Elastic constants isotropic solid

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

Fig. 2.8. Idealized elastic/perfectly plastic solid behavior results in a stress tensor in which there is a constant offset between the hydrostatic (isotropic) loading and shock compression. Such behavior is only an approximation which may not be appropriate in many cases.

Amorphous solids and polycrystalline substances composed of crystals of arbitrary symmetry arranged with a perfectly disordered or random orientation are elastically isotropic macroscopically (taken as a whole). They may be described by nine elastic constants, which may be reduced to two independent (effective) elastic constants. 

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) ... 

In a similar fashion, the rigidity modulus, G, for an elastically isotropic solid is given by 0-4/84 = C44 = 0-5/85 = C55 = cTs/8g = cgg = i(cn - C12) = /r, or C44, which represents a shape change without a volume change. Therefore, the second Lame constant (fi) is the shear modulus for an elastically isotropic body. The Lame constants may also be related directly to the bulk modulus, B, for an elastically isotropic body, which can be obtained through the relations /r = ( )(B - A) and = B - ( )G. 

The force constants of single beam cantilevers (normal spring constant kn, torsional spring constant kv, and lateral spring constant kjJ can be calculated, assuming levers of constant thickness, based on measured cantilever dimensions, from continuum elasticity mechanics of isotropic solids [4-6]... 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

To construct the elastic Green function for an isotropic linear elastic solid, we make a special choice of the body force field, namely, f(r) = fo5(r), where fo is a constant vector. In this case, we will denote the displacement field as Gik r) with the interpretation that this is the component of displacement in the case in which there is a unit force in the k direction, fo = e. To be more precise about the problem of interest, we now seek the solution to this problem in an infinite body in which it is presumed that the elastic constants are uniform. In light of these definitions and comments, the equilibrium equation for the Green function may be written as... 

Eqn (2.92) is the culmination of our efforts to compute the displacements due to an arbitrary distribution of body forces. Although this result will be of paramount importance in coming chapters, it is also important to acknowledge its limitations. First, we have assumed that the medium of interest is isotropic. Further refinements are necessary to recast this result in a form that is appropriate for anisotropic elastic solids. A detailed accounting of the anisotropic results is spelled out in Bacon et al. (1979). The second key limitation of our result is the fact that it was founded upon the assumption that the body of interest is infinite in extent. On the other hand, there are a variety of problems in which we will be interested in the presence of defects near surfaces and for which the half-space Green function will be needed. Yet another problem with our analysis is the assumption that the elastic constants... 

Material Constants, Elastic wave velocities have been obtained for oil shale by ultrasonic methods for various modes of propagation. Elastic constants can be inferred from these data if the oil shale is assumed to be a transversely isotropic solid (9). This is a reasonable approximation considering the bedded nature of the rock. Many of the properties of oil shale depend on the grade (kerogen content), which in turn is correlated with the density ( 10). The high pressure behavior of oil shale under shock loading has been studied in gas-gun impact experiments (11). 

The three-dimensional stresses in a flowing, constant-density newtonian fluid have the same form as the three-dimensional stress in a solid body that obeys Hooke s j law (i.e., a perfectly elastic, isotropic solid). 

The Navier-Stokes equations are the differential momentum balances for a three-dimensional flow, subject to the assumptions that the flow is laminar and of a constant-density newtonian fluid and that the stress deformation behavior of such a fluid is analogous to the stress deformation behavior of a perfectly elastic isotropic solid. These equations are useful in setting up momentum balances for three-dimensional flows, particularly in cylindrical or spherical geometries. 

The response of an isotropic, homogeneous solid to a force is expressed in terms of the elastic constants or elastic moduli. [Unfortunately, a standard set of symbols for these constants is not in use.] Four elastic constants are frequently defined but, as they are interrelated, the elastic properties of a solid can be defined in terms of any two. They are most conveniently defined with respect to the stress, which is the force per unit area applied to the body, and the strain, which is the deformation of the body produced by the force. 

Find the curvature modulii, k and k, of an anisotropic solid film composed of layers where the in-plane symmetry is hexagonal (use the form of the compressional and shear energy expansion for anisotropic media in Ref. 4), in the limit that the in-plane shear elastic constant vanishes. Assume that the film and its elastic constants are uniform throughout its thickness. Compare this case with the isotropic solid film in the limit of vanishing shear modulus. 

Two distinct types of macroscopic theoretical model for the low strain mechanical behaviour of oriented solid polymers will be considered in this chapter. First, models which predict the changes in elastic constants with the development of orientation these will be referred to as orienting element models. Secondly, models which seek to explain the mechanical behaviour of both isotropic and oriented polymers in terms of a two phase material with separate components representing crystalline and amorphous fractions these we shall call composite structure models. 

Fig. 4.12. (a) Temperature dependence of azimuthal Wtp dots) and zenithal Wo (squares) anchoring coefficients for nematic 5CB on rubbed Nylon with Tni being the transition temperature into the isotropic phase (b) A comparison of the ratios of the two anchoring coefficients W jW p (circles) with the ratio of the corresponding Frank elastic constants KijK2 [Q5](solid line) [64]. 

For isotropic elastic solids there are only two independent elastic constants, or compliances. While Young s modulus E and the shear modulus // are the most widely used, we shall choose as the two physically independent pair of moduli the shear modulus /i and the bulk modulus K, where the first gauges the shear response and the second the bulk or volumetric response. However, in stating the linear elastic response in the equations below we still choose the more compact pair of E and //. Thus, for the six strain elements we have... 

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. 

That this form does indeed hold had been demonstrated for the cavitation of glassy polypropylene (Mott et al. 1993) in a computational study furnishing the validity of this extension of the universal binding-energy relation to symmetrical bulk response. The additional attraction of this expression is that it points out directly that application of a pressure produces symmetrical elastic compaction in an isotropic solid. However, more interestingly, one notes that, when dilatation is imposed, the bulk modulus monotonically decreases and eventually, at a dilatation of 1 /p, vanishes. This also leads to the observation that, if dilatation results from thermal expansion in response to a temperature increase, the bulk modulus also decreases. This simple observation represents the essence of the temperature dependence of all other elastic constants in anisotropic solids, beyond the mere effect on the bulk... 

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. 

The JKR theory, similar to the Hertz theory, is a continuum theory in which two elastic semi-infinite bodies are in a non-conforming contact. Recently, the contact of layered solids has been addressed within the framework of the JKR theory. In a fundamental study, Sridhar et al. [32] analyzed the adhesion of elastic layers used in the SFA and compared it with the JKR analysis for a homogeneous isotropic half-space. As mentioned previously and depicted in Fig. 5, in SFA thin films of mica or polymeric materials ( i, /ji) are put on an adhesive layer Ej, I12) coated onto quartz cylinders ( 3, /i3). Sridhar et al. followed two separate approaches. In the first approach, based on finite element analysis, it is assumed that the thickness of the layers and their individual elastic constants are known in advance, a case which is rare. The adhesion characteristics, including the pull-off force are shown to depend not only on the adhesion energy, but also on the ratios of elastic moduli and the layers thickness. In the second approach, a procedure is proposed for calibrating the apparatus in situ to find the effective modulus e as a function of contact radius a. In this approach, it is necessary to measure the load, contact area... 

The first stage in the calculation is to choose a constitutive equation that relates the applied stresses to the resulting strains. For an elastic material, the behavior is described by two independent elastic constants, such as the shear modulus G and the bulk modulus K. The constitutive equation for an isotropic linear elastic solid has the form (28)... 

For an isotropic solid, there are only two independent elastic constants. These two can be taken to be K and G as above, but it is sometimes convenient to use other elastic constants, such as Young s modulus E and Poisson s ratio a. These constants can be calculated from K and G using the standard relations... 

All other terms are zero. In a liquid, there is only one elastic constant K. Note that most polymeric solids are isotropic, either because they are amorphous or because they are polycrystalline, with a random orientation of the crystallites (see Amorphous Polymers Semicrystalune Polymers). 

Solids are characterized by a nonzero static shear elastic constant, whereas a liquid will not support static shear. Conventional viscometric techniques (i.e., capillary flow and falling balls), which now seem rather crude considering the fragile nature of the blue phase lattice, initially showed a large viscosity peak at the helical-isotropic transition [89], [90], [91]. Viscosities and other transport coefficients associated with the pretransitional region of the isotropic phase were addressed by light scattering [92], [93], [94], [95]. 

The Debye temperature, O j, obtainable experimentally from the lattice contribution to the low-temperature specific heat, may be regarded as the iiltimate embodiment of all the elastic constants. ANDERSON [And63] has described how a reliable value of D for an elastically isotropic solid can be caloalated from the monocrystalline moduli, using computed values of the macroscopic moduli as an intermediate step. 

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