Second-order elastic constants

The fundamental quantities in elasticity are second-order tensors, or dyadicx the deformation is represented by the strain thudte. and the internal forces are represented by Ihe stress dyadic. The physical constitution of the defurmuble body determines ihe relation between the strain dyadic and the stress dyadic, which relation is. in the infinitesimal theory, assumed lo be linear and homogeneous. While for anisotropic bodies this relation may involve as much as 21 independent constants, in the euse of isotropic bodies, the number of elastic constants is reduced lo two. 

The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

The second- and higher-order elastic constant studies in single crystals with large Hugoniot limits have provided an examination of elastic behavior... 

An isotropic and non-magnetie amorphous alloy has only two independent second order elastic constants. The other elastic moduli are related through the equations [30] ... 

Determine the form of the matrix C of second-order elastic constants for crystals with the point group 0. 

Now we consider the valence angle bending force field as it appears from the DMM picture. For this end the geometry variation given by the vectors eq. (3.139) must be inserted in eq. (3.72) and the required elasticity constant can be obtained by extracting the second order contribution in vectors 6[Pg.260]

The five distinguishable second-order elastic moduli in a hexagonal crystal are Cn, Cn, Ci3, C33 and C44. There are reports of neither measured nor calculated values but, since each depends principally on the lattice constants [21] which vary by only about 10% across the nitrides, values for AIN (q.v.) may be used as a first approximation. The comparability of bulk moduli in indium, gallium and aluminium nitrides supports this approach. Estimates of the principal transverse and longitudinal elastic constants Ct and Ci are given in TABLE 2. 

We note that when the losses are too large, free oscillations cannot be excited. For this reason it is compulsory to use the elastic auxiliary element in order to get information on the viscoelastic functions. A stiff elastic element, with constant k, can be added to reduce the loss. When the loss of the system is sufficiently small, the discrepancies between the results obtained from the former theory and the solution based on the classical second-order differential equation [see Eq. (7.49), for example]... 

F. Jahnig and F. Brochard, Critical Elastic Constants and Viscosities above a Nematic-Smectic A Transition of Second Order, J. Phys. 35 (1974) 301 ... 

Figure 3. Spontaneous strains and elastic properties at the 422 < i> 222 transition in Te02. (a) Spontaneous strain data extracted from the lattice-parameter data of Worlton and Beyerlein (1975). The linear pressure dependence of (e - (filled circles) is consistent with second-order character for the transition. Other data are for non-symmetiy-breaking strains (e + 62) (open circles), 63 (crosses), (b) Variation of the symmetry-adapted elastic constant (Cn - Cu) at room temperature (after Peercy et al. 1975). The ratio of slopes above and below Po is 3 1 and deviates from 2 1 due to the contribution of the non-symmetry-breaking strains. (After Carpenter and Salje 1998).

The next step is to calculate the constant of proportionality between the stress and the strain, the elastic compliance matrix. This is the inverse of the elastic constant matrix (the second derivative of energy with respect to strain), which is determined by again expanding the lattice energy to second order ... 

Equations (A.7) also show that, in general, the prediction of a property that depends on a tensor of rank I will require knowledge of orientation averages of order /. The elastic constants of a material are fourth-rank tensor properties thus the prediction of their values for a drawn polymer involves the use of both second- and fourth-order averages, in the simplest case P ico O)) and P (x>s6)), and thus provides a more severe test of the models for the development of orientation. The elastic constants are considered in section 11.4. 

There are two possible sources for the discrepancies between measured and predicted elastic constants when the orientation averages used are determined from models for molecular orientation the models may be incorrect and neither Voigt nor Reuss averaging may be appropriate. In order to examine the second of these possibilities more directly it is necessary to determine the orientation averages experimentally. 

---