Complex compliance modulus

In the third place the question arises how the time and frequency scales have to be joined together to connect results of creep or relaxation to those of vibration experiments. It will be clear that here the superposition principle can be applied, since with vibrations the imposed stress (or the strain) varies continuously. Using this principle and using the complex E modulus (or the complex compliance D), it can be shown that, with a few simplifications, t and 1/ffl can be considered as the same parameter, while co = 2nv (v is the frequency in cycles/sec). It has been... 

We note that, in principle, the main physical discussions related to filler networking in this paper do not change if a sinusoidal tensile or uniaxial compres-sional stress (amplitude 0) is imposed on the rubber material. In some examples the complex dynamic modulus is then denoted with E = E + iE" and the compliance with C = C - iC". All theoretical considerations use the shearing modulus G. ... 

Figure 6.3 Vectorial components of (a) the complex relaxation modulus G and (b) the complex compliance function J. ...

According to Eqs. (6.19), the relationships between the components of the complex compliance function and those of the complex relaxation modulus... 

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). 

Accordingly, the loss compliance function presents a maximum in the frequency domain at lower frequency than the loss relaxation modulus. This behavior is illustrated in Figure 8.18, where the complex relaxation modulus, the complex creep compliance function, and the loss tan 8 for a viscoelastic system with a single relaxation time are plotted. Similar arguments applied to a minimum in tan 8 lead to the inequalities... 

Thus either G (to) or G"(co) as a function of to gives the information equivalent to that included in G(t) as a function of t. The complex modulus is experimentally a more convenient quantity to describe the linear viscoelasticity of low-viscosity fluids than the relaxation modulus (1). The complex modulus is related to the complex viscosity / (co) and the complex compliance J (to) ... 

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... 

Derive an expression for the complex modulus of a generalized Maxwell model subjected to a sinusoidal strain. Show that the complex compliance is not obtainable... 

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> ). 

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... 

Dynamic mechanical measurements the complex modulus and complex compliance... 

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. ... 

Figure 6.1 Complex shear modulus (a) and complex shear compliance (b) for standard polyisobutylene reduced to 25 °C. Points from averaged experimental measurements curves from a theoretical model for viscoelastic behaviour. (Reproduced with permission from Marvin and Oser, J. Res. Natl. Bur. Stand. B, 66,171 (1962))...

The two components of the complex dynamic modulus (or of the complex compliance) are weighted quite differently with respect to the long-time and short-time contributions to H (or L). For example, as is evident from equations 26 and 27 of Chapter 3, J at a particular frequency o> is determined primarily by the spectral contributions for which o)t < 1, whereas J" is determined by those for which cot CSC 1. If for any reason ar is not the same for long retardation times and short retardation times, not only will the curves for J and J" fail to match in shape, but any attempt at a forced fit will provide one set of apparent ar values for J and a different set for J". Whenever this occurs, the method of reduced variables in its simple form as given above must be rejected no master curves can be drawn without subjecting the data to a more complicated analysis. 

Alternatively, we can take the reciprocal of the complex compliance, set the rubber modulus to zero, and change from T to t, when we get the result ... 

Using Fourier transforms (see Appendix B), it can be shown that the relaxation modulus and creep compliance can be found from the complex modulus and the complex compliance respectively, by the equations. 

Loss Compliance The imaginary part of the complex compliance. See Compliance and Complex Modulus. 

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