Page 448 - Introduction to Continuum Mechanics

432 Integral Formulation of General Principles

Example 7.2.2

Referring to Example 7.2.1, also show that the resultant moment, about a fixed point O, of
the distributive forces on S is given by

where x is the position vector of the particle with volume dV from the fixed point O and V is
the axial (or dual) vector of the antisymmetric part of T (see Sect. 2B16).
Solution. Let m denote the resultant moment about O. Then

Let m{ be the components of m, then

Using the divergence theorem, Eq. (7.2.3), we have

Now,

Noting that -e/^T^p are components of twice the dual vector of the antisymmetric part of T
dT-
are
e
[see Eq. (2B16.2b)], and ij&j(-jr^) components of [xxdivT], we have
OJtn

Example 7.2.3

Referring to Example 7.2.2, show that the total power (rate of work done) by the stress
vector on S is given by,