Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 179

and since this must hold for all constant vectors c , it follows that
þ
V r

þ

udv ¼ n uds : (7.11)

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The final result in the representation for h E i is achieved immediately by recalling the

T

definition of the small strain tensor E ¼ð 1 = 2 Þððr u Þ þr u Þ ,thusfrom( 7.9 )

þ

1
V RVE

1
2 ð

hi¼ E n u þ u n Þ ds : (7.12)

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With the representations ( 7.8 ) and ( 7.12 ) for the global or macro stress and strain

, respectively, the effective anisotropic elastic constants _ eff tensors, h E i and h T i are

defined by the relation

_ eff _ _ h i h i: (7.13)

This formula provides the tool for the evaluation of the effective material elastic
constants of a composite in terms of the material parameters of its constituent
components or phases and the arrangement and geometry of the constituent
components. In the next section results obtained using this formula are recorded
in the cases of spherical inclusions in a matrix material and aligned cylindrical
voids in a matrix material.
As a second example of this averaging process for material parameters the
permeability coefficients in Darcy's law are considered. In this case the vectors
representing volume averages of the mass flow rates hqi and the pressure gradient
hr

have to be expressed in terms of surface integrals over the RVE. Obtaining
such a formula for the pressure gradient

p i

pc ,
where c is a constant vector, into the divergence theorem in the form (A183), then
remove the constant vector from the integrals, thus

hr p i is straightforward. Substitute r ¼

0
@

1
A ¼

þ
V r

þ
@V ð

c pdv npdv 0 ; (7.14)

then, since ( 7.14 ) must hold for all vectors c , it follows from ( 7.14 ) and ( 7.1 ) that

þ

1
V RVE

hr p i¼ npds : (7.15)

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