Stress sinusoidal

If the material being subjected to the sinusoidal stress is elastic then there will be a sinusoidal variation of strain which is in phase with the stress, i.e. 

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. 

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. 

Note 4 For linear viscoelastic behaviour, a sinusoidal stress (o) results from the sinusoidal strain with... 

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... 

The application of sinusoidal stress to a body leads inevitably to the complex modulus G , where... 

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 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. 

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... 

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". 

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 ... 

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. 

Figure 16. Time profile of an applied sinusoidal stress wave and the corresponding resulting sinusoidal strain wave as they apply to small deformation rheological testing.

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. 

Time to Break for FNMA Under Sinusoidal Stress (0.16A Hz)... 

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... 

By considering a sinusoidal stress applied to a viscoelastic body, show that in-phase strain results in conservation of energy (work) whereas out-of-phase strain results in energy dissipation. 

Dynamic Experiments. We will now consider the response of a Maxwell element subjected to a sinusoidal stress, as in a controlled-stress dynamic mechanical analyzer. In such a case the strain will also be sinusoidal but out of phase with the stress by the angle 8, as discussed in Chapter 2. Thus... 

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. 

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 plotted in Figure 7-2. They are formally identical to the compliance response of a Voigt element in series with a spring when the entire model is subjected to a sinusoidal stress. The complex dielectric constant is thus the analogue of the complex compliance, with the electric field playing the role of stress and the electric displacement 4 7rcr playing the role of strain. 

The gel point in materials represents the point where behavior changes from viscous (liquid-like) to elastic (solid-like). The conditions under which this occurs are critical to such food constituents as wheat-soya solutions used as setting agents within reconstituted meat products. In this case, oscillatory-controlled stress experiments, in which a small sinusoidal stress is applied to the material, provide a convenient method for evaluating elastic and viscous properties without destroying the delicate structure of soft semisolids. 

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 ... 

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. 

The response of a sinusoidal stress signal applied to a material will depend on the viscoelastic nature of that material, hence a sample of spring steel will be almost totally elastic, whereas a tub of grease or honey will be predominantly viscous. Hooke s "True theory of elasticity" said. The power of any spring is in the same proportion with the tension thereof, i.e. if one power stretch will bend it one space, two will bend it two. three will bend it three and so forth." Hence force F = A(A.v). where delta v is the displacement. Euler refined this to include the cross-sectional area A and the original length Z, but it was not until about 1800 that Thomas Young published it more widely as ... 

Starting with a sinusoidal input of strain in a Maxwell element (see Chapter 3), we derive the resulting sinusoidal stress. First we let the strain he a function of a maximum or peak strain o and time t with a frequency w ... 

If we consider the bone being driven by a strain at a frequency u, with a corresponding sinusoidal stress lagging by an angle 5, then the complex Young s modulus E (a>) may be expressed as... 

E9.9 Corrosion fatigue testing is performed using a sinusoidal stress function. The stress cycle is characterized by the stress ratio, R, of minimum to maximum stress,... 

---