Elastic stiffness coefficient

Equation (5.2) also implies that a crystalline solid becomes mechanically unstable when an elastic constant vanishes. Explicitly, for a three-dimensional cubic solid the stability conditions can be expressed in terms of the elastic stiffness coefficients of the substance [9] as... 

Composites provide an atPactive alternative to the various metal-, polymer- and ceramic-based biomaterials, which all have some mismatch with natural bone properties. A comparison of modulus and fracture toughness values for natural bone provide a basis for the approximate mechanical compatibility required for arUficial bone in an exact structural replacement, or to stabilize a bone-implant interface. A precise matching requires a comparison of all the elastic stiffness coefficients (see the generalized Hooke s Law in Section 5.4.3.1). From Table 5.15 it can be seen that a possible approach to the development of a mechanically compatible artificial bone material... 

The group-theoretical stiffness parameters can be expressed in the conventional (cubic) elastic stiffness coefficients ... 

Because stress and strain are vectors (first-rank tensors), the forms of Eqs. 10.5 and 10.6 state that the elastic constants that relate stress to strain must be fourth-rank tensors. In general, an wth-rank tensor property in p dimensional space requires p" coefficients. Thus, the elastic stiffness constant is comprised of 81 (3 ) elastic stiffness coefficients,... 

Unlike stress and strain, which are field tensors, elasticity is a matter tensor. It is subject to Neumann s principle. Hence, the number of independent elastic coefficients is further reduced by the crystal symmetry. The proof is beyond the scope of this book (the interested reader is referred to Nye, 1957), here the results will merely be presented. For example, even with triclinic crystals, the lowest symmetry class, there are only 21 independent elastic-stiffness coefficients ... 

TABLE 10.3. The Independent Elastic-Stiffness Coefficients for Each Crystal Class. If an Unlisted Coefficient is not Related to a Listed One by Transpose Symmetry (c,y= Cy/), it is Zero-Valued... 

Upon inspection of Table 10.3, it can be seen that there are twelve nonzero-valued elastic-stiffness coefficients. Some of these are related by the crystal class and some by transpose symmetry, with the result that there are only six independent coefficients Cn = C22 C12 C13 = C23 C33 C44 = C55 Cee- All other components are zero-valued. Hence, the matrix with all the nonzero independent coefficients designated as such is straightforwardly written as ... 

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) ... 

By using the relations between the elastic-stiffness coefficients in the cubic class from Table 10.3 in Eq. 10.19, the Voigt approximation of the Young s modulus is obtained for a material with cubic symmetry ... 

Using the relations between the elastic-stiffness coefficients from Table 10.3 in Eqs. 10.20 and 10.26, one may also derive the Voigt and Reuss approximations for the rigidity modulus of a cubic monocrystal. These are given by Eqs. 10.35 and 10.36, respectively ... 

The EAM analytical potentials (Eq. 10.46) are multi-variable functions. Their second derivatives yield accurate estimates for the elastic-stiffness coefficients. However, calculating the second derivative of a potential with terms beyond the pair... 

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. 

TABLE 47.2 Elastic Stiffness Coefficients for Various Human and Bovine Bones... 

Table 48.3 shows primary and calculated secondary pyroelectric coefficients of PVF2 films, together with thermal expansion coefficients, and elastic stiffness coefficients, obtained by Kepler and Anderson [18]. The corresponding piezoelectric coefficients were earlier shown in Table 48.1. 

TABLE 48.3. Pyroelectric coefficients p, and pf, thermal expansion coefficients an, and elastic stiffness coefficients Cmn of PVF2 films [18],... 

Elastic Coefficients The elastic stiffness coefficients Cy can be calculated from the measured velocity of propagation of bulk acoustic ultrasonic waves, according to the Papadakis method (quartz transducer with center frequency of 20MHz) (Papadakis, 1967), on differently oriented bar-shaped samples using the equations given by Truell et al. (1969) and corrected for the piezoelectric contributions (Ljamov, 1983 Ikeda, 1990). The samples were oriented in axial directions XYZ, and 45° rotated against the X- and Y-axes, respectively. In order to obtain optimized values for the elastic materials parameters, the elastic stiffness coefficients Cy were used to calculate and critically compare the results of surface acoustic wave (SAW) measurements. 

For continuous media, the propagation velocities for longitudinal and transverse waves is given by v-l = fCuJp and Vj = sJCu/p and the dispersion relation is ct) = vjc. For a monatomic chain of atoms, the dispersion relation is given by Equation 16.7, (o = sin(fai/2). The ratio of /3/m can be related to the elastic stiffness coefficient... 

---