Sinusoidal stress and strain

Figure 18.20. An illustrative relationship between sinusoidal stress and strain waves. Reproduced from Wetton et al. (1991), by permission of Elsevier, Ltd.

When a sinusoidal (harmonic) sound wave propagates through a viscoelastic material, the stresses and strains in the material vay sinusoidally. Eg.22 predicts, in this case, a phase lage between the stress and the strain, which leads to conversion of acoustic energy to heat. From the Fourier transform of Eg.22 it follows that the sinusoidal stress and strain are related by complex, freguency-dependent elastic moduli as follows. 

The application of sinusoidal stress and strain is similar to that for a Maxwell body. The results are summarized in Table 3-1 along with the previously derived results for a Maxwell element. Figure 3-6 displays the frequency dependence of D and D" for the Voigt element in tension. The response in shear would be identical with J replacing D. 

In this case, the shear stress and the strain are 90° out of phase. The response of viscoelastic materials falls between these two extremes. It follows that the sinusoidal stress and strain for viscoelastic materials are out of phase by an angle, say 8. The behavior of these classes of materials is illustrated in Figure 13.7. 

This 90 ° phase difference between sinusoidal stress and strain in liquids is the key to the use of DMA as a tool for the characterization of viscoelastic materials. Since a viscoelastic material has properties intermediate between those of an ideal solid and an ideal liquid, it exhibits a phase lag somewhere between 0° (ideal solid) and 90° (ideal liquid), also shown in Fig. 5.9. Thus, DMA applies a given strain and measures the resulting stress as well as the relative amplitudes of stress and strain (the modulus) and the phase lag, which is a measure of the relative degree of viscous character to elastic character. 

At the bottom, a typical sample behavior is sketched for a DMA experiment. Sinusoidal stress and strain show a phase difference, 6, as can be seen from Eqs. (1) and (2) of Fig. 6.19. 

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... 

The simplest dynamic system to analyse is one in which the stress and strain are changing in a sinusoidal fashion. Fortunately this is probably the most common type of loading which occurs in practice and it is also the basic deformation mode used in dynamic mechanical testing of plastics. 

Fig. 2.53 Sinusoidal variation of stress and strain in viscoelastic material...

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... 

Dynamic mechanical testers apply a small sinusoidal stress or strain to a small sample of the polymer to be examined and measure resonant frequency and damping versus temperature and forced frequency. Instrument software computes dynamic storage modulus (G ), dynamic loss modulus (G") and tan delta or damping factor. Measurements over a wide range of frequency and temperature provide a fingerprint of the polymer with sensitivity highly superior to DSC. 

Fig. 23.3 Sinusoidally varying stress and strain in a dynamic mechanical experiment.

The response of a material to an applied stress after very short times can be measured dynamically by applying a sinusoidally varying stress to the sample. A phase difference, which depends on the viscoelastic nature of the material, is set up between stress and strain. 

This instrument operates by applying an oscillatory, sinusoidal stress and records the strain (Figure 17.16). The solid line corresponds to the applied stress, controlled by the instrument, and the sample s response strain appears as the dotted line. The rheometer measures the variation in strain as a function of applied stress and reports... 

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) ... 

In a dynamic experimeni, the stress will be directly proportional to the strain if the magnitude of the strain is small enough. Then, if the stress is applied sinusoidally the resulting strain will also vary sinusoidally. In special cases the stress and the strain will be in phase. A cross-linked, amorphous polymer, for example, will behave elastically at sufficiently high frequencies. This is the situation depicted in Fig, 1 l-13a where the stress and strain are in phase and the strain is small. At sufficiently low frequencies, the strain will be 90° out of phase wilh the stress as shown in Fig. 11-13c. In the general case, however, stress and strain will be out of phase (Fig. 11-13b). 

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. 

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. 

Treatments similar to those used in equations (3-32) and (3-33) can be applied to the generalized Maxwell model undergoing sinusoidal stress or strain... 

These equations are often used in terms of complex variables such as the complex dynamic modulus, E = E + E", where E is called the storage modulus and is related to the amount of energy stored by the viscoelastic sample. E" is termed the loss modulus, which is a measure of the energy dissipated because of the internal friction of the polymer chains, commonly as heat due to the sinusoidal stress or strain applied to the material. The ratio between E lE" is called tan 5 and is a measure of the damping of the material. The Maxwell mechanical model provides a useful representation of the expected behavior of a polymer however, because of the large distribution of molecular weights in the polymer chains, it is necessary to combine several Maxwell elements in parallel to obtain a representation that better approximates the true polymer viscoelastic behavior. Thus, the combination of Maxwell elements in parallel at a fixed strain will produce a time-dependent stress that is the sum of all the elements ... 

DMA is the most useful technique to study the viscoelastic properties of polymers [21], The sample is mounted in a temperature-controlled chamber. A sinusoidal stress is applied to the sample, and the resulting strain is measured for complex modulus analysis. For purely elastic materials, the stress and strain will be perfectly in phase, while for purely viscous material, there will be a 90° phase angle. The storage and loss moduli of the sample can be obtained. The storage modulus is the elastic part (i.e., stored energy), while the loss modulus is the viscous part (i.e., dissipated energy). The parameters obtained from DMA are listed in Table 20.1. 

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. 

The viscoelastic behaviour of concentrated suspensions can be studied using several different methods (4, 7). The most widely used method consists of subjecting the material to a continuously oscillating strain over a range of frequencies and then measuring the peak value of the stress, ao, and the phase difference between the stress and strain, 8. A sinusoidal deformation is usually employed. 

Similarly, phase angle, 5, is the phase difference between stress and strain in sinusoidal harmonic oscillation, expressed in degrees, to the nearest 0.1°. Figure 4.8 shows a schematic representation of the phase angle under stress control conditions. 

According to CEN EN 12697-26 (2012), complex modulus, E, is defined as the relationship between stress and strain for a linear viscoelastic material submitted to a sinusoidal load wave form at time t, where applied stress o x sin (co x t) results in a strain e x sin (o) x (t - rf>)) that has a phase angle O, with respect to stress (co = angular speed, in radians per second). The amplitude of strain and the phase angle are functions of the loading frequency, f, and the test temperature, . 

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. 

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