In mathematics, in the fields of multilinear algebra and representation theory, invariants of tensors are coefficients of the characteristic polynomial of the tensor A:
Contents
- Properties
- Calculation of the invariants of symmetric 33 tensors
- Engineering application
- Other invariants
- References
where
The first invariant of an n×n tensor A (
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
Properties
The first invariant (trace) is always the sum of the diagonal components:
The nth invariant is just
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:
(the sum of principal minors)
where
Because of the Cayley–Hamilton theorem the following equation is always true:
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
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.
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
Therefore,