Sinusoidal strain, definition

A depiction of the origin of these definitions is provided by Figure 2-13, which shows graphically the nature of the two sinusoidal signals vs. time. For this example, the strain, which is fixed by the instrument, is given by the equation ... 

In the above considerations, a sinusoidal shear strain is applied to the sample. It should be clear that a sinusoidal shear stress could also be applied resulting in corresponding compliance functions J and J". The former results from the deformation in phase with the stress, while the latter corresponds to the out-of-phase deformation. The value of tan 5 remains the same, as can be seen from the curves in Figure 2-13, where we can easily imagine the stress as the applied variable and strain as the measured variable. Tensile stress is equally applicable and definitions of E (co), E" (o), D"(co), D co), etc. are completely analogous to the derived shear parameters. At a given frequency, the value of tan 8 is always the same for any of these quantities, i.e., tan 8 = E"/E = D"/D . 

Mechanical testing procedures in common use involve other patterns of stress history than the simple creep and relaxation experiments on which the definitions of the transient viscoelastic functions are based, and the sinusoidally varying stress which is inherent in the definitions of the so-called dynamic properties. Certain relations between the behavior under coniplicated conditions and the basic viscoelastic functions are presented here together with some related problems. They are limited to linear viscoelastic systems and hence small strains, but in some cases could be extended to describe larger deformations, especially for simple extension. 

Figure 7.13 Sinusoidally varying stress and strain in a dynamic mechanical experiment. Construction shows definition of E = C B and E" =DfB.

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