Linear continued viscoelasticity

Dependence of relaxation modulus upon time for linear continuous line] and cross-linked broken line) polymers, showing four regions of viscoelastic behavior... 

An important and sometimes overlooked feature of all linear viscoelastic liquids that follow a Maxwell response is that they exhibit anti-thixo-tropic behaviour. That is if a constant shear rate is applied to a material that behaves as a Maxwell model the viscosity increases with time up to a constant value. We have seen in the previous examples that as the shear rate is applied the stress progressively increases to a maximum value. The approach we should adopt is to use the Boltzmann Superposition Principle. Initially we apply a continuous shear rate until a steady state... 

Figure 6.3 Plot of a simple non-linear viscoelastic response for (a) the stress relaxation as a function of the applied strain, (b) stress as a function of time at a shear strain y = 1 and (c) viscosity as a function of shear stress. (r (0) = 33Pas, rj(co) = 3 Pas, a = 1, P = 0.1, m = 0.35 and t = Is). Continued overleaf...

Let the linear viscoelastic body be represented by a continuous spectrum of relaxation times,... 

Apply the Boltzmann superposition principle for the case of a continuous stress application on a linear viscoelastic material to obtain the resulting strain y(t) in terms of J(t — t ) and ih/dt, the stress history. Consider the applied stress in terms of small applied At,-, as shown on the accompanying figure. 

While the dynamic experiments described above are to be conducted in the linear viscoelastic range, another experiment can be conducted in which the results obtained in the non-linear range are useful. With a controlled-stress rheometer, one can conduct an experiment in which the stress is increased continuously at a constant oscillatory frequency, say 1 Hz. Results obtained in such an experiment are shown schematically in Figure 3-40. As the stress is increased continuously, initially, G and G" remain relatively constant until at a critical value of stress, Oc, the magnitude of G decreases sharply and that of G" also decreases not as sharply after a slight inerease. One may also use the value of the applied stress at which the curves of G and G" intersect... 

In this book, we review the most basic distinctions and similarities among the rheological (or flow) properties of various complex fluids. We focus especially on their linear viscoelastic behavior, as measured by the frequency-dependent storage and loss moduli G and G" (see Section 1.3.1.4), and on the flow curve— that is, the relationship between the "shear viscosity q and the shear rate y. The storage and loss moduli reveal the mechanical properties of the material at rest, while the flow curve shows how the material changes in response to continuous deformation. A measurement of G and G" is often the most useful way of mechanically characterizing a complex material, while the flow curve q(y ) shows how readily the material can be processed, or shaped into a useful product. The... 

Chapters 5 and 6 discuss how the mechanical characteristics of a material (solid, liquid, or viscoelastic) can be defined by comparing the mean relaxation time and the time scale of both creep and relaxation experiments, in which the transient creep compliance function and the transient relaxation modulus for viscoelastic materials can be determined. These chapters explain how the Boltzmann superposition principle can be applied to predict the evolution of either the deformation or the stress for continuous and discontinuous mechanical histories in linear viscoelasticity. Mathematical relationships between transient compliance functions and transient relaxation moduli are obtained, and interrelations between viscoelastic functions in the time and frequency domains are given. 

Ageing plays a key role in the non-linear viscoelastic behaviour of polymers. When left at rest and at constant temperature, there is continuous stiffening. However, when the aged material is slightly heated or mechanically deformed, it is deaged and softened (Struik, 1978, 1983). 

Because of the interaction of the two complicated and not well-understood fields, turbulent flow and non-Newtonian fluids, understanding of DR mechanism(s) is still quite limited. Cates and coworkers (for example, Refs. " ) and a number of other investigators have done theoretical studies of the dynamics of self-assemblies of worm-like micelles. Because these so-called living polymers are subject to reversible scission and recombination, their relaxation behavior differs from reptating polymer chains. An additional form of stress relaxation is provided by continuous breaking and repair of the micellar chains. Thus, stress relaxation in micellar networks occurs through a combination of reptation and breaking. For rapid scission kinetics, linear viscoelastic (Maxwell) behavior is predicted and is observed for some surfactant systems at low frequencies. In many cationic surfactant systems, however, the observed behavior in Cole-Cole plots does not fit the Maxwell model. 

Dynamic mechanical properties of all pure components and blends were measured as a function of percent strain and indicated a linear viscoelastic region up to approximately 30-35 percent. Therefore, all rheological experiments were conducted at a strain rate of 20 percent. In cases where thermal degradation occurred (as seen in time sweep), the heating chamber was continuously purged with liquid nitrogen. Frequency sweeps, and in some cases frequency-temperature sweeps, were performed on all pure components and blends. 

The second model (Figure 4.15d) describes the complicated viscoelastic behaviour of bitumen. Upon application of stress, the model immediately presents elastic deformation and continues to deform at a non-linear rate. Thus, for a given temperature, if a constant stress (oi) is applied, the strain (e) after time (t) could be calculated using the Burgers model by the following equation ... 

When the solid feature dominates the mechanical response of a shear deformation, the shear stress cr is proportional to the shear strain y, and the proportionality coefficient is the shear modulus E. On the other hand, when the liquid feature dominates the response, the shear stress cr is proportional to the shear rate y, the proportionality coefficient is the shear viscosity 77. Maxwell equation of linear viscoelasticity can be applied to describe the continuous switching between the solid and the liquid (Maxwell 1867),... 

Conformational relaxation of polymers at temperatures below their glass transition temperature is retarded by lack of segmental motions. The conformation and free volume at the glass transition temperature continues at lower temperatures since equilibrium cannot be attained over typical experimental times. Cooperative relaxation towards conformational equilibrium depends upon temperatures, relaxation time spectrum, and the disparity between the actual and equilibrium states. The approach of the vitrified polymer to equilibrium is called thermal aging. Aging is both non-linear and nonexponential and several descriptions and models have been proposed. One model is based on a concept of temporary networks where the viscoelastic... 

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