Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 175

Fig. 7.1 An illustration
of the stress concentration
(by a factor of 3) associated
with the hole in a circular
plate in a uniaxial field
of otherwise uniform stress,
S . From Cowin ( 2002 )

resolve the paradox described above is to observe that the modeler has decided how
big a hole is no hole by choosing to recognize a hole of a certain size and selecting a
value of L RVE to be much less. Let the size of the largest hole in the plate to be d L
and assume that the plate has a dimension of 100 d L , thus d L

L M . Let d S
denote the dimension of the largest of the other holes in the plate. Thus any hole
whose dimension is less than L M ¼

L RVE

d S will not appear in the model although it is in
the real object. The interpretation of the solution to the problem illustrated by
Fig. 7.1 is that there is only one hole in the model of radius d L , no holes of a size less
than d L and greater than L M ¼

d S , and all holes in the real object of a size less than

L M ¼ d S have been “homogenized” or averaged over.

The macro or continuum properties that are employed in continuum models are
micro material properties that have been averaged over an RVE. Let denote the

microdensity field and T the micro stress field; then the average or macrodensity
field

r

field are obtained by volume averaging over
the microscale. The averaging integral operator

hri and the macro stress tensor h T i

h f i of the micro field f is given by

þ

1
V RVE

h f i¼ fdv : (7.1)

V

Equation ( 7.1 ) represents the homogenization of the local or micro material
parameter fields. That is to say, in the volume V RVE , the average field replaces

the inhomogeneous field f in the RVE. The length scale over which the homogeni-
zation is accomplished is L RVE or the cube root of the volume V RVE , which is
intended to be the largest dimension of the unit cell over which the integration ( 7.1 )
is accomplished. A plot of the values of the macro density

h f i

as a function of the
size of the RVE is sketched in Fig. 7.2 . Note that as the size of the volume V RVE or
the L RVE is decreased, the value of the density

hri

begins to oscillate because the
small volume of dense solid material in the volume V RVE is greatly influenced by
the occurrence of small voids. On the other hand, as the size of the volume V RVE or
the L RVE is increased, the value of the density tends to a constant, stable value.
As another illustration of these ideas consider a cross-section of trabecular bone
shown in Fig. 4.3. As was noted in Sect. 4.2, the white regions are the bone

hri