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I have a doubt with the integral of equation 1 shown in the picture and what's the meaning of Im in the integral. I don't have a good knowledge about visco-elastic theory so a simple explanation would be appreciated. I want to know how the equation is transformed and how to calculate energy dissipation using this theory?It is a screenshot of persson's paper on theory of rubber friction

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  • $\begingroup$ Is your only question what Im means? $\endgroup$
    – G. Smith
    Apr 23, 2020 at 20:24

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The elastic energy is given by $$ \mathcal{E} = \int_V \sigma_{ij} \varepsilon_{ij} dV \,.$$

The energy loss can be seen as the change in energy over a specific time interval. If this is written in the infinitesimal theory, we obtain $$ \Delta \mathcal{E} = \int_t \left[\int_V \sigma_{ij} \varepsilon_{ij} dV \right] dt\,.$$ Then, you have to take the Fourier transform to obtain the form of equation (1) in the document you shared.

In visco-elastic theory, a material is lossless if it has a real stiffness tensor (a real stiffness constant). On the other hand, if the stiffness tensor (or stiffness constant) is complex, then the complex part introduces losses. In the document you shared, one tries to calculate the dissipation, so we are only interested in the contribution from the complex part of the stiffness. Therefore, the "Im" is introduced which takes the imaginary part of the stiffness constant.

The real part of $E(\omega)$ is used to calculate the amount of energy that resonantly oscillates in the system whereas the imaginary part has to be used to calculate the energy loss in the system.

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