Page 204 - Introduction to Continuum Mechanics

Equations of Motion - Principle of Linear Momentum 189

These are the equations that must be satisfied for any continuum in motion, whether it be solid
or fluid. They are called Cauchy's equations of motion. If the acceleration vanishes, then Eq.
(4.7.2) reduces to the equilibrium equations

or,

Example 4.7.1
In the absence of body forces, does the stress distribution

where vis a constant, satisfy the equations of equilibrium?
Solution. Writing the first(/ =1) equilibrium equation, we have

Similarly, for i = 2, we have

and for 1 = 3

Therefore, the given stress distribution does satisfy the equilibrium equations

Example 4.7.2

Write the equations of motion if the stress components have the form 7^- = -pfyj where
p = p(xi^ 2^t)