Energy Considerations under Sinusoidal Deformation

The elastic energy stored at time t is given by (1.3.2). Consider this quantity for the sinusoidal strain history given by (1.5.5). Strictly, (1.3.2) is not applicable since e(-oo) does not vanish. However, it averages to zero at large times. Transformation of the time variable gives 

On decomposing the trigonometric product into a sum and averaging over a period In/oj, we find that the average level of stored elastic energy is 

Applying essentially the same argument and using (1.3.9) and (1.5.4b), it may be shown that 

Since D is non-negative, it follows that fi2((o) is also. Furthermore, the average level of stored energy in the body under sinusoidal deformation will be non-negative, so that the same is true for from (1.5.25). It follows that 

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