Shear compliance functions

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... 

Then the Laplace transform of the shear compliance function is given by... 

Figures 3.21(a) and (b) show the same data replotted as the steady shear viscosity r](y) and the steady shear compliance function J° y), the latter defined as...

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. 

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. 

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). 

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. 

In the above considerations, a sinusoidal shear strain is applied to the sample. It should be clear that a sinusoidal shear stress could also be applied resulting in corresponding compliance functions J and J". The former results from the deformation in phase with the stress, while the latter corresponds to the out-of-phase deformation. The value of tan 5 remains the same, as can be seen from the curves in Figure 2-13, where we can easily imagine the stress as the applied variable and strain as the measured variable. Tensile stress is equally applicable and definitions of E (co), E" (o), D"(co), D co), etc. are completely analogous to the derived shear parameters. At a given frequency, the value of tan 8 is always the same for any of these quantities, i.e., tan 8 = E"/E = D"/D . 

When the strains or the strain rates are sufficiently small, the creep response is Unear. In this case, when the time-dependent strain is divided by the fixed stress, a unique creep compUance curve results that is, at each time there is only one value for this ratio, which is the compliance—y(t)lao = J t). The unique shear creep compliance function J t) (Pa or cm /dyne, 1 Pa = 0.1 cm /dyne) obtained for an amorphous polymer has the usual contributions... 

FIGURE 5.17 Logarithmic presentation of the recoverable shear compliance, Jr(t), of Epon 100 IF as a function of the logarithm of time I at nine temperatures as indicated. Dramatic loss of long-time viscoelastic mechanisms is evident when temperature is decreased toward Tg. 

For orthorhombic symmetry on the other hand, tensile creep and lateral compliance measurements on specimens cut from oriented sheet will yield only 6 of the 9 required creep functions those not accessible by this method being Suit), Sssit) and S66(t)- The two shear compliances 555(1) and Seeit) can be obtained by torsional creep experiments, but these need to be carefully designed and involve complex experimental procedures. The only possibility for measurement of Su(t) on sheet appears to be by compressive creep techniques, however, one would expect substantial experimental difficulties largely associated with strain measurement and specimen geometry. There appears to be no reported evaluation of the full characterisation of creep for the case of orthorhombic synunetry. 

A certain pipe-grade PVC deforms in shear with a creep compliance function at 20°C of the form... 

With all of the viscoelastic functions it is important to note the limiting values or forms which are qualitatively independent of the molecular structure. For a viscoelastic liquid, lini, /(f) = Jg, lim, /(f) = tlr], and lim,, Jr t) = J t) -thr] = Jg + Jd = /j.The last Umiting value Js is called the steady-state recoverable shear compliance. It is the maximum recoverable strain per unit stress, which reflects the maximum configurational orientation achievable at the present stress. 

These relations enable one to relate the shear viscoelastic functions to their tensile counterparts. At high compliance levels, rubbers are highly incompressible, and the proportional relation between the tensile and shear moduli and compliances holds. However, at lower compliances approaching Jg, the Poison ratio fi (which in an elongational deformation is -(Mw/dM, where w is the specimen s width and / is its length) is less than Eqs. (28) and (29) are then no longer exact. For a glass ju T. When G(t) = K(t), E t) = 2.25 Gif). 

The Lo(t) that was obtained is shown in Fig. 12.22 with the retardation spectrum obtained from the shear creep compliance function, J(t). The levels of the short time behavior are matched. Three feamres should be noted. The functionality at short times is the same within experimental uncertainty. The 1/3 slope of log L(t) at short time indicates that the response is dominated by motions that contribute to Andrade creep. [66,85-87,141,142]... 

In both equations the shear rate dependence of die swell is directly evident. The possibility of die swell going through a maximum by changing the MW and MWD as described above is evident from the steady shear compliance (/g). This value is known to show a maximum for polymers - each having different MW and narrow MWD - as a function of the blend ratio. Most commercial pol)aners can also be considered as blends of narrowly distributed molecular weight fractions. 

The Viscoelastic Material Functions. In linear viscoelasticity, the moduli discussed for the elastic case can be recast as time- or fi equency-dependent functions. The same is true for the compliance functions that are discussed here. For simplicity, consider the shear modulus G which becomes G(t) or G (a>) in the case of the viscoelastic material. An important point here is that the viscoelastic modulus functions all exhibit time (frequency) dependence. Hence, one will have functions for K(t) and E(t) [or, eg, G t) and v i)] and these are required in the case of a three-dimensional strain or stress field. 

The notation used to describe the contacts is shown in Figure 1. P t) is the time dependent applied load, S P,t) the deformation, a(P,t) the contact radius, and R and Ri the radii of curvature of the two bodies at the point of contact. We consider only flat substrates so that R R and R2 = >. Each elastic material is described by its Young modulus E, Poisson ratio v, and is assumed to be isotopic so that the shear modulus is G = Ejl + v). Viscoelastic materials are assumed to be linear with stress relaxation functions E t) and creep compliance functions J t), All properties are assumed to be independent of depth. 

From knowledge of the shear relaxation modulus, the memory function, or the creep compliance function of a particular material, its stress-strain relations for... 

As an example of bulk viscoelastic behavior, data for a poly(vinyl acetate) of moderately high molecular weight are shown in Fig. 2-9. Measurements by McKinney and Belcher of the storage and loss bulk compliance B and B" at various temperatures and pressures are plotted after reduction to a reference temperature and pressure of 50°C and 1 atm respectively (see Chapter 11). The complex bulk compliance is formally analogous to the complex shear compliance, but the two functions present several marked contrasts. 

The characteristic function in this integral which specifies the relaxation properties of the system is Lj(logr), being called the retardation time spectrum of the shear compliance J ... 

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