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A scalar function that depends entirely on the principal invariants of a tensor is objective, i.e., independent of 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.
However, the stress tensor itself is a physical quantity and as such, it is independent of the coordinate system chosen to represent it. There are certain invariants associated with every tensor which are also independent of the coordinate system. For example, a vector is a simple tensor of rank one.
[1] are a set of tensor invariants that span the space of real, symmetric, second-order, 3-dimensional tensors and are isomorphic with respect to principal stress space. This right-handed orthogonal coordinate system is named in honor of the German scientist Dr. Walter Lode because of his seminal paper written in 1926 describing the effect of ...
The left Cauchy–Green deformation tensor is often called the Finger deformation tensor, named after Josef Finger (1894). [5] The IUPAC recommends that this tensor be called the Green strain tensor. [4] Invariants of are also used in the expressions for strain energy density functions.
In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature.These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors formed from these by the operations of taking dual contractions and covariant differentiations.
Carminati-McLenaghan invariants, for a set of polynomial invariants of the Riemann tensor of a four-dimensional Lorentzian manifold which is known to be complete under some circumstances. Curvature invariant, for curvature invariants in a more general context.
A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.
As a result, these modified ρ(a k) are still G-invariants (because every homogeneous component of a G-invariant is a G-invariant) and have degree less than d (since deg i k > 0). The equation x = ρ( a 1 ) i 1 + ... + ρ( a n ) i n still holds for our modified ρ( a k ), so we can again conclude that x lies in the R -algebra generated by i 1 ...