Compliance functions

Johnson [111] addressed the problem of viscoelastic flow by attempting to modify the JKR equation. In his approach, he postulated a creep compliance function... 

The creep (/) at time / depends on the compliance function J(t), which is a characteristic of the polymer at a given temperature, and on the initial stress time scale has to be employed in J (i.e., ( -0 the time over which that stress was applied. Furthermore, while (0 for any load is given by the product AO17, the stress of concern is the incremental added stress or... 

Integral representations for the time dependent compliance and modulus may be written down similarly as above (9). The creep compliance function is given by... 

Dynamic storage and loss compliance function for the cubic array are presented in Eqs. (T 11) and (T 12), respectively. 

Quality assessments may be delegated to a separate compliance function in the department, but QA should always be apprised of compliance and quality trends. Quality assurance is the flywheel at the center of all the company operations (Fig. 2). 

This is because although 0 = (10), in general, cr(10) oQ (it will usually be less). In principle, the quantities we have defined, E(t), Dit), Gif), and J(i), provide a complete description of tensile and shear properties in creep and stress relaxation (and equivalent functions can be used to describe dynamic mechanical behavior). Obviously, we could fit individual sets of data to mathematical functions of various types, but what we would really like to do is develop a universal model that not only provides a good description of individual creep, stress relaxation and DMA experiments, but also allows us to relate modulus and compliance functions. It would also be nice to be able formulate this model in terms of parameters that could be related to molecular relaxation processes, to provide a link to molecular theories. 

Since the compliance function is a monotonous increasing function of time, /(t -f 0) > J t) if steady state is not reached. In this case one obtains... 

On the other hand, it is possible to relate the shear compliance function to the relaxation modulus by using the ramp experiment described above. Actually, Eqs. (5.35) and (5.52) lead to the expression... 

S.8 LAPLACE TRANSFORM RELATIONSHIPS BETWEEN TRANSIENT RELAXATION MODULI AND TRANSIENT COMPLIANCE FUNCTIONS... 

An apparently easy way to relate transient relaxation moduli and transient compliance functions is by applying Laplace transforms to Eqs. (5.35) and (5.45). By taking into account the convolution theorem, one obtains (see Appendix)... 

In other words, independently of the viscoelastic history in the linear region, the tensile compliance function can readily be obtained from both the shear and bulk compliance functions. For viscoelastic solids and liquids above the glass transition temperature, the following relationships hold when t oo J t) t/T[ [Eq. (5.16)], D t) = y Jt [Eq. (5.21)], and D t)J t)/ >. These relations lead to r 3t that is, the elongational viscosity is three times the shear viscosity. It is noteworthy that the relatively high value of tensile viscosity facilitates film processing. 

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... 

TRANSFORMATION OF COMPLIANCE FUNCTIONS FROM THE FREQUENCY DOMAIN TO THE TIME DOMAIN AND VICE VERSA FOR VISCOELASTIC SOLIDS... 

Therefore these expressions permit us to transform the compliance function from the time domain to the frequency domain. The relationships of Eqs. (6.24) can also be written in terms of sine and cosine Fourier transforms ... 

The inverse of the Fourier transforms of Eqs. (6.25) permits transformation of the compliance function from the frequency domain to the time domain. The pertinent equations are... 

This is the expression commonly found in the literature that relates the real component of the complex compliance in the frequency domain with the compliance function in the time domain. 

ANALYSIS OF COMPLEX CREEP COMPLIANCE FUNCTIONS AT LOW FREQUENCIES... 

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