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Diany2011-ORingRelaxation-JJMIE -91-10.pdf
JJMIE
Volume 5, Number 6, Dec. 2011
ISSN 1995-6665
Pages 478 - 482
Jordan Journal of Mechanical and Industrial Engineering
Finite element analysis for short term O-ring relaxation
Mohammed Diany*,a , Hicham Aissaouib
a
Mechanical Engineering Dept, Faculté des Sciences et Techniques de Béni Mellal, Mghila BP 523 Béni Mellal, 23000 Morocco
b
Electrical Engineering Dept, Faculté des Sciences et Techniques de Béni Mellal, Mghila BP 523 Béni Mellal, 23000 Morocco
Abstract
O-rings are used in machine devices like the seals components. They are inexpensive, and have simple mounting
requirements. In this article, an axisymmetric finite element model is proposed to study the O-ring relaxation during the first
day of its installation in the unrestrained axial loading case. The results of the numerical model are compared with those of an
analytical approach based on the classical Hertzian theory of the contact. The contact stress profiles and the peak contact
stresses are determined versus the time relaxation in order to specify the working conditions thresholds.
© 2011 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved
Keywords: O-ring; contact pressure; analytical modelling;relaxation; FEA
Nomenclature
1. Introduction
F
e
d
D
C
R
b
The elastomeric O-ring gaskets are widely used in
hydraulic and pneumatic equipments to ensure the sealing
of shafts, pistons and lids. The correct operation is due to
the good tightening of the joint that generate a contact
pressures able to confine the fluids inside a rooms or to
prevent their passage from one compartment to another.
Several studies are carried out to model the O-ring
behaviour but without taking in account the effect of the
relaxation and creep phenomena. The equations developed
until today to determine analytically the distribution and
the values of the contact pressure are deduced from the
conventional Hertzian theory of the contact [1]. The
correct operation of the O-ring is conditioned, on the one
hand, by the maximum value of the contact pressure
created during the O-ring compression and on the other
hand by maintaining in operating stage a minimal
threshold value below which the sealing of the joint is
blamed. So the evaluation of the maximum value of
contact pressure evolution in time has a primary
importance to ensure the correct O-ring function during its
nominal lifespan. In this article, it is proposed to study the
O-ring relaxation during the first hours of its installation in
the unrestrained axial loading case, figure 1.
po
Erelax
Ej
j
j 
F
e
d
D
C
R
b
x
po
Erelax
Ej
j
j 


*
Corresponding author. e-mail: [email protected]
(a)
F
(b)
D
F
2e
F
d
x
total compression load (N)
initial O-ring axial displacement (mm)
the O-ring cross-section diameter (mm)
the O-ring mean diameter (mm)
the ratio e/d
the axial compression ratio
the contact width between the gasket and plat (mm)
radial position compared to the vertical axis of
the O-ring cross-section (mm)
maximum contact pressure value or peak contact
stress (MPa)
relaxation modulus (MPa)
elastic modulus for gasket (MPa)
coefficient
relaxation time (s)
total compression load (N)
initial O-ring axial displacement (mm)
the O-ring cross-section diameter (mm)
the O-ring mean diameter (mm)
the ratio e/d
the axial compression ratio
the contact width between the gasket and plat (mm)
radial position compared to the vertical axis of
the O-ring cross-section (mm)
maximum contact pressure value or peak contact
stress (MPa)
relaxation modulus (MPa)
elastic modulus for gasket (MPa)
coefficient
relaxation time (s)
Radial
position, x
Figure 1: Unrestrained axial loading assembly.
D
F
479
© 2011 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 5, Number 6 (ISSN 1995-6665)
2. Background
Several teams were interested in O-ring assembly used
in various industrial services. A temporal reading of
published works on this subject can be classified on three
categories. An analytical approach based in all cases on the
Hertzian classical theory, an experimental part using
various assemblies allowing to characterize the O-ring
itself in traction and compression loads and to model its
real behaviour. In the third shutter, finite elements models
are developed to numerically simulate assemblies with the
O-ring.
George and al. [2] used a finite elements model to
study the behaviour of the O-ring compressed between two
plates. The gasket characteristics were introduced into the
program according to parameter defining the total
deformation energy or by using the Neo-Hookean model.
The results of this analysis were compared with those of
several experimental studies and analytical approaches
based on the Hertzian theory. Dragoni et al. [3] proposed
an approximate model to study the O-ring behaviour
placed in rectangular grove. The influence of the grove
dimensions variation and the friction coefficient was
investigated.
The work of I. Green and C. English [4] reviewed the
majority of used O-rings configurations. A finite elements
Models were developed considering hyperelasticity
behaviour. The results of these models were confronted
with those of empirical studies. New relations expressing
the maximum contact pressure and the width of contact
were proposed. Rapareilli et al. [5] present a validation of
the experimental results by a numerical model which
regarded the joint as an almost incompressible elastic
material. The effects of the fluid pressure as well as the
friction effect between the gasket and the shaft are studied.
The two part of the study were in perfect agreement. In an
experimental study [6], the authors tried to determine the
influence of the fluid pressure on the contact pressure
which ensures of sealing as well as the ageing
deterioration of the joint. Kim et al. [7, 8] tried to find an
approximate solution for the mechanical behaviour of the
O-ring joints in several configurations. The influence of
the friction coefficient is highlighted. An experimental
study was carried out to find more realistic elastic modulus
values for elastomeric O-ring. They compared their results
with those obtained in experiments and by the finite
element analysis. They found that the values given by the
Lindley [9, 10] to calculate the compressive force are
similar to those determined by the finite elements model.
The O-ring relaxation was treated by K.T. Gillen et al.
[11]. In this study, the O-ring degradation is caused by
oxidation or nuclear irradiation. The authors describe
several improvements to the methods used in there
previous studies like substituting the O-ring segments for
difficult-to-prepare a mini-disk samples.
In this article, a 2D axisymmetric model is developed
to simulate the O-ring relaxation when it is axially
compressed by initial tightening between two rigid plans.
The effect of the temporal variation of the longitudinal
elasticity modulus as well as the influence of the axial
compression ratio will be analyzed. The model of the
classical contact theory will be confronted with the results
of the numerical study. The O-ring material behaviour is
similar to the stuffing-box packings one used to study the
creep and relaxation phenomena [12].
3. Conventional analytic Theory
Most of the work dedicated to the study of the O-ring
gasket behaviour used the same analytical model based on
the Hertzian theory of the pressure contact. By adopting
this classical theory, Lindley [9, 10] developed a simple
approximate formula, relation (1), expressing the
compressive force, F, according to the ratio of initial
compressed displacement by the cross-section O-ring
diameter, C  e .
d
3
(1)
F  DdE(1.25.C 2  50.C 6 )
The same theory allowed determining the contact
width, b, and the maximum value of the contact stress po,
according to the formulas (2) and (3).
b  d.
6

3
(1.25.C 2  50.C 6 )
(2)
3
(3)
(1.25.C 2  50.C 6 )
po  4.E.
6
The contact pressure distribution according to the radial
position on the gasket is given by the equation (4).
 2x 
p ( x )  p0 1   
 b 
2
(4)
These formulas do not utilize the mechanical
characteristics of the plates in contact with the joint. Only
the O-ring longitudinal elasticity modulus, E, are used. By
consequence, the same equations remain valid for the
evolution study of the O-ring behaviour according to time
but using a time varying Young modulus, called relaxation
modulus Erelax. The viscoelastic behaviour of the gasket is
given by the modified Maxwell model [13], presented in
figure 2.

Einf
E1
E2
1
2
E3
3
E4
Ej
4
j

Figure 2: A generalized Maxwell model.
The relaxation modulus is defined by the following
equation:
Erelax (t )  E   E j e
j

t
j
(5)
© 2011 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 5, Number 6 (ISSN 1995-6665)
With
j 
480
Table 1: O-ring characteristics
Ej
E
and E0  E 
E
O-ring
(6)
j
j
The relaxation modulus of the equation (5) becomes:
1
6.98
D (mm)
123.19
E (MPa)
(7)
The initial elasticity modulus, E0, and the eight
coefficients j, called Prony series coefficients, are
deduced from the experimental data of the reference [14].
The relaxation study aims to evaluate the variation of
the contact stress versus time, when an initial axial
displacement, e, characterized by an axial compression
ratio R, given by the equation (8), is imposed to the gasket.
For each axial compression ratio, R, the variation of the
contact pressure distribution as well as the change of the
contact surface width are recorded.
e
 200.C
D
(8)
4. Finite element analysis
In order to characterize the O-ring relaxation, an
axisymmetric finite elements model, showed in figure 3,
was developed using ANSYS software [15]. The O-ring is
compressed between two rigid plates.
Upper O_ring surface
2
d (mm)
2.82

t



j
Erelax (t )  E0 1    j (1  e )


j
R  100  2.
Subscript
46.20
0.49967
The value of the vertical displacement imposed on the
upper surface of the joint is calculated by the axial
compression ratio, R, which varied between 7.5 and 25 %
compared to the O-ring cross-section diameter. Thereafter,
the distribution of the contact pressure is recorded
according time.
5. Results and discussions
The suggested analytical model calculates the
maximum contact pressure, in the assembly sealing
conditions, according to initially imposed displacement. In
addition, the finite elements model in the same working
conditions is used to investigate the O-ring material
proprieties effect on the contact pressure values. Figure 4
presents the contact pressure distribution according to the
radial position for a compression ratio of 15% with various
intervals of operating time. It is noticed that the contact
pressure is maximum in the average diameter position. All
the curves have the same appearance and admit the middle
diameter like a symmetrical position. The relaxation speed
is more important at the beginning and becomes null after
18 operating hours.
0.8
Node
Relaxation times
0.7
Rigide plate
Figure 3: Finite elements model.
Since the problem is axisymmetric and the median
horizontal plane cutting the O-ring in two equivalent parts
is a symmetry plane, the joint is modelled by a half-disc
with 2D plane elements having four nodes. The O-ring
material is regarded as viscoelastic characterized by the
Prony coefficients. The plates are modelled by rigid
elements whose displacements are constrained in all
directions. The geometric and mechanic characteristics of
the O-ring joint are summarized in table I. In order to
check the influence of the O-ring rigidity two initial
Young modulus values are considered. The mesh
refinement is optimized to have the convergence while
using less memory capacity.
Contact pressure (MPa)
Element
E01=2.82 MPa
0
0.6
1h
0.5
4h
0.4
8h
0.3
12 h
0.2
18 h
0.1
24 h
0
Percentage compression set : R=15%
-0.1
-0.2
56
58
60
62
64
66
Radial position (mm)
Figure 4: Contact pressure distribution for R=15%.
To inspect the effect of the rigidity of the O-ring, two
values of the longitudinal modulus of elasticity were used.
Figure 5 compares the contact pressure distributions for
the two cases. For a given position, the contact pressure
value depends on the value of the corresponding elasticity
modulus. When E is larger the contact pressures are
higher. When the contact pressure is divided by the initial
elastic modulus, the curves depend only on the relaxation
time and the compression ratio as illustrated in figure 6.
Consequently, we can conclude that the effect of the
481
© 2011 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 5, Number 6 (ISSN 1995-6665)
gasket rigidity does not appear when the curve of the ratio
p/EO is represented according to the radial position for
several cases.
8
E01=2.82 MPa
E02=46.2 MPa
7
Relaxation times
0h
1h
0.2
4h
24 h
0.1
6
5
4
3
2
1
Contact pressure for E02 (MPa)
0.3
Percentage compression set : 7.5%
Contact pressure for E01 (MPa)
0.4
At t=0 sec
0.3
Relative contact stress, P/E0
0.5
0.35
Axial compression ratio R
20%
0.25
Solid line : FE model
Dashed line : Analytical model
0.2
0.15
0.1
At t=4 h
0.05
0
-0.05
0
0.2
0.4
0.6
0.8
1
Relative radial position
0
0
-0.1
15%
Figure 7: FE and Analytical models comparison.
-1
-1
-0.5
0
0.5
Figure 8 compares the influence of the compression
ratio on the speed and the values of contact pressure due to
the viscoelastic relaxation. It is clear that in all the cases
the maximum contact pressure loses a great percentage of
its initial value with time. This loss is very fast in the first
operating hours.
1
Relative radial position
Figure 5: Contact pressure in the two elastic modulus cases.
0.37
E01=2.82 MPa
E02=46.2 MPa
0.32
Axial compression ratio : R=20%
Relaxation times
p(x)/E0i=1;2
0.22
0h
1h
0.17
4h
24 h
0.12
0.07
0.02
-0.03
-1.5
-1
-0.5
0
0.5
1
1.5
Maximum Contact pressure (MPa)
1.2
0.27
1
Axial compression ratio R
0.8
7.50%
15%
20%
25%
Solid line : FE model
Dashed line : Analytical model
0.6
0.4
0.2
Relative radial position
Figure 6: Initial elastic modulus effect.
During the installation of the joint, the analytical model
envisages the same stresses distributions as the finite
elements model for any imposed axial displacement value
as shown in figure 7. The surface of contact and the
maximum contact pressure are larger when the
compression ratio or the relaxation time are more
significant. The difference between the results of the two
models is rather negligible and does not exceed 10%. It
can be affirmed that the analytical model deduced from the
classical theory of the contact pressure remains valid even
for the study of the O-ring relaxation.
0
0
20000
40000
60000
80000
100000
Relaxation time (sec)
Figure 8: Relaxation of maximum contact pressure in the first
material case.
6. Conclusion
This study shows that the classical theory of contact,
developed initially for steady operation, remains valid for
the relaxation case but with some modifications on the Oring mechanical characteristics. In addition, the finite
elements model developed produces the same results as the
analytical model.
In order to generalize these remarks, other cases might
be regarded as the radial loading and the grooves
configuration.
© 2011 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 5, Number 6 (ISSN 1995-6665)
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