Straining sinusoidal

The general mode of operation in dynamic tests is to vary the stress sinusoidally with time. A viscoelastic solid in which the viscous element is that of a Newtonian liquid (as defined earlier) responds with a sinusoidal strain of identical oscillation frequency. However, because of the time-dependent relaxation processes taking place within the material, the strain lags behind the stress, as illustrated in Figure 7.9. 

We mostly chose to probe each frequency individually to minimize the strain on the material and to expand the available frequency window. The experimental time can be reduced by simultaneously applying the sinusoidal strains of the lowest frequencies [120] and then quickly adding the higher frequency part of the spectral probing. 

Dynamic theological tests were used to monitor the evolving cross-linking structure during UV curing. In dynamic tests, a sinusoidal strain, y, deformation... 

Deformation of a material by the application of a small sinusoidal strain (y) such that fo cos a>t... 

Note 4 For linear viscoelastic behaviour, a sinusoidal stress (o) results from the sinusoidal strain with... 

Fig. 18 Strain dependence of G of CR vulcanizates filled with different kinds of nanofillers. The measurements were done by a moving die rheometer (Scarabaeus SIS V50) applying sinusoidal strain at constant frequency of 0.1 Hz at 60°C...

Dynamic testing is especially useful in characterizing the viscoelactic behavior in wide time or/and temperature ranges. For the case of a sinusoidal strain ... 

Dynamic Mechanical Testing - Film properties such as impact resistance and the cure response of thermosetting resins are conveniently investigated by dynamic measurements in which an oscillatory or torsional strain is applied to the sample with the stress and phase difference between the applied strain and measured stress being determined. In the present study, a Rheovibron Viscoelastometer was used which employed a sinusoidal strain at a... 

The first case considered is solute desorption during unconfined compression. We consider a two dimensional plane strain problem, see Fig. 1. A sinusoidal strain between 0 and 15 % is applied at 0.001 Hz, 0.01 Hz, 0.1 Hz and 1 Hz. To account for microscopic solute spreading due to fluid flow a dispersion parameter is introduced. Against the background of the release of newly synthesized matrix molecules the diffusion parameter is set to the value for chondroitin sulfate in dilute solution Dcs = 4 x 10 7 cm2 s-1 [4] The dispersion parameter Dd is varied in the range from 0 mm to 1 x 10 1 mm. The fluid volume fraction is set to v = 0.9, the bulk modulus k = 8.1 kPa, the shear modulus G = 8.9 kPa and the permeability K = lx 10-13m4 N-1 s-1 [14], The initial concentration is normalized to 1 and the evolution of the concentration is followed for a total time period of 4000 s. for the displacement and linear discontinuous. For displacement and fluid velocity a 9 noded quadrilateral is used, the pressure is taken linear discontinuous. 

The steady and dynamic drag-induced simple shear-flow rheometers, which are limited to very small shear rates for the steady flow and to very small strains for the dynamic flow, enable us to evaluate rheological properties that can be related to the macromolecular structure of polymer melts. The reason is that very small sinusoidal strains and very low shear rates do not take macromolecular polymer melt conformations far away from their equilibrium condition. Thus, whatever is measured is the result of the response of not just a portion of the macromolecule, but the contribution of the entire macromolecule. 

Thus, according to the result just given, the response of a linear viscoelastic hody to a sinusoidal strain (a) lags in time behind the applied strain, and (b) is composed of purely elastic and purely viscous parts. Figure E3.3 illustrates these features. 

Sinusoidal strain and stress with phase angle 8. 

When a sinusoidal strain is imposed on a linear viscoelastic material, e.g., unfilled rubbers, a sinusoidal stress response will result and the dynamic mechanical properties depend only upon temperature and frequency, independent of the type of deformation (constant strain, constant stress, or constant energy). However, the situation changes in the case of filled rubbers. In the following, we mainly discuss carbon black filled rubbers because carbon black is the most widespread filler in rubber products, as for example, automotive tires and vibration mounts. The presence of carbon black filler introduces, in addition, a dependence of the dynamic mechanical properties upon dynamic strain amplitude. This is the reason why carbon black filled rubbers are considered as nonlinear viscoelastic materials. The term non-linear viscoelasticity will be discussed later in more detail. 

When the phase angle in sinusoidal strain cycles is small (i.e., sinS=tanS), one gets from Eqs. (52) and (53) the loss factor ... 

In contrast, the shear required to produce sinusoidal strain in a Newtonian fluid would be 90° or TtH out of phase with the strain, as Equations 13-74 would indicate. 

For the case of sinusoidal strain history Equation l can be transformed to yield an expression for the complex modulus, G (jw) ... 

Thus, dynamic mechanical viscoelastic properties may be measured in tests with sinusoidal strain input at fixed frequency. Such tests have to be repeated at different frequencies over the range of interest to completely characterize the material. 

Figure 16. Time profile of an applied sinusoidal stress wave and the corresponding resulting sinusoidal strain wave as they apply to small deformation rheological testing.

A simple linear viscoelastic measurement that has become very easy to implement with the advent of modern electronics is oscillatory shear. A sinusoidal strain with angular frequency oj is applied to a sample in simple... 

A more sensitive rheological techniques for following the stability of multiple emulsions is to use oscillatory techniques. In this case, a sinusoidal strain or stress is applied to the sample, which is placed in the gap of the concentric cylinder or cone-and-plate geometry the resulting stress or strain sine wave is followed at the same time. For a viscoelastic system, as is the case with multiple emulsions, the stress and strain sine waves oscillate with the same frequency, but out of phase. 

For the dynamic experiment, most will agree that the stress response resulting from perfect sinusoidal strain input is likely to be sinusoidal and have the same frequency. However, it is far from obvious that the response will always be this simple. In fact, with real materials a perfectly sinusoidal stress response is achieved only at vanishingly low values of strain, y0. The response at higher strain will still be periodic, but will be mixed with higher frequency components.+ f The relative amplitude of these components will increase with... 

Consider the application of a sinusoidal strain, which may be represented by... 

Here we derive expressions for D and D" of a Voigt-Kelvin model consisting of z elements assuming a sinusoidal strain application. Applying equation (3-22) to the Voigt-Kelvin model experiencing a strain in the yth element given by... 

Derive an expression for the complex modulus of a generalized Maxwell model subjected to a sinusoidal strain. Show that the complex compliance is not obtainable... 

The response of a polymeric material to an applied stress shows both an elastic and a viscous component i.e. a polymer behaves visco-elastic. DMA equipment measures dynamically the E or G moduli the polymer samples are assumed to behave linearly visco-elastic i.e. the stress/strain relation is only a function of time. An oscillating (sinusoidal) strain,... 

The elastic nature of a fluid is characterized by dynamic mechanical or stress relaxation techniques. Dynamic mechanical (oscillatory) testing is a procedure in which a sample is sinusoidally strained and the resultant stress is measured. The shear stress T varies with the same frequency as the shear rate... 

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