Biomedical Engineering Reference
In-Depth Information
2
2
sin 2
2
2
2
sin 4
0 ¼ða bÞ
ðl
ðlp=
2 ÞÞ
a
ðl
ðlp=
2 ÞÞ
2
sin 2
2 ÞÞ sin 2
(3.16)
þ 2 aða bÞðl
ðlp=
ðlp=
2 Þ
þ sin 2
2 Þð 1 sin 2
ðlp=
ðlp=
2 ÞÞ
Following the logic described above for the Williams solution, solutions are
sought for which the stresses are singular at r ¼ 0( l <
1, cf. ( 3.13 )) but
displacements are bounded ( l 0, cf. ( 3.14 )). In these cases, one concludes that
the stress field is singular. One can show from ( 3.16 ) that the stress field is for pairs
( a , b ) in the lighter shaded region of Fig. 3.6 , which corresponds to aða 2 bÞ>
0.
The hatched region corresponding to attachment of tendon and bone lies well within
the singular range.
What does this mean for attachment of engineering materials? A broad literature
exists that is relevant to relating H and l to a failure criterion that predicts loads at
which the tendon and bone would become debonded. Much of this literature derives
from the study of debonding of thin semiconductor films from silicon substrata
[ 21 , 24 ], in addition to the broad literature on structural materials. The overall idea
is to first check whether an interface will present a singularity ( l <
1), and then, if
so, whether the value of H , which increases monotonically with the tractions
applied to the tendon, is sufficient to cause debonding. We address the first problem
in this section and defer the second to the next section.
Suga et al. [ 25 ] compiled data for a broad range of engineering material
mismatches and found them to be clustered over a fairly narrow range; represen-
tative regions are sketched in Fig. 3.6b for several classes of engineering
attachments; note that since all data are antisymmetric about a ¼ 0, only the
region a >
0 is shown in Fig. 3.6b . Common engineering material pairs are found
predominantly along the lines b ¼ a /4 and b ¼ 0, both of which lie well within
the singular range. A few metal/metal, metal/glass, and ceramic/ceramic
interfaces can be found in the non-singular range, but attachment of tendon to
bone is firmly in the singular range.
A singularity between two materials in a butt joint can be eliminated by an
interlayer between them. Interlayers are common in protective coatings on turbine
blades, where one goal is to provide improved adhesion of layers that provide
chemical and thermal protection to the underlying metal blade. In the example we
describe here, the goal is to find a material that will not present a singularity when
paired in a butt joint with either of the two materials to be joined.
As an example, we explore an interlayer between tendon and bone in Fig. 3.8 .
The set of all mechanical property pairs ( E interlayer , n interlayer ) that can be paired in a
butt joint with tendon without a singular stress field arising is represented by the
lightly shaded region; the set of all interlayer properties that can be paired in a butt
joint with bone without a singular stress field arising is represented by a darker
shaded region. In both cases, the appropriate interlayer material properties are not
symmetric about the case of identical material properties, with a broader range of
options available for interfaces with a more compliant interface with a lower
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