Inverse

2nd order tensors

Calculation

The inverse of 2nd order tensors a\boldsymbol{ a} in 2d and 3d are

2d: [aij]1=1det(a)[a22a12a21a11]3d: [aij]1=1det(a)[a22a33a23a32a13a32a12a33a12a23a13a22a23a31a21a33a11a33a13a31a13a21a11a23a21a32a22a31a12a31a11a32a11a22a12a21]\begin{aligned} \text{2d: }[a_{ij}]^{-1} &= \frac{1}{\mathrm{det}(\boldsymbol{ a})} \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{bmatrix} \\ \text{3d: }[a_{ij}]^{-1} &= \frac{1}{\mathrm{det}(\boldsymbol{ a})} \begin{bmatrix} a_{22} a_{33} - a_{23} a_{32} & a_{13} a_{32} - a_{12} a_{33} & a_{12} a_{23} - a_{13} a_{22} \\ a_{23} a_{31} - a_{21} a_{33} & a_{11} a_{33} - a_{13} a_{31} & a_{13} a_{21} - a_{11} a_{23} \\ a_{21} a_{32} - a_{22} a_{31} & a_{12} a_{31} - a_{11} a_{32} & a_{11} a_{22} - a_{12} a_{21} \end{bmatrix} \end{aligned}

as for 2x2 and 3x3 matrices.

Properties

  • aa1=I\boldsymbol{ a}\boldsymbol{ a}^{-1} = \boldsymbol{ I}

  • a:aT=3\boldsymbol{ a}:\boldsymbol{ a}^{-\mathrm{T}} = 3

  • [ab]1=b1a1\left[\boldsymbol{ a}\boldsymbol{ b}\right]^{-1} = \boldsymbol{ b}^{-1}\boldsymbol{ a}^{-1}

  • [ka]1=1ka1\left[k\boldsymbol{ a}\right]^{-1} = \frac{1}{k}\boldsymbol{ a}^{-1}

  • [a+b]1\left[\boldsymbol{ a} + \boldsymbol{ b}\right]^{-1}\quad has no simple formula

Differentiation

a1a=a1aT\begin{aligned} \frac{\partial \boldsymbol{ a}^{-1}}{\partial \boldsymbol{ a}} = -\boldsymbol{ a}^{-1}\overline{\otimes}\boldsymbol{ a}^{-\mathrm{T}} \end{aligned}

4th order tensors

Use the Voigt matrix representation and linear algebra rules. However, some special cases exist, e.g.

  • A: A1= I\textbf{\textsf{ A}}:\textbf{\textsf{ A}}^{-1}=\textbf{\textsf{ I}}

  • [ A: B]1= B1: A1\left[\textbf{\textsf{ A}}:\textbf{\textsf{ B}}\right]^{-1} = \textbf{\textsf{ B}}^{-1}:\textbf{\textsf{ A}}^{-1}

  • [k A]1=1k A1\left[k\textbf{\textsf{ A}}\right]^{-1} = \frac{1}{k}\textbf{\textsf{ A}}^{-1}

  • [ab]1=a1b1\left[\boldsymbol{ a}\overline{\otimes}\boldsymbol{ b}\right]^{-1} = \boldsymbol{ a}^{-1}\overline{\otimes}\boldsymbol{ b}^{-1}