Step shear strain

Figure 10.3 Response of a Maxwell liquid to a shear step strain input.

Rheometric Scientific markets several devices designed for characterizing viscoelastic fluids. These instmments measure the response of a Hquid to sinusoidal oscillatory motion to determine dynamic viscosity as well as storage and loss moduH. The Rheometric Scientific line includes a fluids spectrometer (RFS-II), a dynamic spectrometer (RDS-7700 series II), and a mechanical spectrometer (RMS-800). The fluids spectrometer is designed for fairly low viscosity materials. The dynamic spectrometer can be used to test soHds, melts, and Hquids at frequencies from 10 to 500 rad/s and as a function of strain ampHtude and temperature. It is a stripped down version of the extremely versatile mechanical spectrometer, which is both a dynamic viscometer and a dynamic mechanical testing device. The RMS-800 can carry out measurements under rotational shear, oscillatory shear, torsional motion, and tension compression, as well as normal stress measurements. Step strain, creep, and creep recovery modes are also available. It is used on a wide range of materials, including adhesives, pastes, mbber, and plastics. 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

The step-strain experiments discussed above furnish the simplest example of a strong flow. Many other flows are of experimental importance transient and steady shear, transient extensional flow and reversing step strains, to give a few examples. Indeed the development of phenomenological constitutive equations to systematise the wealth of behaviour of polymeric liquids in general flows has been something of an industry over the past 40 years [9]. It is important to note that it is not possible to derive a constitutive equation from the tube model in... 

Transient birefringence measurements were used by Larson et al. [112] to test the validity of the Lodge-Meissner relationship for entangled polymer solutions. This relationship states that the ratio of the first normal stress difference to the shear stress following a step strain is simply Nx/%xy - y, where y is the strain. Those authors found the relationship was valid, except for ultrahigh molecular weight materials. 

When reptation is used to develop a description of the linear viscoelasticity of polymer melts [5, 6], the same underlying hypothesis ismade, and the same phenomenological parameter Ng appears. Basically, to describe the relaxation after a step strain, for example, each chain is assumed to first reorganise inside its deformed tube, with a Rouse-like dynamics, and then to slowly return to isotropy, relaxing the deformed tube by reptation (see the paper by Montfort et al in this book). Along these lines, the plateau relaxation modulus, the steady state compliance and the zero shear viscosity should be respectively ... 

Measurements of the nonlinear relaxation modulus G(t,y) have also been carried out using the plate-plate geometry. Various step strains were applied on the sample and the stress relaxations were recorded. Since the shear strain is known to be inhomogeneous in such a geometry, a correction of the apparent relaxation Ga(t,y) modulus has to be taken into accoimt to get the real relaxation modulus for the maximum strain in the disk sample. This procedure is very similar to that proposed by Rabinowitch in Poiseuille flow, wherein the shear rate is also non-homogeneous, and has already been described by Soskey and Winter [36]. The correction factor is ... 

Firstly, it has been shown that there may be many experimental problems in a direct determination of the experimental fimction. In shear, damping functions obtained from step strain and from step strain rate experiments do not match each other. This poses an important question on the validity of the separability assumption in the short time rai e. Significant departures from this factorization have already been observed in the case of narrow polystyrene fractions by Takahashi et al. [54]. These authors found that time-strain superposition of the linear and nonlinear relaxation moduli was only possible above a cert2un characteristic time. It is interesting to note that this is predicted by the Doi-Edwards theory [10] and according to this theory, this phenomena is attributed to an additional decrease of the modulus connected to a tube contraction process and time-strain separability may hold after this equilibration process has been completed. Other examples of non-separability were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions. 

Thus far we have imposed a constant strain (the step strain experiment) and constant shear rate (the steady shear experiment). Another simple... 

Transient shear is defined as when a material is subject to an instantaneous change in deformation and the response as a function of time is measured. For example. Figure 3.72 shows an instantaneously applied step-strain test. 

When the slab is subject to a single-step shear history y(t) = yoH(t), where H(t) is the Heaviside unit step function, zero for negative t and unit for t zero or positive, the stress response can be used to characterize the rheological properties. When the materials are subjected to a step strain as shown in Fig. 11a, the different stress responses are obtained as shown in Fig. lib. If the material were perfectly elastic, the corresponding stress history would be of the form t(i) = to//(i), constant for t positive (curve a in Fig. lib). If the material were an ideal viscous fluid, the stress would be instantaneously infinite during the step and then zero for all positive t, like a Dirac delta, 5(t) = H t) (curve b in Fig. 11b). For most real materials, like semisolid foods, the stress response shows that neither of these idealizations is quite accurate. The stress usually decreases from its initial value... 

After imposition of a step strain, the ratio of the first normal stress difference, Ni, to the shear stress is equal to the strain, or... 

In Eq. (16.12), / represents a large number of iterations. In the simulation, all the six combinations of Jafsit) (with a / ) may be used for averaging in the calculation of the time-correlation function the xy component is used as the representative of the shear stress in all the discussions. The simulation result as obtained through Eqs. (16.10) and (16.11) may be referred to as the step strain-simulated Gs t), as opposed to the equilibrium-simulated Gs t) obtained through Eq. (16.12). 

Next we consider a polymer melt of high molecular weight in which entanglement is very important. To calculate G(t)> it is convenient to consider the stress relaxation after a step strain. Suppose at t = 0 a shear strain y is applied to the system in equilibrium. The strain causes the deformation of the molecular conformation, and creates the stress, which relaxes with time as the conformation of polymers goes back to... 

We consider the inextensible chain model. Figure 7.24 eiqilains the change of polymer conformation under the double step strain. Figure 7.24a shows the undeformed state just before the first deformation. Figure 7.24h represents the state immediately after the deformation the primitive chain is deformed by the shear Yi. Figure 7.24c indicates the state just before the second deformation the inner part AB still remains in the deformed tube, while the outer parts are in the undeformed tube. Now when the second deformation is applied, the inner part AB is deformed by the shear Yi + Yi from the equilibrium state, while the outer part is deformed by the shear Yi- It is important to note that the second shear stretches the contour length of the outer part by the factor < (72), but that of the inner part by the factor... 

Below the yield stress 0 the rate of shear deformation (strain) (d /dt) is zero in all three nonlinear models. This is also true of the shear strain /for the Saint-Venant and Bingham models, whereas in the Prandtl-Reuss model /increases slowly with the shear stress t from zero to the yield stress 0. At this point, the value of /increases limitless as a step function, as it does in the two other models (Figure 2.15). 

Step-strain stress-relaxation measurements have been frequently used to determine Sr(X) for polymer melts > . Equation (6) shows that if separability of time and strain effects is possible for the melt under consideration, the stress after a step elongational strain can be factored into a time-dependent function, the linear shear relaxation modulus G(t), and a strain-dependent function, the nonlinear strain measure Sr(X). Also other types of experiment may be oerformed to obtain Sr(X), such as constant-strain-rate experiments "", creep under constant stress and constant-stretching-rate experiments but these methods require more involved analytical and/or numerical calculations. 

In a linear viscoelastic material, the moduli and compliances (t), G(t), D(t), and J t) (tensile and shear modulus and tensile and shear comphance respectively) are functions of t, although they remain independent of stress or strain. Development of constitutive equations for such materials is generally based on the idea that the effects of small increases in stress or strain are additive, which is known as the Boltzmann superposition principle [4, 10, 11]. A strain e(t) may be considered to result from a sum of step strains applied at time u and maintained for a time t-u[Eq.(31)j. 

Actual adhesive shear stress-strain curves like those in Fig. 13 are simplified for analysis purposes. The most widely used model is the linearly elastic, perfectly plastic model developed by the author originally under contract to NASA Langley in the early 1970s. This model is described in Fig. 15 and is the basis of the A4E. series of computer codes, of which the A4EI code for stepped-lap joints and doublers covering variable adhesive properties and adhesive porosity and voids is perhaps the best known. This particular code was developed under contract to the USAF at the Wright Laboratories in Dayton (see [14]). 

Figure 2.19 Showing the viscoelastic mechanical response of an amorphous polymer via the time dependence of the shear modulus following a step strain. Different regimes are indicated. The extent of the plateau where the material exhibits a rubbery response, increases with molar mass...

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

Fig. 17 Shear stress relaxation modulus for unfilled LDPE and a series of LDPE/LDH nanocomposite melts after a step strain...

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