where
I
1
¼
3k
4
1
þ 1
4k
2
1
; I
2
¼
k
4
2
þ 3
4k
2
2
ð3Þ
k
1
and k
2
are the principal stretches in the longitudinal and
circumferential directions, respectively. c
1
is a constant
material parameter. k
1Coll
, k
2Coll
are two material parame-
ters describing the stiffening properties of the collagen
network. (k
1LM
, k
2LM
) and (k
1CM
, k
2CM
) are the material
parameters accounting for the exponential increase in stress
with stretch ratio due to the longitudinal and circular
muscle layers, respectively. Equations (1) and (2) are
expressed as
r
l
¼ H
l
k
1
ðÞ
r
c
¼ H
c
k
2
ðÞ
ð4Þ
for simplification.
2.3 Interaction Model Between CR and Intestine
The interaction model, which contains the geometric model
and the mechanical model, has a significant influence on
the accuracy of the analytical friction model. It is required
not only to approach to reality, but also to be simple
enough for the analytic calculation. When CR moves
inside, the small intestine is forced to expand. Based on
this, the geometric model between CR and intestine is
shown in Fig. 2. The central angle corresponding to the
connection area between the front end of CR and the small
intestine is a. CR is wrapped up by the small intestine
except part of its ends, and the whole geometrical shape is
axisymmetric. The mechanical model is also shown in
Fig. 2. CR is forced to move with a constant velocity v
c
. d
and D are the average wall thickness and inner diameter of
the small intestine. The interactive force of CR contains the
friction f and the circumferential pressure P. The direction
of the former is opposite to CR’s moving direction, while
that of the latter is perpendicular to CR’s shell. The contact
pressure on the tail end is neglected.
For further development of the model, the following
assumptions are made.
1. The material of the small intestine is incompressible.
2. The small intestine deforms symmetrically toward its
radial direction and its cross-sectional area is constant.
3. The deformation of the small intestine is the same as
the external shape of the contact surface of CR.
4. The small intestine deforms at a constant velocity.
3 Analytical Model of the Fluctuant Friction
It has been proved that the friction of CR is fluctuant when
CR moves in the small intestine with a constant velocity.
The whole process can be simplified to a simple physical
model, which is shown in Fig. 3 [29]. An infinitely long flat
car is placed on a smooth surface. One side of the car is
connected to the wall by a nonideal spring, which may be
nonlinear and include dampers. A slider is placed on the
flat car. The friction between them is f . The slider moves at
v
c
. The spring is used to express the stain and stress of the
small intestine in the longitudinal direction, while the
friction f represents the frictional resistance between CR
and the small intestine. The immediate cause is that the
relative sliding between CR and the small intestine appears
periodically. That is to say, relatively static state and rel-
ative motion alternate periodically when CR moves. The
root cause is that the small intestine is a soft tube, whose
material is anisotropic and hyperelastic. When CR moves,
the small intestine resists it; meanwhile, the small intestine
stretches. The critical condition of the relative motion is the
frictional resistance between CR and the small intestine
that equals the tension resulted from the stretch of the small
intestine. Therefore, relatively static state and relative
motion should be analyzed separately.
3.1 Relatively Static State
There is no relative sliding between CR and the small
intestine in the relatively static state. More precisely, no
relative sliding occurs on the front end of CR, but the small
intestine may be stretched or in the initial state. When the
Fig. 2 Interaction model between CR and intestine Fig. 3 Equivalent physical model
Tribol Lett (2016) 64:39 Page 3 of 11 39
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