Generation of Heat in Rapid Oscillating Deformations

From equation 6 above, the energy dissipated per second in small oscillating deformations is (per unit volume of material) 

The dissipated energy causes a rise in temperature, whose magnitude of course depends on the heat capacity of the system. The temperature may reach a steady-state value for continuous sinusoidal deformation, depending on the rate of heat loss to the surroundings. Equation 10 can be used to estimate heat production in various experimental procedures, where the strains are often purposely kept very small at high frequencies to prevent temperature rise as well as to insure linear viscoelastic behavior. It can also be used to estimate heat production in practical situations of cyclic deformations, such as the performance of automobile tires. Values can be compared on a relative basis even though the stress distribution in a loaded tire is complicated and the strains exceed the limitations of linear viscoelasticity and the cyclic deformation does not follow a simple sinusoidal pattern. 

Since in a tire under operating conditions the peak stress is specified, J is the relevant function in equation 10. It has been pointed out that, although minimum heat dissipation at the frequency of wheel rotation is desired, higher losses at low frequencies may be beneficial in providing a smoother riding vehicle. Thus it would be desirable to operate on the left side of the maximum in Fig. 14-16. The losses should be compared not at the ambient temperature but at the steady-state temperature during operation. 

As an example of a numerical calculation, a lightly cross-linked natural rubber with J = 1.0 X 10 cm /dyne under a sinusoidal deformation at 10 Hz and a peak shear stress of 10 dynes/cm would dissipate 0.0038 calorie/cc/sec. Since the heat capacity is about 0.5 calorie/deg/g, the temperature would rise about 0.008 deg/sec if no heat were lost by conduction. Other examples are given by Kramer and Ferry.  

For deformation in flexure, the relevant loss functions are E and D . The relation of E to the steady-state temperature rise AT in continuous cyclic deformation as measured in a Goodrich flexometer, taking into account nonlinearity of response and the effect of added filler in a rubbery polymer, has been studied extensively by Maekawa, Ninomiya, and collaborators. A simplified equation which satisfactorily represented many experimental results is 

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