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Finite Element Stress Analysis of the Crowns of Normal and Restored Teeth A.L. Yettram, K.W.J. Wright and H.M. Pickard J DENT RES 1976 55: 1004 DOI: 10.1177/00220345760550060201 The online version of this article can be found at: http://jdr.sagepub.com/content/55/6/1004

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Finite Element Stress Analysis of the Crowns of Normal and Restored Teeth A. L. YETTRAM, K. W. J. WRIGHT, and H. M. PICKARD Depar-tment of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, England; Department of Biomedical Engineering, Institute of Orthopaedics, University of London; and Department of Restorative -Dentistry, Royal Dental Hospital, London

Stress distributions are presented for a normal and a restored mandibular second premolar under masticatory-type forces. These were obtained using the finite element method of stress analysis applied to twodimensional models. The effect of the relative stiffness of the materials is examined in each instance.

would arise in attempting a structural model of a tooth. With the photoelastic and other materials that are conveniently available, it is virtually impossible to proportion the model stiffnesses in the correct manner. The problems associated with direct methods of measuring surface stresses in actual teeth in vivo are many and obvious because of the vitality of the tooth, its size, and difficulties of

Normal mastication generates considerable reactionary stresses in teeth and their supporting tissues. Until quite recently, the main method for determining stress values and distributions has been by means of indirect experimental techniques. For example, Hoppenstand and McConnell] used a model simulation to study the mechanical failure of Class I type amalgam restorations for two different cement bases. Mahler et a12 used a similar technique to investigate design aspects of Class II restorations. Traditional methods of experimental stress analysis, including transmission and reflection two-dimensional photoelasticity,3-5 three-dimensional photoelasticity,6,7 brittle lacquers,8 and electrical resistance strain gauge techniques9 have all been used in dental stress analysis. The structure of a human tooth and its supporting tissues is a complex assemblage of materials of various mechanical properties. The stiffnesses of some of the elements are reasonably well known, those of others are very much in doubt. Stress distribution within a structure is a function of both its shape and the distribution of stiffness within it. Because of the latter, great difficulties

access. Classical methods of mathematical stress analysis are extremely limited in their scope and are inappropriate to dental structures that are of an irregular structural form. However, the finite element method,10 a modern technique of numerical stress analysis, has the great advantage of being applicable to solids of irregular geometry and heterogeneous material properties. It is therefore ideally suited to the examination of the structural behavior of teeth. The method has already been used by Thresher and Saito" who have simulated a maxillary central incisor under a lateral load at the tip by means of triangular "plane strain" elements and wlho have presented plots of stress distribution across various sections. Farah, Craig, and Sikarskie12 have also used the finite element method, this time in its axisymmetric form, to study the stress distributions in a restored first molar under axial loading. Farah and Craig13 have presented the results of analyses of an upper central incisor with a porcelain crown, and have also examined a restored first molar.'4 The finite element method hias also been used by Farah, Hood, and Craig'5 to examine the effect of cement bases on the stresses in amalgam restorations. In this report, the finite element method is used to examine the stress distribution

Received for publication January 2, 1976. Accepted for publication April 21, 1976.

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FINITE ELEMENT STRESS ANALYSIS

Vol 55 No. 6

0

rLt 2 L

I

4

III 6 I

X(u) mm

FIG 1.-Finite element mesh of pseudo buccolingual section of crown of second mandibular premolar. D, dentin; E, enamel (directions of enamel prisms shown by lines within elements).

within a second mandibular premolar under occlusal loading. The programs have been written by WVright16 in the Algol language alnd the analyses done on the ICL 1903A computera at Brunel University. A normal tooth is considered first and compared to one of identical form whiclh includes a full crown restoration. Materials and Methods NORMAL TOOTH.-The model used for this problem was a two-dimensional pseudo buccolingual slice of a mandibular second premolar liaving an equivalent thickness of 8.76 mm (Fig 1). Although the general external shape of the tooth was taken from the work of WAheeler,17 the internal structural arrangement was averaged from sections prepared from several normal healthy teeth that liad been extracted for ortlhodontic reasons. The finite element model was restrained in both the X and Y coordinate directions at all the nodes lying on the X axis. This is of course an unnatural fixation (that is, the tootli would normally be intruded or tipped under masticatory forces); however, because the investigation is concerned with the structural behavior over tile enamel-dentin-cusp a

International Computers Ltd., Reading, Eng.

1005

stress vector scale: o

02

o-r MN/rn'

Ftc 2.-Principal stress

distribution for twopoint occlusal-type loading. Enamel assumed to be isotropic.

region, this form of restraint does not affect the overall response in this area. Published values for the modulus of elasticity of enamel vary from 9.65 GN/ meter2 (ref 18) for side material to 84.12 GN/ meter2 (ref19) for cusp material; Haines20 gives a value of Poisson's ratio equal to 0.3. Enamel is well known to be a significantly orthotropic material, which may account for the variations in modulus found by different investigators. Of all the dental tissues, dentin is the one that has been most extensively investigated. Here the range of values presented by different authors is not very wide, varying from 16.62 GN/meter2,(ref2l) down to 6.90 GN/meter2.'S Few experimental values of Poisson's ratio are available; those which are20 indicate that it is quite low. Two analyses were done for the normal tooth. In the first experiment, the enamel was assumed to be isotropic and was ascribed an intermediate value for the modulus of elasticity equal to 46.89 GN/meter2. The Poisson's ratio was made equal to 0.3 and so the modulus of rigidity was calculated from the formula G = E/2 (1 + ,u) , where E de-

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J Dent Res November-December 1976

11'72 70-33 23-U I01 0-3 GN/we GN/m5 GN/na FIG 3.-Orthotropic elastic properties of enamel element; x, direction parallel to prisms; y, direction normal to prisms. notes the modulus of elasticity, G the modulus of rigidity, and ,t the Poisson's ratio. The dentin, an isotropic tissue, was given an E value of 11.76 GN/meter2 and it too was ascribed a Poisson's ratio of 0.3. This value lhas also been used by Thresher and Saito1l and by Farah et al,12 it being a fairly common value for general engineering isotropic materials. Furthermore, it is a fact that the Poisson's ratio does not significantly affect the stress distribution in two-dimensional analyses. Forces were applied to the model at two points as seen in Figure 2. Even though the model is not strictly a buccolingual slice of a normal tooth, this form of loading was devised to simulate the cleavage-type action of active centric occlusion. (In a normal tooth, the points of intercusp contact on the buccal and lingual cusps would not occur in the same buccolingual plane.) Figure 2 also shows the distribution of the principal stress in the composite structure and gives a clear physical appreciation of how the loads flow through the tooth. In this figure, the magnitudes of the maximum and minimum principal stresses are represented by the lengtlhs of the lines and the directions in wlhich they act by their respective orientation. Thick lines represent compressive stresses and tlhin lines tensile stresses. Because of the scale chosen to represent the stress magnitude, values below 16 KN/meter2 are not plotted. The enamel in the second experiment was ascribed orthotropic mechanical properties;

stress vector scale:

0

0-2

0-4MNm-

Fic 4.-Principal stress distribution for twopoint occlusal-type loading. Enamel assumed to be orthotropic.

the long axis of the prisms being taken as the primary or stiffer material direction. The line in the center of the enamel elements in Figure 1 indicates the direction assumed for the prisms in that particular element. Although the mechanical properties of the dentin were maintained at the values used in the previous experiment, the properties ascribed to the enamel were as shown in Figure 3. The elastic moduli have a ratio of 3:1, and so, to obey the Maxwell-Betti reciprocal relationships, the Poisson's ratios were similarly proportioned. Although the primary Poisson's ratio was set at 0.3, the value of Gx, was obtained by halving the smaller of the moduli of elasticity. In the first instance, the structure was loaded in a manner identical to that of the previous experiment, that is, two-point loads simulating intercuspation. This arrangement is seen in Figure 4 which shows the resulting principal stress distribution occurring in the tooth structure. Hence, by comparing Figures 9 and 4, the effect of enamel anisotropy can be ascertained.

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1007

(a) stress vector scale:

o

02

04

MN/rn

FIG 5.-Principal stress distribution for singlepoint occlusal-type loading. Enamel assumed to be orthotropic.

A final experiment was conducted on this model structure with the mechanical properties of the tissues remaining at the values used in the previous test. However, in this instance, the structure was only subjected to a single-point load which was applied to the lingual side of the buccal cusp of the "tooth" as seen in Figure 5. This figure also depicts the resulting principal stress distribution in the structure under this particular loading regime.

Results and Discussion The results of all three aforementioned experiments clearly demonstrate the role played by the enamel in reacting to the forces of mastication. Even in the first experiment where the enamel was assumed to be isotropic, its greater stiffness over that of the dentin enabled it to react to the larger proportion of the applied loads. (This result agrees with the experimental observation made by Lehman and Mleyer6 who noticed this behavior in their work with threedimensional tootlh models composed of an epoxy resin-silicone rubber combination.

FIG 6.-Flow of masticatory forces through enamel dentin tooth structure. a, two-point oc-

cloisal loading; b, single-point occlusal loading. Their enamel-dentin material simulation that lhad a modulus of elasticity ratio of 6:1 was very similar to that used in the finite element isotropic experiment.) Hence, masticatory forces tend to "flow" around the enamel cap

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as seen in Figure 6, although the dentin core remains relatively lightly stressed. This in fact is seen to be the cause for all three experiments for isotropic or orthotropic enamel under single- or two-point loading. As the modular ratio between the enamel and the dentin decreases, the distribution of force between the two components becomes less unequal and the tendency for stress to concentrate at the amelocemental junction is reduced. This effect could be brought about by considering the dentin to have a modulus of elasticity at the upper end of its accepted range. Alternatively, if the loading on the tooth is low, then the effective modulus for the enamel would be lower than that used here. Neumann and DiSalvo,22 for example, showed that enamel behaved as a nonlinear material whose stiffness increased with load as a result of the decrease in the fluid content with increase in pressure. Consequently, the enamel near the amelocemental junction is very highly stressed because the reacted forces have to flow into and through this thin wedge of tissue for them to be transmitted into the root of the tooth and subsequently into the supporting alveolus. It is therefore evident that restorations inserted into the cervical region of the teetlh can be subjected to high compressive stresses even though these areas are not susceptible to the direct contact stresses of mastication. Indeed, it may be that these

hiigli stresses are responsible for the pain often experienced by patients who have received cervically placed restorations. Another important feature demonstrated by the experiments is the relatively high tensile stresses that are generated in the fissure under masticatory-type loading. AltIloughl these stresses do not seem to be as large in magnitude as those induced near the amelocemental junction, they could lhave far me- e seriouis consequJences. It can be seen from Figures 2 and 4 that these tensile stresses would tend to pull the enamel prisms apart in this region and may tlhereby assist the attack of caries in the fissures of premolar and molar teeth once the chemical (lemineralization of the enamel has been initiated. Th-e distraction of the cusps would also tend to open up the margins of any restoration placed between the two occlusal contact points. Any increase in the marginal ci-evice would obviously encourage the seepage of cariogenic material into these spaces

FIG 7.-Finite element mesh of pseudo buccolingual section of mandibular second premolar containing full cast gold crown restoration. D, dentin; C, crown.

and consequently may assist in the breakdown of cavity margins placed in this area. TOOTH XWITH FULL CROWN RESTORATION.-

The removal of the enamel from a tooth and its subsequent replacement with a full cast gold crown is a type of restoration often used when large areas of the crown are affected by caries or by fracture. The mechanical lbelavior of this type of restoration will be compared with the response presented earlier for a normal tooth under identical loading conditions. A scaled computer plot of the full crown model is shown in Figure 7 and its likeness in form to the normal tooth model can easily be seen on comparison witIh Figure 1. The occlusal surface of the fuill crown was made identical to that of the original normal tooth surface. It was designed having 50 axial wall angles and rounded shoulder-type cervical margins. The tlhickness of the gold restoration at the cervical margins was approximately 1.25 mm, a somewlhat severe intrusion into the walls of the pulpal cavity. For the two experiments conducted on this model, botlh the gold alloy and the (lentin were asstumed to l)e isotropic materials. The dentin was ascribed the same valuie for its elastic modulus as was used in the previous case, that is, E= 11.76 GN/ meter' wlhereas the gold was given the value

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FINITE ELEMENT STRESS ANALYSIS

Vol 55 No. 6

stress vector scale:

0

0-2

o04 MN/m-

stress vector scale:

a

1009

0.2 O.4MN/Sa

FIG 8.-Principal stress distribution for twopoint occlusal-type loading; cast full gold crown restoration.

FIG 9.-Principal stress distribution for singlepoint occlusal-type loading; cast full gold crown restoration.

of E= 77.91 GN/meter2. Both materials were given the value of 0.3 for their Poisson's ratio, and their respective shear moduli were sub)sequently calculated using the familiar formula G E/2 (I + p). Forces were applied to the model in the first experiment at two points, the same as for the previous problem simulating active centric occlusion. Figure 8 slhows the resulting principal stress plot under this system of loading. In the second experiment with this model, a "masticatory" load was applied at a single point as before and as slhown in Figure 9. This figure also shows the principal stress distibution resulting from this tipping-type

ing (remarkably well) the role of an improvised enamel. This is perhaps hardly surprising as the stiffness of the gold alloy is practically the same as that of the enamel that it is replacing. To demonstrate the significance of the likeness in the stiffness of the gold and that of the enamel, a further experiment was performed. For this test, the same geometric crown structure under identical loading and support conditions was used. However, for this case the full crown restoration was ascribed a stiffness [approximating that of poly (methyl methacrylate) ] equal to half the value ascribed to that of the dentin core. Figure 10 shows the resulting principal stress distribution for this structure. It can be seen from this figure that the less-stiff crown restoration is not bypassing the occlusal loads around the dentin above the pulp chamber as did the gold crown restoration shown in Figure 8. Instead, more of the load is flowing through the coronal dentin which is consequently more highly stressed than it was with either the gold restoration or with the natural enamel crown.

loading. The principal stress distribution shown in Figtsre 8 for the full cast gold crown restoration under the centric occlusal-type loading is, in many respects, similar to the plot of

Figture 2 for the enamel-dentin structure under identical loading conditions. In both instances, the occlusally applied loads are transmitted around the very much stiffer gold or enamel cap material as illustrated earlier in Figure 6. Hence, the gold crown is play-

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stress vector scale:,

0

02

0-4

MN/ma

FIG 10.-Principal stress distribution for twopoint occlusal-type loading; acrylic resin full crown restoration.

Conclusions The major difference between the cast gold crown restoration and the enamel of the normal tooth structure lies in the design of the gold crown's proximal margins. Whereas the enamel thins down to feather edges, the gold crown model stands on two broad legs over the dentin core. In fact, the design of the full gold crown restoration in this region becomes very critical when the single-point tipping-type load is applied to the lingual aspect of the buccal cusp. It can be seen from Figure 9 that this loading condition is trying to rotate the gold crown in an anticlockwise sense off the dentin core. This is indicated by the high compressive stresses generated at right angles to the shoulder at the proximal margin in the buccal leg of the crown and by the correspondingly high tensile stresses in the lingual leg. The reason why these high stresses are induced in this model is that the finite element analysis technique used here rigidly connects the gold to the dentin at all the nodes along the materials' interface. In the actual tooth, however, if either the tensile or compressive strength of the cemented joint between the

J Dent Res November-December 1976 gold crown and the dentin were below the stress level generated, the margin would of course fail and would consequently allow the ingress of unwanted material. Although studies concerned with the strength of dental cements have been reported (see, for instance, the work of Peyton et a123), no work has been traced that deals with the tensile or compressive strength of complete cemented joints. Obviously, failure of a cemented joint will occur along the weakest path. This could possibly be at the mechanical bond at the gold-cement interface, or at the dentin-cement interface, or alternatively tlhrough the body of the cement. Consequently, investigations concerned with dental cements should be done on complete joints, as well as on the cements themselves. Because the tipping-type load is trying to rotate the gold crown off the dentin core, it would be beneficial from the mechanical point of view if the contour of the occlusal suirface of the crown was flattened, and this is recognized in clinical practice. On a largely empirical basis, the cuspal slopes of crowns made flatter than the anatomic are forms they replace so as to reduce this turning moment an(l also to lessen lateral

freqiuently

periodontal stress. Although the two-dimensional model undloultedlv accentuates the rotational effect, it is evid(ent that the (lesign of the cervical margins of crown restorations, the stabilizing height andl the taper of the axial walls, and the forms of the occlusal surface must be carefully considered. The examination of these various aspects of restoration design presents a problem to which the finite element method is particularly well suited.

References 1. HOPPENSTAND, D.C., and MCCONNELL, D.:

Mechanical Failure of Amalgam Restorations with Zinc Phosphate and Zinc Oxide-Eugenol Cement Bases, J Dent Res 39: 899-905, 1960. 2. MAHLER, D.B.; TERKLA, L.G.; and JOHNSON, L.N.: Evaluation of Techniques for Analyzing Cavity Design for Amalgam Restorations, J Dent Res 40: 497-503, 1961. 3. MAHLER, D.B., and PEYTON, F.A.: Photoelasticity as a Research Technique for Analyzing Stresses in Dental Structures, J Dent Res 34: 831-838, 1955. 4. CRAIG, R.G.; EL-EBRASHI, M.K.; LEPEAK, P.J.; and PEYTON, F.A.: Experimental Stress AnalPart 1. Twoysis of Dental Restorations: Dimensional Photoelastic Stress Analysis of Inlays, J Prosthet Dent 17: 277-291, 1967.

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Vol 55 No. 6 5. EL-EBRASHI, M.K.; CRAIG, R.G.; and PEYTON, F.: Experimental Stress Analysis of Dental Restorations: Part 3. The Concept of the Geometry of Proximal Margins: Part 4. The Concept of Parallelism of Axial Walls. J Prosthet Dent 22: 335-353, 1969. 6. LEHMAN, M.L., and MEYER, M.L.: Relationship of Dental Caries and Stress: Concentrations in Teeth as Revealed by Photoelastic Tests, J Dent Res 45: 1706-1714, 1966.

7. JOHNSON, E.W.; CASTALDI, C.R.; GAU, D.J.;

8.

9.

10.

I1. 12.

13.

and WYSOCKI, G.P.: Stress Pattern Variations in Operatively Prepared Human Teeth, Studied by Three-Dimensional Photoelasticity, J Dent Res 47: 548-558, 1968. SHARRY, J.J.; ASKEw, H.C.; and HOYER, H.: Influence of Artificial Tooth Forms on Bone Deformation Beneath Complete Dentures, J Dent Res 39: 253-266, 1960. TILLITSON, E.W.; CRAIG, R.G.; FARAH, J.W.; and PEYTON, F.A.: Experimental Stress Analvsis of Dental Restorations: Part 8. Surface Strains on Gold and Chromium Fixed Partial Dentures, J Prosthet Dent 24: 174-180, 1970. ZIENKIEWICZ, O.C.: The Finite Element Method in Engineering Science, New York: McGraw-Hill, 1971. THRESHER, R.W., and SAITO, G.E.: Stress Anialysis of Human Teeth, J Biomech 6: 443449, 1973. FARAH, J.W.; CRAIG, R.G.; and SIKARSKIE, D.L.: Photoelastic and Finite Element Stress Analvsis of a Restored Axisymmetric First Molar, J Biomech 6: 511-520, 1973. FARAH, J.W., an1d CRAIG, R.G.: Distribution of Stress in Porcelain-Fused-to-Metal and

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Porcelain Jacket Crowns, J Dent Res 54: 255261, 1975. 14. FARAH, J.W., and CRAIG, R.G.: Finite Element Stress Analysis of a Restored Axisymmetric First Molar, J Dent Res 53: 859-866, 1974. 15. FARAH, J.W.; HOOD, J.A.A.; and CRAIG, R.G.: Effects of Cement Bases on the Stresses in 16.

17. 18.

19.

20. 21.

22. 23.

Amalgam Restorations, J Dent Res 54: 10-15, 1975. WRIGHT, K.W.J.: On the Mechanical Behaviour of Human Tooth Structures: An Application of the Finite Element Method of Stress Analysis, PhD thesis, Brunel University, Uxbridge, Eng. 1975. WHEELER, R.C.: Tooth Form: A Manual on Drawing and Carving, Philadelphia: W. B. Saunders, 1939. STANFORD, J.W.; WEIGEL, K.V.; PAFFENBARGER, C.; and SWEENEY, W.T.: Compressive Properties of Hard Tooth Tissues and Some Restorative Materials, JADA 60: 746-756, 1960. CRAIG, R.G.; PEYTON, F.A.; and JOHNSON, D.W.: Compressive Properties of Enamel, Dental Cements and Gold, J Dent Res 40: 936-945, 1961. HAINES, D.J.: Physical Properties of Human Tooth Enamel and Enamel Sheath Material Under Load, J Biomech 1: 117-125, 1968. CRAIG, R.G., and PEYTON, F.A.: Elastic and Mechanical Properties of Human Dentin, J Dent Res 37: 710-718, 1958. NEUMANN, H.H., and DISALVO, N.A.: Compiessioni of TeeLh Under the Load of Chewinig, J Dent Res 36: 286-290, 1957. PEYTON, F.A.: Restorative Dental Materiats, 2nd ed, Lonidon: H. Kimpton, 1964.

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Finite element stress analysis of the crowns of normal and restored teeth.

Stress distributions are presented for a normal and a restored mandibular second premolar under masticatory-type forces. These were obtained using the...
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