Periodic or Dynamic Experiments

To supplement the above transient i.e., nonperiodic) experiments and provide information corresponding to very short times, the stress may be varied periodically, usually with a sinusoidal alternation at a frequency v in cycles/sec (Hz) or o(= 270 ) in radians/sec. A periodic experiment at frequency to is qualitatively equivalent to a transient experiment at time f = 1 /oj, as will be evident in the examples shown in Chapter 2. If the viscoelastic behavior is linear, it is found that the strain will also alternate sinusoidally but will be out of phase with the stress (Fig. 1-7). This can be shown from the constitutive equation as follows. Let 

Geometry and time profile of a simple shear experiment with sinusoidally varying 

The integrals converge only if (J( ) — 0 as 5 oo otherwise they must be formulated somewhat differently (Chapter 3). It is clear that the term in sin U3t is in phase with 7 and the term in cos o)/ is 90 out of phase a is periodic in to but out of phase with 7 to a degree depending on the relative magnitudes of these terms. The quantities in brackets are functions of frequency but not of elapsed time, so equation 18 can be conveniently written 

It is instructive to write the stress in an alternative form displaying the amplitude o-°(co) of the stress and the phase angle 5(co) between stress and strain. From trigonometric relations, 

It is evident that each periodic, or dynamic, measurement at a given frequency provides simultaneously two independent quantities, either G and G or else tan 5 and o /7 , the ratio of peak stress to peak strain. (Over a sufficiently wide range of frequencies, however, G and G are not independent and can be interrelated as discussed in Chapters 2, 3, and 4.) 

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