Sinusoidal straining strain amplitude

The more viscous behavior of PPV relative to its precursor can also be noticed at a macroscopic level by measuring the phase difference, 5, between the dynamic strain applied and the resulting stress using the built-in stress and strain transducers in the stretcher. Figure 21.19 shows a sample output of the phases of dynamic stress and strain extracted from the data for PPV precursor. The thin solid lines is the dynamic stress, which shows a peak at the applied frequency. The dashed and the thick solid lines are the phase of the dynamic stress and strain, respectively. These were calculated as the arctangent of the ratio of the imaginary and real components of the Fourier transforms of the respective signals. The phase difference is then obtained by simple subtraction at the desired frequency (16 Hz in this case). Thus, a sinusoidal strain amplitude of 68 pm resulted in a dynamic sinusoidal stress with an amplitude of 4.2 N and a tan 5 of 0.18 for the PPV precursor. 

On the other hand, for PPV, a dynamic sinusoidal stress with an amplitude of 4.2 N resulted in a sinusoidal strain amplitude of 64 pm with a tan 5 of 0.21. These values were verified by performing dynamic mechanical analysis (DMA) on the precursor and PPV samples over a range of temperature. Tan 5 values for PPV and its precursor at a modulation frequency of 16 Hz were measured in the temperature range —30 to 50°C over which both the materials are stable the value of tan 5 was found to be constant. The phase difference in PPV measured both using the stretcher and using the DMA was found to be significantly higher than that observed in its precursor. Thus, the phase difference between dynamic stress and strain in... 

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... 

It is clear that this data treatment is strictly valid providing the tested material exhibits linear viscoelastic behavior, i.e., that the measured torque remains always proportional to the applied strain. In other words, when the applied strain is sinusoidal, so must remain the measured torque. The RPA built-in data treatment does not check this y(o )/S (o)) proportionality but a strain sweep test is the usual manner to verify the strain amplitude range for constant complex torque reading at fixed frequency (and constant temperature). 

Rheometric Scientific markets several devices designed for characterizing viscoelastic fluids. These instruments measure the response of a liquid to sinusoidal oscillatory motion to determine dynamic viscosity as well as storage and loss moduli. 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 solids, melts, and liquids at frequencies from 10-3 to 500 rad/s and as a function of strain amplitude 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. 

Figure 9-2. Sinusoidal stain and stress cycles. I strain, amplitude a II in-phase stress, amplitude b III out-of-phase stress, amplitude c IV total stress (resultant of II and III, amplitude d. a is the loss angle...

The results of dynamic tests are dependent on the test conditions test piece shape, mode of deformation, strain amplitude, strain history, frequency and temperature. ISO 4664 gives a good summary of basic factors affecting the choice of test method. Forced vibration, non-resonant tests in simple shear using a sinusoidal waveform are generally preferred for design data as... 

A schematic of the system is illustrated in Figure 1. For dynamic frequency sweeps (refer to Figure 2), the polymer is strained sinusoidally and the stress is measured as a function of the frequency. The strain amplitude is kept small enough to evoke only a linear response. The advantage of this test is that it separates the moduli into an elastic one, the dynamic storage modulus (G ) and into a viscous one, the dynamic loss modulus (G"). From these measurements one can determine fundamental properties such as ... 

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 stress is decomposed into two components the ratio of the in-phase stress to the strain amplitude (j/a, maximum strain) is called the storage modulus. This quantity is labeled G (co) in a shear deformation experiment. The ratio of the out-of-phase stress to the strain amplitude is the loss modulus G"(co). Alternatively, if the strain vector is resolved into its components, the ratio of the in-phase strain to the stress amplitude t is the storage compliance J (m), and the ratio of ihe out-of-phase strain to the stress amplitude is the loss compliance J"(wi). G (co) and J ((x>) are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. Tlie loss parameters G" w) and y"(to) on the other hand reflect the nonrecoverable use of applied mechanical energy to cause flow in the specimen. At a specified frequency and temperature, the dynamic response of a polymer can be summarized by any one of the following pairs of parameters G (x>) and G" (x>), J (vd) and or Ta/yb (the absolute modulus G ) and... 

In linear viscoelastic behavior the stress and strain both vary sinusoidally, although they may not be in phase with each other. Also, the stress amplitude is linearly proportional to the strain amplitude at given temperature and frequency. Then mechanical responses observed under different test conditions can be interrelated readily. The behavior of a material in one condition can be predicted from measurement made under different circumstances. 

Dynamic (oscillatory) measurements A sinusoidal stress or strain with amphtudes (Tjj and is appHed at a frequency a> (rads ), and the stress and strain are measured simultaneously. For a viscoelastic system, as is the case with most formulations, the stress and strain amplitudes oscillate with the same frequency, but out of phase. The phase angle shift S is measured from the time shift of the strain and stress sine waves. From a, y and S, it is possible to obtain the complex modulus j G, the storage modulus G (the elastic component), and the loss modulus G" (the viscous component). The results are obtained as a function of strain ampHtude and frequency. 

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... 

When a viscoelastic material such as tire cord or rubber Is subjected to a small amplitude sinusoidal straining, the resulting stress-strain curve Is an ellipse and the material properties are characterized by the real and Imaginary moduli E and E" or the ratio E /E ( tan[Pg.372]

Shear, compression, or torsion cycles with constant stress or constant strain amplitude and various cycle shapes are all perfectly feasible but not commonly used except for the heat buildup tests below. A procedure for plastics taken from metals testing invoices rotating a cylindrical test piece with its ends constrained by bearings that are misaligned. The result is that each element of the test piece goes through a sinusoidal cycle from tension to compression. 

Linearity implies that the strain is sinusoidal, with an amplitude proportional to that of the stress, but there will be a phase lag 8. The strain and stress at angular frequency a> can thus be represented by... 

Specifically, for sinusoidal strain y(f) = Jq sin at oscillating with a small amplitude Yo at an angular frequency co (= 2nf with / being the frequency in Hz), Equation (3.1) gives... 

Usually, the deformation of a sample undergoing oscillatory shear is monitored by measuring the sinusoidally-varying motion of a transducer-controlled driving smface in contact with the sample. However, in turning to the subsequent calculation of shear strain amplitude in dynamic measurements, it must be recognized that conversion of experimentally determined forces and displacements to the corresponding stresses and strains experienced by a sample can involve consideration of the role of sample inertia. 

The Autovibron system is designed to measure the temperature dependence of the complex modulus (E ), dynamic storage modulus (E ), dynamic loss modulus (E") and dynamic loss tangent (tan 6) of viscoelastic materials at specific selected frequencies (0.01 to 1 Hz, 3.5, 11, 35, 110 Hz) of strain input. During measurement, a sinusoidal tensile strain is imposed on one end of the sample, and a sinusoidal tensile stress is measured at the other end. The phase angle 6 between strain and stress in the sample is measured. The instrument uses two transducers for detection of the complex dynamic modulus (ratio of maximum stress amplitude to maximum strain amplitude) and the phase angle 6 between stress and strain. From these two quantities, the real part (E ) and the imaginary part (E ) of the complex dynamic modulus (E ) can be calculated. 

In a typical case of dynamic mechanical analysis, a small stress oscillates periodically in a sinusoidal mode with amplitude cr and frequency co, and the small strain e follows the modulation with a certain phase lag The sinusoidal stress is the imposed stimulation, and in a complex form. 

Consider a material subjected to an oscillating load of small amplitude that is in the linear viscoelastic range. The angular frequency of the sinusoidal oscillation is sinusoidal stress a will produce a sinusoidal strain b, and vice versa. However, because of the viscous component of the deformation, there will be a phase shift between stress and strain. The pertinent quantities can be represented as follows ... 

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