Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 65

_

_ 0. Equating p to

where the term in the square brackets vanishes because x x ¼ S F

H to

and M as required by the conservation of linear and angular momentum,

respectively, we obtain from ( 3.21 ) and ( 3.24 ) that
ð
O r€

S

ð ð

O r

x d v ¼ t d a þ d d v ; (3.25)

@O

and from ( 3.22 ) and ( 3.24 ) that
ð ð ð

x € x r d v ¼ x t d a þ x r d d v : (3.26)

O @O O

These integral forms of the balance of linear and angular momentum are the
global forms of these principles. The global forms are weaker statements of these
balance principles than are the point forms that we will now derive. We say that the
point forms are stronger because it must be assumed that the stress is continuously
differentiable and that r € d are continuous everywhere in the object in order to

obtain the point forms from the global forms. The point form of the balance of
linear momentum is obtained from ( 3.25 ) by first substituting ( 3.16 ) into the surface
integral in ( 3.25 ),

x and r

ð
O rx d v

ð ð

O r

¼ T n d a þ d d v ; (3.27)

@O

then applying the divergence theorem (A184) to the surface integral in ( 3.27 ),
ð
O rx d v

ð
O r

ð
O r

T T d v

¼ þ d d v ;

(3.28)

The second step in obtaining the point form is to rewrite ( 3.28 ) as a single
integral,

ð
O ðr€

T T

x r r d Þ d v ¼ 0 ; (3.29)

and then employ the same argument as was employed in the transition from ( 3.4 )to
( 3.5 ); see the discussion following ( 3.5 ). It follows then that

T T r€ x ¼r þ r d (3.30)

at each point in the object O . There is one more point to make about ( 3.30 ) before it
is complete. That point is that the stress tensor T is symmetric and thus the