Isotropy, elasticity - orientation

If this process is repeated, one finds only three values of Poisson s ratio are needed, not six. For fiber-reinforced materials, the number of elastic constants may be further reduced if other symmetries appear. For example, in some materials short fibers are randomly oriented in a plane and this gives transverse isotropy. That is, there is an elastically isotropic plane but the stiffness and compliance constants will be different normal to this plane (five elastic constants are needed). 

In this model it is proposed that the polymer can be considered as an aggregate of identical units, which in the unstretched state are oriented randomly. As orientation develops, the units rotate and become completely aligned at the maximum achievable orientation. The elastic properties of the units are those of the mostly highly aligned structure, which could be a fibre with transverse isotropy or a film with orthorhombic s3onmetry [40]. In this chapter the theory will be developed for a fibre, where both the fibre and the structural units possess transverse isotropy. 

The macroscopic nematodynamic equations describe the dynamics of the slowly relaxing variables, which usually are either connected with conservation laws or with the Goldstone modes of the spontaneously broken symmetries. To formulate them we wUl follow the traditional approach [65-67] rather than the one based more directly on the principles of hydrodynamics and irreversible thermodynamics [68]. In the nematic state isotropy is spontaneously broken and the averaged molecular alignment singles out an axis whose orientation defines the director n, i. e. an object that has the properties of a unit vector with n = -n. The static properties are conveniently expressed in terms of a free energy density whose orientational elastic part is given by [69]... 

Much of the treatment contained within this volume is limited by the assumptions that the adhesives, and usually the adherends, are linear elastic, homogenous, and isotropic. For bulk adhesives, the assumption of isotropy is usually justified, although instances do arise where preferred orientation of filler particles or crystalline regions can lead to anisotropic behavior. Common adherends such as fiber-reinforced composites, wood, and cold-drawn metals often exhibit anisotropic behavior that can significantly affect Joint behavior. 

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... 

Stoney s relation follows from this general expression for elastic isotropy in the surface plane with Y = 1 / sii, V = -S12 / Sii for cubic (100) surfaces. The result of the appropriate tensor transformations for some surface orientations is given in Table 1, examples of how tensor transformations are performed for arbitrary surface orientations are given in the literature [85Nye, 97Ibal, 99Sanl]. 

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