Biomedical Engineering Reference
In-Depth Information
For the case of a butt joint between tendon and an isotropic material, the order of
the singularity is a strong function of both modulus E (1) and Poisson's ratio n
(1) .
(1) were studied over a broad range of modulus E (1) (Fig. 3.10b ). For
Two values of n
(1) studied, the order of the singularity approached ð 1 0
both cases of n
26 for
attachment to a relatively stiff isotropic material. The graph shows that this order is
relatively constant for E ð 1 Þ
:
> 2 E ð 2 x , and Fig. 3.10a suggests that this is constant
not just for a butt joint but also for any insertion angle y 2 >
~30 for the case of
tendon attached to a stiff material 1 with y 1 ¼ 90 . Results suggest that a butt joint
between tendon and bone will certainly present a singular stress field, but results
again suggest that attachment to a compliant, possibly splayed interlayer, can
extinguish the free edge singularity.
Results presented here are somewhat unsatisfying, because they are a smattering
of special cases. The literature is full of explorations of special cases because this is
an important problem and because no simple way to explore all interfaces exists
that is analogous to the Dundurs' parameters (e.g., Fig. 3.6 ). However, the results
do indicate that attaching a material like tendon to an isotropic material that is at
least twice as stiff will lead invariably to a singularity over a wide range of insertion
angles: as before, an interlayer or careful shaping is needed to avoid a free edge
singularity.
3.6 Concluding Remarks
So we found a free edge singularity. Now what? Sections 3.4 and 3.5 focused on
situations in which a free edge singularity can arise at the interface between two
dissimilar materials. We conclude by describing what one might do to assess
whether such a singularity will lead to catastrophic failure of the interface between
the two materials, and why researchers who are interested in tendon-to-bone
attachment are not much concerned about this.
Several groups have proposed a framework based upon linear elastic fracture
mechanics. These concepts follow directly from the problems we have addressed in
this chapter: the solution for the stress field around a sharp crack in an infinite solid
can be found by considering the case of tendon y 1 ¼ y 2 ¼ 180 in Fig. 3.7 . For the
case of materials 1 and 2 being the same linear elastic, isotropic material, and for a
remote stress field that is a uniaxial stress s 1 applied in the vertical direction, the
asymptotic stress field near the crack has the form:
K I
2 pr
s ij ¼
p
s ij ðyÞ;
(3.28)
where the stress intensity factor K I is a constant that depends upon geometry and
upon s 1 , and the function
s ij ðyÞ is known (e.g., [ 44 ]). Within certain limits, fracture
in many materials is well modeled by the criterion K I <K I , where K I is a critical
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