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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.
[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 ...
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.
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.
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 derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics.These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.
The principal invariants of tensors do not change with rotation of the coordinate system (see Invariants of tensors). The singular values of a matrix are invariant under orthogonal transformations. Lebesgue measure is invariant under translations. The variance of a probability distribution is invariant under translations of the real line.
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.