Fourth-rank tensor invariants

The quantitites (aap(0)ayS(t)y and yfap(0)PfS(t)y which involve four indices (a, ft, y, S) are fourth-rank tensors. The angular brackets indicate an ensemble average where the ensemble represents a uniform, isotropic (rotationally invariant) liquid or gas. The second-rank tensors aafi and are symmetric in th indices a, p. The fourth-rank tensors in Eq. (7.B.6) are consequently symmetric in the indices (a, p) and (y, S) separately. It follows from very general considerations (Jeffreys, 1961) that the most general isotropic (rotationally invariant) fourth-rank tensor possessing the required symmetry34 in (a, p) and (y, S) is... 

This coefficient is a fourth rank tensor. In the THG case the matrices must be invariant under permutation of the indices k, I, and nr, as a result the notation for the third order nonlinear coefficient can be simplified to C,. The unit of C, is m -V" (in the MKSQ/Sl system). 

It is possible to develop expressions for hyper-Rayleigh and Raman intensity components in terms of sixth-rank tensor invariants, analogous to the familiar fourth-rank invariants given above, together with quantum-mechanical expressions for transition hyperpolarizability tensors. However, these expressions are too complicated to give here the articles by D.A. Long in reference 4] should be consulted for further details. 

The quantities cyku describing the material properties, are called elastic stiffness constants or stiffiiesses. The transformation properties of the cyki under a rotation of the coordinate system is uniquely determined by the requirement of a coordinate invariant formulation of Eq. (3.51). The rank of this material tensor is equal to the sum of the ranks of strain and stress tensor. Accordingly, cyki are the coordinates of a fourth rank tensor. The symmetry of strain and stress tensors reqitires the symmetry of the stiffness tensor with regard to an interchange of i and j as well as k and /... 

Accordingly, invariance of second-, third-, and fourth-rank tensors in a change of coordinate system is fulfilled if the state functions are exact eigenfunctions to a model Hamiltonian, and satisfy the hypervirial theorems for position and second-moment operators. 

The observed apparent decrease in activation volume (increase in slope) with strain level in the Eyring plots can be explained by a variant of the previous model, proposed recently by Buckley [12], which recognises intrinsic anisotropy of segment diffusion because of local correlation of molecular segments. Arm (B) of the model must then be replaced by an aggregate of anisotropic flow units. The quasi-linear relation between the viscous part of the rate of deformation D and bond stretch deviatoric stress s of the previous model for PET [6], D" = Os , where 0 is a scalar function of the stress invariants then takes a new form for each unit, D = 0s , where 0 becomes a fourth rank fluidity tensor. 

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