Biomedical Engineering Reference
In-Depth Information
conditions. The displacements can be written in terms of these constants using ( 3.4 )
and ( 3.1 ):
ððl þ 1 ÞFðy; lÞþð 1 nÞC
1 þ n
E
u r ¼ r l
0 ðy; lÞÞ
ðF 0 ðy; lÞþð 1 nÞðl 1 ÞCðy; lÞÞ
(3.8)
1 þ n
E
u y ¼ r l
where n ¼ n=ð 1 , and
( y ; l ), a function whose form is given by Williams [ 8 ],
contains two of the four unknown constants that are found when determining
F
C
( y ; l ).
The key to characterizing the stress field at the point of attachment on the free
edge (origin, r ¼ 0) is l :if l <
1, the stress at the origin is infinite, and if l 1 the
leading term of the asymptotic expansion is zero. The four boundary conditions that
determine these constants are that (1 and 2) the two displacement components for
points on the bone side of the tendon are zero and (3 and 4) the free edge of the
tendon is free of tractions in the directions normal and parallel to the surface. This
results in a system of equations that has a nontrivial solution only if the following
equation is satisfied:
ð 4 ð 1
1
3 4 n
2
2 sin 2
sin 2
ðly T Þ¼
ðly T Þ
y T Þ
(3.9)
Equation ( 3.9 ) is transcendental in l , and an infinite number of l s therefore
satisfy it. All of these are valid solutions. The one of interest would seem to be the
smallest value, as this suffices to determine whether the stresses at the corner point
are unbounded. However, care is usually taken to ensure that the displacement field
is bounded even if the stress field is unbounded. Since from ( 3.8 ) one can observe
that the displacement field scales as r l , the displacement at r ¼ 0 is bounded only if
l >
0. We therefore look for the smallest l >
0 and are interested in identifying
whether this smallest value is in the range 0
1.
A graph of this minimum l as a function of the tendon insertion angle y T shows
that l decreases monotonically over the range of 0 y T p
< l <
/2 (Fig. 3.3 ). At a
critical angle, l drops below 1 and the stress field near the free edge becomes
singular. This angle is dependent upon Poisson's ratio. In an isotropic tendon, the
threshold tendon insertion angle for the onset of a singularity can be as low as 54 .
The exception is n ¼ 0, for which the stress is non-infinite even for a 90 insertion
angle. Note that in the latter case and in non-singular cases in general, the stresses at
the insertion point are not described adequately by considering only the one
potentially singular term in the equation.
This problem studied here is far removed from the reality of a tendon, but
nevertheless shows the role of tendon morphology in determining the effectiveness
of a tendon-to-bone attachment. An outward splay of a tendon at the insertion
of tendon to bone can have a profound effect on the stress level on the
attachment point.
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