Complex elasticity

From strength of materials one can move two ways. On the one hand, mechanical and civil engineers and applied mathematicians shift towards more elaborate situations, such as plastic shakedown in elaborate roof trusses here some transient plastic deformation is planned for. Other problems involve very complex elastic situations. This kind of continuum mechanics is a huge field with a large literature of its own (an example is the celebrated book by Timoshenko 1934), and it has essentially nothing to do with materials science or engineering because it is not specific to any material or even family of materials. 

A range of complex, elastic-plastic behaviors are observed experimentally they are perhaps the most widely encountered and most typical of shock behaviors, but they are perhaps the least understood of the materials responses. Unfortunately, nonspecialists seldom consider realistic elastic-plastic descriptions of shock processes. This section summarizes the very large body of information available in this area. The metallurgical mud is most viscous in this area. 

In a periodic dilatational experiment, the complex elasticity module is a function of the angular frequency ... 

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,... 

Solution In the case of harmonic motion, for which Sj= joiSj, Equation 2.17 implies that attenuation may be accounted for by representing the elastic constants cjj by complex elastic constants cu + jmr u. (This is analogous to accounting for dielectric loss in electromagnetic and optical waveguides by the well-known method of postulating a complex dielectric constant or a complex index of refraction.) Equation 2.13, the lossless wave equation for this shear wave, becomes... 

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,... 

When the dimensions of the scatterers are much smaller than the wavelength of sound simple expressions for f(9) and c are obtained in terms of the complex elastic moduli of the inclusion and host materials, and the volume fraction of the inclusions. Alternately, static self-consistent mean field models can be used to derive expressions for the complex effective moduli of the composite material in terms of the complex elastic moduli and volume fractions of the component materials [32,33,34,35]. The propagation wavenumber c can then be expressed in terms of the effective complex moduli of the composite using Eqs.9 and 10. Particularly interesting. 

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. 

The models of the complex elasticity, given by Eqs (6.12) and (6.17) are derived for surfactant solutions well below the critical micelle concentration, CMC. Thus it is assumed that no aggregates exist in the solution bulk. Lucassen (1976) derived the respective function for the case of micellar solutions and obtained... 

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]...

An improvement over the Whitney—Nuismer approach was that by Garbo and Ogonowski [21]. They solved the case of a fastener hole using complex elasticity and recovered the case of an open hole as a special case. They still use a characteristic distance as is done in the Whitney—Nuismer approach, but their method is applied to every single ply. In addition, it allows for any type of combined in-plane loading instead of uniaxial tension or compression. 

As q " is strictly independent of the temperature, equation (2) gives in the fast motion (27cfx 1) as well as in the slow motion case (27ifx 1) the refractive index n(T) at the laser wavelength Xq (c.f (9)). In the acoustic relaxation regime D(q ", T) exeeds n(T). In (35) we present different theoretical curves of D(q ", q , T) calculated under the assumption, that the real part of the complex elastic constant c (q, T) can be written in the form c (q, T) = c (T)-Ac/ 1 + 47i (q,T)x (T). For the exponent P<1 this formular describes a Cole davidson function. The relaxation time x was assumed to follow a VFT law. Under these conditions the OADF deviates from n(T) only well above the TGT and... 

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 oscillation behaviour of the interfacial tension can be described by the complex elasticity E(ico) defined by the following equation... 

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