Small-amplitude sinusoidal shear

Aside from the simple shearing motion, the response of visco-elastic materials in a variety of other well-defined flow configmations including the cessation/initiation of flow, creep, small amplitude sinusoidal shearing, etc. also lies in between that of a perfectly viscous fluid and a perfectly elastic solid. Conversely, these tests may be used to infer a variety of rheological information about a material. Detailed discussions of the subject are available in a number of books, e.g. see Walters [1975] and Makowsko [1994]. 

Rheological behavior can be determined with small-amplitude sinusoidal shear, using the cone-and-plate steady-shear test to determine the linear viscoelastic shear strain. A sinusoidal curve is charted to represent the viscous (loss) modulus (out-of-phase segment) and the elastic (storage) modulus (in-phase segment) [2]. 

Viscous and elastic moduli with small-amphtude sinusoidal shear can be determined by using an orthogonal rheometer [2]. Small-amplitude sinusoidal shear, using cone-and-plate or paraUel-plate test methods, can determine rheological behavior for normal stresses in shear flow, as well as for shear strain. 

Many time-dependent flows have been studied by polymer chemists. A particularly important one is the small-amplitude sinusoidal shear flow (Figure 2b) where = Re(y e " y, Vy = 0, v = 0, in which is in general a complex quantity and co is the frequency of oscillation the notation Re means the real part of . Then because the amplitude of the oscillation is vanishingly small, the shear stress is also sinusoidal Ty = Reltyj e, where Zy is complex. We now define a complex viscosity by... 

Crystallization in a thermolropic liquid-crystalline polymer is again a process of nucleation and growth [54]. It has been shown that the process can be followed easily using dynamic mechanical analysis (see Chap. 12) [55], in which we measure the stress response of the material to an imposed small-amplitude sinusoidal shear strain. Differential scanning calorimeter (DSC) data on the kinetics of crystallization show that the process is describable by an Avrami equation [56]. 

Although creep-compliance (Kawabata, 1977 Dahme, 1985) and stress-relaxation techniques (Comby et al., 1986) have been used to study the viscoelestic properties of pectin solutions and gels, the most common technique is small-deformation dynamic measurement, in which the sample is subjected to a low-amplitude, sinusoidal shear deformation. The resultant stress response may be resolved into an in-phase and 90° out-of-phase components the ratio of these stress components to applied strain gives the storage and loss moduli (G and G"), which can be related by the following expression ... 

The rheological behavior of a viscoelastic material can be investigated by applying a small-amplitude sinusoidal deformation. The behavior can be described by a mechanical model, called the Maxwell model [33], consisting of an elastic spring with the Hookean constant, G , and a dashpot with the viscosity, r/<,. The variation of storage modulus (G ) and loss modulus (G") with shear frequency, O), are given by the equations... 

Figure 6.15 The responses cr(y) and N (y) to shear rate y as an input variable in steady-state shear flow, and the responses G (o)) and G"(a>) to small-amplitude sinusoidal strain y (ia>) with an angular frequency as an input variable in oscillatory shear flow.

The conclusion is that Lodge s rheological constitutive equation results in relationships between steady shear and oscillatory experiments. The limits y0 0 (i.e. small deformation amplitudes in oscillatory flow) and q >0 (i.e. small shear rates) do not come from Lodge s equation but they are in agreement with practice. These interrelations between sinusoidal shear deformations and steady shear flow are called the relationships of Coleman and Markovitz. 

When a Hookean solid is subjected to sinusoidally varying shear displacements of small amplitudes, at frequencies low enough so that inertia is not important, the shear stress will remain proportional to the shear strain. Thus stress and strain signals will be in phase. When, however, a Newtonian liquid is subjected to the same kind of sinusoidal deformation, the stress will be proportional to the strain rate. Since the strain rate is the time derivative of the strain, it will be 90 degrees out of phase with the strain. For the Newtonian fluid, therefore, stress and... 

Dynamic Oscillatory Experiments The dynamic rheological properties of a polymeric solution can be determined by small-amplitude oscillation tests [2]. In small amplitude oscillatory measurements, a sinusoidally varying shear stress field is imposed on a fluid and the amplitude of the resulting shear strain and phase angle between the imposed stress and the strain is measured. The test is... 

Small-amplitude oscillatory flow is often referred to as dynamic shear flow. Fluid deformation under d)mamic simple shear flow can be described by considering ttie fluid wiflun a small gap dX2 between two large parallel plates of which the upper one undergoes small amplitude oscillations in its own plane with a frequency velocity field within the gap can be given by d , = ydxj but y is not a constant as in steady simple shear. Instead it varies sinusoidally and is given by... 

A sinusoidally varying shear strain rate with small amplitude, such that 721 = 721 cos cor, evokes a sinusoidally varying normal stress difference symmetry considerations because of the proportionality to 721, is predicted by the phenomenological models previously quoted.5 -54 jj,g oscillatory stress difference is superposed on a constant stress and both are proportional to 721 if 721 is small. The coefficient is now defined as the ratio (ai i — ff22)/(72i) - It is the sum of a constant term and two oscillating terms ... 

Later on, semi-soft elasticity concept has been extended to the dynamical case. The theory, which is based on the separation of time scales between the director and the network, describes the mechanical properties of monodomain NEs in the linear response regime, when the sample is subjected to a sinusoidal shear of small amplitude (Terentjev and Warner 2001). When the shear is applied in a plane containing the director, the theory predicts the existence of a low frequency semi-soft elastic plateau, in addition to the usual rubbery plateau observed at higher frequencies. 

When a small-amplitude oscillatory (sinusoidal) shear strain is imposed on a linear viscoelastic fluid, we expect to observe an oscillatory response in shear stress, which can be represented by... 

Consider a deformation consisting of repeated sinusoidal oscillations of shear strain. The relation between stress and strain is an ellipse, provided that the strain amplitude is small, and the slope of the line joining points where tangents to the ellipse are vertical represents an effective elastic modulus, termed the storage modulus /r. The area of the ellipse represents energy dissipated in unit volume per cycle of deformation, expressed by the equation... 

The cone and plate viscometer can be used for oscillatory shear measurements as well. In this case, the sample is deformed by an oscillatory driver which may be mechanical or electromagnetic. The amplitude of the sinusoidal deformation is measured by a strain transducer. The force deforming the sample is measured by the small deformation of a relatively rigid spring or tension bar to which is attached a stress transducer. On account of the energy dissipated by the viscoelastic polymer system, a phase difference develops between the stress and the strain. The complex viscosity behavior is determined from the amplitudes of stress and strain and the phase angle between them. The results are usually interpreted in terms of the material functions, p, G, G" and others [33-40]. 

Although stress-relaxation and creep measurements are used extensively, measuring oscillatory shear is the most commonly used method for characterizing the linear viscoelastic properties of polymer melts and concentrated solutions. As indicated in Fig. 3.10, the liquid is strained sinusoidally at some frequency co, and in the linear region (small-enough strain amplitude yo)- The stress response at steady state is also sinusoidal, but usually out of phase with the strain by some phase angle steady-state stress signal is resolved into in-phase and out-of-phase components, and these are recorded as functions of frequency ... 

For low amplitudes the film responds like an elastic solid. The displacement of the top wall (Figure 6(a)) is in phase with the applied force (Figure 6(b)). Except for a small transient during the first period, both force and displacement are sinusoidal. The two quantities grow linearly with stage displacement and their ratio gives the total elastic shear modulus of the system. This modulus reflects the stiffriess of both the wall/film interface and the film itself. 

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