Biomedical Engineering Reference
In-Depth Information
Fig. 3.2
A highly simplified model of a tendon-to-bone attachment, with a schematic of how
parameters of the Williams solution relate to this picture. The tendon is linear elastic with elastic
modulus
E
and Poisson ratio
n
. The insertion angle
y
T
of the tendon is shown here as being greater
than 90
, although the graphic at the left suggests that nature is in fact smarter than that
to bone. The solutions are for a thin, isotropic material, taken to be a tendon,
attached to a rigid substratum, taken to be bone. The tendon is loaded at some
position far from the attachment point. The two solutions involve a tendon
connected straight to bone in the absence of toughening mechanisms such as a
graded interface or tendon/bone interdigitation. However, this particular aspect of
the models is not too far distant from the reality of surgical practice, in which all
interfacial tissue is resected away and tendon is sutured directly onto bone that has
been prepared by drawing blood through abrasion, and little else. We note as well
that a tendon is not isotropic. In later sections of this chapter we explore how
tendon anisotropy affects stresses, and for now note simply that the isotropic
approximations provide superior qualitative insight into the mechanics of attach-
ment, and that the mechanical properties of a healing tendon-to-bone insertion site
are believed to be reasonably represented by an isotropic continuum [
7
].
3.3.1 The Williams Free Edge Problem
Williams [
8
] considered a problem that represents the model shown in Fig.
3.2
. The
asymptotic stress field in the cylindrical coordinate system shown in Fig.
3.2
can be
written:
s
rr
¼ r
l
1
00
ðy; lÞþðl þ
1
ÞFðy; lÞÞ
ðF
s
yy
¼ r
l
1
ðlðl þ
1
ÞFðy; lÞÞ
s
ry
¼r
l
1
(3.7)
0
ðy; lÞ
F
where
(
y
;
l
) contains
four unknown constants that must be found along with
l
using the boundary
F
(
y
;
l
) is a function whose form is given by Williams [
8
].
F
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