Stress strain phase angle

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... 

Fig. 17. Viscoelastic material stress ( and strain (---------) ampHtudes vs time where 5 is the phase angle that defines the lag of the strain behind the...

A technique for performing dynamic mechanical measurements in which the sample is oscillated mechanically at a fixed frequency. Storage modulus and damping are calculated from the applied strain and the resultant stress and shift in phase angle. 

The ratio is the tangent of a phase angle 8 by which the strain lags behind the stress. 

If the material is anelastic, the stress and the strain will not coincide. The strain will lag behind by an amount which is determined by the phase angle, ( ). Thus ... 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

Tangent of the phase angle difference (< between stress and strain during forced oscillations. 

Polarizability Permeability Poisson s ratio Dielectric polarization Phase angle between stress and strain Specific resistance Polyamide (nylon)... 

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... 

If the rubber were a perfect spring the stress (t) would be similarly sinusoidal and in phase with the strain. However, because the rubber is viscoelastic the stress will not be in phase with the strain but can be considered to precede it by the phase angle (8) so that ... 

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.

Fig. 2.2 Vector representation of an alternating stress leading and alternating strain by phase angle S. (From ref. [1])...

Sinusoidal strain and stress with phase angle 8. 

Dynamic or oscillatory rheometers measure viscous and elastic modulus in shear or tension. Energy dissipation produces a phase difference, so stress, strain, and phase angle can be used to characterize complex viscosity behavior. 

When using small deformation rheology there are several useful parameters that may be obtained to describe a material the complex modulus (G ), storage modulus (G ), loss modulus (G") and the tangent of the phase shift or phase angle (tan 5). These values must be taken from within the LVR, and are obtained using a dynamic oscillatory rheometer (Rao 1999). Outside the LVR, important information may be obtained such as the yield stress and yield 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". 

Fig. 12. Simple dynamic relationships between stress and strain, and the phase angle.

---