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Aissaoui-ORingRelaxation.pdf
Global Journal of Researches in Engineering
Mechanical and Mechanics Engineering
Volume 12 Issue 4 Version 1.0 July 2012
Type: Double Blind Peer Reviewed International Research Journal
Publisher: Global Journals Inc. (USA)
Online ISSN: 2249-4596 Print ISSN:0975-5861
Numerical Simulation of Radial and Axial Compressed
Elastomeric O-Ring Relaxation
By Hicham Aissaoui, Mohammed Diany & Jaouad Azouz
Universite Sultan Moulay Slimane Beni Mellal
Abstract - The elastomeric O-ring gaskets are often used in pressurized hydraulic and pneumatic
equipments to ensure their sealing. The quality of the O-ring is measured by achieving the
desired tightness level. The satisfaction level of practical operation depends on the consistency
of long-terms mechanical properties located during the useful life and on the conditions of
assembly installation and the O-ring location. Indeed, the gasket is pressed on flat faces or
housed in specially arranged grooves for which the dimensions influence greatly the assembly
behavior. In this article, an axisymmetric finite element model is proposed to simulate the O-ring
relaxation behaviour during the few first days of its installation in both the unrestrained and
restrained radial and axial loading cases. The contact stress profiles and the peak contact
stresses are determined versus the time relaxation in order to specify the working conditions
thresholds.
Keywords : O-ring, contact pressure, relaxation, FEA, Radial and axial loading.
GJRE-A Classification : FOR Code: 091399
Numerical Simulation of Radial and Axial Compressed Elastomeric O-Ring Relaxation
Strictly as per the compliance and regulations of:
© 2012 Hicham Aissaoui, Mohammed Diany & Jaouad Azouz. This is a research/review paper, distributed under the terms of the
Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting
all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Numerical Simulation of Radial and Axial
Compressed Elastomeric O-Ring Relaxation
Hicham Aissaoui α, Mohammed Diany σ & Jaouad Azouz ρ
Keywords : O-ring, contact pressure, relaxation, FEA,
Radial and axial loading.
Nomenclature
F total compression load (N)
e initial O-ring axial displacement (mm)
d the O-ring cross-section diameter (mm)
D the O-ring mean diameter (mm)
C the ratio e/d
R the axial compression ratio
b the contact width between the gasket and plat (mm)
x radial position compared to the vertical axis of the Oring cross-section (mm)
po maximum contact pressure value or peak contact
stress (MPa)
Erelax relaxation modulus (MPa)
Ej elastic modulus for gasket (MPa)
αj coefficient
τj relaxation time (s)
ηj viscosity (MPa.s)
Abbreviation
RAL restrained axial loading
RRL restrained radial loading
UAL unrestrained axial loading
URL unrestrained radial loading
Author α σ ρ : Faculté des Sciences et Techniques de Béni Mellal, BP
523 Béni Mellal, Morocco. E-mails : [email protected];
[email protected]; [email protected]
Introduction
The elastomeric O-ring gaskets are widely used
in hydraulic and pneumatic equipments to ensure the
sealing of the shaft, the pistons and the lids. 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
the preservation in operating stage of a minimal
threshold value below which the sealing of the joint is
blamed. Therefore, 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 useful lifetime.
There are two O-ring types, static and dynamic
seals. The first seals are classified as either axial or
radial squeeze. For the second ones three main
classifications are considered, reciprocating, rotary and
oscillating seals. The O-ring seals are placed in grooves
specially arranged to block their possible displacement
and to assure the maximum contact pressure giving the
maximum performance. The dimensions of these
grooves are provided by the seals manufacturers in
tables which are selected according to the cross section
O-ring diameter. The origin formulas giving the groove
dimensions are not known and the theoretical approach
used by the manufacturers is not published.
Several teams were interested in O-ring
assembly used in various industrial services. The
published works on this subject propose the same
analytical approach based in all cases on the Hertzian
classical theory [1]. Others studies have more
experimental studies using various assemblies to
characterize the O-ring itself in traction and compression
loads and to model his 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] propose 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 treated.
© 2012 Global Journals Inc. (US)
July 2012
I.
1
Global Journal of Researches in Engineering ( A ) Volume XII Issue vIV Version I
Abstract - The elastomeric O-ring gaskets are often used in
pressurized hydraulic and pneumatic equipments to ensure
their sealing. The quality of the O-ring is measured by
achieving the desired tightness level. The satisfaction level of
practical operation depends on the consistency of long-terms
mechanical properties located during the useful life and on the
conditions of assembly installation and the O-ring location.
Indeed, the gasket is pressed on flat faces or housed in
specially arranged grooves for which the dimensions influence
greatly the assembly behavior. In this article, an axisymmetric
finite element model is proposed to simulate the O-ring
relaxation behaviour during the few first days of its installation
in both the unrestrained and restrained radial and axial loading
cases. The contact stress profiles and the peak contact
stresses are determined versus the time relaxation in order to
specify the working conditions thresholds.
July 2012
Numerical simulation of radial and axial compressed elastomeric O-ring relaxation
Global Journal of Researches in Engineering ( A ) Volume XII Issue vIV Version I
2
The work of Green and al. [4] reviews the
majority of used O-rings configurations. 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 and 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.
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 and 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 the reference [11] where the
degradation is caused by oxidation or nuclear
irradiation. The authors describe several improvements
to the methods used in there previous studies.
In this study, axisymmetric finite element
models are proposed to simulate the O-ring relaxation
behaviour during the few first days of its installation in
both the unrestrained and restrained radial and axial
loading cases. Figure 1 presents the studied cases. The
contact stress profiles and the peak contact stresses
are determined versus the time relaxation in order to
specify the working conditions thresholds. The effect of
the temporal variation of the longitudinal elasticity
modulus as well as the influence of the axial
compression ratio will be analyzed.
II.
O-ring Mechanical Characteristics
Most of the previous work dedicated to study
the O-ring gasket behaviour use the same analytical
model based on the Hertzian pressure contact theory.
By adopting this classical theory, Lindley [9, 10]
developed a simple approximate formula, relation (2.1),
expressing the compressive force, F, according to the
ratio of initial compressed displacement by the
crosssection O-ring diameter, C = e .
d
3
F = πDdE (1.25.C 2 + 50.C 6 )
(2.1)
The same theory allowed finding out the contact
width, b, and the maximum value of the contact stress
po, according to the formulas (2.2) and (2.3).
© 2012 Global Journals Inc. (US)
b = d.
6
π
3
(2.2)
(1.25.C 2 + 50.C 6 )
3
(1.25.C 2 + 50.C 6 )
po = 4.E.
6π
(2.3)
The contact pressure distribution according to
the radial position on the gasket is given by the equation
(2.4).
⎛ 2x ⎞
p ( x ) = p0 1 − ⎜ ⎟
⎝ b ⎠
2
(2.4)
These formulas do not utilize the mechanical
characteristics of the components in contact with the
seal. Only the O-ring elastic modulus, E, are used.
Practically, the most used O-ring material has a
hyperelastic or viscoelastic behavior. For the relaxation
studies, the viscoelastic behavior is the best choice to
take into account the effect of hyperelasticity and time
variation of mechanical properties.
In previous study [12], it was confirmed that the
same equations remain valid for the time evolution study
of the O-ring behaviour but using a variable elastic
modulus according to time, called relaxation modulus
Erelax. The viscoelastic behaviour of the gasket is given
by the modified Maxwell model [13], presented in figure
2.
The relaxation modulus is defined by:
Erelax (t ) = E∞ + ∑ E j e
−
t
(2.5)
τj
j
With
τj =
ηj
Ej
,
αj =
Ej
E∞
and
E0 = E∞ + ∑ E j
j
(2.6)
The relaxation modulus in Eq. (2.5) becomes:
t
−
⎤
⎡
τ
Erelax (t ) = E0 ⎢1 − ∑ α j (1 − e j )⎥
⎥⎦
⎢⎣
j
(2.7)
The initial elasticity modulus, E0, and the
coefficients αJ, called Prony series coefficients, are
deduced from the experimental data of the reference
[14]. The time variation of this relaxation modulus is
presented in figure 3.
The relaxation study consists to evaluate the
variation of the contact stress versus time, when an
initial displacement, e, characterized by a compression
ratio R, given by the equation (2.8), 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 determined
with the relations (2.2), (2.3) and (2.4).
R = 100 ×
2.e
= 200.C
d
(2.8)
Finite Element Models
The study of the O-ring relaxation, in the four
cases presented in Fig. 1, when it is compressed by the
application of a constant displacement, consists in
following the time evolution of the contact pressure and
contact width. To achieve this goal, an axisymmetric
finite elements model of each assembly was produced
using ANSYS software [15] as showed for RAL case in
figure 5.
Since the problem is axisymmetric and the
median horizontal or vertical plane, respectively for axial
or radial loading, cutting the O-ring in two equivalent
parts is a symmetry plane, the joint is modeled by a halfdisc with four node’s 2D planes elements. The O-ring
material is regarded as viscoelastic characterized by the
Prony coefficients. The rigid component is modeled by
rigid elements for which the displacements are blocked
in all directions. The geometric and mechanic
characteristics of the O-ring joint are summarized in
table 1. 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 computer memory
capacity.
The value of the imposed displacement on the
free seal surface is calculated by the axial compression
ratio, R, which varied between 5 and 35 % compared to
the O-ring cross-section diameter. Thereafter, the
contact pressure distribution is recorded according
time.
IV.
Results and Discussions
The work presented in this paper is the
continuation of an previous work [12] in which it was
concluded that the classical theory of Hertzian contact,
developed initially for steady operation, remains valid for
the relaxation and it is in good agreement with the used
finite element model. Consequently, we will just use the
finite element analysis to compare the case where the
joint is free to move in the direction perpendicular to that
of the applied compressive stress and the case where
this movement is blocked by placing the seal in a
rectangular groove.
After the application of an initial load on
unrestrained O-ring upper surface (UAR case), an initial
displacement is taking place and will be kept fixe over
the relaxation time. Figure 5 presents the variation of the
contact pressure between the O-ring and the fixed
component when the compression ratio is equal to 5, 15
and 30% for various relaxation times. When the
relaxation time increases the contact pressure
decreases and the relaxation rate can be calculated for
each applied compression ratio. In addition, the contact
area is larger when the compression ratio is greater. On
the other hand, whatever the relaxation time for the
same compression ratio, the contact width remains the
same. Thus, when the compression ratio increases from
5% to 15% and to 30%, the contact width, successively,
is evaluated to 20%, 38% and 62% of the seal cross
section diameter.
For the restrained axial loading case, Fig. 6
shows the contact pressure distribution for R=20% and
for two relaxation time, 8 and 24 hours. The same
observations, as for UAL case, remain valid but the
contact pressure values decreases. The limitation of the
radial displacement by the groove creates a pressure
contact distribution along the contact surface side of the
groove. For each relaxation time, the curve is symmetric
about the seal section center. This shows that the
influence of the radial position is negligible on the
symmetry of contact pressure distribution even in the
groove presence.
In order to perceive the importance of the
groove on the contact pressure, Fig. 7 compares the
two cases of axial loading for different initial elastic
modulus. The contact pressure ratio is represented as a
function of the relative radial position. For the same
compression ratio, the stress value in RAL case is larger
than in the URL case. The axial contact area is almost
identical in the two cases. So it is clear that the primary
advantage of placing the seal in a groove is to increase
the contact pressure which ensures more sealing with
the same compression ratio. However, the chosen
groove dimensions are not optimized to provide better
performance. In aerospace applications, the most used
standard is SAE AS5857A [16] that provide
standardized gland (groove) design criteria and
dimensions for elastomeric seal glands for static
applications.
To know the extent of the pressure contact, the
contact width is shown in Fig. 8 as a function of
compression ratio. It is evident that when the O-ring is
more compressed the contact area is larger. The
analytical values of the contact width given by Eq. (2.2)
are closer to those given by the finite element analysis in
the URL case. Indeed, the used analytical model does
not take into account the presence of seal-groove lateral
contact.
For the piston rod and in the static state, the Oring is solicited radially in the perpendicular axis
direction. This configuration is represented by the RRL
and URL cases in Fig. 5. To illustrate the effect of
compression ratio on the initial contact pressure
distribution, Fig. 9 compares these two cases and
highlights the creation of the contact pressure at the
lateral groove-seal contact surface. At the radial contact,
the pressure is greater when the compression ratio
increases or when the seal is placed in the groove. For
the axial contact, the contact pressure reaches almost
50% of the maximum value recorded at the radial
contact.
Figure 10 presents the distribution of the
contact pressure at contact interface between the
groove and the seal for different time periods. The stress
decreases rapidly versus time for all axial position
except at the contact area. It is to be noted that for all
© 2012 Global Journals Inc. (US)
3
Global Journal of Researches in Engineering ( A ) Volume XII Issue vIV Version I
III.
July 2012
Numerical simulation of radial and axial compressed elastomeric O-ring relaxation
Numerical simulation of radial and axial compressed elastomeric O-ring relaxation
July 2012
time periods and for the same compression ratio, the
contact width is the same.
Figures 11 and 12 compare the reductions in
maximum pressure contact due to the relaxation at
different compression ratio for URL and RRL cases. For
the same compression ratio, the variation of this stress
is more significant at the first hours and it stabilised
after. Indeed, for both cases and whatever the
compression ratio, the ratio of the maximum contact
pressure by the initial elastic modulus decreases with a
rate of 56% from its initial value when the relaxation time
is 12 hours while it is about 57% after 72 hours.
Global Journal of Researches in Engineering ( A ) Volume XII Issue vIV Version I
4
V.
Conclusion
This study shows that the installation of the seal
in a groove reduces the compression ratio needed to
create a contact pressure threshold provided for sealing
the assembly. On the other hand, the first few hours
after installing are critical and must be controlled in
order to ensure the proper O-ring functioning. The
geometric configuration of the groove used in our model
is idealized which requires to verify the results obtained
when the geometrical defects are introduced to the
model. An experimental study is expected to confirm
these results and characterize the mechanical behavior
of industrial O-rings.
References Références Referencias
1. Timoshenko and Goodier, 1934, « Theory of
elasticity », McGraw-Hill.
2. George AF, Strozzi A, Rich JI, Stress fields in a
compressed unconstrained elastomeric O-ring seal
and a comparison of computer predictions and
experimental results, Tribology International, 1987,
Vol. 20, No. 5, pp. 237-247.
3. Dragoni E, Strozzi A, Theoretical analysis of an
unpressurized elastomeric O-ring seal inserted into
a rectangular Groove, Wear 1989; 130:41-51.
4. Green I, English C, Stresses and deformation of
compressed elastomeric O-ring seals, 14th
International Conference on Fluid Sealing, Firenze,
Italy, 6-8 April 1994.
5. Rapareilli T, Bertetto AM and Mazza L, Experimental
and numerical study of friction in an elastomeric
seal for pneumatic cylinders, Tribology International
Vol. 30, No 7, pp. 547-552, 1997.
6. Yokoyama K, Okazaki M and Komito T, Effect of
contact pressure and thermal degradation on the
sealability of O-ring, JSAE 1998; 19: 123-128.
7. Kim HK, Park SH, Lee HG, Kim DR, Lee YH,
Approximation of contact stress for a compressed
and laterally one side restrained O-ring, Engineering
Failure Analysis 2007; 14: 1680-1692.
8. Kim HK, Nam JH SH, Hawong JS, Lee YH,
Evaluation of O-ring stresses subjected to vertical
and one side lateral pressure by theoretical
approximation
comparing
with
photoelastic
© 2012 Global Journals Inc. (US)
9.
10.
11.
12.
13.
14.
15.
16.
experimental results, Engineering Failure Analysis,
doi:10.1016/j.engfailanal.2008.09.028.
Lindley PB., 1967, Compression characteristics of
laterally-unrestrained rubber Oring, J IRI, 1, pp. 22013.
Lindley PB., 1966, Load-compression relationships
of rubber units, J Strain Anal, 1(3), pp. 190-5.
Gillen K.T., Celina M. and Bernstein R., 2003,
Validation of improved methods for predicting longterm elastomeric seal lifetimes from compression
stress-relaxation
and
oxygen
consumption
techniques, Polymer Degradation and Stability, 82,
pp. 25-35.
Diany M. and Aissaoui H., 2011 “Finite Element
Analysis for Short term O-ring Relaxation”, Jordan
Journal of Mechanical and Industrial Engineering,
Volume 5, Number 6, Dec. 2011 pp. 478 - 482
McCrum,N. G., Buckley C. P. and Bucknall C. B.,
2004, Principles of Polymer Engineering, Oxford
University Press, New York.
Christensen, R. M., 1982, Theory of Viscoelasticity An Introduction, 2nd ed., Academic Press, New
York.
ANSYS, 2003, ANSYS Standard Manual, Version
11.0.
SAE, 2010, AS5857A : Gland Design, O-Ring and
Other Elastomeric Seals, Static Applications.
Table 1 : O-ring characteristics
O-ring
Subscript
d (mm)
D (mm)
E0 (MPa)
ν
1
2
6.98
123.19
2.82
46.20
0.48
Figure 1 : Unrestrained Axial Loading (UAL); Restrained Axial Loading (RAL) Unrestrained Radial Loading (URL);
Restrained Radial Loading (RRL)
July 2012
d
Numerical simulation of radial and axial compressed elastomeric O-ring relaxation
5
Einf
E1
E2
η1
η2
E3
η3
E4
Ej
η4
ηj
Global Journal of Researches in Engineering ( A ) Volume XII Issue vIV Version I
σ
ε
Figure 2 : A generalized Maxwell model
1.2
Erelax/E01(MPa)
1
0.8
0.6
0.4
0.2
0
0
20000
40000
60000
80000
100000
Time (sec)
Figure 3 : Relaxation modulus ratio
© 2012 Global Journals Inc. (US)
Numerical simulation of radial and axial compressed elastomeric O-ring relaxation
July 2012
Figure 4 : O-ring axisymmetric finite elements model for RAL case
14
1.4
RAL case and E2
UAL case and E1
1 mn
1
8h
24 h
0.8
Compression ratio R
Solid line : 5%
Dotted line : 15%
No line
: 30%
0.6
0.4
Compression ratio R=20%
12
Relaxation time
Contact pressure (MPa)
Contact pressure (MPa)
1.2
10
8
6
24 h
8h
4
0.2
2
0
0.00
0.20
0.40
0.60
0.80
0
0.00
1.00
-0.50
-1.00
Relative radial position
0.50
1.00
Relative radial position
Figure 5 : Contact pressure variation for UAL case
Figure 6 : Contact pressure variation for RAL case
100
R=15%; t= 18 h
E1
0.12
E2
Axial contact width, b (%)
0.14
Contact pressure Ratio (p/E0)
Global Journal of Researches in Engineering ( A ) Volume XII Issue vIV Version I
6
0.1
0.08
0.06
0.04
UAL
RAL
0.02
0
-1.00
90
UAL
80
RAL
Analytic
70
60
50
40
30
20
10
0
-0.50
0.00
0.50
1.00
Relative radiale position
Figure 7 : Comparison of pressure contact for UAL and RAL
© 2012 Global Journals Inc. (US)
0
10
20
30
Compressive ratio, R (%)
Figure 8 : Axial contact width variation
40
Numerical simulation of radial and axial compressed elastomeric O-ring relaxation
4
4
R=10%
Solid line : RRL case
Dashed line : URL case
R=15%
2
1.5
Radial contact
E2 and R= 25 %
3
Relaxation time
48 h
2.5
8h
2
4h
0
1.5
1
1
0.5
0.5
July 2012
R=5%
2.5
E1
3.5
0
0
0
0.2
0.4
0.6
0.8
0
1
5
Contact pressure (MPa)
10
15
20
25
30
Contact pressure (MPa)
Figure 9 : Initial contact pressure vs compression ratio
Figure 10 : Relaxation stress in RRL case
0.45
URL case
Compression ratio
0.5
5%
10%
15%
20%
25%
0.4
0.3
0.2
0.1
Maximum contact pressure ratio, p0/E01
0.6
RRL case
0.4
Compression ratio
0.35
5%
10%
15%
20%
25%
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0
50000
100000
150000
200000
250000
300000
Relaxation time (sec)
Figure 11 : Relaxation of pressure contact for URL
0
50000
100000
150000
200000
250000
300000
Relaxation time (sec)
Figure 12 : Relaxation of pressure contact for RRL case
© 2012 Global Journals Inc. (US)
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Global Journal of Researches in Engineering ( A ) Volume XII Issue vIV Version I
Compression ratio
Axial position (mm)
Lateral contact
3
Maximum contact pressure ratio, p0/E02
Axial position (mm)
3.5
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