Imaginary compliance

Figure 10.7 Real and imaginary compliance as a function of frequency.

Figure C2.1.14. (a) Real part and (b) imaginary part of tire dynamic shear compliance of a system whose mechanical response results from tire transition between two different states characterized by a single relaxation time X.

We can separate out the real and imaginary parts by multiplying by (a — ib), as appropriate. This demonstrates the more sophisticated relationship between the complex compliances and the complex moduli ... 

As seen from Eq. (8), Im II(kw) is finite for k — 0, whereas cjk vanishes in this limit. Therefore the spin Green s function has a purely imaginary, diffusive pole near the T point, in compliance with the result of the hydrodynamic theory [12]. 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

For the standard solid model, the real and imaginary parts of the complex compliance function are... 

Values of storage modulus G (a)) and loss modulus G"(m) can then be obtained by separating Eq. V-8 into its real and imaginary parts (Eq. V-2). Viscosity and recoverable compliance in the large N limit can be obtained from ... 

The nomenclature of complex moduli and compliances is also often used. Here the out-of-phase component is made the imaginary part of a complex parameter thus the complex shear modulus G and the complex shear compliance J are defined as... 

Equations (7.26) are called dispersion relations and analogous equations can be derived for /, and J2 (see problem 7.7) and for more general models. They can also be derived for the real and imaginary parts of the dieleetrie constant (see section 9.2.4). The limiting values of G, and /, at low frequencies are called the relaxed modulus and compliance, G, and J, and the limiting values at high frequencies are called the unrelaxed modulus and compliance, G and / ... 

These equations are the dielectric equivalents of the equations developed in section 7.3.2 for the real and imaginary parts of the compliance or modulus. Just as the imaginary part of the compliance or modulus is a measure of the energy dissipation or loss per cycle (see section 7.3.2), so is e". The variations of e and e" with co are shown in fig. 9.2. 

The ratio of the stress to the strain is used to define a complex modulus "(iw), the real part of which is the storage modulus and the imaginary part the loss modulus, i.e., E ( )=E+ E. Alternatively, one can define a complex compliance < (iw) as the ratio of the strain to the stress. For this case, real part is the storage compliance and the imaginary part is the negative of the loss compliance (note that E = l/d> ). 

The real parts of the moduli E and G, are the storage moduli, because they are related to the storage of energy as potential energy and its release in the periodic deformation. The imaginary parts of the moduli, E and G , are the loss moduli and are associated with the dissipation of energy as heat when the materials are deformed. These dynamic moduli can also be expressed in terms of a complex compliance ... 

G is termed the real part of the modulus, and G" the imaginary part. Conversely, we may use as a parameter the complex shear compliance... 

Compliance n. The degree to which a material deforms under stress the reciprocal of the modulus. Thus, in each mode of stress, the material is characterized by three moduli and their reciprocals, three compliances. However, when the stress is varying, the real and imaginary parts of the complex compliances are not equal... 

Loss compliance The imaginary part of the complex compliance. 

The real part G (a>) is called the storage modulus, while the imaginary coefficient G"(< )) is called the loss modulus. The reciprocal of the dynamic modulus, known as the dynamic compliance is also broken down into a real part/ (a)) and an imaginary part/ (< )), known... 

The loss modulus G" and compliance are often called the imaginary parts of their complex counterparts by analogy with the language of dielectrics and optics in which such notation is conventional. However, G" and J" are of course real quantities which are the coefficients of the imaginary terms in equations 24 and 26. The complex notation is particularly convenient for interconverting modulus and compliance as expressed by equations 27 to 30. 

Compliance with these constraints may be demonstrated by showing that the real and imaginary components of the transfer function obey the Kramers-Kronig (K-K) transforms (Kramers, 1929 de Kronig, 1926 Van Meirhaeghe et al., 1976 Macdonald, 1987), as discussed in detail by Macdonald and Urquidi-Macdonald (1985) and Urquidi-Macdonald et al. (1986). 

---