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Procedia Engineering 114 (2015) 277 283

1st International Conference on Structural Integrity

Photoelastic and numerical analysis of a sphere/plan contact

problem
Rabah Haciane a *, Ali Bilek b , Said Larbi c , Djebali Said d
a

L.M.S.E Laboratory, Mechanical Engineering Department, Mouloud Mammeri University Algeria,

b
Mechanical Engineering Department, Mouloud Mammeri University, Algeria,

Abstract Stress analysis in the neighborhood of contact zones can help improve the design and the durability of mechanical
components. A numerical solution for a three-dimensional contact problem (sphere/plan), under a normal load, is developed. An
experimental analysis with the same load is conducted on a regular polariscope with a three dimensional model for comparison
purposes with the finite element solution. The stress field is locked inside the model with the help of the stress freezing
technique. The stress field is then determined, on a regular polariscope, for slices cut in the model for comparison purposes with
the finite element solution. The experimental isochromatic and isoclinic fringes are compared with the simulated ones. Another
comparison is made by using stress values; the simulated principal stresses differences are obtained for a slice located along the
load direction. The obtained graph is validated by the experimental data obtained by exploiting the photoelastic fringes recorded
on a white field polariscope for the corresponding slice. Good agreements are observed; for a same slice stresses decrease along
the vertical axis of symmetry as me move away from the contact area.
2015
2015Published
The Authors.
Published
by isElsevier

by Elsevier
Ltd. This
an open Ltd.
access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering.
Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering
Keywords: photoelasticity ; birefringent; isochromatic; isoclinic; contact; stress;

1. Introduction
Stress initiation is mainly controlled by the shear stress mechanism, particularly for metallic parts. It is therefore
very important to determine the type and amplitude of the imposed mechanical stresses. Photoelastic fringes
obtained experimentally with plane polarized light can help designers determine stress fields developed in
mechanical parts, particularly in the neighbourhood of the contact zones. Several studies have been conducted [1-9],

* Corresponding author. Tel.: +0213-772- 787-709; fax: +0 213 -772-56-53-69.

E-mail address: rabahhaciane3@hotmail.fr

1877-7058 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering

doi:10.1016/j.proeng.2015.08.069

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experimentally as well as numerically. In this work a contact problem between a sphere and a rigid parallelepiped
was solved experimentally by using photoelasticity and numerically by using the finite elements analysis. By
unwrapping the photoelastic fringes obtained experimentally, the principal stresses difference can be determined
easily as long as the fringe orders are correctly obtained from the fringe pattern. The principal stresses difference is
used for comparison purposes between the experimental and the numerical analysis. Another comparison is made
between the experimental and the simulated fringes in order to validate the finite element solution as done by other
authors [2, 8].
Nomenclature
f

C
F

1
2

Fringe constant
Light wave length used for the experiment
Optical constant C of the model material
Normal load
Isoclinic parameter
Isochromatic parameter
relative retardation
Poissons ratio of the birefringent model
Poissons ratio of the sphere metal

2. Experimental Procedure
The model in the shape of a parallelepiped (52 x 52 x 120 mm) is cut in a birefringent material. The model is
then mounted on a loading frame inside the oven to freeze the stress field inside the volume of the model (Fig.1).
The load (F= 50 N) was applied to the model via a sphere made of steel (6 mm radius).The temperature of the oven
is increased with a speed of five degrees per hour up to the stress freezing temperature (125 degrees C.) The
temperature is maintained during 10 hours to allow equilibrium. The temperature is then lowered slowly (5
degrees/hour) to room temperature; the load should be maintained during the entire test.
The model is then mechanically sliced to allow fringe analysis on a regular polariscope. We used plane polarized
light to obtain the isochromatic and the isoclinic fringes in order to determine the stress values and the stress
directions.

Fig. 1. Model mounted on the loading frame inside the oven.

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Fig. 2. Light propagation through a photoelastic model.

The light intensity obtained on the analyzer (Fig. 2) is given by the following relation (equation 1) [9] :

sin 2 2D sin 2 M 2

(1)

The terms sin 2D and sin M 2 give, respectively, the principal stresses directions and the values of their
difference with the following relation (equation 2):
2

1  2

N f
e

(2)

Where N is the fringe order obtained experimentally from the isochromatic fringe pattern, e is the slice thickness
and f is the fringe constant which depends on the light wavelength  used and the optical constant C of the model
material ( ). The value of the fringe constant f is determined experimentally with a disc of the same material
as the model. The disc, introduced in the oven with the model, is loaded along the diameter. After cooling the disc is
analyzed with the circularly polarized light in order to obtain the fringe constant which is used then to obtain
stresses [9]. The value obtained is f = 0.43N/mm/fringe.
3. Experimental Results
Figure 3 shows the isochromatic fringe pattern obtained for a slice cut along the load direction. The isochromatic
fringe pattern is obtained with circularly polarized light; two quarter wave plates are used in order to eliminate the
isoclinic fringes. The isochromatic fringes are used to obtain the values of the principal stresses difference in the
model along the vertical axis of symmetry, particularly in the neighbourhood of the contact zone. A closeup of the
contact zone (Fig.3 b) is necessary in order to determine accurately the fringe orders. Since we used a white
background configuration, the fringe order of the first isochromatic is N=0.5, starting from the bottom (Fig. 3 b) The
fringe orders increase to a maximum value of N=32.5 , as we move close to the zone of maximum shear stress,
whereas, at the lower part of the model, stresses are very much lower because the load is distributed over the whole
area of contact of the model with the loading frame.

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Zone of maximum
shear stress

Fig. 3. (a) Experimental isochromatic fringes obtained with an 8 mm slice at z=0 mm; (b) closeup of the contact zone.

The maximum value recorded experimentally for the principal stresses difference is about 1.53 MPa. Stresses
decrease then, as we move away from the contact zone, to lower values and almost vanishes as we get close to the
lower contact surface (Fig. 4).

Fig. 4. Principal stresses difference along the vertical axis of symmetry.

4. Numerical analysis
The material is considered to behave everywhere as a purely elastic isotropic material. Fringe constant
f=0.43N/mm, Youngs modulus (E1=210000 MPa, E2=15.9 MPa) and Poissons ratios (1=0.3, 2=0.45)
respectively for the sphere and the parallelepiped are introduced in the finite element program. The mesh is refined
in the neighbourhood of the contact zone (Fig. 5) in order to achieve better approximation of stresses.
Z

Fig. 5. The finite element meshing.

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4.1 Numerical calculation of the isochromatic fringes

The retardation angle is obtained with the equation M
the following relation (equation 3):

2S N . The different values of

2S e / f  V x  V y 2  4W xy2

can be determined with

(3)

The different values of sin2/2 which represent the simulated isochromatic fringes can then be easily calculated
along the z axis and displayed (Fig. 6). We can see in the first slice at z=0, which corresponds to the direction of the
applied load, a concentration of fringes in the neighbourhood of the contact zone. As we move away, 10mm along
the z direction for each successive slice, the number of fringes decreases. This means that stresses are concentrated
in the neighbourhood of the contact zone.

Fig. 6. Simulated isochromatic fringes along the z direction.

A comparison is made with the isochromatic fringes obtained experimentally (Fig. 7 right) for the slice at z=0
along the direction of the applied load as this slice is the most stressed one. We can see relatively good agreements;
however in the neighbourhood of the contact zone we can see some discrepancies.

a)

b)

Fig. 7. Calculated isochromatic (a) and experimental isochromatics (b).

Another comparison using the principal stresses difference along the vertical axis of symmetry (Fig. 8) shows
relatively good agreement.

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Fig. 8. Experimentally and numerically principal stresses difference along the vertical axis of symmetry.

4.1 Numerical calculation of the isoclinic fringes

The isoclinics represented by the term sin 2 2D are the loci of points for which one of the principal stresses direction
is parallel to the analyzer axis. They are obtained with plane polarized light, the quarter way plate are removed from
the light path.
The different values of the isoclinic parameter can be calculated with (equation 5) which can be obtained
readily from Mohrs circle for stresses. The different values of sin 2 2D give then, readily, the isoclinic fringe
pattern. The software package uses a four color scale to represent the values of the sinus term; the blue color
represents the zero value which corresponds to the dark fringe in the experimental isoclinic pattern. The calculated
isoclinic is obtained for = 45. We see good agreement between the experimental and the numerical isoclinics
(Fig. 9).

arct (2Wxy /(Vx - Vy))

(5)

a)

b)

Numerical isoclinic

Experimental isoclinic

45

Fig. 9.

a) Calculated isoclinics and b) Experimental isoclinics.

5. Conclusion
We have analyzed, experimentally by using photoelasticity and numerically by using the finite elements analysis,
the stress field developed in a birefringent parallelepiped modeled by a sphere made of steel in a birefringent
parallelepiped model. The purpose is to analyze the stress field, particularly in the neighbourhood of the contact
zone. We showed that photoelastic fringes and stresses can be calculated easily and accurately for an isolated slice
with sufficient accuracy. Relatively good agreements between experimental and numerical results are achieved .
One should emphasize the importance of photoelasticity to solve this kind of contact problem where the limit
conditions and the application of the load in the finite elements solution is not an easy task; the shape of the surface
in contact is sometimes difficult to determine because both bodies in contact can deform. It is therefore advisable,

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in these cases, to solve experimentally the problem in order to check the reliability of the finite elements solution.
Photoelassticity is still very much used in automobile industry and aeronautics to solve contact problems.

References
[1] Abodol Rasoul Sohouli, Ali Maozemi Goudarzi, Reza Akbari Alashti, Finite Element Analysis of Elastic-Plastic Contact Mechanic
Considering the Effect of Contact Geometry and Material Propertie, Journal of Surface Engineered Materials and Advanced Technology,
2011, 1, 125-129.
[2] J. A. Germaneau, F. Peyruseigt, S. Mistou, P. Doumalin and J.C. Dupr, Experimental study of stress repartition in aeronautical spherical
plain bearing by 3D photoelasticity : validation of a numerical model, 5th BSSM International Conference on Advances in Experimental
Mechanics, Sept. 2007, University of Manchester, UK.
[3] A. Zenina, J.C. Dupr & A. Lagarde, Separation of isochromatic and isoclinic patterns of a slice optically isolated in a 3-D photoelastic
medium, Eur. J. Mech. A/Solids 18, pp. 633-640, 1999.
[4] L. Kogut & I. Etsion, Elastic-Plastic contact analysis of a sphere and a rigid flat, Journal of Applied Mechanics, V. 69, pp.657 - 662,
2002.
[5] T. L. Nguyen, A. Fatu, D. Souchet, Etude du contact entre le coussinet et le logement dans un palier lisse", 21me Congrs Franais de
Mcanique, bordeaux 2013.
[6] Budimir Mijovicand Mustapha Dzoclo. Numerical contact of a Hertz contact between two elastic solids, Engineering Modeling 13 (2000)
3-4, 111-117.
[7] A. Mihailidis, V. Bakolas, & N. Drivakovs, Subsurface stress field of a dry line Contact. Wear V. 249, I.7, pp 546-556, 2001.
[8] A. Bilek, J.C. Dupr , A. Ouibrahim, F. Bremand, 3D Photoelasticity and numerical analysis of a cylinder/half-space contact problem,
Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII, WIT Press Southampton, Boston 2005,
pp. 173-182.
[9] J. W Dally and F. W. Riley, Experimental stress analysis, McGraw-Hill, Inc, 1991.