Viscoelastic system

Like any dynamic strain instrument, the RPA readily measures a complex torque, S (see Figure 30.1) that gives the complex (shear) modulus G when multiplied by a shape factor B = iTrR / ia, where R is the radius of the cavity and a the angle between the two conical dies. The error imparted by the closure of the test cavity (i.e., the sample s periphery is neither free nor spherical) is negligible for Newtonian fluids and of the order of maximum 10% in the case of viscoelastic systems, as demonstrated through numerical simulation of the actual test cavity." ... 

Polymers are viscoelestic at all temperatures. Nevertheless they are not simple elastic solids and the effect of temperature on the response of viscoelastic systems to the perturbation show a very interesting trend. At T = Tg the strain against time remain approximately constant. As the temperature increases the variation of strain with time increases in a square root shape which is larger the higher temperature. See Fig. 2.3. 

With all these models, the simple ones as well as the spectra, it has to be supposed that stress and strain are, at any time, proportional, so that the relaxation function E(t) and the creep function D(t) are independent of the levels of deformation and stress, respectively. When this is the case, we have linear viscoelastic behaviour. Then the so-called superposition principle holds, as formulated by Boltzmann. This describes the effect of changes in external conditions of a viscoelastic system at different points in time. Such a change may be the application of a stress or also an imposed deformation. 

Poh BT, Ong BT (1984) Dependence of viscosity of polystyrene solutions on molecular weight and concentration. Eur Polym J 20(10) 975-978 Pokrovskii VN (1970) Equations of motion of viscoelastic systems as derived from the conservation laws and the phenomenological theory of non-equilibrium processes. Polym Mech 6(5) 693—702... 

The mechanical properties of fibres and yams are quite complex and have been the subject of much experimental work. A stressed fibre is a very complicated viscoelastic system in which a number of irreversible processes, connected with plasticity, can take place. 

Creep Curves Fig. 4 shows the creep curves for the formulations containing 30 and 45 g dm bentonite. Suspensions containing less than 30 g dm bentonite did not give a measurable creep curve. The creep curves shown in Fig. 4 are typical of those obtained with viscoelastic systems. They consist of three regions (a) directly after the application of the stress one observes a rapid elastic deformation resulting in an elastic compliance (instantaneous shear modulus = T/y =... 

Generalized Stress-Strain Relationships for Viscoelastic Systems 227... 

The study of the response of viscoelastic systems to a tensile strength is very important on practical grounds. For a uniaxial strain the... 

Owing to the entropic changes that take place in a viscoelastic system perturbed by a force field, the response does not vanish when the perturbation field ceases. A consequence of this fact is that the deformation depends not only on the actual stress but also on the previous stresses (mechanical history) undergone by the material in the past. Under the linear behavior regime, the responses to different perturbations superpose. Let us assume that the stresses Aa(0i) and A(t(02) are applied on the material at times 0j and 02, respectively. This stress history is shown schematically in Figure 5.12. The response is given by... 

To gain a better understanding of the effects of memory on the strain of viscoelastic systems, it is convenient to write Eq. (5.37) in terms of the entropic and viscous contributions to J(t). The time dependence of the shear strain can be written as... 

The superposition principle leads to the following generahzed relationship between the strain tensor and the stress tensor for viscoelastic systems ... 

The analysis by Laplace transforms can be extended to any viscoelastic system with independence of its degree of symmetry. The following equations can be written. 

In the analysis of viscoelastic systems by dynamic methods, a linear onedimensional system of second order in terms of force and displacement is customarily used. This approach also requires the introduction of the geometric form factors with the purpose of solving the real problem in a simple way (Fig. 7.1.)... 

As shown, a detailed study of the same problem reveals that on some occasions the factors in question depend on the physical properties of the viscoelastic system. Hence, it is important to analyze carefully the hypothesis on which such a reduction of the problem is based, in order to be able to calibrate the quality of both the approximations achieved and the results obtained. Such a critical analysis must be implemented rigorously in the experiments. The previous considerations become more important the nearer one works to the limit stipulated by the technical specifications of... 

The effect of temperature on the response of viscoelastic systems to perturbation fields can be qualitatively observed in Figure 8.1, where the time dependence of the deformation is schematically represented for a viscoelastic liquid. It can be seen that at the glass transition temperature the deformation remains nearly constant for comparatively long times. However, as the temperature of the system increases, the deformation undergoes a dramatic increase, which is larger the higher the temperature. If the shear stress is canceled out after steady-state conditions are reached, the time... 

Accordingly, the loss compliance function presents a maximum in the frequency domain at lower frequency than the loss relaxation modulus. This behavior is illustrated in Figure 8.18, where the complex relaxation modulus, the complex creep compliance function, and the loss tan 8 for a viscoelastic system with a single relaxation time are plotted. Similar arguments applied to a minimum in tan 8 lead to the inequalities... 

Because st is dimensionless, s has the units of reciprocal time and is commonly expressed as = 1 /x. The parameter x, called relaxation time, is taken to be the time at which t = x, that is, the time at which G(t) = Gq/c. This approximation, however, fails even for rather simple viscoelastic systems. In fact, the curve obtained for G(t) using Eq. (9.2) drops much more rapidly than that corresponding to real systems (see Fig. 9.1). A better description of these systems is achieved by using a sum of exponentials (1-5), for example, Gi exp(-5 t). By assuming that the relaxation times of the viscoelastic mechanisms involved in the relaxation process vary continuously between 0 and oo, the sum can be replaced by an integral in such a way that G t) may be considered the Laplace transform of an unknown function N s) (2). According to this,... 

Illustrative curves showing these inequalities for liquid viscoelastic systems are represented in Figures 9.2 and 9.3. 

The Maxwell and Kelvin-Voigt models are unable to represent conveniently the material response of a viscoelastic system. A better approach to the actual behavior is achieved by using more complex models. 

The shear rate dependence of the viscosity and that of the normal stress difference Ni for a viscoelastic fluid follow opposite trends. Thus the first parameter decreases with increasing values of k, while the second increases. It is noteworthy to remark once more that viscoelastic fluids may present nonlinear effects, expressed by the normal stresses, in regions in which the shear stress is a linear function of the shear rate. However, these viscoelastic systems are still called Newtonian fluids due to the fact that the viscosity is independent of the shear rate. 

Note also that infinitesimal temperature changes are not required in this approach. An interesting particular case is that one which external forces are absent. In this situation, the deformation (expansion or contraction) of the viscoelastic system can be due only to temperature effects. In the case of a temperature jump AT, Eq. (16.44b) leads to the relation... 

Let us return to the proposed problem of calculating the displacement of a viscoelastic thin hollow sphere after a sudden internal pressurization. According to Eq. (16.66a), the determination of the displacement u requires to obtain an expression for (1 — v)/E in the viscoelastic system. From the differential operators of a standard solid and the equations for the tensile modulus and the Poisson ratio developed, respectively, in Problems 16.2 and 16.4 at the end of this chapter, the following expression for (1 —v)/E is obtained ... 

Using the same approach as in the plane indentation of elastic materials, the displacement of a viscoelastic system by the action of a concentrated load is given by... 

Express the equations of motion in terms of displacements for a linear viscoelastic system. 

The Equilibration Time in Viscoelastic Systems. When working with viscoelastic detergent systems we should be aware that the systems may need rather long times to reach equilibrium. The time can be of the order of days, sometimes longer. In order to obtain reproducible... 

FIGURE 8.9 Flow behavior of a viscoelastic system. Regions AB and CD represent viscoelastic behavior region BC represents structural breakdown during steady shear. 

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