Elastically Isotropic Solids

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 first Lame constant (A) has no physical interpretation. However, both Lame constants are related to other elastic moduli. To see this, recall that the Young s modulus, E, is defined as the ratio of normal stress to normal strain. Hence, for an elastically isotropic body, E is given by (cn-Ci2)(cn-b 2ci2)/(cn-I-C12), or /r(3A-b 2/r)/(A-b/r). It should be emphasized that the Young s modulus is anisotropic for all crystal classes, including the cubic class, so this relation would never apply to any monocrystal. 

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. 

In terms of the Lame constants, write the equations relating the six stresses to the strains for a polycrystalline material. 

Using Eq. 10.14 with Table 10.3, it is easily shown that the stresses can be written in terms of the two Lame constants as  

---

This equation relates sM to the orientation of the stress-ellipsoid [cf. eq. (1.3)]. This result is first quoted by Lodge. It differs by a factor of one half from that for the (completely recoverable) simple shear s of a perfectly elastic isotropic solid (50) ... 

According to Reuss, the effective Young s modulus of an elastically isotropic solid is given by ... 

To compute the displacements and strains associated with an inclusion explicitly, we must now carry out the operations implied by eqn (10.7). To make concrete analytic progress, we now specialize the analysis to the case of an elastically isotropic solid. As a preliminary to the determination of the elastic fields, it is necessary to differentiate the elastic Green function which was featured earlier as... 

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. 

C.F. YIng and R. Truell, "Scattering of o Plane Longitudinal Wave by a Spherical Obstacle In an Elastically Isotropic Solid/ J. Appl. Phys. 27,1086 (1956). 

Elastic Modulao. The mechanical behavior is in general terms concerned with the deformation that occurs under loading. Generalized equations that relate stress to strain are called constitutive relations. The simplest form of such a relation is Hooke s Law which relates the stress s to the strain e for rmiaxial deformation of the ideal elastic isotropic solid ... 

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. 

One of the simplest constitutive relations is Hooke s law, which relates the stress a to the strain e for the uniaxial deformation of an ideal elastic isotropic solid. Thus... 

Here, represents the Cauchy stress tensor, p is the mass density, and ft and m, are the body forces and displacements in the i direction within a bounded domain Q. The two dots over the displacements indicate second derivative in time. The indices i and j in the subscripts represent the Cartesian coordinates x, y, and z. When a subscript follows a comma, this indicates a partial derivative in space with respect to the corresponding index. For the special case of elastic isotropic solids, the stress tensor can be expressed in terms of strains following Hooke s law of elasticity, and the strains, in turn, can be expressed in terms of displacements. The resulting expression for the stress tensor is... 

In the case of elastically-isotropic solids there are only two independent elastic constants, Cn and Cn because 2c44 = Cu - C12. Since glassy polymers and randomly oriented semi-crystalline polymers fall into this category it is worth considering how the stiffness constants can be related to quantities such as Young s modulus, E, Poisson s ratio v, shear modulus G and bulk modulus K which are measured directly. The shear modulus G relates the shear stress 04 to the angle of shear 74 through the equation... 

The stress at the surfaces of a solid body leads to a compression (or expansion) of the material. While the deformation is small it becomes measurable for small particles. The measurement of the lattice parameter of small particles may therefore be used to determine the surface stress. For simplicity we consider a sphere of a radius R of an elastically isotropic solid with a compressibility K subject to an isotropic surface stress The incremental work against the surface stress and the bulk elasticity by expanding the body is... 

Figure 3 Deformation of an elastically isotropic solid (a) body under an applied stress a with atomic planes of equilibrium separation aoJ (h) interatomic potential energy as a function of separation x (c) dependence of

Figure 3.8 Edge dislocation in an isotropic elastic body. Solid lines indicate isopotential cylinders for the portion of the diffusion potential of any interstitial atom present in the hydrostatic stress field of the dislocation. Dashed cylinders and tangential arrows indicate the direction of the corresponding force exerted on the interstitial atom.

Elasticity of solids determines their strain response to stress. Small elastic changes produce proportional, recoverable strains. The coefficient of proportionality is the modulus of elasticity, which varies with the mode of deformation. In axial tension, E is Young s modulus for changes in shape, G is the shear modulus for changes in volume, B is the bulk modulus. For isotropic solids, the three moduli are interrelated by Poisson s ratio, the ratio of traverse to longitudinal strain under axial load. 

In Fig. XIV-2 we have a representation of the frequencies of vibration of an elastic solid, under the assumption that the waves are propagated as in an isotropic solid the velocity of propagation being independent... 

The calculation which we have carried out in this section has been limited in accuracy by our assumption that the velocity of propagation of the clastic waves was independent of direction and of wave length. Actually neither of these assumptions is correct for a crystal. Even for a cubic crystal, the elastic properties are more complicated than for an isotropic solid and the velocity of propagation depends on direction. 

The stress tensor a in the perfectly elastic and isotropic solid phase of the porous medium is described by the constitutive equation... 

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. 

Elasticity theory for an isotropic solid shows that the strain along the inclined line OB is... 

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]... 

The elastic Green function is a useful tool in the analysis of a variety of problems in the elasticity of defects in solids. In this problem, we consider the various steps in the derivation of the elastic Green function for an infinite, isotropic solid. In particular, flesh out the details involved in the derivation of eqn (2.90) and carry out the Fourier inversion to obtain eqn (2.91). 

---