Isotropy elasticity

It is the dependence of the spatial constitutive functions on the changing current configuration through F that renders the spatial constitutive equations objective. It is also this dependence that makes their construction relatively more difficult than that of their referential counterparts. If this dependence is omitted, then the spatial moduli and elastic limit functions must be isotropic to satisfy objectivity, and the spatial constitutive equations reduce to those of hypoinelasticity. Of course, there are other possible formulations for the spatial constitutive functions which are objective without requiring isotropy. One of these will be considered in the next section. 

We may then write the arrays for the elastic constants for various symmetries, the two most useful being hexagonal (also transverse isotropy as in fibre symmetry) and isotropic. Hexagonal gives ... 

Because of the asumed transverse isotropy it follows that = 1/2 (Cu - Cn). The terms Qj are the elastic stiffnesses expressed in the contracted (Voigt) notation. 

For most particulate composites the mismatch between the particles and the matrix is more important than the anisotropy of either component (though alumina/aluminium titanate composites provide a notable exception and are described below). The main features of the stresses can therefore be understood in terms of a simple elastic model assuming thermoelastic isotropy and consisting of a spherical particle in a concentric spherical shell of matrix with dimensions chosen to give the appropriate volume fractions. The particles are predicted to be under a uniform hydrostatic stress, ap after cooling. If the particles have a larger thermal expansion coefficient than the matrix, this stress is tensile, and vice versa. For small particle volume fractions the stress... 

Determining mechanical characteristics of fibrous materials is far from simple, mainly because of their small diameter. In particular, in the case of anisotropic fibers such as carbon or aramid, we need to determine five elastic constants, assuming isotropy in the cross-sectional plane. Figure 9.3 shows three of the five elastic constants the longitudinal Young s modulus of fiber, E or E, the transverse Young s modulus E22 or Ej, and the principal shear modulus, or Not shown are the two Poisson ratios the longitudinal Poisson s ratio of... 

The liner is an elastic plastic isotropic material. Besides, the laminate behaviour is different from a layer to another and each layer behaves according to the fibre direction. The fibre is assumed to have a transverse isotropy and equivalent properties in the (2-3) plane which normal axis (1) refers to the fibre longitudinal direction, as shown in Figure 1. 

The solution arrived at in our linear elastic model may be contrasted with those determined earlier in the lattice treatment of the same problem. In fig. 5.13 the dispersion relation along an arbitrary direction in g-space is shown for our elastic model of vibrations. Note that as a result of the presumed isotropy of the medium, no g-directions are singled out and the dispersion relation is the same in every direction in g-space. Though our elastic model of the vibrations of solids is of more far reaching significance, at present our main interest in it is as the basis for a deeper analysis of the specific heats of solids. From the standpoint of the contribution of the thermal vibrations to the specific heat, we now need to determine the density of states associated with this dispersion relation. 

Elastic isotropy considerably simplifies the analyses that we are forced to undertake in our goal of characterizing the deformation fields associated with a dislocation. On the other hand, there are some instances in which it is desirable to make the extra effort to include the effects of elastic anisotropy. On the other hand, because the present work has already grown well beyond original intentions and because the addition of anisotropy is for the most part an elaboration of the physical ideas already set forth above, we refer the reader to the outstanding work of Bacon et al. (1979). 

KLI 88] KLIMANEK P., KUZEL R., X-ray diffraction line broadening due to dislocations in non-cubic materials. I. General considerations and the case of elastic isotropy applied... 

Since the Qj are simply related to the technical elastic moduli, such as Young s modulus (T), shear modulus (G), bulk modulus (iC), and others, it is possible to describe the moduli along any given direction. The full equations for the most general anisotropy are too long to present here. However, they can be found in Yoon and Katz [1976a]. Presented below are the simplified equations for the case of transverse isotropy. Young s modulus is... 

Recently, Kinney etal. [2004] used the technique of resonant ultrasound spectroscopy (RUS) tomeasure the elastic constants (Qj) of human dentin from both wet and dry samples. As (%) and Ac (%) calculated from these data are included in both Table 47.5 and Figure 47.4. Their data showed that the samples exhibited transverse isotropic symmetry. However, the Qj for dry dentin implied even higher symmetry. Indeed, the result of using the average value for Q i and Cu = 36.6 GPa and the value for C44 = 14.7 GPa for dry dentin in the calculations suggests that dry human dentin is very nearly elastically isotropic. This isotropic-lifce behavior of the dry dentin may have clinical significance. There is independent experimental evidence to support this calculation of isotropy based on the ultrasonic data. Small angle x-ray diffraction... 

The motility appears to be due to a passive piezoelectric behavior of the ceU plasma membrane [Kahnec and colleagues, 1992]. Iwasa and Chadwick [1992] measured the deformation of a ceU under pressure loading and voltage clamping and computed the elastic properties of the wall, assuming isotropy. It appears that for agreement with both the pressure and axial stiffness measurements, the ceU wall must be... 

Thus the relationships (6.21) and (6.21a) are compatible with the isotropy and incompressibility of a rubber and reduce to Hooke s law at small strains. Materials that obey these relationships are sometimes called neo-Hookeian solids. Equation (6.21a) is compared with experimental data in fig. 6.6, which shows that, although equation (6.21a) is only a simple generalisation of small-strain elastic behaviour, it describes the behaviour of a real rubber to a first approximation. In particular, it describes qualitatively the initial fall in the ratio of to k that occurs once k rises above a rather low level. It fails, however, to describe either the extent of this fall or the subsequent increase in this ratio for high values of k. 

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). 

The added complication is that olivine is not an isotropic material so using p and v (which automatically implies elastic isotropy) would be a simplification. 

Studies of mechanical anisotropy in polymers have been made on specimens of two distinct types. Uniaxially drawn filaments or films have fibre symmetry, with isotropy in the plane perpendicular to the draw direction. Films drawn at constant width or films drawn uniaxially and subsequently rolled and annealed under closely controlled conditions, show orthorhombic symmetry. For fibre symmetry (also called transverse isotropy) the number of independent elastic constants reduces to five and the compliance matrix is... 

Although assumptions of ideal material properties such as linear elasticity and isotropy were considered during simulations of the 3D reconstructed microstructures, the isotropy of the actual microstructures remained questioned. One of the factors responsible for the fact that the microstructures of the films could not possibly be ideally isotropic was that the films experienced constrained sintering which induced greater lateral shrinkage/densification across the direction normal to the surface than that in the other two directions which might result in non-identical... 

To put this relationship under the form of an impedance, one needs to link with the stress, which can be done easily by assuming the linearity of the elasticity (around a working point) and the isotropy of space, so its Fourier transformation being a scalar, one has... 

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