Biomedical Engineering Reference
In-Depth Information
value (as was the case during the initial, first loading cycle), but at point B, or at the value of
the residual strain ε t2 . The first residual strain provides the initial presupplementation of articular
plateaus with synovial fluid. Fig. 13. manifests that the envelope curve OBDF slightly grows
during cyclic loading to stabilize after a certain time at a steady value characterizing long-term
strain (during the time of cyclic loading) and long-term presupplementation of articular space with
synovial fluid. After cyclic loading stops (i.e. after AC unloading) during the last loading
cycle, as seen in Fig. 1., the strain relaxation follows the convex curve, and strains
asymptotically approach to the time axis t (or zero). After the termination of the last loading
cycle, SF (in the form of synovial sol) is sucked back into the peripheral layer of AC. The
mechanism of viscous strain time growth and viscous strain relaxation creates a highly
efficient protective pump functioning not only to discharge and suck back synovial fluid, but
also to pump (accumulate) it into the articular space.
Stresses in the peripheral zone may be expressed for the Kelvin Voigt model by the
constitutive equation:
dt
ε
()
σ
()
t
=
η
+
E
ε
()
t
(1)
dt
where η is the coefficient of viscosity, E is the modulus of elasticity, ε(t) is the strain of AC
dt
dt
ε
()
and
is the strain rate of cartilage tissue in the peripheral zone.
Equation (1) is a first order linear differential equation for an unknown function ε(t). The
solution to the non-homogeneous equation (1) under the given initial conditions determines
the time related strain of articular cartilage. In our case, it is in the form:
1
t
1
1
Et
e τ
E
ε
()
t
=
e
στ
( )
τ
(2)
η
η
η
t
0
Let us further consider the case where articular cartilage is loaded by a constant load
()
στ== (Fig. 13.) :
const
c
1
σ
Et t
(
)
0
ε
()
t
=
c
1
e
η
(3)
E
Equation (3) implies that the strain of AC is a function of time depending on the magnitude of
the constant stress σ c also (for example by shifting an individual's weight onto one foot).
The presence of residual strain (marked by a thick line in Fig. 13.) ensures the accumulation of
synovial fluid between articular plateaus. It means that during each step (during cyclic
loading) articular plateaus are presupplemented with the lubrication medium - synovial fluid.
The magnitudes of residual strains of AC play a key role in the presupplementation of AC
surface plateaus with synovial fluid. The magnitudes of residual strains may be determined
from the functions expressing strain during the strain time growth and from the functions
expressing strain during the strain relaxation of AC, this may be performed separately for
each loading cycle of cartilage (Fig. 13.).
For the 1 st phase of the first loading cycle, for
t
∈<
t
;
t
> , (Fig. 13.) the concave curve is
01
defined by function (3) for the articular cartilage strain.
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