Definition of Stiffness Components¶
Tensor components: (See [1] page 99 for details.)
\[\begin{split}\begin{align*}
\mathbb{C}
&=
C_{ijkl}
\;
\mathbf{e}_{i}
\otimes
\mathbf{e}_{j}
\otimes
\mathbf{e}_{k}
\otimes
\mathbf{e}_{l}\\
\end{align*}\end{split}\]
Matrix components:
\[\begin{split}\begin{align*}
\mathbb{C}
&=
\begin{bmatrix}
C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\
& C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\
& & C_{33} & C_{34} & C_{35} & C_{36} \\
& & & C_{44} & C_{45} & C_{46} \\
& \text{sym} & & & C_{55} & C_{56} \\
& & & & & C_{66}
\end{bmatrix}_{[\text{Voigt}]} \hspace{-10mm}
\scriptsize{
\boldsymbol{V}_{\alpha} \otimes \boldsymbol{V}_{\beta}
} \\
&=
\begin{bmatrix}
C_{11} & C_{12} & C_{13} & \sqrt{2}C_{14} & \sqrt{2}C_{15} & \sqrt{2}C_{16} \\
& C_{22} & C_{23} & \sqrt{2}C_{24} & \sqrt{2}C_{25} & \sqrt{2}C_{26} \\
& & C_{33} & \sqrt{2}C_{34} & \sqrt{2}C_{35} & \sqrt{2}C_{36} \\
& & & 2C_{44} & 2C_{45} & 2C_{46} \\
& \text{sym} & & & 2C_{55} & 2C_{56} \\
& & & & & 2C_{66}
\end{bmatrix}_{[\text{Mandel6}]} \hspace{-15mm}
\scriptsize{
\boldsymbol{B}_{\alpha} \otimes \boldsymbol{B}_{\beta}
}
\end{align*}\end{split}\]
with
\(\boldsymbol{B}_{\alpha}\) : Base dyad of Mandel6 notation (See
mechkit.notation)\(\boldsymbol{V}_{\alpha}\) : Base dyad of Voigt notation (See [csmbrannonMandel])