# Invariants of tensors

In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor ${\displaystyle \mathbf {A} }$ are the coefficients of the characteristic polynomial[1]

${\displaystyle \ p(\lambda )=\det(\mathbf {A} -\lambda \mathbf {I} )}$,

where ${\displaystyle \mathbf {I} }$ is the identity operator and ${\displaystyle \lambda _{i}\in \mathbb {C} }$ represent the polynomial's eigenvalues.

## Properties

The principal invariants do not change with rotations of the coordinate system (they are objective, or in more modern terminology, satisfy the principle of material frame-indifference) and any function of the principal invariants is also objective.

## Calculation of the invariants of rank two tensors

In a majority of engineering applications, the principal invariants of (rank two) tensors of dimension three are sought, such as those for the right Cauchy-Green deformation tensor.

### Principal invariants

For such tensors the principal invariants are given by:

{\displaystyle {\begin{aligned}I_{1}&=\mathrm {tr} (\mathbf {A} )=A_{11}+A_{22}+A_{33}=\lambda _{1}+\lambda _{2}+\lambda _{3}\\I_{2}&={\frac {1}{2}}\left((\mathrm {tr} \left(\mathbf {A} \right))^{2}-\mathrm {tr} (\mathbf {A} ^{2})\right)=A_{11}A_{22}+A_{22}A_{33}+A_{11}A_{33}-A_{12}A_{21}-A_{23}A_{32}-A_{13}A_{31}=\lambda _{1}\lambda _{2}+\lambda _{1}\lambda _{3}+\lambda _{2}\lambda _{3}\\I_{3}&=\det(\mathbf {A} )=-A_{13}A_{22}A_{31}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}-A_{11}A_{23}A_{32}-A_{12}A_{21}A_{33}+A_{11}A_{22}A_{33}=\lambda _{1}\lambda _{2}\lambda _{3}\end{aligned}}}

For symmetric tensors these definitions are reduced.[2]

The correspondence between the principal invariants and the characteristic polynomial of a tensor, in tandem with the Cayley–Hamilton theorem reveals that

${\displaystyle \ \mathbf {A} ^{3}-I_{1}\mathbf {A} ^{2}+I_{2}\mathbf {A} -I_{3}\mathbf {I} =0}$

where ${\displaystyle \mathbf {I} }$  is the second-order identity tensor.

### Main invariants

In addition to the principal invariants listed above, it is also possible to introduce the notion of main invariants[3][4]

{\displaystyle {\begin{aligned}J_{1}&=I_{1}\\J_{2}&=I_{1}^{2}-2I_{2}\\J_{3}&=I_{1}^{3}-3I_{1}I_{2}+3I_{3}\end{aligned}}}

which are functions of the principal invariants above.

### Mixed invariants

Furthermore, mixed invariants between pairs of rank two tensors may also be defined.[5]

## Calculation of the invariants of rank two tensors of higher dimension

These may be extracted by evaluating the characteristic polynomial directly, using the Faddeev–LeVerrier_algorithm for example.

## Calculation of the invariants of higher order tensors

The invariants of rank three, four, and higher order tensors may also be determined.[6]

## Engineering applications

A scalar function ${\displaystyle f}$  that depends entirely on the principal invariants of a tensor is objective, i.e., independent from rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density, or Helmholtz free energy, of a nonlinear material possessing isotropic symmetry.[7]

This technique was first introduced into isotropic turbulence by Howard P. Robertson in 1940[8] where he was able to derive Kármán–Howarth equation from the invariant principle. George Batchelor and Subrahmanyan Chandrasekhar exploited this technique and developed an extended treatment for axisymmetric turbulence[9][10][11].

## References

1. ^ Spencer, A. J. M. Continuum Mechanics. Longman, 1980.
2. ^ Kelly, PA. Lecture Notes: An introduction to Solid Mechanics (PDF) http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_11_Eigenvalue.pdf. Retrieved 27 May 2018. Missing or empty |title= (help)
3. ^ Kindlmann, G. Tensor Invariants and their Gradients (PDF) https://people.cs.uchicago.edu/~glk/pubs/pdf/Kindlmann-TensorInvariantsGradients-MnV-2006.pdf. Retrieved 24 Jan 2019. Missing or empty |title= (help)
4. ^ Jörg Schröder and Patrizio Neff. Poly-, Quasi- and Rank-One Convexity in Applied Mechanics. Springer, 2010.
5. ^ Jörg Schröder and Patrizio Neff. Poly-, Quasi- and Rank-One Convexity in Applied Mechanics. Springer, 2010.
6. ^ Betten, J. (1987). Irreducible invariants of fourth-order tensors. In Mathematical Modelling (Vol. 8, pp. 29-33).
7. ^ Ogden, R. W. Non-linear elastic deformations. Dover, 1984.
8. ^ Robertson, H. P. (1940, April). The invariant theory of isotropic turbulence. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 36, No. 2, pp. 209-223). Cambridge University Press.
9. ^ Batchelor, G. K. (1946). The theory of axisymmetric turbulence. Proc. R. Soc. Lond. A, 186(1007), 480-502.
10. ^ Chandrasekhar, S. (1950). The theory of axisymmetric turbulence. Royal Society of London.
11. ^ Chandrasekhar, S. (1950). The decay of axisymmetric turbulence. Proc. Roy. Soc. A, 203, 358-364.