Tensor principal invariant

Recall from the discussion in Section 2.5.4 that the stress tensor, like the strain-rate tensor, has certain invariants. For any known stress tensor, these invariant relationships can be used to determine the principal stresses. 

Because the viscosity is a scalar, it can be a function only of the scalar invariants of the rate of deformation tensor. There are three combinations of the components of the rate of deformation tensor (Ajj), which are scalar invariants. They define any state of deformation rate independentiy of the coordinate system. They are referred to as the principal invariants of the rate of deformation tensor ... 

If a fluid can be considered incompressible, the first principal invariant of the rate of deformation tensor will be zero, Ij = 0. The third principal invariant vanishes in many simple flow situations, like axial flow in pipe, tangential flow between concentric cylinders, etc. In more general terms, the third invariant is zero in rectilinear... 

Apart from the principal stresses, a stress tensor has another set of invariants, called the principal invariants Ji, J2, and J3 (see appendix A. 7). They are defined as... 

The von Mises yield criterion uses the principal invariants of the deviatoric stress tensor, J(, J2, and J3. Using equation (3.26), we find... 

Every scalar function lyofa symmetric tensor G which satisfies (27) is called an Isotropic Tensor Function of G. An isotropic scalar-valued function of G is also called a scalar invariant of G. It may easily be checked that the principal invariants of G, defined by... 

If the invariants are known for some arbitrary strain-rate state, then it is clear that the three equations above form a system of equations from which the principal strain rates can be uniquely determined. This analysis is explained more fully in Appendix A. Using the principal axes greatly facilitates subsequent analysis, wherein quantitative relationships are established between the strain-rate and stress tensors. 

If the potential energy of a lattice 0 is expanded in a number of components of small displacements of atoms from their equilibrium position Uj up to a cubic term, then the temperature dependence of the principal values of the LTEC tensor ,(T )are, in view of the translational invariancy of the lattice, described by the following expression ... 

Since the Raman techniques described are sensitive to (fl(t) - fi (0)>, we are interested in the properties of products of two elements of II, subject to the above constraints. Furthermore, it is necessarily true that any tensor elements of n that are related by reversed indices are identical, e.g., nxy = Flyx. This leaves us only two independent elements of R<3) to consider. It is conventional to express these independent elements in terms of rotationally invariant features of n One of these invariants (usually denoted a) is given by one third of the trace of FI (24). Since this invariant measures the average polarizability of the system, it is known as the isotropic component of n. The other invariant is generally denoted ft, and in the principal axis system of FI it is given by (24)... 

The Gas Phase Polarizability Anisotropy. Murphy50 has measured the depolarization ratio for Rayleigh scattering, pR, and analysed the intensity distribution in the rotational Raman spectrum of the vapour at 514.5 nm. The ratio R20 of the invariants of the a,-,aA/ tensor can be determined by fitting the rotational Raman distribution, and a is known (from the Zeiss-Meath formula). Knowledge of the three quantities, a, pR and R2o, allows the polarizability anisotropy, Aa, and the three principal values of the tensor to be calculated. The polarizability anisotropy invariant is numerically equal to the quantity,... 

At first order, it can be shown that only the symmetrized part of the interaction tensor contributes to the frequency shift. The majority of second-order contributions arise from large EFG, the EFG tensor being symmetric by definition. Thus, only symmetric second-rank tensors T can be considered, which can be decomposed into two contributions T = isol3 + AT with D3 the identity matrix. The first term is the isotropic part so = l/3Tr(T) that is invariant by any local symmetry operation. The second term is the anisotropic contribution AT, a symmetric second-rank traceless tensor, which depends then on five parameters the anisotropy 8 and the asymmetry parameter rj that measures the deviation from axial symmetry, and three angles to orient the principal axes system (PAS) in the crystal frame. The most common convention orders the eigenvalues of AT such that IAzzL and defines 8 — Xzz and rj — fzvv i )... 

The value of E depends upon the values of the elonents in the stress and strain tensors. Under plane stress conditions, one of die principal stresses is zero and E is equal to Young s modulus, E. However, under plane strain conditions, the strain in one of the principal axes is zero and E = E/(l — v ) where v is Poisson s ratio. For most polymers 0.3 < v < 0.5 and the values of both Gic and Kic invariably are much greater when measured in plane stress. For the purposes both of toughness comparisons and component design, die plane strain values of Gic and Kic are preferred because th are the minimum values fm any given material. In order to achieve plane strain conditions, the following criteria need to be satisfied ... 

Notation. The symbols a, f, y are used throughout to denote the electric field polarizability, first and second hyperpolarizabilities respectively, suitably qualified by frequency factors where necessary. The magnetizability is denoted by y and the nuclear screening tensor by a. The numerous but well-known acronyms specifying the computational procedures are used without definition. The possibly rather less well-known acronyms for the principal gauge invariant procedures are given in Table 1. 

In isotropic materials, the yield surface must not depend on the orientation of the load. Thus, the function / describing the yield surface can only contain those parts of the deviatoric stress tensor that do not change during coordinate transformations. This is already ensured if the principal stresses are used because the hydrostatic stress (Thyd is also coordinate invariant. 

This general expression first accounts for the principal of causality by stating that the state of stress at a time t is dependent on the strains in the past only. Secondly, by using the time dependent Finger tensor B, one extracts from the fiow fields only those properties which produce stress and eliminates motions like translations or rotations of the whole body which leave the stress invariant. Equation (7.128) thus provides us with a suitable and sound basis for further considerations. 

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