Isotropic Elements

In many cases considerable simplification is possible, because of the constraints imposed on the number of independent elastic constants. For cubic symmetry, for which the elastic stiffness tensor has only three independent constants as given in (6.29), the elements of T, are given in Table 11.1(b), and for hexagonal symmetry the elements are given in Table 11.1(c). If c12 = Cn - 2c44 were to be substituted in Table 11.1(b) the isotropic elements would... 

These results of Walpole61 include as special cases those of Hill47 and of Hashin and Shtrikman48. For anisotropic phases Walpole58 gives bounds on the five elastic moduli of an aligned array of transversely isotropic elements and for randomly oriented fibrous inclusions in an isotropic matrix. For the former case (alignment) the bounds are expressed in terms of phase concentration q and the quantities k, 1, m, n, p defined as follows k - 1/2 (Cjj + C, m — 1/2 (Cn — C22), = C13, n = C33, p = C44 = C55. 

For aligned transversely isotropic elements the self consistent method gives (Walpole58 ) the relations... 

Quantitative evaluations show that the transition to chain models with additional centers of viscous friction in side groups (i.e. the introduction of more isotropic elements of the model) leads to a sharp decrease in the anisotropy, at least for thermodynamically flexible chains. 

Thus, the first term on the right-hand side of (6.3.78) has the same structure as the expression that was obtained with (6.3.66) for an isotropic element. The other terms can be calculated by using the definitions (6.1.8) and the expressions (6.3.79). The cross-section matrix appears in the form... 

The refraction index n is the experimentally measurable quantity which determines the interaction between light and a homogeneous condensed material. From the value of this index, one determines the polarizability a of the polarizable element. In the case of isotropic elements, with concentration Nj V in a homogeneous material, the relation is given by the Lorentz-Lorenz1 formula... 

Unlike the situation embodied in section A2.4.1. in which the theory was developed in an essentially isotropic manner, the presence of an electrode introduces an essentially non-isotropic element into the equations. Neglecting rotational-dependent interactions, we see that the overall partition function can be written... 

Figure 13.6 illustrates the geometry for an array of N isotropic elements arranged along the z axis, having spacings d with equal excitation magnitudes / and electrical phase shift a between pairs of elements. It can be shown that the radiation pattern for this array maybe expressed as... 

FIGURE 13.6 Geometry for an array of N isotropic elements having spacing d, equal excitation magnitudes I, and phase shift a between pairs of elements. 

If the isotropic elements are replaced by elements having radiation pattern function /(0), then it can be shown that the resultant pattern of the array is given by... 

The directivity of an antenna provides a measure of performance compared to a reference antenna. The reference is either the hypothetical isotropic element or the practical half-wave dipole. Specifically, if the antenna under test and the reference are supplied with the same input power, then the directivity would simply compare the ratio of the power densities of the former with the latter. The directivity is used to express the performance in the direction of maximum power density from the antenna under test. It is presumed that aU of the power supplied to the antenna is radiated, that is, it is 100% efficient. The directive gain is similar to directivity except it can be used for any direction and not necessarily the direction of maximum radiation. The power gain takes into account the efficiency of the antenna system since all of the power supplied is not radiated. This can include the losses in the transmission line feed configuration. 

Directivity Provides a measure of performance compared to a reference antenna, such as the isotropic element. 

Brandtzaeg, A. (1929) Failure of a material composed of non-isotropic elements, Det. Kgl. Norske Idenskabers Selskabs Skrifter, No. 2, Trondjem. 

The two-shell model of Kerner [65] conforms to the conditions of the second group of models. The dilatation of a spherical inclusion surrounded by a homogeneous medium is derived subject to the condition that displacements and tractions at the surface of the inclusion are continuous. The homogeneous medium is supposed to have the elastic properties of the composite as a whole. The model interrelates shear (Gj) and compressive (Kj) moduli (or Poisson s ratios p ) of an arbitrary number of isotropic elements with the macroscopic moduli Gc and Kc. 

Since space is isotropic, K (spatial) is a symmetry group of the molecular Hamiltonian v7in that all its elements conmuite with // ... 

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... 

An alternative scheme for extracting all tliree isotropic nonlinear susceptibilities can be fomuilated by examining equation B 1.5.39. By choosing an appropriate configuration and the orientation of the polarization of die SH radiation e 2a) such that the SHG signal vanishes, one obtains, assuming only surface contribution with real elements, ... 

A schematic diagram of the surface of a liquid of non-chiral (a) and chiral molecules (b) is shown in figure Bl.5.8. Case (a) corresponds to oom-synnnetry (isotropic with a mirror plane) and case (b) to oo-symmetry (isotropic). For the crj/ -synnnetry, the SH signal for the polarization configurations of s-m/s-out and p-m/s-out vanish. From table Bl.5.1. we find, however, that for the co-synnnetry, an extra independent nonlinear susceptibility element, is present for SHG. Because of this extra element, the SH signal for... 

A key element in the experimental determination of the stiffness and strength characteristics of a lamina is the imposition of a uniform stress state in the specimen. Such loading is relatively easy for isotropic materials. However, for composite materials, the orthotropy introduces coupling between normal stresses and shear strains and between shear stresses and normal and shear strains when loaded in non-principal material coordinates for which the stress-strain relations are given in Equation (2.88). Thus, special care must be taken to ensure obtaining... 

The simplest situation is one in which a planar substrate lacks any crystallographic structure. Then the confined fluid is homogeneous and isotropic in transverse (x,y) directions. All off-diagonal elements of T vanish, Txx = Tyy = T, and Eq. (5) simplifies to... 

First of all the term stress-induced crystallization includes crystallization occuring at any extensions or deformations both large and small (in the latter case, ECC are not formed and an ordinary oriented sample is obtained). In contrast, orientational crystallization is a crystallization that occurs at melt extensions corresponding to fi > when chains are considerably extended prior to crystallization and the formation of an intermediate oriented phase is followed by crystallization from the preoriented state. Hence, orientational crystallization proceeds in two steps the first step is the transition of the isotropic melt into the nematic phase (first-order transition of the order-disorder type) and the second involves crystallization with the formation of ECC from the nematic phase (second- or higher-order transition not related to the change in the symmetry elements of the system). 

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