Sinusoidal stress, viscoelasticity measurements

Note 2 Viscoelastic properties are usually measured as responses to an instantaneously applied or removed constant stress or strain or a dynamic stress or strain. The latter is defined as a sinusoidal stress or strain of small amplitude, which may or may not decrease with time. 

A general description of the fundamental relationships governing the dynamic response of linear viscoelastic materials may be found in several sources (28, 37, 93). In general, sinusoidally applied strains (stresses) result in sinusoidal stresses (strains) that are out of phase. Measurements may be made under uniaxial, shear, or dilational loading conditions, and the resultant complex moduli or compliance and loss-phase angle are computed. Rotating radius vectors are usually taken to represent the... 

Dynamic mechanical analysis involves the determination of the dynamic properties of polymers and their mixtures, usually by applying a mechanical sinusoidal stress For linear viscoelastic behaviour the strain will alternate sinusoidally but will be out of phase with the stress. The phase lag results from the time necessary for molecular rearrangements and this is associated with the relaxation phenomena. The energy loss per cycle, or damping in the system, can be measured from the loss tangent defined as ... 

A complete description of the viscoelastic properties of a material requires information over very long times. Creep and stress relaxation measurements are limited by inertial and experimental limitations at short times and by the patience of the investigator and structural changes in the lest material at very long times. To supplement these methods, the stress or the strain can be varied sinusoidally in a dynamic mechanical experiment. The frequency of this alternation is u cycles/s or m(= 27ri ) rad/s. An alternating experiment at frequency w is qualitatively equivalent to a creep or stress relaxation measurement at a time t = (I /w) sec. 

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. 

The four commonly used techniques to extract information on the viscoelastic behavior of suspensions are creep-compliance measurements, stress-relaxation measurement, shear-wave velocity measurements, and sinusoidal oscillatory testing (25-27). In general, transient measurements are aimed at two types of measurements, namely, stress relaxation, which is to measure the time dependence of the shear stress for a constant small strain, and creep measurement, which is to measure the time dependence of the strain for a constant stress. 

The apparatus described in this article can be used to measure the modulus and the elastic stress-viscoelastic stress phase angle diflFerence as functions of the phase angle in a sinusoidal straining of a specimen and to obtain the hysteresis loss through an automated integral circuit. 

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 ... 

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. 

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. 

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. 

DMA measures the viscoelastic properties of a sample using either transient or dynamic oscillatory tests. Transient tests include creep and stress relaxation. In creep, a stress is applied to the sample and held constant, while deformation is measured versus time. After some time, the stress is removed and the recovery measured. In stress relaxation, a deformation is applied to the sample and held constant the degradation of the stress required to maintain the deformation is measured versus time. The most common test is the dynamic oscillatory test, where a sinusoidal stress (or strain) is applied to the material and the resultant sinusoidal strain (or stress) is measured. Also measured is the phase difference, 8, between the two sine waves. The phase lag will be 0° for a purely elastic material and 90° for a purely viscous material. Viscoelastic materials such as polymers will exhibit an intermediate phase difference. Since modulus equals stress divided by strain, the complex modulus, E, can be calculated. From E and 8, the storage modulus, E, the loss modulus, E", and tan 8 can be calculated ... 

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. 

In dynamic (oscillator) measurements, a sinusoidal strain, with frequency v in Hz or CO in rad s (concentric cylinder) or plate (of a cone and plate) and the stress is measured simultaneously on the bob or the cone, which are connected to a torque bar. The angular displacement of the cup or the plate is measured using a transducer. For a viscoelastic system, such as the case with a cosmetic emulsion, the stress oscillates with the same frequency as the strain, but out-of-phase [11). Figure 12.4 illustrates the stress and strain sine waves for a viscoelastic system. 

The cone and plate viscometer can be used for oscillatory shear measurements as well. In this case, the sample is deformed by an oscillatory driver which may be mechanical or electromagnetic. The amplitude of the sinusoidal deformation is measured by a strain transducer. The force deforming the sample is measured by the small deformation of a relatively rigid spring or tension bar to which is attached a stress transducer. On account of the energy dissipated by the viscoelastic polymer system, a phase difference develops between the stress and the strain. The complex viscosity behavior is determined from the amplitudes of stress and strain and the phase angle between them. The results are usually interpreted in terms of the material functions, p, G, G" and others [33-40]. 

The viscoelastic properties of a polymeric material can be described by its reversible and irreversible responses to deformation. These can be identified most easily by dynamic mechanical analysis (DMA). Usually, the adhesive is placed between two parallel plates, one of which is oscillating sinusoidally, and the torque is measured. From the amplitude and the phase shift of the sinusoidal stress - strain curve, the elastic component, which is in phase, and the viscous component, which is 90° out of phase, can be derived [211, p. 158 if]. 

At the University of Wisconsin since 19 6, studies of viscoelasticity have evolved from concentrated polymer solutions to undiluted amorphous polymers, dilute solutions, lightly cross-linked rubbers, glassy polymers, blends of different molecular weights, copolymers, cross-linked rubbers with controlled network structures, and so forth. It became evident that each type of system required a different approach. Moreover, in amorphous polymers, the terminal, plateau, and transition zones had to be described separately. Both dynamic (sinusoidal) and transient measurements such as creep and stress relaxation have been utilized. The inderlying theme of this work is the relation of macromolecTilar dynamics—modes of motion of polymer molecules— to mechanical and other physical properties. 

The oscillatory rheological data are very useful to understand the microstracture of the polymeric material while subjected to deformation in the linear viscoelastic region usually with a small amphmde sinusoidal strain, measuring the resultant sinusoidal stress (Khan et al. 1997). Large amplitude oscillatory strain can also be applied. 

Thus, the aim of linear viscoelastic measurements (for incompressible materials) is to experimentally determine the relaxation modulus G(t) or quantities equivalent to G(t). In most cases of actual linear viscoelastic measurements, a sinusoidal shear strain y(t) =yosin(Bt (co is the angular frequency and is equal to 2rr/ with / being the frequency in the unit of Hertz) with the amplitude yo 1 is applied to a material. From eqn [22], the resulting shear stress is expressed as... 

Rheometric Scientific markets several devices designed for characterizing viscoelastic fluids. These instmments measure the response of a Hquid to sinusoidal oscillatory motion to determine dynamic viscosity as well as storage and loss moduH. 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 soHds, melts, and Hquids at frequencies from 10 to 500 rad/s and as a function of strain ampHtude 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. 

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. 

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) ... 

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