Page 471 - Introduction to Continuum Mechanics

Principle of Conservation of Energy 455

Since

Therefore, Eq, (7.10.2) becomes

Thus, at every point, we have

For a symmetry tensor T, this equation can also be written

Equations (7.10.3a) or (7.10.3b) is the energy equation. A slightly different form of
Eq. (7.10.3b) can be obtained if we recall that Vv = D+W, where D, the symmetric part of
Vv is the rate of deformation tensor, and W, the antisymmetric part of Vv, is the spin tensor.
We have

therefore, we rediscover the energy equation in the following form:

On the other hand, if we use the Reynold's theorem in the form of Eq. (7.4.1), we obtain

Equation (10.5) states that the time rate of work done by surface and body forces in a control
volume + rate of heat input = total rate of increase of internal and kinetic energy of the
material inside the control volume + rate of outflow of the internal and kinetic energy across
the control surface