Cartesian order tensor

The time-averaged Wigner rotation matrices Dq (0, ) are the order parameters in a uniaxial phase and can be written in the form of a Cartesian order tensor S which is a symmetric traceless 3x3 matrix. The S has a maximum of five independent, non-zero elements. It is convenient to choose these as 5, Sxx Syy Sxy, Sxzi and Syz- The relations between S and time-averaged Wigner rotation matrices Dq, 0, iP) are [2.10]... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

Cartesian tensors, i.e., tensors in a Cartesian coordinate system, will be discussed. Three Independent quantities are required to describe the position of a point in Cartesian coordinates. This set of quantities is X where X is (x, X2, X3). The index i in X has values 1,2, and 3 because of the three coordinates in three-dimensional space. The indices i and j in a j mean, therefore, that a j has nine components. Similarly, byi has 27 components, Cp has 81 components, etc. The indices are part of what is called index notation. The number of subscripts on the symboi denotes the order of the tensor. For example, a is a zero-order tensor... 

Any or all of these forces may result in local stresses within the fluid. Stress can be thought of as a (local) concentration of force, or the force per unit area that bounds an infinitesimal volume of the fluid. Now both force and area are vectors, the direction of the area being defined by the normal vector that points outward relative to the volume bounded by the surface. Thus, each stress component has a magnitude and two directions associated with it, which are the characteristics of a second-order tensor or dyad. If the direction in which the local force acts is designated by subscript j (e.g., j = x, y, or z in Cartesian coordinates) and the orientation (normal) of the local area element upon which it acts is designated by subscript i, then the corresponding stress component (ay) is given by... 

In noncartesian coordinates the divergence of a second-order tensor cannot be evaluated simply as a row-by-row operation as it can in a cartesian system. Hence some extra, perhaps unexpected, terms (e.g., rrg/r) appear in the direction-resolved force equations. General expressions for V-T in different coordinate systems are found in Section A.ll. 

The (stress or strain-rate) state at a point is a physical quantity that cannot depend on any particular coordinate-system representation. For example, the stress state is the same regardless of whether it is represented in cartesian or cylindrical coordinates. In other words, the state (as represented by a symmetric second-order tensor), is invariant to the particular coordinate-system representation. 

The major notations of scalars, vectors, and tensors and their operations presented in the text are summarized in Tables A1 through A5. Table A1 gives the basic definitions of vector and second-order tensor. Table A2 describes the basic algebraic operations with vector and second-order tensor. Tables A3 through A5 present the differential operations with scalar, vector, and tensor in Cartesian, cylindrical, and spherical coordinates, respectively. It is noted that in these tables, the product of quantities with the same subscripts, e.g., a b, represents the Einstein summation and < jj refers to the Kronecker delta. The boldface symbols represent vectors and tensors. 

Briefly, we have expanded on the work by Challacombe et al., who have shown that Hermite Gaussians have a simple relation to elements of the Cartesian multipole tensor (Challacombe et al., 1996). Once the Hermite coefficients have been determined, they may be employed to calculate point multipoles centered at the expansion sites. Thus, if hctuv represents the coefficient of a Hermite Gaussian of order Atuv, then if this Hermite is normalized we have... 

Namely, we have assumed that we have an idealized prolate-shaped molecule with cylindrical symmetry. If we remove this assumption, the orientation of the k, /, m Cartesian frame fixed to the molecule (see Fig. 5.1) is described by a second-rank tensor, S, the order tensor with five independent elements 5, = (3(cos cos j) —8ij)/2y where the specify the orientation of the i axis relative to n and 8ij is... 

The Cartesian tensor representation can be extended to describe the orientational ordering of biaxial molecules in biaxial phases by introducing [6] a fourth rank ordering tensor ... 

I have tried to expose the tensor monster as really quite a fiiendly and useful little man-made invention for transforming vectors. It greatly simplifies notation and makes the three-dimensional approach to rheology practical. I have tried to make the incorporation of tensors as simple and physical as possible. Second-order tensors, Cartesian coordinates, and a minimum of tensor manipulations are adequate to explain the basic principles of rheology and to give a number of useful constitutive equations. Vi th what is presented in the first four chapters, students will be able to read and use the current rheological literature. For curvilinear coordinates and detailed development of constitutive equations, several good texts are available and are cited where appropriate. 

Notation. We will use boldface italic letters to denote vectors and tensors. We adopt the summation convention for repeated indices, imless stated otherwise. Most often, vectors are denoted by lowercase boldface italic letters, and second-order tensors, or 3x3 matrices, by lowercase boldface Greek letters. Fourth-order tensors are usually denoted by uppercase boldface italic letters. We will make use of a Cartesian coordinate system with an orthonormal basis ei, ej, e. Where it is necessary to show components of a vector or a tensor, these will always be relative to the orthonormal basis e, 2, 3. Throughout this work we will identify a second-order tensors r with a 3x3 matrix. We will always use 1 < / <3, to denote the components of the vector a, and the components of the... 

Although the irreducible spherical tensor approach is valuable fix the definition of the (xdo ing tensors and for their manipulation under rotation it does not always provide a ready undostanding of the physical significance of the various components. This is sometimes available from a Cartesian representation of the ordering tensor. The most familiar example is the Saupe ordering matrix which represents the orientational ordering at the second rank level [8]. It is defined by... 

Although the Cartesian analogue of the second rank ordering tensor is relatively straightforward to manipulate the same is not true for the fourth rank ordering matrix [9] or rather supa matrix. This is defined by... 

It is assumed that the reader is familiar with some basic properties of Cartesian tensors, such as those that may be found in the book by Spencer [256]. In this book we shall define the divergence of a second order tensor Tij to be the tensor... 

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... 

Fig. 7. The nature of information concerning the mean orientation and dynamics of an internuclear vector r, which can be obtained from RDC analysis. Upon diagonali-zation of the Cartesian dipolar interaction tensor R, described in the text, the mean vector orientation, r, will be described by the Euler angles a and /3. The eigenvalues will correspond to the axial and rhombic order parameters which describe the amplitude of motion. If the motion is asymmetric, as reflected in a nonzero rhombic order parameter, then the principal direction of asymmetry is described by the Euler angle y.

Following are some of the important vector and tensor operations which are used in the derivations in Sec. II of this review these relations are given for Cartesian coordinates. The quantities v and F are vectors, v is a second-order symmetric tensor, and T is a scalar. 

It turns out, however, that the state of stress at P can be completely specified by giving the stress vector components in any three mutually perpendicular planes passing through the point. That is, only nine components, three for each vector, are needed to define the stress at point P. Each component can be described by two indices ij, the first denoting the orientation of the surface and the second, the direction of the force. Figure 2.3 gives these components for three Cartesian planes. The nine stress vector components form a second-order Cartesian tensor, the stress tensor8 n. ... 

Such order can be described in terms of the preferential alignment of the director, a unit vector that describes the orientation of molecules in a nematic phase. Because the molecules are still subject to random fluctuations, only an average orientation can be described, usually by an ordering matrix S, which can be expressed in terms of any Cartesian coordinate system fixed in the molecule. S is symmetric and traceless and hence has five independent elements, but a suitable choice of the molecular axes may reduce the number. In principle, it is always possible to diagonalize S, and in such a principal axis coordinate system there are only two nonzero elements (as there would be, for example, in a quadrupole coupling tensor). In the absence of symmetry in the molecule, there is no way of specifying the orientation of the principal axes of S, but considerable simplification is obtained for symmetric molecules. If a molecule has a threefold or higher axis of symmetry, its selection as one of the axes of the Cartesian coordinate system leaves only one independent order parameter, with the now familiar form ... 

The second-order non-linear susceptibility x is a third-rank tensor, which, in Cartesian coordinates, is defined by a set of 27 elements with i.J and k rep-... 

---