Page 458 - Introduction to Continuum Mechanics

442 Integral Formulation of General Principles

In the above equation, everything is a function of the material coordinates^ and t, T 0 is the
first Piola-Kirchhoff stress tensor and n 0 is the unit outward normal. Using the divergence
theorem for the stress vector term, Eq. (7.6.8) becomes

where in Cartesian coordinates,

From Eq. (7.6.9) ,we obtain

This is the same equation derived in Chapter 4, Eq. (4.11.6).

A homogeneous rope of total length / and total mass m slides down from the corner of a
smooth table. Find the motion of the rope and the tension at the corner.
Solution. Let* denote the portion of rope that has slid down the corner at time t. Then
the portion that remains on the table at time t is l-x. Consider the control volume shown as
(V c)i in Figure 7.3. Then the momentum in the horizontal direction inside the control volume
at any time t is, with x denoting dx /dt:

Fig. 73