Conventions for representing tensors

2nd-rank tensors

2nd-rank tensors are represented as vectors using the Mandel representation. Components ordering follow the conventions used by the MFront project described here.

A symmetric 2nd-rank tensor \(\boldsymbol{a}\) stored as follows in 3D:

\[ \{\boldsymbol{a}\} = \begin{Bmatrix}a_{11} & a_{22} & a_{33} & \sqrt{2}a_{12} & \sqrt{2}a_{13} & \sqrt{2}a_{23} \end{Bmatrix}^\text{T} \]

and in 2D:

\[ \{\boldsymbol{a}\} = \begin{Bmatrix}a_{11} & a_{22} & \sqrt{2}a_{12} \end{Bmatrix}^\text{T} \]

A non-symmetric 2nd-rank tensor \(\boldsymbol{a}\) stored as follows in 3D:

\[ \{\boldsymbol{a}\} = \begin{Bmatrix}a_{11} & a_{22} & a_{33} & a_{12} & a_{21} & a_{13} & a_{31} & a_{23} & a_{32} \end{Bmatrix}^\text{T} \]

and similarly in 2D:

\[ \{\boldsymbol{a}\} = \begin{Bmatrix}a_{11} & a_{22} & a_{12} & a_{21} \end{Bmatrix}^\text{T} \]

4th-rank tensors

4th-rank tensors are represented as matrices with components complying with the representation of 2nd-rank tensors.

For instance, a symmetric 4th-order tensor in 2D will be represented as:

\[\begin{split} [\mathbf{C}] &= \begin{bmatrix} C_{1111} & C_{1122} & \sqrt{2}C_{1112} \\ C_{2211} & C_{2222} & \sqrt{2}C_{2212} \\ \sqrt{2}C_{1112} & \sqrt{2}C_{2212} & 2C_{1212} \end{bmatrix} \end{split}\]