16 Structure and Function
a high degree of stiffness due to the collagen
fibers being stretched. The final region is the
point at which the ultimate strength has been
reached, when the ligament is no longer able
to sustain a given force and consequently rup-
tures. The structural properties of a ligament
can be derived from the force–deformation
curve. The linear stiffness is the slope of the lin-
ear portion of the curve; stiffness is an exten-
sive material property, meaning that it depends
both on the material being measured and its size
or boundary conditions. The material’s ultimate
strength and ultimate elongation are the deter-
mined at the point of rupture, and the energy to
failure is the area under the curve prior to fail-
ure. Force–deformation curves can also be used
to determine attachment site behavior as part of
the structure.
Stress–strain curve
Stress is defined as load (MPa) divided by cross-
sectional area, and is calculated using known
load data and a cross-sectional measurement of
the ligament. Strain can be thought of as defor-
mation, and is defined as the change in length
divided by original length; during uniaxial ten-
sile testing, strain can be directly determined.
The stress–strain curve can be divided into three
regions (Figure 2.4). The first is the toe region,
at which point large changes in strain result in
minimal stress being applied; it is in this region
that the “un-crimping” of collagen fibers occurs
as they are being recruited to bear load. The
next is the linear region, which occurs when the
ligament fibers themselves are being stretched.
The slope of this linear region defines Young’s
modulus (a.k.a. elastic modulus or modulus).
Young’s modulus describes the properties of
the composite material, including all of its solid
and fluid constituents. The last region occurs
when the ligament begins to rupture, when
the stiffness is reduced. The point before this
decrease in stiffness defines ultimate stress and
ultimate strain of the tissue. When sufficient
fiber damage has occurred and elongation con-
tinues, the remaining fibers rupture and the lig-
ament breaks. The area under the curve before
failure is called the strain–energy density, and
reflects the energy stored in the elastic ligamen-
tous material.
Ultimate StressUltimate Strain010102020Yield303040405050607080StrainStress (MPa)
Strain-energy
densityYoung’s
ModulusFigure 2.4 Typical stress–strain curve for ligament
during uniaxial tensile testing. A stress–strain curve is
determined by calculating the amount of strain
(deformation) that occurs as a result of varying levels of
tensile (or compressive) loading. The slope of the linear
aspect of the stress–strain curve defines Young’s modulus.
A material’s yield point occurs once the material begins
to plastically deform prior to rupture. The point of rupture
is where the ultimate stress and ultimate strain are
defined.Viscoelasticity
Ligament, like all soft collagenous connective
tissues, is considered a viscoelastic material, in
that it exhibits both viscous and elastic proper-
ties when undergoing deformation, and, there-
fore, has time-dependent mechanical behav-
ior. Two characteristic material properties that
are exhibited by viscoelastic materials arecreep
andrelaxation. Creep describes how a ligament
continues to stretch under a sustained force,
while relaxation describes how the tensile force
required to keep a stretched ligament at a con-
stant length diminishes over time. As a con-
sequence of relaxation, the stiffness of a vis-
coelastic material depends on the rate at which
load is applied. For a fast rate of loading, the
viscoelastic CrCL will be stiffer than if it is
stretched slowly. Another viscoelastic property
ishysteresisor energy dissipation. Ligaments
exhibit hysteresis during loading, which can
be seen when a ligament is loaded and then
unloaded, as the loading and unloading stress–
strain curves do not follow the same path. This
is due to a loss of energy that occurs dur-
ing loading. However, during repetitive loading