Orientational anisotropy, elastic properties

For cubic symmetry materials, three independent elastic properties that are orientation dependent are required to describe the mechanical behavior of the material. This anisotropy effect increases significantly the number of the nonzero elements in the FE stiffness matrix leading to alteration in the calculated stress components and the wave speed. In order to test these anisotropy effects, we plot the wave profiles of three different orientations and compare it with the isotropic behavior with a loading axis in the [001] directions as shown in Fig 8. We observed that under the same loading condition, the peak stress of [111] and [Oil] orientations are slightly higher than those of the [001] which is lower that that of isotropic material. Furthermore, wave speed varies moderately with orientation with the fastest moving wave in the [ 111 ] followed by [011 ], isotropic medium and [001 ] respectively. 

The chapter begins with an overview of elastic anisotropy in crystalline materials. Anisotropy of elastic properties in materials with cubic symmetry, as well as other classes of material symmetry, are described first. Also included here are tabulated values of typical elastic properties for a variety of useful crystals. Examples of stress measurements in anisotropic thin films of different crystallographic orientation and texture by recourse to x-ray diffraction measurements are then considered. Next, the evolution of internal stress as a consequence of epitaxial mismatch in thin films and periodic multilayers is discussed. Attention is then directed to deformation of anisotropic film-substrate systems where connections among film stress, mismatch strain and substrate curvature are presented. A Stoney-type formula is derived for an anisotropic thin film on an isotropic substrate. Anisotropic curvature due to mismatch strain induced by a piezoelectric film on a substrate is also analyzed. 

Elasticity is a macroscopic property of matter defined as the ratio of an applied static stress (force per unit area) to the strain or deformation produced in the material the dynamic response of a material to stress is determined by its viscosity. In this section we give a simplified formulation of the theory of torsional elasticity and how it applies to liquid crystals. The elastic properties of liquid crystals are perhaps their most characteristic feature, since the response to torsional stress is directly related to the orientational anisotropy of the material. An important aspect of elastic properties is that they depend on intermolecular interactions, and for liquid crystals the elastic constants depend on the two fundamental structural features of these mesophases anisotropy and orientational order. The dependence of torsional elastic constants on intermolecular interactions is explained, and some models which enable elastic constants to be related to molecular properties are described. The important area of field-induced elastic deformations is introduced, since these are the basis for most electro-optic liquid crystal display devices. 

Most metals (Table L.2) show elastic anisotropy. Note that a thin film is polycrystalline, that is, formed of monocrystalfine grains, the size and orientation of which depend on deposition conditions. They will exhibit elastic macroscopic anisotropy only when texture is present, that is, when crystallites show a preferential orientation. Tungsten is an archetype since the crystallites are elastically isotropic. Hence tungsten films show microscopic and macroscopic elastic isotropy. For A = 1 the elastic properties are completely defined by two elastic constants E and v and we have... 

As mentioned frequently the mechanical and optical response of molecules — and of their crystallites — is highly anisotropic. Depending on the property under consideration the carriers of the molecular anisotropy are the bond vectors (infrared dichroism), chain segments (optical and mechanical anisotropy), or the end-to-end vectors of chains (rubber elastic properties). For the representation of the ensuing macroscopic anisotropies one has to recognize, therefore, the molecular anisotropy and the orientation distribution of the anisotropic molecular units (Fig. 1.9.). Since these are essentially one-dimensional elements their distribution and orientation behavior can be treated as that of rods such a model had been used successfully to explain the optical anisotropy [78], and the anisotropies of thermal conductivity [79], thermal expansion or linear compressibility [80], and Young s modulus [59,... 

While in the Kuster and Toksoz model randomly distributed cracks are assumed and an isotropic effect results, Hudson s concept results in an anisotropy effect caused by the oriented fractures. For a single crack set, the first correction terms are given in Table 6.17. Please note that in Eq. (6.109), the correction term is added, but Table 6.17 shows that the correction term is negative—thus, elastic properties decrease with fracturing, where As> fis are Lame constants of the solid host material (background material) the crack density is... 

The introduction of large gas phase volumes into the polymer alters the physical characteristics of the material volume weight, permeability to fluids and gases, and physico-mechanical properties. Moreover, the properties of the polymer matrix itself are changed (owing to orientation effects, supermolecular structure of the polymer in the walls, ribs and tension bars of cells), which drives up the value of specific strength on impact, and results in anisotropy of elasticity. 

The present review shows how the microhardness technique can be used to elucidate the dependence of a variety of local deformational processes upon polymer texture and morphology. Microhardness is a rather elusive quantity, that is really a combination of other mechanical properties. It is most suitably defined in terms of the pyramid indentation test. Hardness is primarily taken as a measure of the irreversible deformation mechanisms which characterize a polymeric material, though it also involves elastic and time dependent effects which depend on microstructural details. In isotropic lamellar polymers a hardness depression from ideal values, due to the finite crystal thickness, occurs. The interlamellar non-crystalline layer introduces an additional weak component which contributes further to a lowering of the hardness value. Annealing effects and chemical etching are shown to produce, on the contrary, a significant hardening of the material. The prevalent mechanisms for plastic deformation are proposed. Anisotropy behaviour for several oriented materials is critically discussed. 

Physical properties of liquid crystals are generally anisotropic (see, for example, du Jeu, 1980). The anisotropic physical properties that are relevant to display devices are refractive index, dielectric permittivity and orientational elasticity (Raynes, 1983). A nematic LC has two principal refractive indices, Un and measured parallel and perpendicular to the nematic director respectively. The birefringence An = ny — rij is positive, typically around 0.25. The anisotropy in the dielectric permittivity which is given by As = II — Sj is the driving force for most electrooptic effects in LCs. The electric contribution to the free energy contains a term that depends on the angle between the director n and the electric field E and is given by... 

---