Biomedical Engineering Reference
In-Depth Information
the repeated indices), so that n zx ¼ n yx ¼ 0.2; in the plane of transverse isotropy,
m yz ¼ E y 2 ð 1 þ n yz ÞÞ .
A word about Poisson's ratio is needed here, because, as seen even for the
simplest problems discussed in this chapter, the stress field at the free edge is quite
sensitive to the mismatch in Poisson's ratio between the two adjoining materials.
Those familiar with linear, isotropic elasticity will recognize the value of 2 for n xz
and n xy as being outside the thermodynamic bounds for an isotropic material, but
they are well within the allowable range for a transversely isotropic material
(see, e.g., [ 42 ]). The requirement that stressing of any sort must lead to positive
stored energy requires that each term on the diagonal of the compliance matrix S ij
be positive, and that each term on the diagonal of the inverse of S ij
(the stiffness
matrix, C ij S 1
ij ) be positive. The latter requirement yields four criteria for an
orthotropic material (e.g., [ 43 ]):
1 n ij n ji > 0
(3.23)
and
1 n xy n yx n yz n zy n zx n xz 2 n yx n zy n xz >
0
(3.24)
For the transversely isotropic material studied here, these can be written:
2
1 n yz n zy >
0 ! n
yz <
1
(3.25)
for the transverse plane, and:
2
2
2
2 n yz n
xy ðE y =E x Þ<
1 n
yz 2 n
xy ðE y =E x Þ
(3.26)
and
2
n
xy <E x =E y
(3.27)
for the remaining Poisson ratios. For the numbers chosen to represent tendon,
n yz ¼ 0 satisfies ( 3.25 ), and two criteria emerge for
n xy
from ( 3.26 )to( 3.27 ).
Equation ( 3.27 ) requires that 3
:
<n xy <
:
16
3
16, but ( 3.26 ) is more restrictive in
this case, requiring that 2
:
< n xy <
:
24. Note that a fully isotropic material in
which moduli are the same for all directions of stressing, the above equations
require that 1 < n isotropic < 0 : 5.
The order of the free edge stress singularity at the interface between transversely
isotropic “tendon” and an isotropic material of elastic modulus E (1) and Poisson
ratio n
24
2
(1)
¼ 0.3 is shown in Fig. 3.10a . Note that we plotted 0 for all values of
(1- l )
0, because the stress field is not singular in such cases. In this example, for
two values of E (1) and for an insertion angle y 1 ¼ 90 , the order of the free edge
singularity was studied for a range of tendon insertion angles y 2 . The case of
relevance to our theme of tendon attached directly to bone is that of E 2 100 times
greater than E ðxÞ
1
>
; that is, with the isotropic modulus of bone 100 times greater than
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