Strain wave, stress response

Elastic response This occurs when the maximum of the stress amplitude is at the same position as the maximum of the strain amplitude (no energy dissipation). In this case, there is no time shift between the stress and strain sine waves. Viscous response This occurs when the maximum of the stress is at the point of maximum shear rate (i.e., the inflection point), where there is maximum energy dissipation. In this case, the strain and stress sine waves are shifted by (referred to as the phase angle shift, 5, which in this case is 90°). 

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 ... 

Figure 6, Stress response to the fundamental strain wave superimposed with smaller amplitude strain wave, (top) Fundamental strain wave (middle) superimposed strain wave (Ibottom) resulting stress wave.

Pig. 1. (a) When a sample is subjected to a sinusoidal oscillating stress, it responds in a similar strain wave, provided the material stays within its elastic limits. When the material responds to the applied wave perfectly elastically, an in-phase, storage, or elastic response is seen (b), while a viscous response gives an out-of-phase, loss, or viscous response (c). Viscoelastic materials fall in between these two extremes as shown in (d). For the real sample in (d), the phase angle S and the amplitude at peak k are the values used for the calculation of modulus, viscosity, damping, and other properties. 

If we perform an oscillatory test by applying a sine-wave-shaped input of either stress or strain, we can then, using suitable electronic methods, easily resolve (i.e. separate) the resulting sinusoidal strain or stress output into a certain amoimt of solid-like response, which is in phase with the input, and a corresponding amount of liquid-like response which is n/2 (i.e. 90°) out of phase with the input, see figure 8. 

For controUed-strain rheometers, the shear strain that is a sinusoidal function of time, t, can be expressed as, y(t) = yo(sin cot), where yo is the amplitude of the applied strain and a> is the angular frequency of oscillation (in rad s ). The angular frequency is related to frequency, /, measured in cycles per second (Hz) whereby a> = litf The shear stress resulting from the apphed sinusoidal strain will also be a sinusoidal function, which can be expressed as t(t) = to(sin cot + S) in which to is the amplitude of the stress response and 8 is the phase difference between the two waves. 

On the other hand, for stress-controlled rheometers, the shear stress is applied as t(0 = to(sin cot) and the resulting shear strain is measured as y(t) = yo(sin cot + S). For a purely elastic material, it follows from Hooke s law that the strain and stress waves are always in phase (8 = 0°). On the other hand, while a purely viscous response has the two waves out of phase by 90° (8 = 90°). Viscoelastic materials give rise to a phase-angle somewhere in between (Fig. 3). 

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... 

In this section we deal with perhaps the most conceptually difficult of all the responses observed in linear viscoelastic materials. This is the response of a material to an oscillating stress or strain. This is an area that illustrates why rheological techniques can be considered as mechanical spectroscopy. When a sample is constrained in, say, a cone and plate assembly, an oscillating strain at a given frequency can be applied to the sample. After an initial start-up period, a stress develops in direct response to the applied strain due to transient sample and instrumental responses. If the strain has an oscillating value with time the stress must also be oscillating with time. We can represent these two wave-forms as in Figure 4.6. 

In an experiment when we apply a step strain the rate of application of the strain influences the relaxation of the stress at short times. There are other factors which can influence the response that is observed. For example it is common for elastic samples to resonate with the applied actuator and transmit transient waves through the sample. This can lead to fluctuations in the stress at short times. A typical example is shown in Figure 4.14. 

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 ... 

Let us analyze the response of a material when it is compressed or expands in a single direction, for example along the Xj axis, and deformations along the X2 and X3 axes are not permitted. This situation occurs when a material is compressed in a cylinder or when an acoustic wave propagates through the material. In these conditions, the stress and strain tensors can be written as... 

This indicates a linear dependence of the shear wave on the distance X2 and consequently a time dependence of the strain in phase with the displacement and the stress. In other words, a thin slab is nearly consistent with a linear response. 

The test arrangement is shown diagramatically in Fig. 3. The upper end of the specimen is screwed into a strain-gauged PMMA Hopkinson bar of 19 mm diameter and 1.5 m length, calibrated for short-time response. Pilot tests had shown that the craze lifetime under impact lay within the expected uniaxial stress wave return time of 1.2 ms. 

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. 

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