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A real tensor in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part.
[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 ...
As it is a second order tensor, the stress deviator tensor also has a set of invariants, which can be obtained using the same procedure used to calculate the invariants of the stress tensor. It can be shown that the principal directions of the stress deviator tensor s i j {\displaystyle s_{ij}} are the same as the principal directions of the ...
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.
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 .
comprise a complete set of invariants for the Riemann tensor. In the case of vacuum solutions , electrovacuum solutions and perfect fluid solutions , the CM scalars comprise a complete set. Additional invariants may be required for more general spacetimes; determining the exact number (and possible syzygies among the various invariants) is an ...
where is the first invariant of the stress tensor, is the second invariant of the deviatoric part of the stress tensor, is the yield stress in uniaxial compression, and is the Lode angle given by θ = 1 3 cos − 1 ( 3 3 2 J 3 J 2 3 / 2 ) . {\displaystyle \theta ={\tfrac {1}{3}}\cos ^{-1}\left({\cfrac {3{\sqrt {3}}}{2}}~{\cfrac {J_{3}}{J_{2 ...