Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 178

@
@

x i @

T jk
@

T jk @

x i

x j ð

T jk x i Þ¼

x j þ x j ¼

T ik (7.4b)

@

since @ T jk 0, @x i

@x j ¼ @x j ¼ d ij . The expansion of

rð x T Þ is shown below:

rð x T Þ ¼ ðr x Þ T þ x ðr T Þ¼ 3 T þ x ðr T Þ (7.5)

Substitution of the identity ( 7.3 ) into the definition for h T i yields

þ þ

V fr ð

1
V RVE

1
V RVE

T dv

hi¼ T Tdv ¼ T x Þg ; (7.6)

V

and subsequent application of the divergence theorem (A184) converts the last
volume integral in ( 7.6 ) to the following surface integral:

þ
@V f

1
V RVE

T ds

hi¼ T n ð T x Þg : (7.7)

Finally, employing the Cauchy relation (3.16), t ¼ T n , provides a relationship

h T i

between and the integral over the boundary of the RVE depending only upon the

stress vector t acting on the boundary:

þ

1
V RVE

hi¼ T x tds : (7.8)

@ V

in terms only of
boundary information, the surface tractions t acting on the boundary.
It is even easier to construct a similar representation of E

This is the desired relationship because it expresses T hi

as an integral over the
boundary of the RVE. To accomplish this set the f in ( 7.1 ) equal to

hi

r u and then

employ the divergence theorem to obtain

þ
V ðr

þ
@V ð

1
V RVE

1
V RVE

h r u i ¼ u Þ dv ¼ n u Þ ds : (7.9)

This form of the divergence theorem employed above is obtained by setting

T c in (A184), where u is the displacement vector and c is a constant vector.

The divergence theorem (A184) may then be written in the form
þ

¼ u

þ

c ðr u Þ dv ¼ c ð n u Þ ds ; (7.10)

V @V