Complex modulus defined

We could thus represent these two moduli on a phasor diagram as shown in Fig. 2.54. El leads 2 by 90° (tt/2 radians) and from this diagram it is possible to define a complex modulus, E where... 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

Viscosity is the ratio of a stress to a strain rate [Eq. (I l-9)j. Since the complex modulus G has the units of stress, it is possible to define a complex viscosity t] as the ratio of G to a complex rate of strain ... 

For a specific food, magnitudes of G and G are influenced by frequency, temperature, and strain. For strain values within the linear range of deformation, G and G" are independent of strain. The loss tangent, is the ratio of the energy dissipated to that stored per cycle of deformation. These viscoelastic functions have been found to play important roles in the rheology of structured polysaccharides. One can also employ notation using complex variables and define a complex modulus G (o) ... 

A dimensionless quantity called the Deborah number, De, is defined as the fluid s characteristic relaxation time t divided by a time constant tf characterizing the flow (Reiner 1964). Thus, De = t///. In an oscillatory shearing flow, for example, we might take tf to be the inverse of the oscillation frequency (o, and then De = xo). At high Deborah number, the flow is fast compared to the fluid s ability to relax, and the fluid will respond like a solid, to some extent. Thus, in an oscillatory shearing flow, when De = cur 3> 1 the complex modulus is solid-like, while when De = 1 a liquid-like terminal behavior is... 

There are a great number of techniques for the experimental determination of viscoelastic functions. The techniques most frequently found in the literature are devoted to measuring the relaxation modulus, the creep compliance function, and the components of the complex modulus in either shear, elongational, or flexural mode (1-4). Although the relaxation modulus and creep compliance functions are defined in the time domain, whereas the complex viscoelastic functions are given in the frequency domain, it is possible, in principle, by using Fourier transform, to pass from the time domain to the frequency domain, or vice versa, as discussed earlier. 

The complex modulus will be employed throughout this paper except that the dynamic viscosity >j (to) will be used in some cases. In order to describe the low frequency behavior of a material, the zero-shear viscosity rj and the steady shear compliance Je° are used. They are defined... 

In this section we will define several quantities which are convenient for the discussion of the viscoelasticity of dilute polymer solutions. As mentioned above, the two quantities G and G — o)t s related to the complex modulus as functions of to, are the measurable quantities. In order to make comparisons with theory, one has to extrapolate these quantities to infinite dilution, i.e.,... 

Another often-used viscoelastic response function, the complex modulus, is defined by... 

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 stability and energy of formation are plotted for the four emulsions in Fig. 5. The stability in these figures is the complex modulus divided by the starting-oil viscosity. In summary, the stability, as here defined, was found to be the only single parameter that could be used to describe the emulsions mathema-tically. Furthermore, stability was... 

The complex shear modulus is an indicator of the stiffness or resistance of the bituminous binder to deformation under load. The complex shear modulus and the phase angle define the resistance to shear deformation of the bituminous binder in the linear viscoelastic region. Finally, the complex modulus and the phase angle are used to determine or calculate performance-related criteria in accordance to specifications. 

According to CEN EN 12697-26 (2012), complex modulus, E, is defined as the relationship between stress and strain for a linear viscoelastic material submitted to a sinusoidal load wave form at time t, where applied stress o x sin (co x t) results in a strain e x sin (o) x (t - rf>)) that has a phase angle O, with respect to stress (co = angular speed, in radians per second). The amplitude of strain and the phase angle are functions of the loading frequency, f, and the test temperature, . 

Failure, in this test, is defined as the number of loadings where the normalised complex modulus (NCM or NM) (see Section 7.7.2) obtains the highest value. The number of loadings at failure (Nf) expresses the asphalt s resistance to fatigue. 

The response defines the storage modulus G to), loss modulus G"( ), and the complex modulus G ( ),... 

A quantitative measure of energy dissipation, defined as the ratio of stress 90° out of phase with oscillating strain to the magnitude of strain. The loss modulus may be measured in tension of flexure, E , compression, K , or shear, G (see also Complex modulus ). 

The complex modulus is defined by the proportionality constant between the stress and deformation as... 

A complementary treatment can be developed to define a complex compliance J =J — U2, which is directly related to the complex modulus, as G = U. ... 

The resulting strain signal is analyzed in terms of its amplitude (y) and shift 8. Due to linear viscoelasticity, the stress amplitude is proportional to the strain amplitude, and a complex modulus is defined as the ratio of the sinusoidal stress to the strain. The complex modulus E comprises the contribution of both the elastic component and the viscous component ... 

The complex modulus may be defined as the ratio of the sinusoidal stress to strain ... 

In the same manner as above, a complex modulus can be defined as below ... 

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