Anisotropic elastic solid, mechanical

The mechanical properties of an anisotropic elastic solid for small strains are defined by the generalised Hooke s law... 

The mechanical properties of an anisotropic elastic solid where the stresses are linearly related to the strains are defined by the generalized Hooke s law where each component of stress can relate to all six independent components of strain, and equivalently each component of strain can relate to all six independent components of stress. In the former case we have, for example... 

These matrices define the relationships between stress and strain in a general elastic solid, whose properties vary with direction, that is an anisotropic elastic solid. In most of this book, we will be concerned with isotropic polymers all discussion of anisotropic mechanical properties will be reserved for Chapter 8. 

Although bone is a viscoelastic material, at the quasi-static strain rates in mechanical testing and even at the ultrasonic frequencies used experimentally, it is a reasonable first approximation to model cortical bone as an anisotropic, linear elastic solid with Hooke s law as the appropriate constitutive equation. Tensor notation for the equation is written as ... 

With few exceptions, we shall idealize the elasticity of solids as isotropic, as stated earlier, so as not to burden the discussion of the physical mechanisms with inessential operational detail. We note here, however, that many cubic crystals are quite anisotropic. Tungsten, W, which is often cited as being isotropic, is so only at room temperature. Thus, we shall make use principally of the elastic relations in eqs. (4.15) and (4.16), unless we are specifically interested in anisotropic solids such as some polymer product that had undergone deformation processing. The relationships among various combinations of elastic constants of isotropic elasticity are listed in Table 4.1 for ready reference. 

The mathematical representation of the elastic behavior of oriented heterogeneous solids can be somewhat improved through a more appropriate choice of the boundary conditions such as proposed by Hashin and Shtrikman [66] and Stern-stein and Lederle [86]. In the case of lamellar polymers the formalisms developed for reinforced materials are quite useful [87—88]. An extensive review on the experimental characterization of the anisotropic and non-linear viscoelastic behavior of solid polymers and of their model interpretation had been given by Hadley and Ward [89]. New descriptions of polymer structure and deformation derive from the concept of paracrystalline domains particularly proposed by Hosemann [9,90] and Bonart [90], from a thermodynamic treatment of defect concentrations in bundles of chains according to the kink and meander model of Pechhold [10—11], and from the continuum mechanical analysis developed by Anthony and Kroner [14g, 99]. 

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