Strain with phase angle, variation

A part of this peculiar variation in the stress-strain phase angle difference comes from the variation of elastic modulus with the phase angle. Therefore, to determine the phase angle difference which is caused by the nonelastic contribution, it is necessary to determine and to subtract the contribution from the variation of elastic modulus as a function of the strain. 

The above analysis shows that the nonlinear dynamic viscoelastic behavior of polymers can be resolved into three components the nonlinear elasticity resulting from the variation of modulus with the phase angle or strain during the cycle nonlinear internal friction resulting from strain and strain-rate dependence and eflFects associated with the reversible, strain-induced structural changes. 

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. 

If a specimen is subject to a sinusoidal shear strain, and if steady-state conditions have been established, then the shear stress will also vary sinusoidally, but it will lag behind the stress by some phase angle <5. Thus, if the variation of shear stress with time is written as... 

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