Isotropic material, elastic properties

It is also important to note that although the laminae [ 45] indicates that Ex = Ey =. GN/m, this laminate is not isotropic or even quasi-isotropic. As shown in Chapter 2, in an isotropic material, the shear modulus is linked to the other elastic properties by the following equation... 

When a bar is elongated axially, as in Figure 2-25, it will contract laterally. The negative ratio of the lateral strain to the axial strain is called Poisson s ratio v. For isotropic materials, materials that have the same elastic properties in all directions, Poisson s ratio has a value of about 0.3. 

In general, there are three kinds of moduli Young s moduli E, shear moduli G, and bulk moduli K. The simplest of all materials are isotropic and homogeneous. The distinguishing feature about isotropic elastic materials is that their properties are the same in all directions. Unoriented amorphous polymers and annealed glasses are examples of such materials. They have only one of each of the three kinds of moduli, and since the moduli are interrelated, only two moduli are enough to describe the elastic behavior of isotropic substances. For isotropic materials... 

The big difference between normal isotropic liquids and nematic liquids is the effect of anisotropy on the viscous and elastic properties of the material. Liquid crystals of low molecular weight can be Newtonian anisotropic fluids, whereas liquid crystalline polymers can be rate and strain dependent anisotropic non-Newtonian fluids. The anisotropy gives rise to 5 viscosities and 3 elastic constants. In addition, the effective flow properties are determined by the flow dependent and history dependent texture. This all makes the rheology of LCPs extremely complicated. 

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) ... 

The mechanical properties of a linear, isotropic material can be specified by a bulk modulus, K, and a shear modulus, G. For an ideal elastic solid, these moduli are real-valued. For real solids undergoing sinusoidal deformation, these are best represented as complex quantities [49] K = K jA and G = G -I- jG". The real parts of K and G represent the component of stress in-phase with strain, giving rise to energy storage in the film (consequently K and G are referred to as storage moduli) the imaginary parts represent the component of stress 90° out of phase with strain, giving rise to power dissipation in the film (thus, K" and G" are called loss moduli). 

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. 

As usual, we have invoked the summation convention and in addition have assumed that the material properties are homogeneous. For an isotropic linear elastic solid, the constitutive equation relating the stresses and strains is given by... 

In one study, a model for elastin, the main protein that confers elasticity on solid structures in mammals, had its mobility investigated by examining 1H-13C and 1H dipolar couplings extracted from isotropic-anisotropic correlation experiments.29 The elastic properties of elastin are almost certainly conferred by molecular degrees of freedom, so such studies are important in understanding how this material works in Nature. The motional amplitudes determined from these experiments were found to depend upon the degree of hydration, with the mean square fluctuation angles found to be 11-18° in the dry protein and 16-21° in the 20% hydrated protein. 

For cubic symmetry materials, three independent elastic properties that are orientation dependent are required to describe the mechanical behavior of the material. This anisotropy effect increases significantly the number of the nonzero elements in the FE stiffness matrix leading to alteration in the calculated stress components and the wave speed. In order to test these anisotropy effects, we plot the wave profiles of three different orientations and compare it with the isotropic behavior with a loading axis in the [001] directions as shown in Fig 8. We observed that under the same loading condition, the peak stress of [111] and [Oil] orientations are slightly higher than those of the [001] which is lower that that of isotropic material. Furthermore, wave speed varies moderately with orientation with the fastest moving wave in the [ 111 ] followed by [011 ], isotropic medium and [001 ] respectively. 

All these results apply to a completely general triclinic crystal system whose elastic properties are expressed by the twenty-one independent quantities Cy or Jy. For crystals of higher symmetry there are further relations between the Cy or 5y which reduce their number still further. For the hexagonal and cubic systems these relations are illustrated in fig. 8.1, together with similar relations for a completely isotropic, non-crystalline material. It can be seen that for a hexagonal crystal like ice there are only five non-zero independent elastic constants Jn, i3> % and 44 or the corresponding Cy. 

It is interesting to make an estimate of the extent to which the elastic properties of ice are anisotropic. This can be done with the help of fig. 8.1 and the figures from table 8.i. For an isotropic material = % while for ice these quantities differ by about 10 per cent. Similarly we should have while for... 

Materials reveal their mechanical properties when subjected to forces. The apphcation of a force results in a deformation. The amount of deformation will depend on the magnitude of the force and its direction measured with respect to the crystallographic axes. Both force and deformation are vector quantities. In the discussion below, it will be assumed that all materials are isotropic in this respect and that there is no crystallographic relationship between force and deformation, which are both presumed to be scalars (numbers. See section S4.13). In fact, in much of the discussion, especially of the elastic properties of sohds, the atomic structure is ignored, and the sohds are treated as if they were continuous. This viewpoint caimot explain plastic deformation, and knowledge of the crystal structure of the sohd is needed to understand the... 

The anisotropy of cortical bone tissue has been described in two symmetry arrangements. Lang [1969], Katz and Ukraincik [1971], and Yoon and Katz [1976a,b] assumed bone to be transversely isotropic with the bone axis of symmetry (the 3 direction) as the unique axis of symmetry. Any small difference in elastic properties between the radial (1 direction) and transverse (2 direction) axes, due to the apparent gradient in porosity from the periosteal to the endosteal sides of bone, was deemed to be due essentially to the defect and did not alter the basic symmetry. For a transverse isotropic material, the stiffness matrix [Qj] is given by... 

Another factor of anisotropic design analysis is greater dependence of stress distributions on materials properties. For isotropic materials, whether elastic, viscoelastic, etc., static values often result in stress fields which are independent of material stiffness properties. In part, this is due to the fact that Poisson s ratio is the only material parameter appearing in the compatibility equations for stress. This parameter does not vary widely between materials. However, the compatibility equations in stress for anisotropic materials depend on ratios of Young s moduli for different material axes, and this can introduce a strong dependence of stress on material stiffness. This approach can be used in component design, but the product and material design analysis become more closely related. 

Because the fibers in mat are randomly oriented, mat-reinforced materials have essentially the same strength and elastic properties in all directions in the plane of the plate, that is, they are essentially isotropic in the plane. Consequently, the usual engineering theories and design methods employed for isotropic engineering materials may be applied. It is only necessary to know strength, modulus of elasticity, shearing modulus, and Poisson s ratio of the combined mat and resin. These can be obtained from standard stress-strain measurements made on specimens of the particular combination of fiber and plastic under consideration. 

Moving to a larger scale, let us now look at the influence of microstructure on elastic behavior. As indicated in the previous section, the elastic constants are a fundamental property of single crystals through the geometry and stiffness of the atomic bonds. Thus, one may expect elastic behavior to be controlled simply by the choice of material. By using composite materials, however, one can control the final set of elastic properties with some precision, i.e., by mixing phases with different elastic constants. Clearly, it is useful to be able to predict the elastic constants of a composite from those of its constituents. This has been accomplished for many types of composite microstructures. For this section, however, the emphasis will be on (elastically) isotropic composites, i.e., composites contain-... 

Most polycrystalline solids are considered to be isotropic, where, by definition, the material properties are independent of direction. Such materials have only two independent variables (that is elastic constants) in matrix (7.3), as opposed to the 21 elastic constants in the general anisotropic case. The two elastic constants are the Young modulus E and the Poisson ratio v. The alternative elastic constants bulk modulus B and shear modulus /< can also be used. For isotropic materials, n and B can be found from E and t by a set of equations, and on the contrary. 

Meyer KH, von Susich G, Valko E (1932) Elastic properties of fibres. Kolloid Z 59 208-216 Mooney M (1940) A theory of large elastic deformation. J Appl Phys 11 582-590 Rivlin RS (1949) Large elastic deformations of isotropic materials. V. The problem of flexure. Proc Roy Soc London A 195 463 73... 

As already mentioned, for the fixed direction of the nematic director n the shear modulus is absent because the shear distortion is not coupled to stress due to the material slippage upon a translation. The compressibility modulus B is the same as for the isotropic liquid. New feature in the elastic properties originates from the spatial dependence of the orientational part of the order parameter tensor, i.e. director n(r). It is assumed that the modulus S of the order parameter Qij r) is unchanged. In Fig. 8.4 we can see the difference between the translation and rotation distortion of a nematic. 

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