Elastic behavior anisotropic

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

Wissbrun earlier observed a very long relaxation time and high elasticity for anisotropic melts of aromatic polyesters, as well as several other types of flow anomalies. Unfortunately, in most of these earlier studies, the rheological behavior of liquid crystal melts of polymers could not be directly compared with that of the isotropic phase of the same polymers because of their high clearing temperatures. 

Many composites are elastically anisotropic and, thus, more than the two elastic constants are needed to describe their elastic behavior. This is a very large topic and, for this text, just some of the basic ideas that apply to unidirectional fiber composites will be discussed. Consider a two-phase material, with the two geometries shown in Fig. 3.15, being subjected to a uniaxial tensile stress. For the structure in Fig. 3.15(a), the two phases are subjected to equal strain whereas. 

At temperatures sufficiently below the glass transition and under stresses well below the plastic yield stress to be defined later, all polymers exhibit reversible elastic behavior, which is quite often anisotropic, particularly when it relates to a polymer product that has undergone substantial prior deformation processing. 

McCullough, R. L. Anisotropic Elastic Behavior of Crystalline Polymers, Treatise on Material Science and Technology. New York Academic, pp. 453-540, 1977. 

Blood vessels either delivering oxygenated blood - arteries, arterioles, capillaries or returning with carbon dioxide -veins and venules, display highly nonlinear elastic and anisotropic mechanical behavior and exhibit complex material properties [1,2],... 

For the material models, the following assumptions are made (i) the fibers obey a linear elastic and anisotropic material law, (ii) the matrix follows an isotropic, linear elastic-plastic material law, (iii) the material behavior of the interface elements is described in Section 5.1.2, and (iv) the material model of the anisotropic outer layer is the same as that of the macro-contact model. 

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]. 

Other anisotropic elasticity relations are used to define Chentsov coefficients that are to shearing stresses and shearing strains what Poisson s ratios are to normal stresses and normal strains. However, the Chentsov coefficients do not affect the in-plane behavior of laminaeS under plane stress because the coefficients are related to S45, S46, Equation (2.18). The Chentsov coefficients are defined as... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

The macromechanical behavior of a lamina was quantitatively described in Chapter 2. The basic three-dimensional stress-strain relations for elastic anisotropic and orthotropic materials were examined. Subsequently, those relations were specialized for the plane-stress state normally found in a lamina. The plane-stress relations were then transformed in the plane of the lamina to enable treatment of composite laminates with different laminae at various angles. The various fundamental strengths of a lamina were identified, discussed, and subsequently used in biaxial strength criteria to predict the off-axis strength of a lamina. 

RTR filament-wound pipe is, however, an anisotropic material. That is, its material properties, such as its modulus of elasticity and ultimate strength, are different in each of the principal directions of hoop and longitude. It is here where the design approaches for steel and RTR pipe part company [Fig. 4-2(c)]. This behavior is a result of the construction of filament-wound RTR pipe. 

Microstructural length scales that initially arise from uniform, but unstable, order parameters are readily understood by the perturbation analyses that lead to the amplification factor R(f3) in Eqs. 18.28 and 18.34. When a system is anisotropic such as in a elastically coherent material, the perturbation s behavior may depend on its direction with respect to the material s symmetry axes. 

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