Order tensor properties

Here c[-], which will be called the elastic modulus tensor, is a fourth-order linear mapping of its second-order tensor argument, while b[-], which will be called the inelastic modulus tensor, is a linear mapping of k whose form will depend on the specific properties assigned to k. They depend, in general, on and k. For example, if k consists of a single second-order tensor, then in component form... 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

In this section, well-known properties of second-order positive-definite symmetric tensors and functions involving them will be cited without proof. The principal values and principal vectors (m = 1, 2, 3) of a symmetric second-order tensor A are given by... 

The nth-order property is the nth-order derivative of the energy, d EjdX" (the factor 1 /n may or may not be included in the property). Note that the perturbation is usually a vector, and the first derivative is therefore also a vector, the second derivative a matrix, the third derivative a (third-order) tensor etc. 

Equation (I-l) is the general representation of the dispersion model. The dispersion coefficient is a function of both the fluid properties and the flow situation the former have a major effect at low flow rates, but almost none at high rates. In this general representation, the dispersion coefficient and the fluid velocity are all functions of position. The dispersion coefficient, D, is also in general nonisotropic. In other words, it has different values in different directions. Thus, the coefficient may be represented by a second-order tensor, and if the principal axes are taken to correspond with the coordinate system, the tensor will consist of only diagonal elements. 

Particles subject to Brownian motion tend to adopt random orientations, and hence do not follow these rules. A particle without these symmetry properties may follow a spiral trajectory, and may also rotate or wobble. In general, the drag and torque on an arbitrary particle translating and rotating in an unbounded quiescent fluid are determined by three second-order tensors which depend on the shape of the body ... 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

Characterization of Molecular Hyperpolarizabilities Using Third Harmonic Generation. Third harmonic generation (THG) is the generation of light at frequency 3co by the nonlinear interaction of a material and a fundamental laser field at frequency co. The process involves the third-order susceptibility x 3K-3 , , ) where —3 represents an output photon at 3 and the three s stand for the three input photons at . Since x(3) is a fourth (even) rank tensor property it can be nonzero for all material symmetry classes including isotropic media. This is easy to see since the components of x(3) transform like products of four spatial coordinates, e.g. x4 or x2y2. There are 21 components that are even under an inversion operation and thus can be nonzero in an isotropic medium. Since some of the terms are interrelated there are only four independent terms for the isotropic case. 

Second order NLO properties including SHG arise from the second order NLO susceptibility x tensor in the relationship for the bulk polarization, P, such that (2-3)... 

There is a first-order splitting pattern common to all 3n states, independent of the physical content. All the molecule-dependent physical information is contained in the parameter Aso- These facts are, of course a consequence of the tensor properties expressed in the Wigner-Eckart theorem. 

Equation (2.26) for heat conduction and Eq. (2.3) for momentum transfer are similar, and the flow is proportional to the negative of the gradient of a macroscopic variable the coefficient of proportionality is a physical property characteristic of the medium and dependent on the temperature and pressure. In a three-dimensional transport, Eqs. (2.27) and (2.15) differ because the heat flow is a vector with three components, and the momentum flow t is a second-order tensor with nine components. 

Let us now consider the second-order molecular properties. The static electric dipole-polarizability tensor is given by the expression... 

In this equation, po is the permanent dipole moment of the molecule, a is the linear polarizability, 3 is the first hyperpolarizability, and 7 is the second hyperpolarizability. a, and 7 are tensors of rank 2, 3, and 4 respectively. Symmetry requires that all terms of even order in the electric field of the Equation 10.1 vanish when the molecule possesses an inversion center. This means that only noncentrosymmetric molecules will have second-order NLO properties. In a dielectric medium consisting of polarizable molecules, the local electric field at a given molecule differs from the externally applied field due to the sum of the dipole fields of the other molecules. Different models have been developed to express the local field as a function of the externally applied field but they will not be presented here. In disordered media,... 

Xi f i terms are the components the nth-order electric susceptibility of the medium, is a (n- - l)-order tensor that determines completely the optical (linear and nonlinear) properties of the medium. Conservation of energy requires that... 

In the case of thin films with point symmetry oomm there are 2 nonzero tensor components Xzzz and Xxxz were T is the symmetry axis. For historical reasons often in the literature d" tensor is used to describe the second order NLO properties, with corresponding components defined as d p = jxxxz and dpp = Itzzz - In practice s denotes the polarization of fundamental beam and p of the harmonic one, respectively. 

The notation used in the generic equation is strictly only valid for scalar properties. In the particular case when a vector property is considered the tensor order of the corresponding variables is understood to be adjusted accordingly. Hence, the quantities ipk, 4>k and second order tensors. 

Particle Shape Effect. To this point, we have been dealing only with spherical particle suspensions. When the particles have irregular shapes, the rheological properties are expected to be very different from those of the spherical particle suspensions. Consider, for example, a simple system of cylindrical fibre suspensions. Because the particles are expected to align in the direction of the flow or shear, the viscosity needs to be treated as a second-order tensor, that is, the values of the viscosity under the same condition are different when different directions are referred. Only at the low (zero) shear limit may the particles be randomly distributed and have an isotropic rheological behavior. 

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