Sinusoidal straining viscoelastic response

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... 

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. 

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 Ay is very small (Ay < 0.1%), the stress response caused by the sinusoidal straining given by Equation 1 is approximately sinusoidal, and the viscoelastic behavior falls in the region of linear viscoelasticity. In this case, the phase angle difference 8 between the stress wave and strain wave is constant throughout the cycle, and the stress response can be expressed by ... 

Characteristics of the Hysteresis Loop and Stress Wave in the Nonlinear Viscoelastic Response to the Sinusoidal Straining. Figure 3 is a schematic of a hysteresis loop obtained when a nylon 6 monofilament was subjected to a sinusoidal straining with yo = 1% and Ay = 1% at 90°C under a frequency of 10 cycles per sec. 

According to this representation, the nonlinear viscoelastic stress response resulting from sinusoidal strain ... 

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. 

The d3niamic instrument uses the method of sinusoidal excitation and response. In this case, the applied force and the resulting deformation both vary sinusoidally with time, the rate usually being specified by the frequency f in cycles/sec or w = 2 tt f in radians/sec. For linear viscoelastic behavior, the strain will alternate sinusoidally but will be out of phase with the stress. 

The amplitude of the sinusoidal deformation is made small (1.0 x 10 cm, approximately 0.03 to 0.05 percent strain) to ensure linear viscoelastic response. The frequency is maintained constant for each data point. 

DMA provides the most significant information on the viscoelastic behavionr of polymers in addition to thermal transitions. In principle, a sinusoidal strain or stress is applied to a sample and the response is monitored as a function of frequency and temperature. A viscoanalyser is commonly used to apply a displacement d(w) at the... 

Fig. 13.31. Imposed sinusoidal strain (solid line), and the corresponding sinusoidal stress (broken line) response of a viscoelastic material.

Indeed, viscoelasticity has been observed and measured in diastole through both organ-level and molecular-based studies. For example, Templeton et al. [73] applied a sinusoidal volume variation to an isolated LV chamber and measured its viscoelastic response via the phase delay of the resulting pressure response. Additionally, Rankin et al. [51] found that to fit the diastatic stress-strain relationship, a viscoelastic, rather than purely elastic model is needed. Similar results have been reported by Hess et al. [23] in humans, and other investigators have observed viscoelastic chamber properties in a variety of experimental settings [14,30,44,69,80]. 

It is interesting, from a fundamental point of view, to characterize the viscoelastic behavior of a fluid, which reflects the forces between the particles and hence the structure of the fluid. This characterization is done dynamically by applying on the fluid a sinusoidal stress with frequency N and low amphtude, so as not to break the structure of the fluid (linear viscoelastic mode). The response is a deformation of the same frequency. Elastic systems have a response in phase with the stress. In a Newtonian system, the shear stress is proportional to the strain rate and the sinusoidal strain response is dephased by 90° compared to the sinusoidal stress applied. The phase angle 5 (0 < 8 < 90°) is therefore a characteristic of viscoelastic behavior. We use the formahsm of complex numbers to write ... 

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. 

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... 

Figure H3.2.2 Responses of an ideal elastic, viscous, and viscoelastic material to a sinusoidal deformation. 8, phase angle y, shear strain co, angular frequency o, shear stress.

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. 

Due to the viscoelastic nature of the material the stress response, after the application of the oscillatory shear strain, is also a sinusoidal but out of phase relative to the strain what can be represented by equation (2.7) as... 

We wish to derive the steady state response of a linear viscoelastic body to an externally applied sinusoidal shear strain (dynamic testing) using the constitutive Eq. 3.3-8, which for this viscometric flow reduces to... 

Fig. E3.3 The schematic stress response of elastic, a viscous, and a viscoelastic hody to a sinusoidally applied strain.

Figire 17.16. Applied oscillatory, sinusoidal stress (solid), and sample response strain for pure solid (long dash) (A), pure liquid (short dash) (B) and a viscoelastic material (long short dash). The phase angle (< )) is the raw single used to determine G and G". 

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 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. 

Let us assume that a sinusoidal shear strain z t) = Sq sin cot is imposed on a viscoelastic solid, where 8q and co are, respectively, the amplitude and frequency of the perturbing strain. A dynamic shear strain is illustrated in Figure 6.1. Experimentally one observes that the shear stress (response) is... 

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