Models for the Mechanics of Joining Dissimilar Materials - Structural Interfaces and Attachments in Biology - page 60

value that can be calibrated experimentally. K I can be related to a more physically
intuitive parameter, the fracture energy

G c that was introduced in Sect. 3.1 . For

2

2

Þ=ðK I Þ example, for the case of plane strain G c ¼ð 1 n =E . The available energy

2

2

G ¼ð 1 n G c , and

fracture can be predicted under certain conditions. Central to these conditions is
that the “process zone,” which is the region of material just ahead of the crack tip
that withstands very high strains without failing, is small compared to the
dimensions of the crack and the body in which it is embedded. We will discuss
the process zone more in the context of tendon-to-bone attachment.
The free edge intensity factor H in ( 3.13 ) can serve the role of a stress intensity
factor and serve as an indicator of the likelihood of the onset of fracture. However,
fracture need not be catastrophic, and the problem of whether a crack within an
interface will advance has been studied by Akisanya and Fleck [ 20 ]. The problem
studied involves an edge crack embedded within an interface and loaded with the
singular stress field of ( 3.13 ). The stress field ahead of the crack tip is more
complicated (see ref. [ 45 , 46 ] and the review by Hutchinson and Suo [ 47 ]),
involving rapid oscillation in the traction between the two materials ahead of the
crack tip. However, the concept of a critical fracture energy still holds, with minor
extension to account for the sensitivity of fracture toughness to this oscillatory
stress field. Klingbeil and Beuth [ 21 , 24 ] developed an elegant analysis to account
for the interactions between free edge and interfacial crack singularities when
designing a layered structure, but this is beyond the scope of this chapter.
Why can materials withstand stress singularities, and what does this mean for
attachment of biological tissues? In many metals, the mechanisms are fairly well
understood and involve the phenomenon of plasticity. Beyond a critical stress level
called a yield stress, the stiffness of a metal drops dramatically—this is the end of
the linear region of the uniaxial stress vs. strain response shown in Fig. 3.11 .This
mechanical response results in a reduction of stresses at the crack tip for two
reasons. First, the crack tip “blunts,” meaning that the metal at the crack tip
stretches so as to turn the sharp crack into a blunt crack, thereby eliminating the
otherwise infinite stresses locally. Second, the reduction of stiffness of that material
at higher strains (the stiffness of relevance is the slope of the tangent to the
stress-strain curve) causes stress redistribution away from the crack tip; this occurs
even in brittle matrix ceramics [ 48 , 49 ]. The result is that the material can undergo
major irreversible local changes that reduce the severity of the crack without
endangering a well-designed structure made of the material.
The situation is different, however, for a living tissue. First, such local injury is
undesirable for many tissues. Second, the stress-strain responses of many
biological tissues differ fundamentally from those of engineering materials
(c.f. the data stress-strain data for a tendon in Fig. 3.11 ). In addition to withstanding
larger strains and reaching lower peak stresses, a typical stress-strain curve is concave
up instead of concave down. The consequence is that stiffness ahead of a tear in
a tissue can increase, rather than decrease as in a metal. Stress concentrations in a
biological tissue can therefore become more rather less severe as stresses increase.

ÞðK I Þ =E can be compared to the critical fracture energy,