Sinusoidal shear

Forced sinusoidal nonresonant shear directly applied by pole pieces of electromagnet to sample disk Forced sinusoidal shear strain imposed by mechanical drive of clamped annular plate of propellant... 

Forced sinusoidal shear strain imposed by vibrating outer ring of annular plate of propellant on an electrodynamic shaker Forced sinusoidal uniaxial tension and compression imposed by vibrating weighted rectangular column of propellant on electrodynamic shaker... 

Forced sinusoidal shear imposed by piezoelectric driver to single-lap shear specimen... 

Tanner,R.I., Simmons,J.M. Combined simple and sinusoidal shearing in elastic liquids. Chem. Eng. Sci. 22,1803-1815 (1967). 

Figure H3.1.6 Dependence of the extent of the linear viscoelastic range of a food shortening on the frequency of the applied sinusoidal shear stress. As the stress frequency decreases, the range of linear behavior also decreases.

We wish to derive the steady state response of a linear viscoelastic body to an externally applied sinusoidal shear strain (dynamic testing) using the constitutive Eq. 3.3-8, which for this viscometric flow reduces to... 

If a Maxwell element is subjected to a sinusoidal shear deformation that starts at time t = 0, Eq. (13.77) becomes... 

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. 

Finally, one of the most useful ways of measuring viscoelastic properties is dynamic mechanical analysis, or DMA. In this type of experiment, an oscillating stress is applied to the sample and the response is measured as a function of the frequency of the oscillation. By using different instruments this frequency can be varied over an enormous range. Actually, the sample is usually stretched a little bit and oscillated about this strain also, the stress necessary to produce an oscillatory strain of a given magnitude is the quantity usually measured. If the sample being oscillated happens to be perfectly elastic, so that its response is instantaneous, then the stress and strain would be completely in-phase. If a sinusoidal shear strain is imposed on the sample we have (Equation 13-72) ... 

The TICA specimen preparation procedure has been described elsewhere (4). The mechanical measurements were made with the Rheometrics Mechanical Spectrometer (RMS) which measures the in-phase and out-of-phase stress response (a and b component respectively) of a specimen being subjected to a sinusoidal shear strain. The instrumental set-up was reported by Lee (6). The frequency of the strain function was kept constant at 1.6 Hz (10 rad/sec). All temperature scan experiments were scanned at 2 C/min rate. The temperature was scanned down at the same rate when the maximum temperature was reached. 

Let us assume that a sinusoidal shear strain z t) = Sq sin cot is imposed on a viscoelastic solid, where 8q and co are, respectively, the amplitude and frequency of the perturbing strain. A dynamic shear strain is illustrated in Figure 6.1. Experimentally one observes that the shear stress (response) is... 

The work performed per cycle by a material that undergoes sinusoidal shear deformation 8 = 8oIm exp(/cot) is given by (1)... 

The response of a material to a sinusoidal shear stress a = ao sin cot is delayed an angle 6 with regard to the perturbation, and the relaxation between the shear deformation (response) and the shear stress is given by... 

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 ... 

Figure 2-13. The response of a sample to a sinusoidal shear strain y(t) is a sinusoidal shear stress a t) that leads the strain by a phase angle 8. Arrows show the physical meaning of the stresses

In the above considerations, a sinusoidal shear strain is applied to the sample. It should be clear that a sinusoidal shear stress could also be applied resulting in corresponding compliance functions J and J". The former results from the deformation in phase with the stress, while the latter corresponds to the out-of-phase deformation. The value of tan 5 remains the same, as can be seen from the curves in Figure 2-13, where we can easily imagine the stress as the applied variable and strain as the measured variable. Tensile stress is equally applicable and definitions of E (co), E" (o), D"(co), D co), etc. are completely analogous to the derived shear parameters. At a given frequency, the value of tan 8 is always the same for any of these quantities, i.e., tan 8 = E"/E = D"/D . 

Figure 2.9 Blending of the left and right halves of the domain, marked black and white, by chaotic advection in the alternating sinusoidal shear flow (Eq. (2.66)), a = UT/L = 0.8. Time runs from left to right and then from top to bottom.

A simple numerical example of chaotic advection in a piecewise steady sinusoidal shear flow periodically alternating along the x and y direction is shown in Fig. 2.9, where the velocity field is defined as... 

Figure 2.21 Development of the strange eigenmode in the sinusoidal shear flow (Eq. (2.66)) with random phase and a = 0.6. Time increases from left to right and top to bottom. Since the amplitude of the concentration fluctuations decreases in time the grayscale has been rescaled for each snapshot to cover the range between minimum and maximum concentrations.

Figure 6.4 Filamental (left) and smooth (right) distributions of a decaying tracer in the chaotic sinusoidal shear flow with source distribution of the form S(x,y) = sin(27rx/L) sin(27ry/L) for two different values of the decay rate. Lower panels show the fluctuations of the concentration along one-dimensional sections for both cases.

The properties of viscoelastic materials can also be described in terms of the responses to sinusoidal inputs. In a sinusoidal shear test, the applied strain or stress to the sample is sinusoidal and, in general, the response of the stress or strain is dependent on both shear frequency and the rate of shear strain (Fig. 13). 

ISO 4664 1998 Rubber—Determination of dynamic properties of vulcanisates for classification purposes (by forced sinusoidal shear strain). 

Reinforced vulcanized samples generally present a marked viscoelastic behavior that is usually studied by dynamic viscoelastic measurements. In this experiment, a sample is subjected to periodic sinusoidal shear strain y... 

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]. 

The viscoelastic behavior of biomaterials is typically measured using DMA. In rheological terms, viscoelastic is the concomitance of viscous (fluid-like) and elastic (solid-like) elements. The proportion of viscous and elastic properties is depending on the used material as well as on the measuring conditions such as the temperature. In DMA measurements, a sinusoidal shear load is applied to the sample while measuring the shear stress (cr ) with a stress transducer. The strain induced... 

In the frequency sweep test, a repeated sinusoidal shear loading is applied at 10 frequencies and at a given temperature while a varying axial load is applied to prevent dilation of the specimen. The loads and deformations are used to calculate the complex shear modulus, G, and phase angle, 5, of the specimen at each frequency. 

Sinusoidal stresses or strains of constant frequency are applied to a sample until a steady sinusoidal strain or stress results, with a fixed phase angle between the input and the output. For example, for a sinusoidal shear strain. 

Another deformation of interest is to impose a sinusoidal shear strain... 

Reinforced vulcanized samples generally present a marked viscoelastic behavior that is usually studied by dynamic viscoelastic measurements. In this experiment, a sample is subjected to periodic sinusoidal shear strain y (at defined frequency (o and temperature T). Its dynamic shear modulus G is complex and can be written as the sum of the storage modulus G, and the loss modulus G". 

A very good way to characterize and differentiate between elastomers and rigid plastics is by the measurement of dynamic mechanical properties. A most convenient method to study dynamic mechanical properties is to impose a small, sinusoidal shear or tensile strain and measure the resulting stress. Dynamic mechanical properties are most simply determined for a small sinusoidally varying strain, for which the response is a sinusoidally varying stress. An increase in frequency of the sinusoidal deformation is equivalent to an increase in strain rate. 

Analysis of the distribution of lifetimes for the bridges can be used to deduce their affect on the shear stress relaxation after a unit shear strain [40]. A similar approach has been used to study the dynamic response of triblock copolymers, adsorbed via their terminal blocks between two parallel plates, when they are subjected to step and sinusoidal shear [41]. 

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. 

Now consider what happens if a sinusoidal shear strain is applied to an ideal liquid ... 

FIG. 1 -8. Vectorial resolution of components of complex modulus and compliance in sinusoidal shear deformations. 

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