Models for the Mechanics of Joining Dissimilar Materials - Structural Interfaces and Attachments in Biology - page 49

critical energy release rate G c = 2 E T t 0 : 05 . In this case, the two terms are

' ¼ 25 .
The final wrinkle in this solution that we will mention here is that discussed by
Williams and Kauzlarich [ 15 ] for the case in which the tendon exists in a state of
pre-strain. The pre-strain value is given by a uniaxial strain level e o , which
corresponds to an axial force F required to stretch the portion of the tendon that
is adhered to the bone to e o :

approximately equal at

2

f c

w þ ð f c F Þ

G c ¼ð 1 cos 'Þ (3.12)

2 E T w 2 t

The result is that a pretensioning of the attached region actually increases the
force f needed to cause the tendon to debond. Is this relevant to an idealized
tendon-to-bone attachment? This is unclear. Tendons certainly exist in a state of
pretensioning, and while this pretensioning is well characterized in the
midsubstance, it is not well characterized within the attachment of tendon to
bone. We note that pretensioning in the midsubstance adds to f c and reduces
resilience. For pretensioning to be an effective contributor to attachment, the
level of pretensioning in the region of tendon adhered to bone would have to exceed
that of the tendon at midsubstance.
These solutions have been worked out for arbitrary monotonically increasing
nonlinear uniaxial constitutive laws for the tendon and for large strain [ 16 ], but
these are beyond the scope of the current chapter other than to note that the basic
principles remain unchanged.

3.4 Idealized Tendon Attached to Isotropic, Elastic Bone

The next moderate step towards realism is to model the bone with realistic isotropic
mechanical properties. This is still a significant approximation. Bone is sufficiently
stiff relative to tendon that it is well modeled with the small strain assumptions of
linear elasticity, and it is much better modeled as isotropic than tendon is. However,
as described later, bone does present anisotropy over the length scale of hundreds of
micrometers (osteons), and this length scale is relevant to tendon-to-bone

attachment.
The problem of interest was considered by, among others, Bogy [ 17 ], Hein and
Erdogan [ 18 ], and Akisanya and Fleck [ 19 , 20 ], with important later contributions
by many others including Klingbeil and Beuth [ 21 ]. Here, an isotropic tendon is
attached to an isotropic bone. The insertion angles of the tendon ( y 1 ) and the bone
( y 2 ) are both important (Fig. 3.6 ). The stress at the corner point again has the form:

m

ij ¼ Hr l 1 F ij ðy; lÞ

s (3.13)