Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 66

transpose notation in ( 3.30 ) is not necessary. In the next paragraph the conservation
of angular momentum is used to show that the stress tensor T is symmetric.
The arguments to show that the stress tensor T is symmetric are algebraically
simpler if we replace the statement of the conservation of angular momentum given
above, ( 3.26 ), by the equivalent requirement that the skew-symmetric part of Z ,

ð ð ð

Z ¼ x x r

d v

x t d a x r d d v ;

(3.31)

O @ O O

be zero:

Z T Z ¼ 0 : (3.32)

Equation ( 3.32 ) is equivalent to ( 3.26 ). The rationale for this equivalence is
that the components of the cross-product of two vectors, say a b , are equal to

the components of the skew-symmetric part of the open product of the two vectors,
a b . Since ( 3.32 ) requires that Z be symmetric, it follows that the skew-
symmetric part of Z is zero. Equation ( 3.26 ) is equivalent to the skew-symmetric
part of Z , and the equivalence is established.
The sequence of steps applied to the conservation of linear momentum in the
paragraph before last are now applied to the expression ( 3.31 ) for Z . The point form
of ( 3.31 ) is obtained by first substituting ( 3.16 ) into the surface integral in ( 3.31 ),

ð ð ð

Z ¼ x € x r d v x Tn d a x r d d v ; (3.33)

O @O O

then applying the divergence theorem (A184) to the surface integral in ( 3.33 ),

ð ð

O fðr

ð

Z ¼ x € x r d v x Þ T þð x r T Þg d v x r d d v : (3.34)

O O

This result is simplified by observing that

∇

x ¼ 1 and collecting all the

remaining integrals containing x together, thus

ð ð

T T

fr €

Z ¼ T d v þ x x r r d Þ d v : (3.35)

O O

The second integral in ( 3.35 ) is exactly zero because its integrand contains the
point form statement of linear momentum conservation ( 3.30 ); thus ( 3.35 ) and
( 3.32 ) show that

ð
O ð

T T T Þ d v ¼ 0 : (3.36)