Stress sinusoidal variation

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. 

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

In dynamic mechanical analysis of plastics, the material is subjected to a sinusoidal variation of stress and the strain is recorded so that 1, 2 and S can be determined. The classical variation of these parameters is illustrated in Fig. 2.55. 

Consider a sinusoidal variation of the applied stress on the SLS model, i.e., a=aQCOS(ot. The differential equation can be found from Eq. (5.50) and the solution is... 

In contrast, stress and strain can be 90° out of phase when sufficiently low fl e-quencies are used, a situation that is characteristic of liquid bodies. At intermediate frequencies, the phase difference between stress and strain is less pronounced. Sinusoidal variations of the stress can be represented as a rotating vector (OA) (Figure 12.6) whose projection (OB) on the vertical axis corresponds to the stress applied at a given time in that direction. 

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... 

It is useful to see how the Voigt linear viscoelastic models of Section 7.2 behave with a sinusoidal strain input. When the strain variation equation (Eq. 7.21) is substituted in the model constitutive equation (Eq. 7.2), the stress is given as... 

The molecular organization in the banded texture is understood in some detail as a result of optical microscopy [49, 50, 83, 110-112, 115-122, 126], transmission electron microscopy [49, 76, 89, 90, 115, 127] and scanning electron microscopy [50]. It has been shown that the orientation variation of the molecules in this texture is near sinusoidal about the shear direction with most of the misorientation within the shear plane. There is a high degree of register of the molecules between each of the layers [50]. The influence of a number of parameters on the formation of bands has been studied the molecular weight [96], the sample thickness [110,120-122], the shear rate [109], the time during which the shear stress was applied [ 122] and the total shear deformation [121]. 

In general, fatigue results are generated under sinusoidal loading conditions, where the time variation of the three stress components is given by... 

Figure 6.3 The stress variation for shear in a perfect crystal can be assumed to be approximately sinusoidal. The maximum shear stress is termed the theoretical shear strength.

In the case of the slip coefficient variation given in Figure 7, the pressure follows a sinusoidal curve with an average value equal to 5 MPa and average amplitude of around 0.55 MPa. The stress distribution around the caliper can be calculated with the Solidworks software, using the average value of pressure applied to the cahper. The maximum stress value obtained is equal to 196.2 MPa. 

The use of photon-coimting techniques in combination with sinnsoidal variation of stress-induced polarization modulation imposes quantifiable lirtritatiorrs on the accnracy and precision of CPL measurements. We consider first the optical characteristics of the PEM. The phase difference q>) between the two orthogonal crystal axes of the PEM is related to the sinusoidally varying periodic stress (sin at) thronghthe following Bessel fnnction... 

FIGURE 10.1 Schematical representation of the sinusoidal stress load subjected to a polymer sample and of the corresponding sinusoidal strain response. The linear gray line represents the variation of the load in the case of a quasistatic mechanical experiment. 

The treatment presented here does not follow the traditional approach of guessing a sinusoidal expression for the shear stress (see for example the treatment by Hirth and Lothe, mentioned in the Further reading section). Instead, we adopt a more general point of view based on a variational argument for the total energy of the dislocation, and introduce the sinusoidal behavior as a possible simple choice for the displacement potential. The essence of the resulting equations is the same as in traditional approaches. 

Rheological studies show some similarities with chemical relaxation studies. For instance, a rectangular shear rate is applied and the relaxation of the stress is monitored. This directly yields the stress relaxation time(s). One can also apply a sinusoidal deformation or strain of angular frequency 0). The response of the system is a two-component sinusoidal shear stress. The first component is in phase with the strain and corresponds to the elastic (storage) properties of the system. The second component is out of phase with the strain with a phase angle 5, and corresponds to the viscous loss in the system. These quantities give access to the storage (elastic) modulus G (co) and to the loss (viscous) modulus G"(o)), with G" ((o)/G (co) = tg5. As in the case of chemical relaxation methods with harmonic perturbation, the variations of G (w) and G" (co) with co yield the relaxation time(s) of the system. 

---