Isochronal viscoelastic functions

It is difficult to compare the shapes of the viscoelastic functions for single crystal mats and bulk crystalline polymer from the data of Figs. 16-1, 16-2, 16-S, and 16-6 without extensive recalculation under circumstances where approximation methods give poor precision because the functions vary so slowly with time or frequency cf. equation 40 of Chapter 4). Comparisons of the two types of samples will be shown for isochronal viscoelastic functions in Section B below. 

Fig. 35. Typical djaiamic mechanical behavior for viscoelastic functions and G and G" and log damping decrement = tt tan S (isochronal plot, constant frequency = 1 Hz) for a styrene-butadiene copolymer (78). To convert Pa to dynes/cm, multiply by 10.

From a single isochrone, neither the temperature dependence of or nor any of the basic viscoelastic functions can be determined. However, if ar(T ) is known from another source e.g., dielectric measurements under circumstances where a judicious identification can be made) or if a reasonable assumption of its form can be made, the isochrone can be transformed to an effective isotherm simply by plotting against war. and the other viscoelastic functions can be obtained by the methods of Chapter 4. In particular, in the transition zone, ar(T ) can be estimated from the WLF equation in one or another of its forms provided Tg is known. 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

Rheological measurements also show that PS-PI diblock and PS-P1-PS triblock copolymers with /< 0.2 (for either block) exhibit a liquid-like viscoelastic response, even at temperatures below the ODT (Adams et al. 1994 Han et al. 1995 Sakamoto et al. 1997). Han et al. (1995) and Sakamoto et al. (1997) have observed that the ODT cannot be located for these samples based on a discontinuity in the isochronal shear moduli as a function of temperature but can be obtained from plots of logG versus logG" (Fig. 2.4(c)). 

The most obvious problem of non-linearity is the definition of a modulus. For a linear viscoelastic material we need to define not only a real and an imaginary modulus but also a spectrum of relaxation times if we are fully to describe the material - although it is more usual to quote either an isochronous modulus or a modulus at a fixed frequency. We must, for a full description of a non-linear material give the moduli (and relaxation times) as a function of strain as well this will not usually be practicable so we satisfy ourselves by quoting the modulus at a given strain. The question then arises as to whether this... 

Torsional Experiments. The geometry and equations for torsion of an elastic cylinder are presented above. For the viscoelastic K-BKZ material, the equations look similar. For isochronal values of the strain potential function, one can define what looks like a time-dependent strain-energy ftinction Wi(Ii, I2, t) ... 

The strain-clock term (eq. 120) is a function of the entire deformation history. McKenna and Zapas used the strain-clock formalism to describe the torsional response of a PMMA material in two-step strain histories (112). The difficulty arises because the determination of material parameters requires at least the data for both the first and second step responses. Furthermore, McKenna and Zapas also assumed that the clock-form for the torque response and for the normal force response was the same. Their results were consistent with this assumption, as shown in Figure 62. However, that work also indicates that considerable experimental data are required to use the model—a constant issue in nonlinear viscoelasticity. One other interesting thing that came from the work of McKenna and Zapas (112-114) was the verification of equation 60 for the normal force. This is seen in Figure 63, where the normal force response after the half-step history is plotted against the duration of the first step for two different isochrones. As seen, for times beyond 1677 s and for both short and long isochrones, the response is both independent of the duration of the first step and the same as if the material had... 

Typical examples of tensile (isochronous) linear and nonlinear stress-strain diagrams for elastic and viscoelastic materials are shown in Fig, 10.1. For elastic materials, the response is time independent, so there is a single curve for multiple times and the nonlinearity is apparent as a deviation of the stress-strain response from linear. For linear viscoelastic materials, the isochronous response is linear, but the effective modulus decreases with time so that the stress-strain curves at different times are separated from one another. When a viscoelastic material behaves nonlinearly, the isochronous stress-strain curves begin to deviate from linearity at a certain stress level. Fig. 10.2 shows creep compliance data for an epoxy adhesive as a function of stress level for various time intervals after initial loading. 

One important issue in dealing with the nonlinear viscoelastic response of materials is the amount of data needed to determine the material parameters in the models. As noted above, even the general finite elasticity theory requires significant work to obtain the material parameters over the full three-dimensional deformation space. This is one reason that the VL framework is so attractive, when it works. Therefore, it is of interest to investigate whether or not the model can be extended to include compressibility. Pesce and McKenna (146) performed torsional tests on polycarbonate as described above. They then asked whether the VL function could be used to predict the tension and compression responses of the material. An important assumption in their approach was that the VL function determined from the torsional measurements using equations 45, 46, 47, 48, 49, 50, 51 (described immediately above) could be used to predict uniaxial data. When the incompressible equations 50 were applied to try to estimate the uniaxial stress-deformation data (isochronal), these equations did not work. However,... 

It is common to measure some viscoelastic quantity at constmt frequency (or isochronally) over a series of temperatures, thus efficiently gaining a view of all viscoelastic mechanisms. However, since the relaxation function is temperature dependent, employing temperature as the independent variable convolutes the time and temperature dependencies, yielding data that are not amenable to theoretical analysis. The convenience of sweeping temperature at fixed test frequency comes at the expense of rigor. 

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