Invariants of tensors

In mathematics, in the fields of multilinear algebra and representation theory, invariants of tensors are coefficients of the characteristic polynomial of the tensor A:^[1]

\ p(\lambda):=\det (\mathbf{A}-\lambda \mathbf{E})

,

where $\mathbf {E}$ is the identity tensor and $\lambda \in {\mathbb {C}}$ is the polynomial's indeterminate (it is important to bear in mind that a polynomial's indeterminate $\lambda$ may also be a non-scalar as long as power, scaling and adding make sense for it, e.g., $p(\mathbf {A} )$ is legitimate, and in fact, quite useful).

The first invariant of an n×n tensor A ( $I_{A}$ ) is the coefficient for $\lambda^{n-1}$ (because the coefficient for $\lambda^n$ is always 1), the second invariant ( $II_A$ ) is the coefficient for $\lambda^{n-2}$ , etc., the nth invariant is the free term.

Properties

The first invariant (trace) is always the sum of the diagonal components:

\ I_A=A_{11}+A_{22}+ \cdots + A_{nn}=\mathrm{tr}(\mathbf{A}) \,

The nth invariant is just $\pm \det \mathbf{A}$ , the determinant of $\mathbf {A}$ (up to sign).

The invariants do not change with rotation of the coordinate system (they are objective). Obviously, any function of the invariants only is also objective.

Calculation of the invariants of symmetric 3×3 tensors

Most tensors used in engineering are symmetric 3×3. For this case the invariants can be calculated as:

\mathrm{I}_A= \mathrm{tr}(\mathbf{A}) = A_{11}+A_{22}+A_{33} = A_1+A_2+A_3 \,

\mathrm {II} _{A}={\frac {1}{2}}\left((\mathrm {tr} \mathbf {A} )^{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}

(the sum of principal minors). This expression holds for non-symmetric tensors. ^[2]

\begin{align} & = A_1A_2+A_2A_3+A_1A_3 \\ \mathrm{III}_A & = \det (\mathbf{A})= A_1 A_2 A_3 \end{align}

where $A_{1}$ , $A_{2}$ , $A_{3}$ are the eigenvalues of tensor A.

Because of the Cayley–Hamilton theorem the following equation is always true:

\ \mathbf{A}^3 - \mathrm{I}_A \mathbf{A}^2 +\mathrm{II}_A \mathbf{A} -\mathrm{III}_A \mathbf{E}= 0

where E is the second-order identity tensor.

A similar equation holds for tensors of higher order.

Engineering application

A scalar valued tensor function f that depends merely on the three invariants of a symmetric 3×3 tensor $\mathbf {A}$ is objective, i.e., independent from rotations of the coordinate system. Moreover, every objective tensor function depends only on the tensor's invariants. Thus, objectivity of a tensor function is fulfilled if, and only if, for some function $\ g:R^3\to R$ we have

\ f(\mathbf A)=g(I_A,II_A,III_A). \,

A common application to this is the evaluation of the potential energy as function of the strain tensor, within the framework of linear elasticity. Exhausting the above theorem the free energy of the system reduces to a function of 3 scalar parameters rather than 6. Within linear elasticity the free energy has to be quadratic in the tensor's elements, which eliminates an additional scalar. Thus, for an isotropic material only two independent parameters are needed to describe the elastic properties, known as Lamé coefficients. Consequently, experimental fits and computational efforts may be eased significantly.

This technique was first introduced into isotropic turbulence by Howard P. Robertson in 1940^[3] and he was able to derive Kármán–Howarth equation from the invariant principle. George Batchelor and Subrahmanyan Chandrasekhar exploited this technique by extending to axisymmetric turbulence^[4]^[5]^[6].

Other invariants

Since the invariants are constant in any reference, functions of invariants are also constant. Some sources define the three invariants of the 3×3 tensors as

I_{1}=tr(\mathbf {A} )=A_{ii}

I_{2}=tr(\mathbf {A} ^{2})=A_{ij}A_{ji}

This expression only holds for symmetric tensors.

I_{3}=tr(\mathbf {A} ^{3})=A_{ij}A_{jk}A_{ki}

Therefore,

I_{A}=I_{1}

II_{A}={\frac {1}{2}}(I_{1}^{2}-I_{2})

III_{A}={\frac {1}{6}}(I_{1}^{3}-3I_{1}I_{2}+2I_{3})

References

↑ SPENCER, A. J. M. Continuum Mechanics. Longman, 1980.
↑ 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)
↑ 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.
↑ Batchelor, G. K. (1946). The theory of axisymmetric turbulence. Proc. R. Soc. Lond. A, 186(1007), 480-502.
↑ Chandrasekhar, S. (1950). The theory of axisymmetric turbulence. Royal Society of London.
↑ Chandrasekhar, S. (1950). The decay of axisymmetric turbulence. Proc. Roy. Soc. A, 203, 358-364.

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[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] 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.

[4] Batchelor, G. K. (1946). The theory of axisymmetric turbulence. Proc. R. Soc. Lond. A, 186(1007), 480-502.

[5] Chandrasekhar, S. (1950). The theory of axisymmetric turbulence. Royal Society of London.

[6] Chandrasekhar, S. (1950). The decay of axisymmetric turbulence. Proc. Roy. Soc. A, 203, 358-364.