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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.
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 ...
In continuum mechanics, a Mooney–Rivlin solid [1] [2] is a hyperelastic material model where the strain energy density function is a linear combination of two invariants of the left Cauchy–Green deformation tensor.
[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 ...
This model has the general form and the isotropic form respectively =: = +. where : is tensor contraction, is the second Piola–Kirchhoff stress, : is a fourth order stiffness tensor and is the Lagrangian Green strain given by = [() + + ()] and are the Lamé constants, and is the second order unit tensor.
The model was proposed by Ronald Rivlin in 1948 using invariants, though Mooney had already described a version in stretch form in 1940, and Wall had noted the equivalence in shear with the Hooke model in 1942. In contrast to linear elastic materials, the stress–strain curve of a neo-Hookean material is not linear. Instead, the relationship ...
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.
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 ...