Elastic compliance coefficients

Generally, the elastic properties of crystals should be described by 36 elasticity constants Cit but usually a proportion of them are equal to zero or are interrelated. It follows that in crystals, the tensors (2.6) and (2.7) are symmetric tensors, owing to which the number of elastic compliance coefficients is reduced, e.g., in the triclinic configuration, from 36 to 21 (Table 2.1). With increasing symmetry, the number of independent co-... 

The six independent strain components can likewise be given, as a function of stress, in terms of 36 elastic-compliance coefficients, Sy ... 

TABLE 10.4. The Independent Elastic-Compliance Coefficients for Each Crystal Class. If an Unlisted Coefficient is not Related to a Listed One by Transpose Symmetry (s,y= Sy/), it is Zero-Valued... 

Likewise, using the relations between the elastic-compliance coefficients from Table 10.4 in Eq. 10.25, gives the Reuss approximation of the Young s modulus of a cubic crystal ... 

Other explicit equations for the relationship between the elastic-stiffness and elastic-compliance coefficients in the various crystal classes can be found in Nye s book (Nye, 1957). 

These are called the first piezoelectric equations, Sij are elastic compliance coefficients, are dielectric constants and dmi are piezoelectric coefficients (or piezoelectric moduli). Ifwe taking E and [S] as variables, the second piezoelectric equations will be as follows ... 

Equations (6) and (7) express these relationships. are the elastic compliance constants OC are the linear thermal expansion coefficients 4 and d jj,are the direct and converse piezoelectric strain coefficients, respectively Pk are the pyroelectric coefficients and X are the dielectric susceptibility constants. The superscript a on Pk, Pk, and %ki indicates that these quantities are defined under the conditions of constant stress. If is taken to be the independent variable, then O and are the dependent quantities ... 

The coefficients Cn are called elasticity constants and the coefficients Su elastic compliance constants (Azaroff, 1960). Generally, they are described jointly as elasticity constants and constitute a set of strictly defined, in the physical sense, quantities relating to crystal structure. Their experimental determination is impossible in principle, since Cu = (doildefei, where / i, and hence it would be necessary to keep all e constant, except et. It is easier to satisfy the necessary conditions for determining Young s modulus E, when all but one normal stresses are constant, since... 

An excellent reference describing appropriate ways of measuring the piezoelectric coefficients of bulk materials is the IEEE Standard for Piezoelectricity [1], In brief, the method entails choosing a sample with a geometry such that the desired resonance mode can be excited, and there is little overlap between modes. Then, the sample is electrically excited with an alternating field, and the impedance (or admittance, etc.) is measured as a function of frequency. Extrema in the electrical responses are observed near the resonance and antiresonance frequencies. As an example, consider the length extensional mode of a vibrator. Here the elastic compliance under constant field can be measured from... 

Not all the tensor components are independent. Between Eqs (6.29a) and (6.29b) there are 45 independent tensor components, 21 for the elastic compliance sE, six for the permittivity sx and 18 for the piezoelectric coefficient d. Fortunately crystal symmetry and the choice of reference axes reduces the number even further. Here the discussion is restricted to poled polycrystalline ceramics, which have oo-fold symmetry in a plane normal to the poling direction. The symmetry of a poled ceramic is therefore described as oomm, which is equivalent to 6mm in the hexagonal symmetry system. 

Donor doping in PZT would be expected to reduce the concentration of oxygen vacancies, leading to a reduction in the concentration of domain-stabilizing defect pairs and so to lower ageing rates. The resulting increase in wall mobility causes the observed increases in permittivity, dielectric losses, elastic compliance and coupling coefficients, and reductions in mechanical Q and coercivity. 

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. 

Equations (6) and (7) express these relationships. Sy are the elastic compliance constants are the linear thermal expansion coefficients and are the direct... 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

Tables 4.4-3-4.4-21 are arranged according to piezoelectric classes in order of decreasing symmetry (see Table 4.4-2), and alphabetically within each class. They contain a number of columns placed on two pages, even and odd. The following properties are presented for each dielectric material density q, Mohs hardness, thermal conductivity k, static dielectric constant Sij, dissipation factor tanS at various temperatures and frequencies, elastic stiffness Cmn, elastic compliance s n (for isotropic and cubic materials only), piezoelectric strain tensor di , elastooptic tensor electrooptic coefficients r k (the lat-...

The coefficients syki are called elastic coefficients or elastic compliances. Obviously they are the coordinates of a tensor inverse or reciprocal to the stiflffiess tensor. This is expressed by the relation... 

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