Linear viscoelasticity experimental data

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

In this approach the reviews concerned the rheology involving the linear viscoelastic behavior of plastics and how such behavior is affected by temperature. Next is to extend this knowledge to the complex behavior of crystalline plastics, and finally illustrate how experimental data were applied to a practical example of the long-time mechanical stability. 

There are several other comparable rheological experimental methods involving linear viscoelastic behavior. Among them are creep tests (constant stress), dynamic mechanical fatigue tests (forced periodic oscillation), and torsion pendulum tests (free oscillation). Viscoelastic data obtained from any of these techniques must be consistent data from the others. 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

The four variables in dynamic oscillatory tests are strain amplitude (or stress amplitude in the case of controlled stress dynamic rheometers), frequency, temperature and time (Gunasekaran and Ak, 2002). Dynamic oscillatory tests can thus take the form of a strain (or stress) amplitude sweep (frequency and temperature held constant), a frequency sweep (strain or stress amplitude and temperature held constant), a temperature sweep (strain or stress amplitude and frequency held constant), or a time sweep (strain or stress amplitude, temperature and frequency held constant). A strain or stress amplitude sweep is normally carried out first to determine the limit of linear viscoelastic behavior. In processing data from both static and dynamic tests it is always necessary to check that measurements were made in the linear region. This is done by calculating viscoelastic properties from the experimental data and determining whether or not they are independent of the magnitude of applied stresses and strains. 

The derivation of fundamental linear viscoelastic properties from experimental data obtained in static and dynamic tests, and the relationships between these properties, are described by Barnes etal. (1989), Gunasekaran and Ak (2002) and Rao (1992). In the linear viscoelastic region, the moduli and viscosity coefficients from creep, stress relaxation and dynamic tests are interconvertible mathematically, and independent of the imposed stress or strain (Harnett, 1989). 

Time-temperature superposition was first suggested by H. Leaderman who discovered that creep data can be shifted on the horizontal time scale in order to extrapolate beyond the experimentally measured time frame (9-10). The procedure was shown to be valid for any of the viscoelastic functions measured within the linear viscoelastic range of the polymer. The time-temperature superposition procedure was first explicitly applied to experimental data by... 

The experimental ranges of strain rates (or strains) are summarized in Table 2 for the various types of experiments. Time-temperatiire superposition was successfully applied on the various steady shear flow and transient shear flow data. The shift factors were foimd to be exactly the same as those obtained for the dynamic data in the linear viscoelastic domain. Moreover, these were found to be also applicable in the case of entrance pressure losses leading to an implicit appUcation to elongational values. 

Figure 3.10 Predictions of the temporary network model [Eq. (3-24)] (lines) compared to experimental data (symbols) for start-up of uniaxial extension of Melt 1, a long-chain branched polyethylene, using a relaxation spectrum fit to linear viscoelastic data for this melt. (From Bird et al. Dynamics of Polymeric Liquids. Vol. 1 Fluid Mechanics, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

These predictions of the Zimm model are compared with experimental data on dilute polystyrene solutions in two -solvents in Fig. 8.7. The Zimm model gives an excellent description of the viscoelasticity of dilute solutions of linear polymers. 

The described universal scenario of shear-molten glass and shear-thinnig fluid makes up the core of the MCT-ITT predictions derived from (11-14). Their consequences for the nonlinear rheology will be discussed in more detail in the following sections, while the MCT results for the linear viscoelasticity were reviewed in Sect. 3. Yet, the anisotropy of the equations has up to now prevented more complete solutions of the MCT-ITT equations of Sect. 2. Therefore, simplified MCT-ITT equations become important, which can be analysed in more detail and recover the central stability equations (20, 22). The two most important ones will be reviewed next, before the theoretical picture is tested in comparison with experimental and simulations data. 

Figure 20 shows the result for a flow curve, where a small positive separation parameter was necessary to fit the flow curve and the linear viscoelastic moduli simultaneously. The data are compatible with the (ideal) concept of a yield stress, but fall below the fit curves for very small shear rates. This indicates the existence of an additional decay mechanism neglected in the present approach [32, 33]. Again, the A-formula describes the experimental data correctly for approximately four decades. For higher shear rates, an effective Herschel-Bulkley law... 

The theory behind linear viscoelasticity is simple and appealing. It is important to realize, however, that the applicability of the model for fluoropolymers is restricted to strains below the yield strain. One example comparing predictions based on linear viscoelasticity and experimental data for PTFE with 15 vol% glass fiber in the very small strain regime is shown in Fig. 11.4. 

The figure shows that the model predictions are in reasonable agreement with the experimental data. For example, the model quantitatively captures the unloading behavior and hysteresis. An illustration of what happens when linear viscoelasticity is applied to a deformation history that goes past yield is illustrated in Fig. 11.5. This figure shows that linear... 

Figure 11.5 Comparison between experimental data for PTFE (15 vol% glass fiber) and prediotions from linear viscoelasticity theory.

More recently, a new, viscoelastic-plastic model for suspension of small particles in polymer melts was proposed [Sobhanie et al., 1997]. The basic assumption is that the total stress is divided into that in the matrix and immersed in it network of interacting particles. Consequently, the model leads to non-linear viscoelastic relations with yield function. The latter is defined in terms of structure rupture and restoration. Derived steady state and dynamic functions were compared with the experimental data. 

Thus, once the four parameters of Eq 7.42 are known, the relaxation spectrum, and then any linear viscoelastic function can be calculated. For example, the experimental data of the dynamic storage and loss shear moduli, respectively G and G , or the linear viscoelastic stress growth function in shear or uniaxial elongation can be computed from the dependencies [Utracki and Schlund, 1987] ... 

Boltzmann superposition principle A basis for the description of all linear viscoelastic phenomena. No such theor) is available to serve as a basis for the interpretation of nonlinear phenomena—to describe flows in which neither the strain nor the strain rate is small. As a result, no general valid formula exists for calculating values for one material function on the basis of experimental data from another. However, limited theories have been developed. See kinetic theory viscoelasticity, nonlinear, bomb See plasticator safety. 

Extensional measurements involve a simple loading process, but with compression and torsion measurements simplifying assumptions and/or corrections need to be applied to experimental data. In compression it is necessary to assume plane strain," and Poisson s ratio measurements may be performed either at constant strain or constant stress. For a non-linear viscoelastic polymer these two methods are not equivalent. 

The response functions g t—x) and h(t—x) can be written in terms of a, b and c in (19), and can further be used to describe completely the creep response through the analogue of (20). In other words, the complete non-linear viscoelastic response should be describable in terms of these two functions only. Certain features of this representation can be checked very quickly against the experimental data. For example, it can be shown that for a creep experiment with or = 0 for r < 0, and 0, then... 

We have now discussed in turn, the stresses required to produce yield, the relationship between stress and plastic strain increment, the structural reorientation occurring as a result of yield, and the relationship between constant strain-rate yield and features of non-linear recoverable creep deformation. Theoretical models to describe the behaviour have ranged from single crystal plasticity through to the oriented continuum ideas of plasticity and viscoelasticity. On many points both the experimental data and the interpretations appear almost contradictory and it is therefore helpful to see if any common ground can be established. 

Comparison between linear viscoelasticity (one-term Prony series) and experimental data for GUR 1050 (30 kGy, y-Nj). 

Experimental data (-0.02/s, -0.05/s, -0.10/s) Linear Viscoelasticity (2-term Prony series)... 

Entanglements of flexible polymer chains contribute to non-linear viscoelastic response. Motions hindered by entanglements are a contributor to dielectric and diffusion properties since they constrain chain dynamics. Macromolecular dynamics are theoretically described by the reptation model. Reptation includes fluctuations in chain contour length, entanglement release, tube dilation, and retraction of side chains as the molecules translate using segmental motions, through a theoretical tube. The reptation model shows favourable comparison with experimental data from viscoelastic and dielectric measurements. The model reveals much about chain dynamics, relaxation times and molecular structures of individual macromolecules. 

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

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