T. Tamiya,^ A. Matsumoto/^ K. Sato,^ K. Konosu, ® T. A. Stolarski · 2014. 5. 14. · EMail: Tadeusz.Stolarski @ brunel. ac. uk Abstract The experimental method of caustics has been

Caustic image simulation based on boundary

element method

T. Tamiya,̂ A. Matsumoto/̂ K. Sato,̂ K. Konosu, ® T. A. Stolarski ™^ Graduate School, Chiba University, 1-33 Yayoi-cho Inage-ku Chiba-shChiba, JapanEMail: [email protected], [email protected]^ Department of Urban Environment Systems, Chiba UniversityEMail: [email protected]^ Department of Electric and Mechanical Engineering, Chiba UniversityEMail: kouken @meneth. tm. chiba-u. ac.jp^ Department of Mechanical Engineering, Brunei University, Uxbridge,Middlesex, UBS 3PH, U.K.EMail: Tadeusz.Stolarski @ brunel. ac. uk

Abstract

The experimental method of caustics has been applied to rolling contact fatiguefor determining the crack growth mechanism. Caustic images formed from stressfields under rolling contact with a crack are very complicated. Therefore, thetheoretical background of the experimental caustic images formation has beenstudied through caustic image simulations. The simulation has been done by thescheme combining directly caustics method with boundary element method. Thesize and shape of simulated caustic images, which are formed from the contactregion and crack tip in disk-on-disk rolling contact fatigue, was coincident withexperimental caustic images. In the present paper, the computer program wasextended to simulate not only caustic images but also their initial curves that areloci of points in the disk through where the light rays passed. This program isutilized to discuss the mechanism of damage formation in rolling contact fatigueand to examine experimental results.

1 Introduction

In order to study the mechanisms of crack initiation and propagation under

Transactions on Engineering Sciences vol 24, © 1999 WIT Press, www.witpress.com, ISSN 1743-3533

380 Computational Methods in Contact Mechanics

rolling contact fatigue, it is very important to measure the magnitude andamplitude of the complex stress fields produced by rolling contact. The methodof caustics is a powerful technique to evaluate such stress fields. The method isan optical and a non-contact technique, and it is well known to be applicable tomeasure static and dynamic stress intensity factors of cracks.

Theocaris* has discussed the theoretical aspect and geometry of causticimages formed by a concentrated load. Sato et al.̂ have applied the theory tomeasuring contact forces in the disk-on-plate contact. Tamiya and Satô haveexpanded the method to the disk-on-disk contact. The above application andcaustic image simulation has been based on the Hertz's contact theory. In orderto apply this method to more complicated problems, Sato* has proposed ascheme of caustic images simulation with boundary element method, and hasshown that the images simulated by the scheme are in good agreement with theexperimental images under the disk-on-plate contact.

The main purpose of this investigation is to apply the method of caustics tothe complex stress fields in rolling contact fatigue with a crack. In this paper,therefore, the authors have developed a new simulation program of causticimages using the boundary element method, and has expanded it to solve thecontact problem. Furthermore, to consider the backgrounds of caustic imagesformation, the initial curve, that is a locus of points where the light rays passed,has been calculated and represented by contours after calculating the values ofJacobian Jby the program.

2 Caustic image simulation by boundary element method

2.1 The theory of caustics method

Figure 1 shows an example of caustic images formed from the stress fields underrolling contact with a crack. The white curves are the caustic images. The causticimages are formed by concentration of light rays. There are two caustic imagesin figure 1. A cusp-like image is formed from the contact region and a circularimage is formed from the crack tip. The shape and size changes with optical,material, and contact conditions.

Figure 1: An example of the Caustic Image.


Computational Methods in Contact Mechanics 381

Figure 2: Schematic illustration representing caustic image formationand vector relations.

Figure 2 shows the vector relationship for representing caustic imageformation. Now we consider PMMA disk to steel disk contact, and the light rayimpinges the point A at the PMMA disk. When disks are in non-contact the lightray passes thorough the point A arrives at the point A ' on the screen, but the disksare in contact, the ray arrives at point B deviated from point A '. When the vectorfrom the origin O to the point A on PMMA disk is the vector r, the vector fromthe origin O' to the point B on steel disk is the vector W, and the vector from thepoint A' to the point B is the vector w, the relationship of the vectors areexpressed by

(1)

where /î(=(z,+zj/zj is an optical magnification factor, C(=ẑ c/) is a constantdepending on an optical set up, z, and z^ are the distance from the specimen to thefocus and the distance from the specimen to the screen respectively, c, is anoptical constant of the material and Ms a thickness of the material. Laser raysforming the caustic image passed through the points on a curve called 'initialcurve'. The initial curve satisfies following equation, which means zeroing of theJacobian determinant J=0 of eqn (1),

J = = 0. (2)



2.2 Combination of the caustics method with BEM

In caustic image simulation, it is necessary to calculate the coordinates ofyj corresponding to many points of A(x, y) in eqn (1). This is done bycalculating the gradient of stress sum grad(o,+oy) at many points of A on thespecimen. In boundary element method, stresses q, and oy are calculated fromthe following equation using displacements u^ and Uy and surface forces ̂ and tyon boundary elementŝ :

(3)

where D,& and Ŝ are coefficient matrices created from surface forces and surfacedisplacements, / represent x and y, and A: is a suffix to the rule of sum totalexpression. Then the stress sum (ô +oy) is given by,

Therefore, eqn (1) is given by, using eqns (3) and (4),

W = ̂r + Cgrad[

= (̂ + Ĉ!L)/ox ay

Furthermore, for obtaining the initial curves Jacobian J shown by eqn (2)was programmed. The equation coded in the program is expressed by thefollowing, from eqns (2) and (5),

(6)

In the scheme of analyses, eqns (5) and (6) were coded direct in a programof boundary element method, and then the vector W and Jacobian J wereanalyzed directly from surface forces and surface displacements, not afterobtaining the stresses. Although a set of points filling J=0 gives an initial curve,it is difficult to calculate as the approximate equation analytically. Then weobtained the initial curves by contouring the all the values of Jacobian J by othersoftware.



Specimen : PMMA£=3.7GPa He-Ne Laser

Figure 3: Optical set up used in Caustic Experiments.

3 Method of experiment and simulation

3.1 specimen and method for experiments

The specimen was a PMMA disk of 100mm in diameter and 4mm in thickness.Its Young's modulus was £=3.2GPa and Poisson's ratio was v=0.35. Thespecimen had a radial crack of 2.4mm long started from its rolling surface. Thespecimen was contacted with a steel disk of 290mm in diameter and 5mm inthickness. Its Young's modulus was £=206GPa and Poisson's ratio was y=0.33.The above conditions are the same as in rolling contact fatigue tests carried outby Tamiya and Sato*. Applied normal and tangential contact forces, P and Q,were 75N/mm and 7.5N/mm, that is, the coefficient of tangential force was /

Figure 3 shows the optical setup used for the caustic experiments. The lightsource was a He-Ne gas laser of 632.8nm in wavelength. The laser rays wereexpanded by collimator, and changed divergent light rays by a convex lens.When the light rays were irradiated around a contact region, caustic images wereformed on screen, and were captured in a PC using a CCD camera. The opticalmagnification was 7.74, because of z,=235mm and z/=1585mm.

3.2 Method for analyses

The boundary element model used is shown in figure 4. The lower body is thePMMA disk and the upper body is the steel disk. The PMMA disk had a radialcrack of 2.4mm-long inclining an angle of 0to the contact point. The boundaryelement analysis was carried out for the crack positions 9 ranging from -6 to +6degrees at every 0.5 degree.

The bottom surface of PMMA disk was fully constrained. On the upperboundary of the steel disk, normal force of />=75N/mm was applied. Tangentialforce Q was applied to both sides as friction force. The value of coefficient of



tangential force was selected to be f(=Q/P)=0.l as an adopted experimentalcondition, although the value of coefficient of friction is //=0.3. The elementsnear the contact point and crack faces were meshed finer. To calculate the valuesof vector W and Jacobian J, many internal points, over 3,000, were inputtedaround the contact point and the crack tip automatically.

4 Results and discussions

4.1 Comparison between experiment and simulationFigures 5(a)-(d) show the typical results by experiments and simulations. Theyalso show the Jacobian contours calculated: the top figures (1) are experimentalcaustic images, the middle figures (2) are simulated caustic images, and thebottom figures (3) are contours of Jacobian J.

We can divide caustic images in figures 5(a) and (b) into two types: one isthe cusp-like image formed from the contact region, and another is the circularimage formed from the crack tip. Their shape and size changes when the crack isapproaching the contact point. When the crack approaches to the contact point asshown in figure 5(c), the circular image coalesces with the cusp-like image.

— linitial curve (.-20

-30

0

1

-40 -30 -20 -10 0 10 20 30 40 '-40 -30 -20 -10 0 10 20 30 40;,, mm %,, mm

0 1 2 3 4 - 4 - 3 - 2 - 1 0 1 2 3 4x, mm x, mm

(d) 0=0 deg

Figure 5: Continued


385

^

)

f v Y N

A

"̂

' 1 >

«3

' 1 \, j, ,f \

-̂*

^>

o

Crack

Inner point for simulatingcaustic image

Figure 4: Model and Boundary Element meshes for BEM analysis.

— linitial curve (J=0)

"

%, mm

(b)0=-3.f

Figure 5: Experimental and analytical results: (1) experimental causticimages, (2) simulated caustic images, and (3) J contours. Thelines of J-0 correspond to the initial curve.



Furthermore, when the crack tip is just under the contact point, only the cusp-likeimage is formed in the contact region, and the caustic image formed from thecrack tip disappears due to crack closure and symmetrical deformation.

We can clearly observe the above changes of caustic images in rollingcontact process. This means that the method of caustics is able to observe thestress and deformation behavior in rolling contact fatigue, such as damagingprocess, crack open/close behavior, crack shear deformation, etc. Although thecaustic images in rolling contact are very complicated because of complexity instress fields, the simulated caustic images agree well with the experimentalimages. The BEM program presented in this paper is useful to simulate theexperimental behavior and to study the crack initiation and propagationmechanism under rolling contact fatigue.

4.2 The relationship between caustic images and initial curves

At the bottom in figure 5, the initial curves are shown by curves of 7=0. Thecaustic images are formed by light lays passing through the points on the initialcurves in the specimen. Initial curves can be seen clearly. The elliptical initialcurve is around the contact region, and the circular one is around the crack tip.

When the crack is far from the contact region, as the case in figure 5(a),both initial curves separate from each other, and the circular and the cusp-likecaustic images are separate. When the crack closes to the contact region, as thecase in figure 5(c), two initial curves connect and the corresponding causticimages are connected. In figure 5(b), we can see a region where the value of Jequals nearly zero, marked with Arrow A, then the corresponding brighter imageforms, marked with Arrow B. Furthermore, in figure 5(d), in which the crack isjust under the contact point, the initial curve around the crack tip vanishes andthe caustic image formed is only from the contact region.

From the above relationship between caustic images and initial curves, it isconcluded that the shape and size of caustic images is directly influenced byinitial curves. The fact that initial curve could be clearly calculated and related tothe caustic image, means usefulness of the program to study the informationobtained from caustic images.

5 Conclusions

The simulation program of caustic images was developed for applying themethod of caustics to study rolling contact fatigue with a crack. The applicabilityof the program was examined experimentally. The main conclusions were asfollows:

1. The program has been developed by the scheme directly are combiningthe theories of the method of caustics and boundary element method.

2. The initial curves, which are relating to the caustic image, are alsoobtained through calculating the values of Jacobian J, in boundaryelement analysis.

3. Circular and cusp-like caustic images, formed in rolling contact



experiments, can be well simulated using the program.4. The shape and size of these caustic images formation is discussed in the

light of initial curves changes.5. This program is useful to simulate caustic images formed from complex

stress fields and to understand the background of caustic image formation.

References

1. Theocaris, P.S., Mechanics of Fracture, Vol.7, Martinus Nijhoff Pub,Hague, pp. 189-252, 1981.

2. Sato, K., Sato, T, Itoga, H. and Namaizawa, Y., Measurement of ContactLoad by the Method of Caustics, Journal of Japan Society NonDestructive Inspection, 39-8, pp. 648-653, 1990.

3. Sato, K., Sato, T, Kojima, E. and Dohi, A., Measurement of Contact Loadby the Method of Caustics (Part II), Journal of Japan Society NonDestructive Inspection, 40-12, pp. 798-803, 1991.

4. Sato, K. and Ito, T, A Method for Measuring Tractions in Disk-To-PlateContact Using Caustic Images, Journal of Japan Society ofTribologists,40-9, pp. 762-767, 1995.

5. Tamiya, T, Sato, K., Applications of the Method of Caustics to Studies ofRolling Contact Fatigue, Journal of Japan Society ofTribologists, 43-8,pp. 723-729, 1998.

6. Aliabadi, M. H. & Samartin, A. (eds.), Computational Methods in ContactMechanics III, Computational Mechanics Publications, Southampton andBoston, pp. 331-340, 1997.

7. Yuki, Y. & Kisu, H., Elastic Analysis by Boundary Element Method,Baifu-kan, Tokyo, pp. 73-75, 1987.

8. Tamiya, T, Sato, K., Surface-Initiated Crack Growth in Rolling ContactFatigue, Journal of Japan Society of Mechanical Engineering, 65-632(A),pp. 833-839, 1999.


Documents

T. Tamiya,^ A. Matsumoto/^ K. Sato,^ K. Konosu, ® T. A. Stolarski · 2014. 5. 14. · EMail: Tadeusz.Stolarski @ brunel. ac. uk Abstract The experimental method of caustics has been