Biomedical Engineering Reference
In-Depth Information
Discrete strain at the time t
0
is
ε
t
=
0
, at the time t
1
discrete strain is:
0
⎡
−
1
⎤
σ
Et t
(
−
)
10
ε
=
1
−
e
(4)
c
η
⎢
⎥
t
1
E
⎣
⎦
For the 2
nd
phase of the first loading cycle (for
t
∈<
t
;
t
> ) (Fig. 13.), the convex curve AB
12
is defined by the function for articular cartilage strain:
−
1
Et t
(
−
)
ε
()
t
=
ε
e
1
(5)
η
t
1
Discrete strain at the time
t
0
is
ε
t
=
0
, at the time
t
1
discrete strain is:
0
−
1
Et t
(
−
)
εε
=
η
t
e
21
(6)
t
2
1
The magnitudes of strains during cyclic loading at the starting points of loading and
unloading of articular cartilage may be expressed by recurrent relations. For the time t
i
with
an odd index, the strain at the respective nodal points is:
σ
⎡
−
E
kl
⎤
(1)
+
ε
=
1
−
e
η
(7)
c
⎢
⎥
t
E
(2
k
+
1)
⎣
⎦
k
=
0 ,1 ,2...
where
l
is the length of the time interval <t
i
; t
i+1
> . For the time t
i
with an even index, the
strain is :
⎡
−
E
−
E
⎤
σ
l
(1)
k
+
l
ε
=
c
e
η
−
e
η
(8)
⎢
⎥
t
E
2
k
⎣
⎦
k
=
0,1,2...
where
l
is the length of the time interval
<
t
i
; t
i+1
>
,
i = 0, 1, 2, … During long-term cyclic
loading and unloading, for k → ∞ the strain ε
t(2k+1)
asymptotically approaches the steady
−
E
l
σ
state σ
c
/E ; for k → ∞ the strain ε
t2k
asymptotically approaches the steady state
. It
c
e
η
E
is evident that for k → ∞ it holds true that:
−
=>=
η
σ
σ
E
l
ε
ε
e
(9)
c
c
t
E
t
E
(2
k
+
1)
2
k
2.2.1 The strain rate of articular cartilage in peripheral zone during the strain time
growth
Strain
ε(t)
of AC during the strain time growth in the interval of t∈〈t
0
; t
1
〉 is given by
equation (3). Because
dt
ε
()
=
ε
() 0
t
>
(in the indicated interval) the function
ε(t)
is
dt
increasing. The strain rate of AC during the strain-time growth in interval of
t
∈〈
t
0
; t
1
〉 is
given by equation (10):
