Tensor isotropic

As we note at the end of Sect. 3.6 aU this and the subsequent results follow if the assumption of linearity has been used in a constitutive relation of a nonsimple fluid with viscosity and heat conduction (3.127), (3.146) (i.e., before application of admissibility principle). These constitutive relations are scalar, vector and symmetric tensor isotropic functions (3.128) (including /) which are linear in vector g, h and synunetrical tensor D. 

In this book, we confine ourselves only to the special case of fluids mixture (4.128) which is linear in vector and tensor variables.We denote it as the chemically reacting mixture of fluids with linear transport properties or simply the linear fluid mixture [56, 57, 64, 65]. Then (see Appendix A.2) the scalar, vector and tensor isotropic functions (4.129) linear in vectors and tensors (symmetrical or skew-symmetrical) have the forms ... 

FIGURE 20.14. Schematic CSA tensor powder pattern for an axially asymmetric (a) and axially symmetric (b) tensor. Isotropic chemical shifts (o-,) are indicated by dashed lines, (c) Typical Andrews [37] sample holder (rotor) rotating on air bearings within a stator (shaded). 

There are now three cubic classes that do not have a center of symmetry. We should therefore expect these three - 23 (T), 43m (Tjj), and 432 (O) - to be piezoelectric, but only the first two are. The latter has three fourfold rotation axes perpendicular to each other, making the piezoelectric tensor isotropic in three dimensions. The piezoeffect would not then depend on the sign of the stress, which is only possible for all... 

Consider an isotropic medium that consists of independent and identical microscopic cln-omophores (molecules) at number density N. At. sth order, each element of the macroscopic susceptibility tensor, given in laboratory Cartesian coordinates A, B, C, D, must carry s + 1 (laboratory) Cartesian indices (X, Y or Z) and... 

For an isotropic material, all orientations are equally probable and all such products that have an odd number of Tike direction cosines will vanish upon averaging-. This restricts the nonvanishing tensor elements to those such as xVaaa abba - Similarly for the elements Such orientational averaging is crucial in... 

Equations (8.20) are not sufficiently specific for practical purposes, so it is important to consider special cases leading to simpler relations. When the pore orientations are isotropically distributed, the second order tensors k, 3 and y are isotropic and are therefore scalar multiples of the unit tensor. Thus equation (8.20) simplifies to... 

An isotropic tensor is one whose components are unchanged by rotation of the coordinate system. 

For cubic crystals, which iaclude sUicon, properties described by other than a zero- or a second-rank tensor are anisotropic (17). Thus, ia principle, whether or not a particular property is anisotropic can be predicted. There are some properties, however, for which the tensor rank is not known. In addition, ia very thin crystal sections, the crystal may have two-dimensional characteristics and exhibit a different symmetry from the bulk, three-dimensional crystal (18). Table 4 is a listing of various isotropic and anisotropic sUicon properties. Table 5 gives values for the more common physical properties and for some of the thermodynamic properties. Figure 5 shows some thermal properties. 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

This is the condition (A.85) that the elastic limit function / be an isotropic tensor functions of its arguments. By analogy with the hypoelastic constitutive equation, the name hypoinelasticity suggests itself for this formulation. 

This is the condition (A.85) for to be isotropic, as in the hypoinelastic formulation (5.116i). It is therefore clear that, when the moduli c and b, as well as the elastic limit function do not include the special dependence on F indicated in (5.155) and (5.160), then objectivity demands that they be isotropic tensor functions of their arguments, and the spatial formulation reduces to the hypoinelastic formulation. 

A scalar-valued function f(A, B) of two symmetric second-order tensors A and B is said to be isotropic if... 

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

Fig. 2.8. Idealized elastic/perfectly plastic solid behavior results in a stress tensor in which there is a constant offset between the hydrostatic (isotropic) loading and shock compression. Such behavior is only an approximation which may not be appropriate in many cases.

FIG. 20 (a) Density profiles p(z) vs z for e = —2 and four average bulk densities (f> as indicated, (b) Surface excess vs density in the bulk for four choices of e. (c) Profiles for the diagonal components of the pressure tensor and of the total pressure for (p = l.O and e = —2. Insert in (c) shows the difference between P, and Px to show that isotropic behavior in the bulk of the film is nicely obtained, (d) Interfacial tension between the polymer film and the repulsive wall vs bulk density for all four choices of e. Curve is only a guide for the eye [18]. 

Bilayers have received even more attention. In the early studies, water has been replaced by a continuous medium as in the monolayer simulations [64-67]. Today s bilayers are usually fully hydrated , i.e., water is included exphcitly. Simulations have been done at constant volume [68-73] and at constant pressure or fixed surface tension [74-79]. In the latter case, the size of the simulation box automatically adjusts itself so as to optimize the area per molecule of the amphiphiles in the bilayer [33]. If the pressure tensor is chosen isotropic, bilayers with zero surface tension are obtained. Constant... 

Now, in order for us to recover standard hydrodynamical behavior, we require that the momentum flux density tensor be isotropic i.e. invariant under rotations and reflections. In particular, from the above expansion we see that must be isotropic up to order... 

These conditions show us immediately that in the case of the four-neighbor HPP lattice (V = 4) f is noni.sotropic, and the macroscopic equations therefore cannot yield a Navier-Stokes equation. For the hexagonal FHP lattice, on the other hand, we have V = 6 and P[. is isotropic through order Wolfram [wolf86c] predicts what models are conducive to f lavier-Stokes-like dynamics by using group theory to analyze the symmetry of tensor structures for polygons and polyhedra in d-dimensions. 

In quantum theory as in classical theory the isotropic Raman spectrum is expressed in terms of the average value of the polarizibility tensor a(0) = (1/3) Sp a randomly changing in time due to collisions ... 

The summations are over all nuclei in radicals A and B for which aj and are finite. The a-values are tensor quantities but are treated as isotropic here. 

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