Page 223 - Introduction to Continuum Mechanics

208 Stress

Now, if the boundaries of the body are kept at fixed temperature, then when the steady-state
is reached, the net rate of heat flow into any element in the body must be zero. Thus, the
desired equation is

For constant /c, this reduces to the Laplace equation

4.14 Energy Equation
Consider a particle with a differential volume dVat the position x at time t. Let U denote
its internal energy, KE the kinetic energy, Q c the net rate of heat flow by conduction into the
particle from its surroundings,^ the rate of heat input due to external sources (such as
radiation) and P the rate at which work is done on the particle by body forces and surface
forces (i.e., P is the mechanical power input). Then, in the absence of other forms of energy
input, the fundamental postulate of conservation of energy states that

Now, using Eq. (4.12.6) and Eq. (4.13.1), we have

thus, Eq. (4.14.1) becomes

If we let u be the internal energy per unit mass, then

In arriving at the above equation, we have used the conservation of mass principle

Thus, the energy equation (4.14.1) becomes