Biomedical Engineering Reference
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of
F
and
M
coincide, only the shape of the ellipsoid changes.
M
is a positive
definite symmetric tensor because it represents an ellipsoid. The fabric tensor or
mean intercept ellipsoid can be measured using the techniques described above for
a cubic specimen. On each of three orthogonal faces of a cubic specimen of
cancellous bone an ellipse will be determined from the directional variation of
mean intercept length on that face. The mean intercept length tensor or the fabric
tensor can be constructed from these three ellipses that are the projections of the
ellipsoid on three perpendicular planes of the cube.
7.8 The Stress-Strain-Fabric Relation
If the porous medium may be satisfactorily represented as an orthotropic, linearly
elastic material, the associated elasticity tensor
C
will depend upon the porous
architecture represented by the fabric tensor
F
. The six-dimensional second rank
elasticity tensor
C
relates the six-dimensional stress vector to the six-dimensional
infinitesimal strain vector in the linear anisotropic form of Hooke's law, (4.36 H).
The elasticity tensor
C
completely characterizes the linear elastic mechanical
behavior of the porous medium. If it is assumed that all the anisotropy of the porous
medium is due to the anisotropy of its solid matrix pore structure, that is to say
that the matrix material is itself isotropic, then a relationship between the
components of the elasticity tensor
C
and
F
can be constructed. From previous
studies of porous media it is known that the medium's elastic properties are strongly
dependent upon its apparent density or, equivalently, the solid volume fraction of
matrix material. This solid volume fraction is denoted by
n
and is defined as the
volume of matrix material per unit bulk volume of the porous medium. Thus
C
will
be a function of
as well as
F
. A general representation of
C
as a function of
and
F
was developed based on the assumption that the matrix material of the porous
medium is isotropic and that the anisotropy of the porous medium itself is due only
to the geometry of the microstructure represented by the fabric tensor
F
. The
mathematical statement of this notion is that the stress tensor
T
is an isotropic
function of the strain tensor
E
and the fabric tensor
F
as well as the porosity
n
n
j
. Thus
the tensor valued function
T
¼
T
ð';
E
;
F
Þ;
(7.35)
has the property that
QTQ
T
QEQ
T
QFQ
T
¼
T
ð';
;
Þ;
(7.36)
for all orthogonal tensors
Q
. The most general form of the relationship between the
stress tensor and the strain and the fabric tensors consistent with the isotropy
assumption (
7.37
)is
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