Strain Oscillation

So far we have seen that if we begin with the Boltzmann superposition integral and include in that expression a mathematical representation for the stress or strain we apply, it is possible to derive a relationship between the instrumental response and the properties of the material. For an oscillating strain the problem can be solved either using complex number theory or simple trigonometric functions for the deformation applied. Suppose we apply a strain described by a sine wave  

Now in order to apply the Boltzmann Superposition Principle (Equation 4.60) we need to express this as a strain rate. Differentiating with respect to time gives us 

Now the Boltzmann superposition integral is given by Equation (4.60). Substituting for the strain and replacing t by t in Equation (4.88) gives 

For convenience we can change variables and the integral limits so that 

Simple trigonometry allows the cosine term to be rearranged  

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We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

The electromagnetic spectrum measures the absorption of radiation energy as a function of the frequency of the radiation. The loss spectrum measures the absorption of mechanical energy as a function of the frequency of the stress-strain oscillation. 

Figure 11.11. Effect of fat level (half-fat, HF, 17%, w/w full-fat, FF, 32%, w/w) and assay temperature (4 or 40°C) on the elastic shear modulus of Cheddar cheeses measured using low-amplitude strain oscillation at a frequency of 0.1( ), 1 ( ) or 10 (g), Hz.

When the chaotic dispersion is slow the populations of the three strains oscillates periodically in time, each strain becoming temporarily dominant in a cyclic fashion (Fig. 8.8). The chaotic dispersion also affects the spatial structure, by stretching the patches occupied by different strains into elongated filaments (Fig. 8.9). Thus the effect of chaotic dispersion is to synchronize the local oscillations over the whole system. The amplitude of the oscillations increases with the dispersion rate and eventually leads to the extinction of two of... 

The early work on viscoelasticity was performed on silk, mbber, and glass, and it was concluded that these materials exhibited a delayed elasticity manifest in the observation that the imposition of a stress resulted in an instantaneous strain, which continued to increase more slowly with time. It is this delay betweai cause and effect that is fundamental to the observed viscoelastic response, and the three major examples of this hysteresis effect are (1) creep, where there is a delayed strain response afto the rapid application of a stress, (2) stress-relaxation (Section 13.15), in which the material is quickly subjected to a strain and a subsequent decay of stress is observed, and (3) dynamic response (Section 13.17) of a body to the imposition of a steady sinusoidal stress. This produces a strain oscillating with the same frequeney as, but out of phase with, the stress. For maximum usefulness, these measurements must be carried out over a wide range of temperature. 

Buzza et al. (105) have presented a qualitative discussion of the various dissipative mechanisms that may be involved in the small-strain linear response to oscillatory shear. These include viscous flow in the films. Plateau borders, and dispersed-phase droplets (in the case of emulsions) the intrinsic viscosity of the surfactant monolayers, and diffusion resistance. Marangoni-type and marginal regeneration mechanisms were considered for surfactant transport. They predict that the zero-shear viscosity is usually dominated by the intrinsic dilatational viscosity of the surfactant mono-layers. As in most other studies, the discussion is limited to small-strain oscillations, and the rapid events associated with T1 processes in steady shear are not considered, even though these may be extremely important. 

The incorporation of fillers to elastomeric compounds strongly modifies the viscoelastic behavior of the material at small strains and leads to the occurrence of a non-linear behavior known as Payne effect [49] characterized by a decrease in the storage modulus with an increase in the amplitude of small-strain oscillations in dynamic mechanical tests. This phenomenon has attracted considerable attention in the past decade on account of its importance for industrial applications [50-54]. The amplitude AG = G q—G ) of the Payne effect, where G q and G aie the maximum and minimum values of the storage modulus respectively, increases with the volume fo-action of filler as shown in silica-filled PDMS networks (Figure 4.7a). At a same silica loading, the PDMS network filled with untreated silica displays a much higher G value than the treated one and is much more resistant to the applied deformation (Figure 4.7b). 

The phenomenon utilized in the quartz clock is called the piezoelectric effect. When a crystal is subjected to alternate compressive and tensile strains, oscillating (+, -) electric charges appear on opposing faces of the crystal. Similarly, when a crystal is subjected to oscillating charges, it undergoes expansion and contraction. 

However, the software of advanced stress controlled instruments allows for running an experiment at variable strain amplitudes. In this operation mode, several iterative cycles have to be measured before the actual measurement. In these iterations, the applied torque is adjusted to produce the desired strain amplitude [27]. In contrast to the classical way of amplitude adjustment, new operating modes of stress controlled rheometers (termed Direct Strain Oscillation or Continuous Oscillation) use a feedback control to compare the current strain signal y(t) at time t to the desired pure sinusoidal signal yd t) = y o sin(control loop then adjusts the torque accordingly in order to minimize the difference Yd t + At) — y (t -I- At)I for the next step at t -I- At. This deformation control enables a stress controlled rheometer to mimic a strain controlled experiment [27]. This holds true even beyond the linear regime where nonlinear contributions to the strain wave are compensated for and are then transferred into the stress wave, as the control loop tries to make the appropriate adjustments to the torque within minimum time. 

LAOS measurements for two samples, a polyisoprene melt (abbreviated PI-84k, Mw = 84,000 g/mol, PDI = 1.04) and a 10 wt% solution of poly isobutylene (abbreviated PIB, Af = 1.1 xlO g/mol) in oligoisobutylene, were conducted on four different rheometers. The first two were separated motor transducer(SMT)-rheometers, namely the ARES-G2 (TA Instruments) and the ARES-LS (TA Instruments) with a IKFRTNl transducer. The DHR-3 (TA Instruments) and the MCR501 (Anton Paar) are in principle stress controlled instruments, but can be used for strain controlled experiments when using the deformation control feedback option (called continuous oscillation for DHR-3 and direct strain oscillation for MCR501). 

Lhuger J, Wollny K, Huck S (2002) Direct strain oscillation a new oscillatory method enabling measurements at very small shear stresses and strains. Rheol Acta41 356-361... 

In dynamic mechanical analysis (DMA [27]) of a polymer film, a sample with the same dimensions as in the tensile stress-strain analysis described above is slightly pre-tensioned and then subjected to a low-amplitude and low-frequency sinusoidal deformation (typically 0.1 % and 1 Hz respectively). As the measurement is performed below the material s elastic limit, the stress follows the strain in a sinusoidal manner. The amplitude ratio and the phase difference between the stress and strain oscillations enables the dynamic elastic modulus E to be calculated ... 

The relaxation and creep experiments that were described in the preceding sections are known as transient experiments. They begin, run their course, and end. A different experimental approach, called a dynamic experiment, involves stresses and strains that vary periodically. Our concern will be with sinusoidal oscillations of frequency v in cycles per second (Hz) or co in radians per second. Remember that there are 2ir radians in a full cycle, so co = 2nv. The reciprocal of CO gives the period of the oscillation and defines the time scale of the experiment. In connection with the relaxation and creep experiments, we observed that the maximum viscoelastic effect was observed when the time scale of the experiment is close to r. At a fixed temperature and for a specific sample, r or the spectrum of r values is fixed. If it does not correspond to the time scale of a transient experiment, we will lose a considerable amount of information about the viscoelastic response of the system. In a dynamic experiment it may... 

Suppose an oscillating strain of frequency co is induced in a sample ... 

Figure 3.14 General representations of stress and strain out of phase by amount 5 (a) represented by oscillating functions and (b) represented by vectors.

A technique for performing dynamic mechanical measurements in which the sample is oscillated mechanically at a fixed frequency. Storage modulus and damping are calculated from the applied strain and the resultant stress and shift in phase angle. 

An interesting improvement from the classical treatment of the bond under stress was proposed by Crist et al, [101], Considering the chain as a set of N-coupled Morse oscillators, these authors determined the elongation and time to failure as a function of the axial stress. The results, reported in Fig. 20, show a decreasing correlation between the total elastic strain before failure and the level of applied force with the chain length. To break a chain within some reasonable time interval (for example <10-3s) requires, however, the same level of stress (a0.7 fb) as found from the simpler de Boer s model. 

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. 

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... 

Figure 7 shows the schematic construction of Figure 8. Phase displacement between an oscillating rheometer. stress and strain.

The linear visco-elastic range ends when the elastic modulus G starts to fall off with the further increase of the strain amplitude. This value is called the critical amplitude yi This is the maximum amplitude that can be used for non-destructive dynamic oscillation measurements... 

In driven dynamic testing an oscillating strain (or stress) is applied to a specimen. This is almost always sinusoidal for ease of analysis. In this case... 

A development of the moving die rheometer where the operation of the unit is fully computer controlled. The rate of oscillation, temperature and level of strain can all be run through a series of options. The torque measurements are also highly sophisticated. As a consequence, the unit can be set up to monitor processing parameters, then the cure behaviour and finally the finished dynamic properties of the cured material. It is manufactured by Alpha Technologies. 

The degree of vulcanisation of a rubber compound is assessed technically by the indefinite terms of undercure, correct cure, optimum cure and overcure. It may be given precision by (a) measurement of stress-strain relationship of a range of cures, (b) measurement of the modulus at 100% elongation, (c) measurement of the volume swelling in benzene, or (d) by the use of instruments such as the oscillating disc rheometer and the moving die rheometer. 

In an elastic material medium a deformation (strain) caused by an external stress induces reactive forces that tend to recall the system to its initial state. When the medium is perturbed at a given time and place the perturbation propagates at a constant speed (or celerity) c that is characteristic of the medium. This propagating strain is called an elastic (or acoustic or mechanical) wave and corresponds to energy transport without matter transport. Under a periodic stress the particles of matter undergo a periodic motion around their equilibrium position and may be considered as harmonic oscillators. 

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