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]

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

which are functions of the principal invariants above.

Mixed invariants

Furthermore, mixed invariants between pairs of rank two tensors may also be defined.[4]

Calculation of the invariants of order 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.[5]

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.[6]

This technique was first introduced into isotropic turbulence by Howard P. Robertson in 1940 where he was able to derive Kármán–Howarth equation from the invariant principle.[7] George Batchelor and Subrahmanyan Chandrasekhar exploited this technique and developed an extended treatment for axisymmetric turbulence.[8][9][10]

See also

• Symmetric polynomial
• Elementary symmetric polynomial
• Newton's identities
• Invariant theory

References

1. Spencer, A. J. M. (1980). Continuum Mechanics. Longman. ISBN 0-582-44282-6.
2. Kelly, PA. "Lecture Notes: An introduction to Solid Mechanics" (PDF). Retrieved 27 May 2018.
3. Kindlmann, G. "Tensor Invariants and their Gradients" (PDF). Retrieved 24 Jan 2019.
4. Schröder, Jörg; Neff, Patrizio (2010). Poly-, Quasi- and Rank-One Convexity in Applied Mechanics. Springer.
5. Betten, J. (1987). "Irreducible Invariants of Fourth-Order Tensors". Mathematical Modelling. 8: 29–33. doi:10.1016/0270-0255(87)90535-5.
6. Ogden, R. W. (1984). Non-Linear Elastic Deformations. Dover.
7. Robertson, H. P. (1940). "The Invariant Theory of Isotropic Turbulence". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press. 36 (2): 209–223. Bibcode:1940PCPS...36..209R. doi:10.1017/S0305004100017199.
8. Batchelor, G. K. (1946). "The Theory of Axisymmetric Turbulence". Proc. R. Soc. Lond. A. 186 (1007): 480–502. Bibcode:1946RSPSA.186..480B. doi:10.1098/rspa.1946.0060.
9. Chandrasekhar, S. (1950). "The Theory of Axisymmetric Turbulence". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 242 (855): 557–577. Bibcode:1950RSPTA.242..557C. doi:10.1098/rsta.1950.0010. S2CID 123358727.
10. Chandrasekhar, S. (1950). "The Decay of Axisymmetric Turbulence". Proc. Roy. Soc. A. 203 (1074): 358–364. Bibcode:1950RSPSA.203..358C. doi:10.1098/rspa.1950.0143. S2CID 121178989.