Viscoelasticity nonlinear

As many nonlinear approaches are beyond the intended level and scope of this text, the focus will be on single integral mathematical models which are an outgrowth of linear viscoelastic hereditary integrals and lead to an extended superposition principle that can be used to evaluate nonlinear polymers. The emphasis will be on one-dimensional methods but these can be readily extended to three dimensions using deviatoric and dilatational stresses and strains as was the case for linear viscoelastic stress analysis as discussed in Chapters 2 and 9. 

In Chapter 4, it was noted that linear viscoelastic behavior is observed only in deformations that are very small or very slow. The response of a polymer to large, rapid deformations is nonlinear, which means that the stress depends on the magnitude, the rate and the kinematics of the deformation. Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties. There exists no general model, i.e., no universal constitutive equation or rheological equation of state that describes all nonlinear behavior. The constitutive equations that have been developed are of two basic types empirical continuum models, and those based on a molecular theory. We will briefly describe several examples of each type in this chapter, but since our primary objective is to relate rheological behavior to molecular structure, we will be most interested in models based on molecular phenomena. The most successful molecular models to date are those based on the concept of a molecule in a tube, which was introduced in Chapter 6. We therefore begin this chapter with a brief exposition of how nonlinear phenomena are represented in tube models. A much more complete discussion of these models will be provided in Chapter 11. 

Although it is a powerful means of investigating molecular structure and of basic characterization, and provides a general indication of the influence of M on flow behavior, the restrictions imposed by linear viscoelasticity make it inapplicable to a wide variety of practical problems. Nonlinearity is often associated with the phenomenon of shear thinning , that is, a reduction of the viscosity with shear rate in steady flow, characteristic of many polymer melts at intermediate shear rates [15]. This contrasts with the Newtonian behavior implied by the Boltzman superposition principle for steady flow [Eq. (54), in a liquid, G(s) must vanish as s coj. 

Newtonian regimes are nevertheless widely observed in polymer melts in the high and low shear rate limits, where the viscosities are designated by and respectively. This is reflected by the empirical expressions widely used in engineering practice to describe the response to steady shear flow, an example being the Cross model [Eq. (55)], which reduces to the well known power law of Eq. (56) when tj tj , with n = l/(m+l) between 10 and 20 for most polymer melts. 

This section considers the behavior of polymeric liquids in steady, simple shear flows - the shear-rate dependence of viscosity and the development of differences in normal stress. Also considered in this section is an elastic-recoil phenomenon, called die swell, that is important in melt processing. These properties belong to the realm of nonlinear viscoelastic behavior. In contrast to linear viscoelasticity, neither strain nor strain rate is always small, Boltzmann superposition no longer applies, and, as illustrated in Fig. 3.16, the chains are displaced significantly from their equilibrium conformations. The large-scale organization of the chains (i.e. the physical structure of the liquid, so to speak) is altered by the flow. The effects of finite strain appear, much as they do when a polymer network is deformed appreciably. 

If a liquid is sheared at a constant shear rate y, the stress that results will eventually reach a steady-state value. In the parallel-plate illustration in Fig. 3.17, the upper plate moves at constctnt velocity, F, in direction 1, and a constant shear stress, a = F A, acts in direction 1 on all planes of the liquid that are normal to direction 2. 

The deformation is homogeneous the shear rate y = V/H is the same everywhere in the liquid, and the components of velocity are as follows  

The subscripts 1, 2, and 3 refer to the flow, velocity gradient, and neutral directions, respectively, and X2 is the vertical distance measured from the fixed plate. Apart from pressure, the forces acting on each element of the liquid are also the same everywhere. 

Suppose that we could isolate a very small bit of the liquid in this simple shearing flow at some instant and examine the forces acting upon it. Consider for example a small cubic element with faces parallel to one of the three coordinate directions, as sketched in Fig. 3.18. For a Newtonian liquid, the components of force that act normally to the six faces of the cube have the same magnitude, originating from the pressure. The force acting on some of the faces also has a shear component. The shear forces, which are equal in magnitude but opposed in direction, as is needed for mechanical equilibrium, originate from the viscosity and are directly proportional to the shear rate. 

Viovy et argued that as the contour length of the surrounding primitive chain contracts, there will be an extra relaxation by the release of the topological constraints. They proposed a theory which gives a slightly different relaxation behaviom for Though this proposal 

As in the case of linear viscoelasticity, the effect of the tube reorganization will play an important role in the molecular weight distribution. The problem of how it affects the nonlinear behaviour is interesting but unsolved. 

Neutron scattering. In a series of experiments, Bou6 et al. have studied by neutron scattering the conformational relaxation of the labelled chain after the stepwise deformation. In the short time-scale, the observed relaxation is well described by the Rouse dynamics. In the long time-scale (near t/ ), no clear indication has been found so far for the contraction of the contour length. Various reasons for this behaviour are conceivable such as the limited range of the scattering wave vector or polydi rsity of the sample. On the other hand the results may indicate the importance of the tube deformation in the nonequilibrium state.  

Theoretical calculations of the scattering intensity based on the reptation dynamics are given in refs 81-83. 

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Schuler, K.W., Nunziato, J.W., and Walsh, E.K., Recent Results in Nonlinear Viscoelastic Wave Propagation, Internat. J. Solids and Structures 9, 1237-1281... 

P. E. Rouse. The theory of nonlinear viscoelastic properties of dilute solutions of scaling polymers. J Chem Phys 27 1273-1280, 1953. 

A. Khatory, F. Lequeux, F. Kern, S. J. Candau. Linear and nonlinear viscoelasticity of semidilute solutions of wormlike micelles at high-salt content. Langmuir 9 1456-1464, 1993. 

Sui, C., McKenna, G.B., and Puskas, J.E. Nonlinear viscoelastic response of dendritic (arborescent) polyisobutylenes in single- and reversing double-step shearing flows, J. Rheol, 51, 1143, 2007. 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

Strain Sweep Test Protocols for Nonlinear Viscoelasticity Investigations.826... 

Nonlinear Viscoelasticity along Rubber Processing Lines.830... 

Stream Effects and Nonlinear Viscoelasticity in Rubber Processing Operations... 

With respect to stream rheological effects in rubber processing, and despite all the restrictions discussed above, it seems nevertheless that the key information is how the nonlinear viscoelasticity is related to the processing behavior of mbber compounds. Such information can be deduced from the appropriate test procedure with the RPA, providing one considers the capabilities of the instmment to provide nonlinear viscoelastic data. 

FT rheometry is a powerful technique to document the nonlinear viscoelastic behavior of pure polymers as observed when performing large amplitude oscillatory strain (LAOS) experiments. When implemented on appropriate instmments, this test technique can readUy be applied on complex polymer systems, for instance, filled mbber compounds, in order to yield significant and reliable information. Any simple polymer can exhibit nonlinear viscoelastic properties when submitted to sufficiently large strain in such a case the observed behavior is so-called extrinsic... 

Strain Sweep Test Protocols eor Nonlinear Viscoelasticity Investigations... 

Odd torque harmonics become significant as strain increases and are therefore considered as the nonlinear viscoelastic signature of tested materials, only available through FT rheometry. Figure 30.10 shows the typical pattern of corrected relative TTHC (i.e., T(nw/lw)) versus strain y that, in most cases, can be modeled with a simple equation, i.e.. 

NONLINEAR VISCOELASTICITY ALONG RUBBER PROCESSING LINES... 

Figure 30.14 shows an interesting aspect of RPA-FT experiments, i.e., the capability to quantify the strain sensitivity of materials through parameter B of ht Equation 30.3. As can be seen, curatives addition strongly modifies this aspect of nonlinear viscoelastic behavior, with furthermore a substantial change in strain history effect. Before curatives addition, mn 2 data show very lower-strain... 

FIGURE 30.21 Mixing silica-filled compound overall cfianges in nonlinear viscoelastic character along the mixing fine. 

FIGURE 30.22 Mixing silica-filled compound Complex modulus versus strain amplitude as modeled with Equation 30.3 typical changes in nonlinear viscoelastic features along the mixing hne. 

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