Isotropic and anisotropic materials

Extension of linear isotropic elastic analysis to allow for anisotropy is treated in some standard texts. The range of standard formulae is much more restricted than that for isotropic materials. Some computer software use FEA that include the use of anisotropic elements, so that anisotropic analyses can be used. However, it requires materials data. Thus, although the procedures for isotropic and anisotropic materials are the same, the latter may be limited by available formulae. However, material nonlinearity is less likely to be encountered with RP materials (Figure 7.35). 

Sandhu, R.S., A survey of failure theories of isotropic and anisotropic materials, Technical Report, AFFDL-TR-72-71, Jan. 1972. 

Infrared ellipsometry is typically performed in the mid-infrared range of 400 to 5000 cm , but also in the near- and far-infrared. The resonances of molecular vibrations or phonons in the solid state generate typical features in the tanT and A spectra in the form of relative minima or maxima and dispersion-like structures. For the isotropic bulk calculation of optical constants - refractive index n and extinction coefficient k - is straightforward. For all other applications (thin films and anisotropic materials) iteration procedures are used. In ellipsometry only angles are measured. The results are also absolute values, obtained without the use of a standard. 

The inherent anisotropy (most often only orthotropy) of composite materials leads to mechanical behavior characteristics that are quite different from those of conventional isotropic materials. The behavior of isotropic, orthotropic, and anisotropic materials under loadings of normal stress and shear stress is shown in Figure 1-4 and discussed in the following paragraphs. 

There are two principal optical classes isotropic and anisotropic. In an isotropic material, light travels at the same speed no matter which direction it goes. In anisotropic materials, the velocity of the light depends on which direction it is going relative to the crystal structure. 

Both E, in ideal solids, and rj, in ideal liquids, are material functions independent of the size and shape of the material they describe. This holds for isotropic and homogeneous materials, that is, materials for which a property is the same at all directions at any point. Isotropic materials are so characterized because their degree of symmetry is infinite. In contrast, anisotropic materials present a limited number of elements of symmetry, and the lower the number of these elements, the higher the number of material functions necessary to describe the response of the material to a given perturbation. Even isotropic materials need two material functions to describe in a generalized way the relationship between the perturbation and the response. In order to formulate the mechanical behavior of ideal solids and ideal liquids in terms of constitutive equations, it is necessary to establish the concepts of strain and stress. 

The scattering component of OCintrinsic is caused by the heterogeneities in a material, such as local fluctuations in its density or its refractive index. It can be subdivided into isotropic and anisotropic scattering components, each of which can be estimated by using semiempirical expressions based on theoretical considerations but containing empirical parameters. 

There are two types of conductive adhesives conventional materials that conduct electricity equally in all directions (isotropic conductors) and those materials that conduct in only one direction (anisotropic conductors). Isotropically conductive materials are typically formulated by adding silver particles to an adhesive matrix such that the percolation threshold is exceeded. Electrical currents are conducted throughout the composite via an extensive network of particle-particle contacts. Anisotropically conductive adhesives are prepared by randomly dispersing electrically conductive particles in an adhesive matrix at a concentration far below the percolation threshold. A schematic illustration of an anisotropically conductive adhesive interconnection is shown in Fig. 1. The concentration of particles is controlled such that enough particles are present to assure reliable electrical contacts between the substrate and the device (Z direction), while too few particles are present to achieve conduction in the X-Y plane. The materials become conductive in one direction only after they have been processed under pressure they do not inherently conduct in a preferred direction. Applications, electrical conduction mechanisms, and formulation of both isotropic and anisotropic conductive adhesives are discussed in detail in this chapter. 

In this section, only the optical constants of isotropic films determined by the multiwavelength approach in IRRAS will be discussed. The optical constants are assumed to be independent of the film thickness, and any gradient in the optical properties of the substrate (Section 3.5) is ignored. This undoubtedly lowers accuracy of the results. Anisotropic optical constants of a film are more closely related to real-world ultrathin films. At this point, it is worth noting that approaches to measuring isotropic and anisotropic optical constants are conceptually identical An anisotropic material shows a completely identical metallic IRRAS spectrum to the isotropic one if the complex refractive index along the z-direction for the anisotropic material is equal to that for the isotropic one [44]. However, to... 

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