Linear viscoelastic response

At sufficiently low strain, most polymer materials exhibit a linear viscoelastic response and, once the appropriate strain amplitude has been determined through a preliminary strain sweep test, valid frequency sweep tests can be performed. Filled mbber compounds however hardly exhibit a linear viscoelastic response when submitted to harmonic strains and the current practice consists in testing such materials at the lowest permitted strain for satisfactory reproducibility an approach that obviously provides apparent material properties, at best. From a fundamental point of view, for instance in terms of material sciences, such measurements have a limited meaning because theoretical relationships that relate material structure to properties have so far been established only in the linear viscoelastic domain. Nevertheless, experience proves that apparent test results can be well reproducible and related to a number of other viscoelastic effects, including certain processing phenomena. 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

Factorizability has also been found to apply to polymer solutions and melts in that both constant rate of shear and dynamic shear results can be analyzed in terms of the linear viscoelastic response and a strain function. The latter has been called a damping function (67,68). 

The term y(t,t ) is the shear strain at time t relative to the strain at time t. The use of a memory function has been adopted in polymer modelling. For example this approach is used by Doi and Edwards11 to describe linear responses of solution polymers which they extended to non-linear viscoelastic responses in both shear and extension. 

Another approach we can use to describe the stress relaxation behaviour and all the linear viscoelastic responses is to calculate the relaxation spectrum H. Ideally we would like to model or measure the microstructure in the dispersion and include the role of Brownian diffusion in the loss of structural order. The intermediate scattering... 

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...

When Zotefoam HDPE materials of density 98 kg m" were subjected to a single major compressive impact (419), after recovery at 50 °C for 1 hour, the performance, defined as the energy density absorbed before the compressive stress reached 2.5 MPa was back to 75% of the initial value. Further severe impacts caused a further deterioration of the performance of the recovered foam. Peak compressive strains of 80 to 90% caused some permanent buckling of the cell walls of HDPE foams. The recovery is much slower than the 0.1 second impact time, so is not a conventional linear viscoelastic response. It must be driven by the compressed air in internal cells in the gas, with some contribution from viscoelasticity of the polymer. Recovery of dimensions had slowed to a very low rate after 10 seconds at 20 °C or after 10 seconds at 50 °C. 

PP bead foams of a range of densities were compressed using impact and creep loading in an Instron test machine. The stress-strain curves were analysed to determine the effective cell gas pressure as a function of time under load. Creep was controlled by the polymer linear viscoelastic response if the applied stress was low but, at stresses above the foam yield stress, the creep was more rapid until compressed cell gas took the majority of the load. Air was lost from the cells by diffusion through the cell faces, this creep mechanism being more rapid than in extruded foams, because of the small bead size and the open channels at the bead bonndaries. The foam permeability to air conld be related to the PP permeability and the foam density. 15 refs. 

It appears that the formal theories are not sufficiently sensitive to structure to be of much help in dealing with linear viscoelastic response Williams analysis is the most complete theory available, and yet even here a dimensional analysis is required to find a form for the pair correlation function. Moreover, molecular weight dependence in the resulting viscosity expression [Eq. (6.11)] is much too weak to represent behavior even at moderate concentrations. Williams suggests that the combination of variables in Eq. (6.11) may furnish theoretical support correlations of the form tj0 = f c rjj) at moderate concentrations (cf. Section 5). However the weakness of the predicted dependence compared to experiment and the somewhat arbitrary nature of the dimensional analysis makes the suggestion rather questionable. 

The stresses used in a creep test are chosen in two ways. First, a value is chosen from oscillatory tests (specifically stress or strain amplitude sweeps at 1 Hz or 10 rad/sec unit hs.i) to define the linear region. Using two to five different values, the sample is taken from a linear viscoelastic response to the onset of... 

The viscoelastic response of polymer melts, that is, Eq. 3.1-19 or 3.1-20, become nonlinear beyond a level of strain y0, specific to their macromolecular structure and the temperature used. Beyond this strain limit of linear viscoelastic response, if, if, and rj become functions of the applied strain. In other words, although the applied deformations are cyclic, large amplitudes take the macromolecular, coiled, and entangled structure far away from equilibrium. In the linear viscoelastic range, on the other hand, the frequency (and temperature) dependence of if, rf, and rj is indicative of the specific macromolecular structure, responding to only small perturbations away from equilibrium. Thus, these dynamic rheological properties, as well as the commonly used dynamic moduli... 

The linear viscoelastic models (LVE), which are widely used to describe the dynamic rheological response of polymer melts below the strain limit of the linear viscoelastic response of polymers. The results obtained are characteristic of and depend on the macromolecular structure. These are widely used as rheology-based structure characterization tools. 

Experimentally a variety of quantities are used to characterise linear viscoelasticity (Ferry 1980). There is no need to consider all the characteristics of linear viscoelastic response of polymers which are measured under different regimes of deformation in linear region, they are connected with each other. The study of the reaction of the system in the simple case, when the velocity gradients are independent of the co-ordinates and vary in accordance with the law... 

The master curves and shift factors of transient and dynamic linear viscoelastic responses are calculated for linear, semi-crystalline, and cross-linked polymers. The transition from a WLF dependence to an Arrhenius temperature dependence of the shift factor in the vicinity of Tg is predicted and is related to the temperature dependence of physical aging rate. 

It is not clear why this transition should occur at such a higher level of arm entanglement for polystyrene stars than for other star polymers. This observation is in direct conflict with the standard assumption that through a proper scaling of plateau modulus (Go) and monomeric friction coefficient (0 that rheological behavior should be dependent only on molecular topology and be independent of molecular chemical structure. This standard assumption was demonstrated to hold fairly well for the linear viscoelastic response of well-entangled monodisperse linear polyisoprene, polybutadiene, and polystyrene melts by McLeish and Milner [24]. 

For oscillatory strains less than approximately 0.03-0.04, the suspensions showed linear viscoelastic response with the plateau modulus. Go, being independent of frequency (Chow and Zukoski, 1995b). Because Go increases exponentially with volume... 

Doi and Edwards (1978a, 1979, 1986) developed a constitutive equation for entangled polymeric fluids that combines the linear viscoelastic response predicted by de Gennes... 

The rheological properties of glassy liquids are dominated by one or more very long relaxation times and a high modulus. The detailed linear viscoelastic response varies somewhat with the type of liquid. Some inorganic glassy liquids, such as zinc alkali... 

Extremely limited range of linear viscoelastic response... 

In the range of frequencies low enough that the interfacial terms become important, one can usually assume that both components of the blend are in their terminal regimes, and thus behave as Newtonian liquids. Following this reasoning, Gramespacher and Meissner (1992) divided the linear viscoelastic response of a blend of polystyrene (PS) and poly(methylmethacrylate) (PMMA) into bulk and interfacial terms, as in Eq. (9-35). The interfacial contributions were taken from the Choi-Schowalter theory for Newtonian liquids. Eq. f9-37h with ju, T], and Xj from Table 9-1. while the bulk contributions were obtained from Eq. (9-41), With these expressions for G ui and Gramespacher and... 

Linear Viscoelasticity-Response of Materials to Transient Experiments 198... 

LINEAR VISCOELASTICITY-RESPONSE OF MATERIALS TO TRANSIENT EXPERIMENTS... 

Experimental materials characterisation. Linear and non linear viscoelastic response, simple shear, extensional flow and mixed shear behaviour. 

The linear viscoelastic response of the two fractions is very different. Fig. 7 shows the temperature dependence of the complex moduli at 1 rad/sec for both fractions. Each fraction shows sharp drops in modulus when their respective melting temperatures are reached (120°C for BP6Ls and 160°C for BP6Li, see Fig. 6). 

An example of the linear viscoelastic response in oscillatory shear for a nearly monodisperse linear polybutadiene melt is shown in Fig. 1.2%. Extrapolation of the limiting power laws oiG uP and G" u (the dashed lines in Fig. 7.28) to the point where they cross has special significance. The intersection of the power laws G = J qrj uP and G = t]uj using the above two equations allows us to solve for the frequency where they cross uj= l/(/)/eq), which is the reciprocal of the relaxation time r [Eq. (7.132)]. The modulus level where the two extrapolations cross, obtained by setting a = 1/r = 1 /(/ /eq) iti either equation, is simply the reciprocal of the steady... 

The various experimental methods of linear viscoelasticity are summarized in Table 7.1. All information for linear viscoelastic response can, in principle, be obtained from each method. The oscillatory methods are particularly useful because they directly probe the response of the system on the time scale of the imposed frequency of oscillation 1 juj. Commercial rheometers can accomplish this with either applied stress or applied strain,... 

Recall that Fig. 9.3 showed the linear viscoelastic response of a polybutadiene melt with MjM = 68. The squared term in brackets in Eq. (9.82) is the tube length fluctuation correction to the reptation time. With /i = 1.0 and NjN = 68, this correction is is 0.77. Hence, the Doi fluctuation model makes a very subtle correction to the terminal relaxation time of a typical linear polymer melt. However, this subtle correction imparts stronger molar mass dependences for relaxation time, diffusion coefficient, and viscosity. 

In many studies it is presumed that linear viscoelastic behaviour always occurs, but this is not the case for many reactive systems. Conventional experimental rheology utilizes a dynamic strain sweep, which examines the dynamic rheological response to varied strain amplitudes, at a fixed frequency. If the system shows an effect of strain amplimde on dynamic properties (such as G or G") the system is said to be exhibiting a non-linear (viscoelastic) response. If the properties are independent of strain amplitude, then the system is said to be exhibiting linear viscoelastic behaviour. Figure 4.2 shows the response of an industrial epoxy-resin moulding compound (approximately 70 wt.% silica) at 90 °C at strain amplitudes of 0.1% to 10% for frequencies of 1, 10 and lOOrad/s. 

It is commonplace for rheologists to investigate the linear viscoelastic properties of materials and, indeed, this is certainly the case in the pharmaceutical and related sciences. Bird et al. (24) and Barnes et al. (2) have suggested several reasons for this, including the ability to derive speculative molecular structures of materials from their rheological response in the linear viscoelastic region and, additionally, the ability to relate the parameters derived from the linear viscoelastic response to quality control procedures and, in some instances, to clinical response (7,9,19,25-27). Furthermore, the mathematical principles associated with the linear viscoelastic response are less complex than those for nonlinear viscoelasticity, thus ensuring relatively simple interpretations of results. 

Any of equations (2-45), (2-46), (2-49), or (2-50) is sufficient as a statement of the Boltzmann superposition principle for linear viscoelastic response of a material. Often in particular applications, however, it is more convenient to use one form than another. All can be extended to three dimensions by using the same forms with the strains given by equation (2-18). Thus, for example, equation (2-46) becomes ... 

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