Hooke’s Tensors

c \epsilon = \sigma

s \sigma = \epsilon

  • The math objects \epsilon and \sigma are second rank tensors
  • The math objects c and s are fourth rank tensors.

This should make sense. In the world of matrix algebra it takes a 3×3 to operate on an old 3 to make a new 3 (matrix on vector to make vector).

Similarly it takea a 3x3x3x3 to operate on an old 3×3 to make a new 3×3.

Appendix A

\begin{bmatrix} \begin{bmatrix} c_{1111} & c_{1112} & c_{1113} \\ c_{1121} & c_{1122} & c_{1123} \\ c_{1121} & c_{1122} & c_{1123} \end{bmatrix} & \begin{bmatrix} c_{1211} & c_{1212} & c_{1213} \\ c_{1221} & c_{1222} & c_{1223} \\ c_{1221} & c_{1222} & c_{1223} \end{bmatrix} & \begin{bmatrix} c_{1311} & c_{1312} & c_{1313} \\ c_{1321} & c_{1322} & c_{1323} \\ c_{1321} & c_{1322} & c_{1323} \end{bmatrix} \\ \\ \begin{bmatrix} c_{2111} & c_{2112} & c_{2113} \\ c_{2121} & c_{2122} & c_{2123} \\ c_{2121} & c_{2122} & c_{2123} \end{bmatrix} & \begin{bmatrix} c_{2211} & c_{2212} & c_{2213} \\ c_{2221} & c_{2222} & c_{2223} \\ c_{2221} & c_{2222} & c_{2223} \end{bmatrix} & \begin{bmatrix} c_{2311} & c_{2312} & c_{2313} \\ c_{2321} & c_{2322} & c_{2323} \\ c_{2321} & c_{2322} & c_{2323} \end{bmatrix} \\ \\ \begin{bmatrix} c_{3111} & c_{3112} & c_{3113} \\ c_{3121} & c_{3122} & c_{3123} \\ c_{3121} & c_{3122} & c_{3123} \end{bmatrix} & \begin{bmatrix} c_{3211} & c_{3212} & c_{3213} \\ c_{3221} & c_{3222} & c_{3223} \\ c_{3221} & c_{3222} & c_{3223} \end{bmatrix} & \begin{bmatrix} c_{3311} & c_{3312} & c_{3313} \\ c_{3321} & c_{3322} & c_{3323} \\ c_{3321} & c_{3322} & c_{3323} \end{bmatrix} \end{bmatrix}

The above math object operating on

\begin{bmatrix}\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\ \epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\ \epsilon_{31} & \epsilon_{32} & \epsilon_{33}\end{bmatrix}

will produce

\begin{bmatrix}\sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33}\end{bmatrix}

Every component like \sigma_{13} is a linear combination of the compoents { \epsilon_{11}, \epsilon_{12}, \epsilon_{13}, \epsilon_{21}, \epsilon_{22}, \epsilon_{23}, \epsilon_{31}, \epsilon_{32}, \epsilon_{33} }