Biomedical Engineering Reference
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π
a
3
i
B
i
−1
(
4
M
=
(5-18)
2
π
γ
)
a
3
j
d
γ
(
a
1
/
a
2
)cos
2
(
a
2
/
a
1
)sin
2
γ +
γ
0
where
a
3
i
and
a
3
j
are the direction cosines of the angles between the
indentation direction and the elastic stiffness matrix. The expression
must be solved numerically and therefore is not easily implemented in
routine data analysis. Commonly reported are simply the effective
indentation modulus for different orientations, in which the results are
not converted back to components of the stiffness tensor (
C
ij
). Overall,
the indentation modulus varies less with orientation than does the
effective elastic modulus, where the maximum indentation modulus is
less than the maximum apparent modulus in tension and the minimum
indentation modulus is greater than the apparent modulus in tension.
17
Figure 5-7. Projected contact area for an isotropic and anisotropic material following
spherical or pyramidal indentation. For the isotropic case, the contact radii in
perpendicular directions are equal (
a
1
=
a
2
=
a
) whereas for the anisotropic case
a
1
a
2
.
5.2
. Inhomogeneous materials
Two features differentiate the indentation responses of composite
materials (
i.e
. materials with two or more distinct phases) compared with
homogeneous materials. First, a length-scale is introduced, in the form of
feature size is the particle radius
r
). Second, there is a critical dependence
on indenter tip placement in terms of the observed indentation response,
especially at small indentation depths where the indentation contact
radius
a
or contact depth
h
c
is comparable in scale to the composite
feature size
r
. Several recent studies have explicitly examined
indentation of materials with substantial heterogeneity.
18-20
The general
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