Anisotropic Elasticity

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. 

More serious errors may result when the grain-size of a specimen is small compared with the size of an indentation. Then, since all crystals are elastically anisotropic a rigid indenter will produce differing amounts of elastic strain in the grains depending on their orientations. This will create an effective roughening of the surface and increase the friction coefficient. This may result in overestimates of hardnesses. For example, this may underlie reports of nanocrystalline materials being harder than diamond. 

Fine-scale, spatially periodic microstructures are characteristic of spinodal decomposition. In elastically anisotropic crystalline solutions, spinodal microstructures are aligned along elastically soft directions to minimize elastic energy. Microstructures resulting from continuous ordering contain interfaces called antiphase boundaries which coarsen slowly in comparison to the rate of the ordering transformation. 

The craze is modeled by an elastic anisotropic medium with Young s moduli i and 2 corresponding to the stiffness of the cross-tie and the main fibrils ( 1 2). As the strip thickens along the direction 2 (Fig. 4b), the elastic energy density W is stored so that the energy release rate at failure... 

Brown suggested the use of Eq. 17 together with the energy release rate estimated for an edge-loaded elastic anisotropic specimen as [46,47]... 

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) ... 

As shown schematically in Figure 13.4c, the strain can change among different crystallites, e.g. as a consequence of plastic deformation in an elastically anisotropic medium, but can also vary across each crystallite, e.g. because of the presence of dislocations (the two terms are sometimes referred to as strain of the second and third kind, respectively). Notably, a strain broadening effect can be observed even if the macrostrain (mean value of the strain distribution) is zero (Figure 13.4c), as in a powder sample. Otherwise, the simultaneous effect of macrostrain and microstrain results in a shift and broadening of the diffraction profile (Figure 13.4d). 

A stability analysis has shown that twist disclinations are less favourable than wedge disclinations in elastically anisotropic media. This may explain why the former are so rarely seen in ordinary nematics. 

Real nematics are, of course, elastically anisotropic. In certain situations, as for example at temperatures close to the nematic-smectic transition, the anisotropy becomes very large and certainly cannot be ignored. We shall now investigate some of the consequences of elastic anisotropy on the properties of disclinations. 

We have so far assumed Kxx=K22=K =K. Real nematics are of course, elastically anisotropic. From the values of the energies of the disclinations calculated for the case where Ki =K22 K22 it follows that the wedge disclinations are more stable than the twist disclinations if A 22>A =Af i=A 33, and vice versa. Anisimov and Dzyaloshin-skii [38] have shown that lines of half-integral strength may be stable against three-dimensional perturbations if the twist elastic constant (22 i/2(A, j-i-A"3 3). More precisely,... 

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. 

Various methods have been proposed (for example, Wolcott, 1959, and Anderson, (1963)) for calculating a mean, or isotropic, acoustic wave velocity from elastic anisotropic crystals. In this chapter, Anderson s (1963) averaging scheme was employed where d s were computed from Q s because it is inherently suited to such calculations. In his scheme, Anderson approximates the mean velocity, v, by which is defined in the following expression... 

Otto JW, Vassiliou JK, Frommeyta G (1998) Nonhydrostatic compression of elastically anisotropic poly crystals. L Hydrostatic limits of 4 1 methanol-ethanol and paraffin olL Phys... 

Compressive stresses were applied to the bicrystal (constant stress arises from the application of constant strain along the X- and Y-directions, as illustrated in Fig. lb, where X, Y- and Z-axes define laboratory coordinates and [100], [010], and [001] define crystallographic axes, respectively), giving rise to a change in the elastic free energy of each grain (the crystals are elastically anisotropic) and a thermodynamic driving force for GB displacement. Computations details are described in our previous papers [21,22]. 

Ligament/tendon -Tension -Elasticity -Anisotropic, aligned uniaxially Ultimate stress = 50-150 MPa -Tensile modulus = 1.2-1.8 GPa... 

The results for characterizing stress in elastically anisotropic cubic single crystal films presented in Section 3.5 and in this section can also be adapted, under appropriate conditions, for textured polycrystalline films. An equi-biaxial mismatch strain in the (001) or (111) textured thin film of a cubic crystal results in an equi-biaxial mismatch stress. Because the response is transversely isotropic, the biaxial moduli derived for cubic crystals of these orientations can also be used to characterize polycrystalline thin... 

In all of the present theories about the excitation of nematic or cholesteric liquids by an electric field, the mesomorphic material is treated as a continuous elastic anisotropic medium. The Oseen -Frank elastic theory is used to describe the interaction between the applied field and the fluid. The application of an electric field causes the liquid crystal to deform. For a material with a positive dielectric anisotropy, Ae = > 0, the director aligns in the direction of... 

Every, A. G., 1980, General closed-form expressions for acoustic waves in elastically anisotropic solids, Phys. Rev. B 22,1746-1760. 

R.W. Armstron, A.K. Head, Dislocation queueing and fracture in an elastically anisotropic... 

Let us formulate the relaxation equations of state of a nonlinear anisotropic viscoelastic liquid. The introduction of a term which contains the time derivative of the stress is one of the simplest ways of modifying the equation of state of a viscous anisotropic liquid for taking the elastic anisotropic properties into consideration ... 

The elastic properties of many intermetallic compounds have been investigated extensively since 1967, the year in which the first edition of Intermetallic Compounds, edited by Westbrook, was published. In the present chapter, the elastic constants of single-crystal intermetallic compounds have been summarized, and the elastically anisotropic behaviors of single crystals have been discussed. The elastic properties of dislocations as calculated from the anisotropic elasticity have been shown. The elastic moduli of polycrystalline materials have also been summarized. Finally, the effects of various factors like composition, temperature, etc. on the elastic moduli were described. 

The fact that most ordered alloys are elastically anisotropic (see Chapter 37 by Nakamura in this volume) prevents one from comparing forces or energies analytically and, except for specific crystal directions... 

Under certain conditions on overall isotropy, there are narrower but algebraically more complicated bounds usually referred to as the Hashin-Shtrikman bounds (see, e.g.. Ref. 1). However, for not too elastically anisotropic single crystals, one is usually satisfied with bulk and shear moduli given by the Voigt-Reuss-Hill approximation, which has the forms... 

In elastically anisotropic rocks, the velocity of wave propagation depends on the direction of propagation. Obvious anisotropy exists in sedimentary rocks with layering (bedding) and schists. Crampin and Lovell (1991) list five possible sources of seismic anisotropy ... 

In an anisotropic material, coupling can exist between all stresses and all strains. Hence, the generalized Hooke s law for a linearly elastic anisotropic material, expressed using matrix algebra, has no zeros appearing in the compliance matrix ... 

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