Complex modulus definition

Here t is the resulting shear stress, 6 is the phase shift often represented as tan(d), and (O is the frequency. The term 6 is often referred to as the loss angle. The in-phase elastic portion of the stress is To(cosd)sin(wt), and the out-of-phase viscous portion of the stress is To(sind)cos(complex modulus and viscosity, which can be used to extend the range of the data using the cone and plate rheometer [6] ... 

It must be pointed out that the definition of the damping term depends on the method used. It is, therefore, necessary to express damping in terms of a complex modulus in order to be able to make comparisons. Thus ... 

Basic Damping Concepts and Definitions The complex modulus, E, can be expressed as... 

Detailed comparison of dynamic properties predicted by various theories may be performed by comparing the plots of reduced intrinsic complex modulus [G j as a function of reduced angular frequency coR [see Eq. (1.18) for definition]. The effect of varying h in the Zimm theory as evaluated by Tschoegl (59) is shown in Fig. 2.2. It is obvious... 

Mukherjee and Kushnick s definitions were as follows the interfaeial tension increment, dy, per unit fractional area change, dA/A, is equated to the complex modulus, e (f) ... 

Equations (3.60) through (3.62) can be tested on the basis of a formal blending law of the complex modulus, = Xx=a,b Gx (ra) with Gx (co) being the bare (nonnormalized) complex modulus of the component X in the blend. (This definition of Gx (co) is identical to thaf utilized in Equation 3.58.) As explained earlier, the CR effect on the viscoelastic relaxation changes on... 

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 each case the attenuation of sound can be formally represented by defining a complex wavenumber, where the sound attenuation coefficient is the imaginary part of the wavenumber. The complex wavenumber also leads to the definition of a complex sound speed and a complex dynamic elastic modulus. 

If the catalyst particles are not completely wetted by the liquid phase and the pores consequently not completely filled with liquid phase (static holdup gives some indication of whether this is the case or not), the situation is considerably more complex. In addition to being a function of the Thiele modulus, the catalytic effectiveness will now depend on the fraction of external wetting, rjcs, and the fraction of pore volume filled with liquid, rji. Dudokovic [M.P. Dudokovic, Amer. Inst. Chem. Eng. Jl., 23, 940 (1977)] proposed a reasonable approach that accounts for all three factors. If the reaction proceeds only on the catalyst surface effectively wetted by the liquid phase and components of the reaction mixture are nonvolatile, then one can in principle modify the definition of the Thiele modulus to... 

Rheology is not just about viscosity, but also about another important property, namely the elasticity. Complex fluids also exhibit elastic behaviour. Similar to the viscosity defined above being similar to the definition of a Newtonian viscosity, the elasticity of a complex material can be defined similar to its idealised counterpart, the Hookean solid. The modulus of elasticity is defined as... 

The quantities E and G refer to quasi-static measurements. When cyclic motions of stress and strain are involved, it is more convenient to use dynamical mechanical moduli. The complex Young s modulus is then defined as = " + iE", where E is the storage modulus and " the loss modulus. The storage modulus is a measure of the energy stored elastically during deformation the loss modulus is a measure of the energy converted to heat. Similar definitions hold for G, J, and other mechanical properties. 

The quantities E and G refer to quasistatic measurements. When cyclical or repetitive motions of stress and strain are involved, it is more convenient to talk about dynamic mechanical moduli. The complex Young s modulus has the formal definition... 

According to Eq. (11) the cell ensemble should be characterised by the universal mean relaxation mode time , i.e. by the mean relaxation time = Tkin (Eq. 9). Hence, the distortions of coc produced during cell growth should bring about a defined and typical frequency dispersion . To describe this we define the complex density n(a>c) (in analogy to the definition of a complex shear modulus) by... 

---