aspects concerning contact stresses …...64 the annals of university “dunĂrea de jos“ of...

64THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS“ OF GALATI

FASCICLE VIII, 2010 (XVI), ISSN 1221-4590, Issue 2TRIBOLOGY

ASPECTS CONCERNING CONTACT STRESSES FROM A DENTALARTICULATOR JOINT

Florina CIORNEI, Stelian ALACI, Ilie MUSCĂ,Gheorghe Frunză, Luminiţa Irimescu, Delia CERLINCĂ

“Ştefan cel Mare” University of Suceava, [email protected]

ABSTRACTA dental articulator is a mechanical device, which simulates the temporo-

mandibular joint (TMJ). The articulator is important because it replicates the basicrevolve action of the upper and lower mandibles, as well as translational motions.In the present paper, the stresses from an articulator TMJ modelled as a sphere intoa spherical/cylindrical cavity are analyzed by two methods, first applying theHertzian contact theory and secondly using a FEA simulation.

Keywords: Hertzian contact, elastic material, dental articulator, FEA

1. INTRODUCTION

An articulator is a mechanical device whichsimulates the temporo-mandibular joint (TMJ) andjaw members to which maxillary and mandibularcasts may be attached to represent jaw movements.The entire assembly attempts to reproduce themovements of the mandible and the variousintercuspidian relationships that accompany thosemovements (Fig. 1) [19]. Dental models taken withimpressions and poured in dental stone are placed onthe machine either for examination and diagnosis, orto construct dental appliances. Special records aretaken to accurately position the dental models on thearticulator. The facebow record is a measurementfrom the upper teeth to the joints. The centric relation,[1], or bite record is a measurement of where the teethare positioned with the joints positioned correctlybefore the teeth actually come into contact, [20]. Adental articulator assists in the fabrication ofremovable prosthodontic appliances (dentures), fixedprosthodontic restorations (crowns, bridges, inlaysand onlays) and orthodontic appliances. A dentalmodel articulator is used to correlate upper and lowerdental models in the forming and adjustment of thedental prosthesis. The model is used by a technicianto determine the optimum shape and position for the

prosthetic to work in conjunction with the patient'sother teeth and normal jaw motion.

The articulator is important because it replicatesthe basic pivot action of the upper and lowermandibles, as well as translational motions. Aprosthetic can then be inserted into the model toconfirm that its proposed design and intendedinstalled position should work properly in thepatient's mouth.

In order to reproduce the individual parametersof the patient the articulator must be adjustable. Thesettings are measured on the patient and using a facebow, the relative location of the occlusal plane istransferred from the patient to the mechanical dentalarticulator (Fig. 2) [21]. Dental articulators are usedfor more than 120 years [14], and different types ofarticulators were constructed (Fig. 3) [22, 23]. Theirstructures contain upper and lower bodies and theTMJ-s, which are the most important part of thearticulators, present a great variety of configurations.

In the present paper the stresses from anarticulator TMJ modelled as a sphere into aspherical/cylindrical cavity are analyzed by twomethods, first applying the Hertzian contact theoryand secondly using a FEA simulation.

mailto:[email protected]



Fig. 1. Temporo-Mandibular Joint Anatomy

Fig. 2 Human skull and dental articulator

a) Keystone Deluxe b) Brussels – A c) Yamahachi OHTAK 1Fig. 3. Different types of dental mechanical articulators

2. STUDIED TMJ MODEL

The studied model, from a Yamahachi dentalarticulator [23], that is a standard arcon typearticulator with a ball slot type path, designed inCATIA, is presented in Fig.4 and consists of a sphere(1), guided by an unfigured pin, placed into aspherical/cylindrical groove, (2) and pressed onto it inthe necessary position by a screw (3). The mandibledoes not act like a simple pivot but it rotates aroundthree reciprocal orthogonal axes. Recent researchesstudied and modelled the human jaw and chewingbehaviour, [5, 8, 12, 13, 17]. A model should achieve,in terms of kinematics, the human-like chewingmovements. For a spatial mechanism where at leastone of the elements presents a general spatial motion,the mobility can be obtained using Kutzbach’scriterion [16]:

5

1

16k

kck)n(M (1)

where n is the total number of elements (including the

ground), kc is the number of joints restrictioning k

elementary motions and k is the number of kc joints.

For the studied TMJ model, 4n , namely: theground, the spheres and the mobile jaw;

22 c between the spheres and channel and 24 c

between spheres hole and pins and it results:64222146 )()()(M (1a)

Fig. 4. Model of a dental articulator TMJ

Therefore, the model presents 6 degrees offreedom, likewise the real TMJ, ensuring a completepositioning of the cast from the mobile jaw.In the modelled TMJ, several Hertzian contacts areidentified:

1

2

3



Sphere and spherical cavity, for a symmetricanatomical position;

Sphere-cylindrical cavity, for asymmetriclower jaw anatomy;

Sphere-flat screw end; Cylindrical shaft-inner ball hole.In the present model, the following simplifying

assumptions were made, in order to avoid toocomplex computations in FEA:

the mandible is in a symmetrical position,therefore the contact between the sphere and thegroove occurs in the spherical cavity;

the screw end has a concave surfaceconforming the sphere and the normal force is appliedon the flat top of the screw, as a distributed pressure;

the linear contact between the pin and thesphere was not considered;

the finite element analysis was performed onlyfor one-half of the model with the purpose to reducethe computing time and better allocate the computerresources as the symmetrical loading and designinggeometry allow it.

3. ELEMENTS OF HERTZIANCONTACT THEORY

Assuming that the sphere and the groove satisfythe hypothesis shown by Johnson [6, 7, 11], thecontact can be described by Hertzian theory. When anormal load is applied, the bodies deform and acontact area occurs. On this area, the unknown stressdistribution is found by applying superpositionprinciple in Boussinesq problem.

A numerical program Mathcad was used forcomputing contact parameters and contact pressure ofspheres with different elastic characteristics in normalcontact, using the relations from classical Hertziantheory.

The geometry of contact is determined by theradii of curvature for the sphere, yx RR 11 and radii

of curvature for the groove (seat), that is x2R ,

0R y2 for cylinder or 0RR y2x2 for spherical

cavity. The curvatures of the contacting bodies are:

y2y2

x2x2

y1y1

x1x1 R

1k;

R

1k;

R

1k;

R

1k (2)

and the contact curvature is found with therelation:

yxyx kkkkk 2211 . (3)

The contact rigidity depends on the elasticcharacteristics of contacting materials:

2

22

1

21

E

1

E

1

, (

4)where 21 E,E are the Young moduli for sphere and

cylinder materials respectively and 21 , are the

Poisson coefficients for sphere and cylinder materialsrespectively.

The contact parameter found with the relation:

)2cos(kkkk2

kkkk

k

1

y2x2y1x1

2y2x2

2y1x1

(5)

is used in determining the eccentricity e of thecontact area. The eccentricity of ellipse is the root ofthe equation 0)e(F , where:

;d)][sin(e1)e(E2/

0

22

.d)][(sin(e

)e(K

/

2

0221

1

(6)

;e

)e(E)e(K)e(D

2

;14

kB,A

(7)

;B

A

)e(D)e(K

)e(D)e1()e(F 2

(

8)

The contact parameters are:

- maximum contact pressure:312

0 2

3/

p Qk

np

;

(9)

- contact ellipse semimajor axis:31

2

3/

a k

Qna

;

(10)

- contact ellipse semiminor axis:31

2

3/

b k

Qnb

;

(11)

- normal approach:31

2 2

4

9

2

/

kQn

. (

12)

4. RESULTS

The contact geometry in the studied TMJ modelis given by the radius of sphere and the radius of thecavity: mm.R;mm.R cs 78575 , respectively. TheYoung moduli of the materials for sphere and channel

are 211s m/N101.1E , 211

c m/N102E and

the Poisson coefficients are 26603410 .;. cs respectively. In the human chewing process, themaximum forces vary, but can attain even 652N,[Sp89]. Here, a regular load was considered,

N.Q 89 . The results obtained applying therelations (2-12) and using a Mathcad programme aregiven in Table 1.



Table 1. Results obtained applying Herzian theory

Contact parameterHertz theory

Symbol Sphere in sphericalcavity

Sphere in cylindricalgroove

maximum contact pressure 0p MPa.1541 MPa.2247

contact area major one-half axis a mm.3380 mm.5340

contact area minor one-half axis b mm.3380 mm.0360

normal approach mm. 4107662 mm. 4105654 contact area eccentricity e 0 9980.

The results obtained by finite element analysiswere found using Generative Structural Analysismodule from CATIA software package developed byDassault Systems and they are presented in thefollowing next figures. The refined meshing can beobserved in the vicinity of contacting surfaces forcomponents of TMJ in Fig. 5 and in Fig. 6, for theassembly. The elements of the mesh are lineartetrahedrons. The contact pressure obtained by FEAis shown in Figures 7-9 from different views or

with/without mesh visualisation. In Figure 10, theglobal maximum von Mises stress is highlighted andin Figure 11 the extreme assembly contact pressure(blue) is seen. Figures 12 and 13 present the contactpressures in the two individual contacting parts whilein Figures 14 and 15 the equivalent von Misesstresses are presented for the two parts. Figures 16and 17 present sections made with a plane normal tothe channel axis, for a more complete visualisation ofstresses and mesh visualisation.

Seat Sphere ScrewFig. 5. Refined meshing on contacting surfaces of parts from TMJ model

Fig. 6. TMJ assembly meshing Fig. 7. FEA contact pressures with mesh



visualisation

Fig. 8. FEA contact pressures Fig. 9. FEA contact pressures with meshvisualisation

Fig. 10. Von Mises stresses for the TMJ assembly with the global maximum value and detail.

Fig. 11. Extreme values for z stress tensor component in the TMJ assembly and detail.



Fig. 12. Extreme value for z (contact pressure) stress tensor component in the sphere

Fig. 13. Extreme values for z (minima is the contact pressure) stress tensor component in the groove and

detail

Fig. 14. Extreme Von Mises stresses in the sphere



Fig.15 Extreme Von Mises stresses in the channel

Fig. 16. Contact pressures in a section Fig. 17. Von misses stresses in section

From the FEA analysis, the maximum Hertzianpressure for the sphere (Fig. 12), was

MPa15.41p FEA0 and the error computed for the

maximum contact pressure was found:

%1847.1100p

pp

Hz0

FEA0Hz0

This is considered a very good error value,allowing for the rather coarse meshing performed.The element was chosen linear and the dimension ofthe local mesh size is 0.3mm, having thus the samemagnitude order as the contact area. A parabolicmeshing element and a refined meshing should offermore precise results [18], but they require powerfulcomputing performances.

The maximum contact pressure from the cavitymust be very carefully analyzed. The FEA programshows the extreme stress, but this value is obtained, ata detailed view, Fig. 13, on the discontinuity regionbetween spherical and cylindrical grooves. Themaximum contact pressures have in fact the samevalues in the sphere and in the channel. Thus, aparticular situation occurs when the sphere-channelcontact happens on the boundary between sphericaland cylindrical surfaces. In this case, the classicalHertzian theory relations cannot be applied becausethe surface has a second order discontinuity due todifferent curvature radia on the sphere-cylinderborder in the symmetry plane of the groove and,therefore, special methods must be applied [Gl99].



In order to find the stresses for the steel cavityapplying contact theory, one must consider thedissimilarity between contacting materials, [9, 10].From the authors’ previous work [2, 3, 4], the sphere-halfspace contact considering the friction betweendissimilar materials is a rather elaborate task. Theactual problem for the sphere in a cavity consideringboth friction and materials dissimilarity is therefore afuture research goal.

Once proven the validity of the FEA model byvalidation of maximum contact pressure, the majoradvantage provided by finite element analysis ishighlighted, as it provides the stress tensorcomponents and the equivalent von Mises stress,together with the strain field, for both contactingbodies, from a unique analysis. The graphs of thefields of stresses and strains shown on the geometryof the elements give also an overall image of whatactually happens when the force is applied.

5. CONCLUSIONS

Contact stresses from an articulator TMJ werestudied by classical Hertzian theory and by finiteelement analysis. In a model TMJ occur severalHertzian contacts but in the present paper only thesphere in spherical cavity was solved by bothmethods above mentioned.

The results were obtained in FEA for the globalTMJ; an accurate analysis assumes considering onlythe neighbourhood around contact, in order to allowfor a fine meshing and the discretization element sizeto be much smaller than the contact area characteristicdimension. For regions situated remote from contact,as the Saint-Venant principle states, a qualitativeagreement was found between stress field lines andthe theory.

The values for contact parameters obtainedapplying Hertzian theory were rapidly found, usingMathcad.

The FEA maximum contact pressure fromsphere-channel contact validates the Hertzian contactpressure from Mathcad calculus made applying theanalytical relations.

The FEA analysis is time and computingresources consuming. The meshing was rather coarsedue to computer hardware limitations. Once theanalysis made with finite element method in CATIA,the stress fields from the contacting bodies arevisualised rapidly, for stress tensor components or forequivalent von Mises stresses and can provide ageneral image.

The FEA method is useful for a preliminary stepin design, when the whole assembly, with loads, isenvisaged, for emphasizing the most stressed regions

of the parts from the assembly. A further analysis isrequired in the local region, for contact stress analysisas the trial to diminish the size of the mesh element isa poor technique, consuming time and hardwareresources.

REFERENCES

1. Burlui V., Forna N., Ifteni G., 2001, Clinica si terapiaedentatiei partiale reduse, Ed. Apollonia (in Romanian).2. Ciornei F.C., Alaci S., Irimescu L., Cerlinca D., 2008, Surl’etat des contraintes en fretting radial, ACME 2008, Iaşi, , TomLIV (LVIII) Fasc 1, Secţia Construcţii de Maşini.3. Ciornei F.C., Musca I., Irimescu L., Alaci S., Frunza Gh.,2008, Upon pressure distribution on fingerpad during grasping,VAREHD 14, Suceava, Romania.4. Ciornei F.C., Alaci S., Irimescu L. Frunză G., Cerlinca D.,2009, Some remarks on contact models of fingerpad - part II:surface stresses, Annals of the Oradea University, Fascicle ofManagement and technological Engineering, CD-ROM, VolVIII(XVIII).5. Daumas B., Xu W.L., Bronlund J., 2005, Jaw mechanismmodeling and simulation, Mech. Mach. Theory, 40, pp. 821-823.6. Diaconescu E.N., 1993, Decissive stresses for rolling contactfatigue, Scientific report for EEC, INSA Lyon.7. Glovnea M.L., 1999, Efectul discontinuităţilor geometrice desuprafaţă asupra contactului elastic, PhD Thesis (in Romanian),University Suceava, Romania.8. Hannam, A.G., Stavness I., Lloyd J.E., Fels S., 2008, Adynamic model of jaw and hyoid biomechanics during chewing, J.Biomechanics, 41, pp. 1069-1076.9. Hills D.A. Sackfield A., 1987, The stress field induced bynormal contact between dissimilar spheres, J. of AppliedMechanics, ASME, Vol. 54, March, pp. 8-14.10. Hills D.A., Nowel D., Sakfield A., 1993, Mechanics of elasticcontacts, Butterworth-Heinemann Ltd.11. Johnson, K.L., 1985, Contact Mechanics, CambridgeUniversity Press.12. Koolstra J.H., van Eijden T.M.G.J., 2005, Combined finiteelement and rigid body analysis of human jaw joint dynamics, J.Biomechanics, 38, pp. 2431-2439.13. Koolstra J.H., 2002, Dynamics of the human masticatorysystem, Crit. Rev. Oral. Biol. Med., 13, 366.14. Solaberrieta E., EtxanizO., Minguez R., Muniozguren J.,Arias A., 2009, Design of a virtual articulator for the simulationand analysis of mandibular movements in dental CAD/CAM, Proc.19th CIRP Design Conference-Competitive Design, CranfieldUniversity, 323.15. van Spronsen P.H. et al., 1989, Comparison of jaw musclebite force cross sections obtained by means of magnetic resonanceimaging and high resolution CT scanning, J. Dent. Res., 68; 1765.16. Uiker J.J.jr., Pennock G.R., Shigley J.E., 2003, Theory ofmachines and mechanisms, Oxford University Press, New York.17. Xu W.L., Lewis D, Bronlund J.E., Morgenstern M.P., 2008,Mechanism, design and motion control of a linkage chewing devicefor food evaluation, Mech. Mach. Theory, 43, pp. 376-389.18. Zienkiewicz O.C., Taylor, R.L., 2005, The Finite ElementMethod for Solid and Structural Mechanics, Butterworth-Heinemann.19. http://www.aaoms.org/tmj.php20. http://www.qualitydentistry.com/dental/information/articulator.html21. http://www.e-strategy.ubc.ca/22. http://www.flanders-dentistry.eu/en/welkom23. http://www.yamahachi-dental.co.jp/en/products/08articulator/01correct_articulator/

http://www.aaoms.org/tmj.php

http://www.qualitydentistry.com/dental/information/articulator

http://www.e-strategy.ubc.ca/

http://www.flanders-dentistry.eu/en/welkom

http://www.yamahachi-

aspects concerning contact stresses …...64 the annals of university “dunĂrea de jos“ of...

Documents