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:

\boldsymbol{U}=\alpha \boldsymbol{T_1} \otimes \boldsymbol{T_1}+\boldsymbol{T_2} \otimes \boldsymbol{T_2}+\boldsymbol{T_3} \otimes \boldsymbol{T_3} (180)

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

\boldsymbol{U}=\alpha \boldsymbol{T_1} \otimes \boldsymbol{T_1}+\alpha \boldsymbol{T_2} \otimes \boldsymbol{T_2}+\alpha \boldsymbol{T_3} \otimes \boldsymbol{T_3} = \alpha \boldsymbol{I} (181)

representing an isotropic expansion. Equation () 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}. (182)

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.