# Visco-elasticity and dissipation of energy

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?

• Is your only question what Im means? Commented Apr 23, 2020 at 20:24

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.
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.