymbol{w_0}+[\alpha_0 \boldsymbol{R} (\boldsymbol{\theta_0}) - \boldsymbol{I}] \cdot (\boldsymbol{p}-\boldsymbol{q})-\boldsymbol{u_0}.$ (200)

Consequently, a knot resulting from a shell expansion is characterized by 3 translational degrees of freedom, 3 rotational degrees of freedom and 1 stretch degree of freedom.