Shear stress relaxation model

In other cases, several discrete relaxation times or distributions of relaxation times can be found [39]. This is typically the case if the stress relaxation is dominated by reptation processes [42 ]. The stress relaxation model can explain why surfactant solutions with wormlike micelles never show a yield stress Even the smallest applied stress can relax either by reptation or by breakage of micelles. For higher shear rates those solutions typically show shear thinning behaviour and this can be understood by the disentanglement and the orientation of the rod-like micelles in the shear field. 

Appendix. Fractal Model of Shear Stress Relaxation... 

The shear stress relaxation modulus for one element in the Maxwell model is given by... 

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

For dynamical studies of diffusion, conformational and transport behavior under shear stress, or kinetics of relaxation, one resorts to dynamic models [54,58,65] in which the topological connectivity of the chains is maintained during the simulation. 

The effect of driving shear stresses on the dislocations are studied by superimposing a corresponding homogeneous shear strain on the whole model before relaxation. By repeating these calculations with increasing shear strains, the Peierls barrier is determined from the superimposed strain at which the dislocation starts moving. 

In a further development of the continuous chain model it has been shown that the viscoelastic and plastic behaviour, as manifested by the yielding phenomenon, creep and stress relaxation, can be satisfactorily described by the Eyring reduced time (ERT) model [10]. Creep in polymer fibres is brought about by the time-dependent shear deformation, resulting in a mutual displacement of adjacent chains [7-10]. As will be shown in Sect. 4, this process can be described by activated shear transitions with a distribution of activation energies. The ERT model will be used to derive the relationship that describes the strength of a polymer fibre as a function of the time and the temperature. 

If the applied shear stress varies during the experiment, e.g. in a tensile test at a constant strain rate, the relaxation time of the activated transitions changes during the test. This is analogous to the concept of a reduced time, which has been introduced to model the acceleration of the relaxation processes due to the deformation. It is proposed that the reduced time is related to the transition rate of an Eyring process [58]. The differential Eq. 123 for the transition rate is rewritten as... 

It is fairly clear that as re approaches rd the role of Rouse relaxation is significant enough to remove the dip altogether in the shear stress-shear rate curve. As the relaxation process broadens, this process is likely to disappear, particularly for polymers with polydisperse molecular weight distributions. The success of the DE model is that it correctly represents trends such as stress overshoot. The result of such a calculation is shown in Figure 6.23. 

The slower rise and decay of normal stress transients compared to shear stress arises quite simply and directly from the polydispersity of relaxation times (78), and probably has no direct bearing on entanglement mechanisms per se. Likewise, the depression of the superimposed moduli at low frequencies follows from rather non-specific continuum models, the loss moduli by Eqs.(8.49) and (8.50) from the simple fluid model, and the storage moduli from the following properties of the more specific but still quite general BKZ model (366,372) ... 

We will first consider the parameters we are trying to model. Let us start with stress relaxation, where it is usual to describe properties in terms of a relaxation modulus, defined in Table 13-5 for tensile [ (r)] and shear [G(r)] experiments. The parameter used to describe the equivalent creep experiments are the tensile creep compliance [D(r)] and shear creep compliance [7(0]. It is important to realize that the modulus and the compliance are inversely related to one another for linear, tune-independent behavior, but this relationship no longer holds if the parameters depend on time. 

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. 

Table 26.1 Cross model parameters is the zero shear viscosity, t critical shear stress, A relaxation time and n power exponent, where ES20 represents 20 wt% styrene ESI and MI denotes the melt index...

The temporary network model predicts many qualitative features of viscoelastic stresses, including a positive first normal stress difference in shear, gradual stress relaxation after cessation of flow, and elastic recovery of strain after removal of stress. It predicts that the time-dependent extensional viscosity rj rises steeply whenever the elongation rate, s, exceeds 1/2ti, where x is the longest relaxation time. This prediction is accurate for some melts, namely ones with multiple long side branches (see Fig. 3-10). (For melts composed of unbranched molecules, the rise in rj is much less dramatic, as shown in Fig. 3-39.) However, even for branched melts, the temporary network model is unrealistic in that it predicts that rj rises to infinity, whereas the data must level eventually off. A hint of this leveling off can be seen in the data of Fig. 3-10. A more realistic version of the temporary network model... 

Problem 3,9 Integrate the Doi-Edwards equation (3-71) using the Currie expression for the Q tensor, Eq. (3-75), for steady-state shearing, for yr = 0.1, 0.3, 1.0, 3.0, and 10.0, using only one relaxation time in the spectrum. Plot the values of dimensionless shear stress OnjG versus yr on the same plot as in Problem 3.8. How close is the prediction of the approximate differential model to that of the exact integral model ... 

The purpose of this paper is to explore various aspects of the rheological behaviour of lyotropic liquid crystalline systems. Lyotropics are often used as model systems for thermotropics because their viscoelastic behaviour seems to be quite similar (1) and solutions are much more easier to handle and can be studied more accurately than melts. The emphasis is on transient data as these are essential for verifying viscoelastic models but are hardly available in the literature. Transient experiments can also provide insight in the development of flow—induced orientation and structure. The reported experiments include relaxation of the shear stress and evolution of... 

The main disadvantage of the Maxwell model is that the static shear modulus p0 vanishes in this model, while the drawback of the Kelvin-Voigt model is that it cannot describe the stress relaxation. The Zener model [131] lacks these disadvantages. This model combines the Maxwell and Kelvin-Voigt models and describes strains closely approximating the actual physical process. The elasticity equation for the Zener model taking account of anomalous relaxation effects can be written as [131]... 

Calculate the stress relaxation modulus of the Rouse model (Eq. 8.55) by showing that after a small step shear strain 7 at time t 0 the correlation function of normal modes decays as Xpx t)X y(t)) — i kTjkp) exp (- tjxp). 

---