Invariants of tensors
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2008-04-25T10:05:49Z
Tomeasy
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/* Application for engineering */ renamed section
{{Expert-subject|Mathematics}}
In [[mathematics]], in the fields of [[multilinear algebra]] and [[representation theory]], '''invariants of tensors''' are coefficients of the [[characteristic polynomial]] of the [[tensor]] ''A'':
:<math>\ \det (\mathbf{A}-\lambda \mathbf{E}) = 0 </math>
The first invariant of an ''n''×''n'' tensor A (<math>I_A</math>) is the coefficient for <math>\lambda^{n-1}</math> (coefficient for <math> \lambda^n </math> is always 1), the second invariant (<math>II_A</math>) is the coefficient for <math>\lambda^{n-2}</math>, 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 (linear algebra)|trace]] of a tensor <math>A</math> is usually denoted as <math>I_A</math> in the textbooks on rheology.
==Properties==
The first invariant (trace) is always the sum of the diagonal components:
:<math>\ I_A=A_{11}+A_{22}+ \dots + A_{nn}=\mathrm{tr}(\mathbf{A})</math>
The n-th invariant is just <math>\pm \det \mathbf{A}</math>, the determinant of <math>\mathbf{A}</math> (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 [[Application of tensor theory in engineering science|tensors used in engineering]] are symmetric 3×3.
For this case the invariants can be calculated as:
:<math>\ I_A= tr(\mathbf{A}) = A_{11}+A_{22}+A_{33}=A_1+A_2+A_3</math>
:<math>\ II_A= \frac{1}{2} \left( tr(\mathbf{A})^2 - \mathbf{A}^T \cdot \mathbf{A} \right) = 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</math>
:<math>\ III_A=\det (\mathbf{A})= A_1 A_2 A_3</math>
where <math>A_1</math>, <math>A_2</math>, <math>A_3</math> are the [[eigenvalues]] of tensor ''A''.
Because of the [[Cayley-Hamilton theorem]] the following equation is always true:
:<math>\ \mathbf{A}^3 - I_A \mathbf{A}^2 +II_A \mathbf{A} -III_A \mathbf{E}= 0</math>
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 <math>\mathbf{A}</math> is objective, i.e., independent from rotations of the coordinate system. Thus objectivity is fulfilled if
:<math>\ f\ (A_{ij})=f(I_A,II_A,III_A)</math>.
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 (physics and chemistry)|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.
==See also==
*[[Symmetric polynomial]]
*[[Elementary symmetric polynomial]]
*[[Newton's identities]]
*[[Invariant theory]]
==References==
{{unreferenced|date=October 2007}}
<references/>
[[Category:Tensors]]
[[Category:Invariant theory]]
[[Category:Linear algebra]]
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