Compliance coefficient

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). 

To describe the response of the medium on a stress suddenly applied to a solid and held constant, it was introduced unrelaxed (subscript V) and relaxed (subscript R) stiffness and compliance coefficients. The unrelaxed quantities relate to immediate response while the relaxed ones are the coefficients after relaxation occurs. The process of relaxation is characterized with the relaxation time r. Proposed by Zener... 

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. 

The compliance coefficient Sij for silicon, germanium, and diamond are given in Table 8.1. 

Table 8.1. Compliance coefficients (GPa 1) for silicon [55,112], germanium [28] and diamond [5]...

Corresponding matrices can be written for the compliance coefficients, the Sy, based on the inverse equation to Equation 47.2 ... 

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 ... 

When d is substituted as outlined above and the compliance coefficients associated with the induced strain coefficients d are used, then the formulation turns into the upper part of the constitutive equations given on the right-hand side of Eqs. (4.10a). In addition, the actual thermal coefficients may be taken into consideration by the vector a. Thus, supplying specialized finite elements also capable of capturing anisotropic thermal effects with the constitutive coefficients and electric field strength of the electromechanically coupled problem, as given by Eq. (4.16), is a convenient procedure for the case of static actuation. 

Measurements of lateral strains in thin films can be carried out using interferometric techniques [43]. Alternatively, methods have been developed for measuring the appropriate compliance coefficients to enable 21, 031 and 0 2 fo be calculated [41]. For PVDF, values of a again depend on the manufacturer one set of results reported values of 0.25, 0.57 and 0.45 for O21, 0 1 and 0 2 respectively [43]. 

C = creep compliance coefficient T = retardation time fir) = distribution of retardation times... 

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