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":

:$det\; (mathbf\{A\}-lambda\; mathbf\{E\})\; =\; 0$

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 throughout 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.

Properties

The first invariant (trace) is always the sum of the diagonal components::$I\_A=A\_\{11\}+A\_\{22\}+\; dots\; +\; A\_\{nn\}=mathrm\{tr\}(mathbf\{A\})$The n-th 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::$I\_A=\; tr(mathbf\{A\})\; =\; A\_\{11\}+A\_\{22\}+A\_\{33\}=A\_1+A\_2+A\_3$:$II\_A=\; frac\{1\}\{2\}\; left(\; tr(mathbf\{A\})^2\; -\; mathbf\{A\}^T\; cdot\; mathbf\{A\}\; ight)\; =\; A\_\{11\}A\_\{22\}+A\_\{22\}A\_\{33\}+A\_\{11\}A\_\{33\}-A\_\{12\}^2-A\_\{23\}^2-A\_\{13\}^2\; =A\_1A\_2+A\_2A\_3+A\_1A\_3$:$III\_A=det\; (mathbf\{A\})=\; A\_1\; A\_2\; A\_3$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::$mathbf\{A\}^3\; -\; I\_A\; mathbf\{A\}^2\; +II\_A\; mathbf\{A\}\; -III\_A\; mathbf\{E\}=\; 0$where E is the 2nd 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. Thus objectivity is fulfilled if

:$f\; (A\_\{ij\})=f(I\_A,II\_A,III\_A)$.

Moreover, the mere dependence on this tensor's invariants is a necessary condition for the objectivity of the function. Reason behind is that the tensor's six degrees of freedom are reduced by three degrees of freedom for arbitrary rotation in three dimensional space. Therefore, the objective function depends at most on three degrees of freedom.

A common application to this is the evaluation of the potential energy as function of the deformation tensor (stress and strain tensors). Exhausting the above theorem the energy potential reduces to a function of 3 scalar parameters rather than 6. Consequently, experimental fits and computational efforts may be eased significantly.

ee also

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

References