Invariants of tensors
where is the identity operator and represent the polynomial's eigenvalues.
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 tensorsEdit
For such tensors the principal invariants are given by:
For symmetric tensors these definitions are reduced.
The correspondence between the principal invariants and the characteristic polynomial of a tensor, in tandem with the Cayley–Hamilton theorem reveals that
where is the second-order identity tensor.
which are functions of the principal invariants above.
Furthermore, mixed invariants between pairs of rank two tensors may also be defined.
Calculation of the invariants of rank two tensors of higher dimensionEdit
Calculation of the invariants of higher order tensorsEdit
The invariants of rank three, four, and higher order tensors may also be determined.
A scalar function 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.
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. George Batchelor and Subrahmanyan Chandrasekhar exploited this technique and developed an extended treatment for axisymmetric turbulence.
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