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'':
:$det\; (A-lambda\; E)\; =\; 0$
The first invariant of an ''n''×''n'' tensor A ($I\_A$) is the coefficient for $lambda^\{n-1\}$ (coefficient for $lambda^n$ is always 1), the second invariant ($II\_A$) is the coefficient for $lambda^\{n-2\}$, etc., the n-th invariant is the free term.
The definition of the ''invariants of tensors'' and specific notations used through out the article were introduced into the field of Rheology by Ronald Rivlin and became extremely popular there. In fact even the trace of a tensor $A$ is usually denoted as $I\_A$ in the textbooks on rheology.

Contents

Properties

Calculation of the invariants of symmetric 3×3 tensors

Engineering applications

See also

Properties

The first invariant (trace) is always the sum of the diagonal components:
:$I\_A=A\_\{11\}+A\_\{22\}+\; dots\; +\; A\_\{nn\}=mathrm\{tr\}(A)$
The n-th invariant is just $pm\; det\; A$, the determinant of 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:
:$I\_A=A\_\{11\}+A\_\{22\}+A\_\{33\}=A\_1+A\_2+A\_3$
:$II\_A=A\_\{11\}A\_\{22\}+A\_\{22\}A\_\{33\}+A\_\{11\}A\_\{33\}-A\_\{12\}^2-A\_\{23\}^2-A\_\{13\}^2\; =A\_1A\_2+A\_2A\_3+A\_1A\_3$
:$III\_A=det\; (A)=\; A\_1\; A\_2\; A\_3$
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:
:$A^3\; -\; I\_A\; A^2\; +II\_A\; A\; -III\_A\; E=\; 0$
where E is the 2nd order Identity Tensor.
A similar equation holds for tensors of higher order.

Engineering applications

Any function of invariants of a tensor is independent from rotations of the coordinate system (objective). For the symmetric 3×3 tensors these are the only possible objective functions (a symmetric 3×3 tensor has six degrees of freedom and there are three degrees of freedom for rotation in three dimensional space, thus only three degrees of freedom are left for the objective functions).
Thus, all objective (independent form the coordinate system) scalar functions 'f' of 3×3 symmetrical tensors can depend only upon its invariants.
:$f(A\_\{ij\})=f(I\_A,II\_A,III\_A)$
So if we want to evaluate an objective scalar function of a symmetric 3×3 tensor (e.g. potential energy as function of the deformation tensor), then instead of an unknown function of 6 independent parameters, we have an unknown function of only three independent parameters, that is much easier to fit empirically. Often additional considerations help to further reduce the number of the independent parameters.

See also

★ Symmetric polynomial

★ Elementary symmetric polynomial

★ Newton's identities

★ Invariant theory