Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 20

The global form of the linear momentum balance equation for a fixed region V bounded by the closed surface S having outward unit normal n is written as

(2.32)

where σ is the stress tensor and where b is the body force per unit mass acting on the medium. Such body forces usually arise from the effects of gravity. On using (2.27) and the divergence theorem, the linear momentum balance equation (2.32) may be written as

(2.33)

As relation (2.33) must be satisfied for any region V of the system, it follows that the local form of the linear momentum balance equation has the form

(2.34)

As

(2.35)

it follows from (2.34) on using (2.22) that

(2.36)

On using (2.31), relation (2.36) reduces to the well-known equation of motion

(2.37)

which must be satisfied at every point in the medium for all times t > 0.

It is worth noting that by considering the balance of angular momentum it can be shown that in the absence of couple stresses, the stress tensor σ must be symmetric.