Complex elasticity modulus

However, in contrast to the cases of complex elastic modulus G and dielectric constant e, the imaginary part of the piezoelectric constant, e", does not necessarily imply an energy loss (Holland, 1967). In the former two, G"/G and e"/e express the ratio of energy dissipation per cycle to the total stored energy, but e"/e does not have such a meaning because the piezoelectric effect is a cross-coupling effect between elastic and electric freedoms. As a consequence, e" is not a positive definite quantity in contrast to G" and e". In a similar way to e, however, the Kramers-Kronig relations (Landau and Lifshitz, 1958) hold for e ... 

In the following, we calculate K and H for the model of polymer film in which part of the amorphous phase is connected in series with the crystalline phase and part in parallel (Fig. 10). The amorphous phase is characterized by a complex elastic modulus,... 

Eg.8 has the same form as the relation between the (real) elastic modulus and sound speed in a lossless medium. From Eqs.6-8 it follows that the sound attenuation coefficient is related to the complex elastic modulus,... 

To model the exchange of matter at the interface of a mixed surfactant solution the same principle can be used as for the system containing only one surfactant. However, one term for each of the r surface active compounds in the system is needed. Garrett Joos (1976) derived the complex elasticity modulus which is given by... 

Only for a generalised linear adsorption isotherm the adsorption of the components i are independent and consequently the functions ai. Assuming a generalised Langmuir type adsorption isotherm Eq. (2.47) the following complex elasticity modulus results. 

Figure 10.18. Evolution of the slope of the complex elastic modulus with changing temperature and formic acid content in the solution, 0 and 40% v/v [FEL 09]...

FIGURE 4 Dependence of Lysozyme complex elasticity modulus on MR concentration at frequency 0.007 and 0.625 rad/s. The drop lines indicate the interval of change corresponding parameters for pure LYS. 

The apparent complex elasticity modulus can obviously be decomposed into components according to... 

From equations 17-19 it follows that the sound attenuation coefficient is related to the complex elastic modulus,... 

This type of behavior is well described by means of complex numbers, and the stress response can be decomposed m two components, an in-phase component and an out of-phase component. In this sense, a complex elasticity modulus may be defined as... 

The imaginary part is related to the real part of the elastic modulus G as rj" = G l(0, where the complex elastic modulus is defined as ... 

A full description of the mechanical properties of the solution requires a study of its complex elastic modulus as a function of frequency. Such a description has been made in detail for polymer melts in the classic book by Ferry (see also Volume 2, Chapter 8 of this work). We limit our discussion here to the macroscopic viscosity. 

To describe the dynamic mechanical properties of two-phase systems, mechanical models may be used. The most distributed is the Takayanagi model developed for two-phase polymer blends [ 176]. Two mechanical models describe the elastic properties of the heterogeneous systems, namely, the complex elastic modulus. The corresponding expressions for the moduli are ... 

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