The elastic moduli of enamel and dentine

Clinical Materials 14 (1993) 35-39

The Ebtic

Moduli of Enamel and Dentine

J. S. Rees & P. H. Jacobsen Department of Conservative CF4 4XY, UK

Dentistry,

(Received 16 February

Dental

School,

University

of Wales College of Medicine,

Heath Park, Cardiff

1992; sent for revision 4 October 1992; accepted 11 November 1992)

Abstract: The finite element method was used to model an in-vitro tooth loading

system. The elastic modulus values of enamel and dentine used in the analysis were altered in order to replicate the movement of the in-vitro system. It was found that a dentine modulus of 15 GPa and an enamel modulus of 40-80 GPa gave the best replication of cuspal movement.

involved placing a steel sphere in the occlusal surface of a derooted premolar cemented to a brass platten. A load of 30 kg (294 N) was applied to the cusps of the teeth via the splhere and the horizontal deformation of both cusps was recorded by linear variable displacement transducers (LVDTs) placed against the outer surface of t’he cusps at the level of the base of the occlusal fissure (Fig. 1). The cuspal movement for an intact tooth and for teeth with various cavity designs was recorded. If this system was modelled using the FEM, the present authors reasoned that changing the modulus of enamel and dentine would result in changes in the outward movement of the cusps. Further, the values of the moduli of enamel and dentine which corresponded with the experimental data of Hood would give a good indication of the moduli of tooth substance in function.

INTRODUCTION The finite element method (FEM) is becoming an increasingly popular tool in dental research since it is well suited to modelling the stress patterns in complex composite structures such as teeth.‘-3 One of thie essential factors required for an accurate analysis is the elastic modulus of each of the component parts of the structure. If these values are not accurately known then the analysis will be a poor approximation at best. Unfortunately, there is considerable variation in the reported moduli of enamel and dentine and as a result it is difficult to know which value to choose when unldertaking a finite element analysis of a tooth or a restorative system. This problem is highlighted in Table 1) which presents a range of modulus values used in some published dental finite element analyses. Studies; on the elastic modulus of dentine are numerous and the values obtained extend over quite a wide range. However, studies on enamel are rather more scarce due to the difficulty in preparing and testing extremely small specimens. The literature on the physical properties of teeth has been reviewed by both Waters16 and Braden.17 Unfortunately, only a few new papers have been published. since these reviews. The current state of knowledge is presented in Tables 2 and 3. The aim of this study was to model the in-vitro tooth loading system of Hood.34 His method

MATERIALS AND METHODS The exact dimensions of the teeth used by Hood were not quoted, so a model was constructed from the mean dimensions of a large population sample.36 A two-dimensional plane strain model consisting of 767 eight-noded isoparametric elements was constructed (Fig. 2). The model was loaded by a point load of 147N on each cusp at the points estimated to be the contact points of the sphere in the original system. The load is shown 35

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1993

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36 Table 1. A representative sample of the modulus enamel and dentine used in FEM studies

values

-_.._----Reference

Enamel

Dentine

89.00 89.00 46.89 82.50 49.00 41.00 50.00 50.00 83.36 46.90 46.89 8BGl

IS.60 18.60 11.76 18.60 11.77 19.00 13.00 13.00 20”59 15.40 BB.76 20.98

Farah et aL4 Farah et al.’ Wright & Yettram Farah et al.’ Takahashi et al.’ Rubin et al9 Peters & Poort” de Vree et al.‘] Khera et al.‘* Morin et 01.‘~ Reesr4 Yang & Thompson1s

by the arrows in Fig. 2. The analysis was carried out on a Sun Sparcstation (Sun Microsystems Inc., Mountain View, CA) using the Nisa II (EMRC, Troy, MI) finite element analysis suite. An intact tooth and a tooth with a minimal occlusal cavity involving just the fissure system of the enamel were modelled in an attempt to replicate the in-vitro system used by Hood. The values of the cuspal movement which the authors were trying to replicate are given in Table 4. Note that the movements are not quite symmetrical. The enamel modulus was varied from 40 000 to 80000 MPa in 1000MPa increments, while the dentine modulus was varied from 12000 to 18 000 MPa in 3000 MPa increments The Poisson’s ratio for enamel was O-3 and fsr dentine 0~31.16>17 The horizontal movement of the three nodal points on the outer surface of the buccal and palatal cusps which corresponded to the area covered by the

Table 2. Published

Peyton et al.” Neumann & Di Salvo” Stanford et cdzO Stanford et al.” Craig & Peytonz2 Rensonz3 Hainesz4 Bowen & Rodriguez” Lehmanz6 Tyldeslyz7 Hannah** Renson & Braden Watts et aL3’

IIJelhod

J%Xtic ~gQ~~~~~~ (C&z)

Elastic modl.dus (GPa)

Reference

Reference

values of enamel modulus _-_-_--..-.-.-.-___l--

of

Static

Stanford

ef n1.2’ Compressaon

Craig et al.“’ Tyldeslyz7 annah2”

Compression

Molar ! L4.l-46.2 Incisor 2Of-474 78G84.2 !3”1

FkXUR

Compression

35,96

Acoustic Acoustic

765 “4 0

DJWfGC

eich et al ” Katz3’

The

impedance impedance

variousco

and for the class remolar, the ran model was 8~5-13. ments was 13~7-2103pm.

values of dentine modulus Method

Compression Compression Compression Compression Compression Compression Compression Tension Tension Flexure Compression Shear Compression

Elastic

modulus (GPa) 11% lG11-7 11.0 Il.7713% 166-18.5 12% 11.0 19.3 11.0 12.3 42.96 11.1-19.3 13.26

Fig. 1. The tooth ioading system described by Hood.‘” loading piatten; B, stainless steel sphere; C; tooth crown;

A, D,

The elastic moduli of enamel and dentine

37

Fig. 2. The finite element mesh (the right side of the figure is tbe palatal surface of the tooth),

it is interesting to note that the movements are slightly asymmetrical, with the larger movement taking place in the palatal direction. This is exactly the same as the finding of Hood34 and suggests that the present authors’ finite element model is an accurate representation of the in-vitro system. The range of cuspal movements of this finite element model, compared with the experimental data of Hood, is small. For the intact premolar, the range of movements for the FEM model was 8.5-13~ 1 pm (depending on the modulus values used), compared to Hood’s value of 11 pm. For the class I cavity the range of movements was 13~7-21..3 pm, compared to Hood’s value of 16pm. When the cuspal movement values of the present analysis are compared with the experimental results ood for the intact tooth, a dentine/enamel of

Table 4. Cuspal deformation

Intact toot.h Class 1 cavity

(pm) determined

by Hood34

Buccul movement

Palatal movement

Total

54 14

6.0 94

Il.0 16.0

modulus combination of 15 and 40 IGPa in the present analysis gives buccal and pal&al movements of 5.1 and 5.8pm respectively; total 10.9pm (case 6). This is almost identical to Hood’s data of buccal/ palatal movements of 5 and 6 pm, respectively. Considering the class I cavity, a dentine modulus of 15 GPa together with an enamel modulus of 7080 GPa gives displacement values closest to Hood’s data. Two enamel modulus values are suggested because the value of 70 GPa (case 24) gave a palatal displacement (8.8 pm) closest to ood’s data (9 km) while the value of 80 GPa (case 25) gives a buccal displacement (7.3 pm) closest to Hoods’ data (7 pm). Furthermore, the total displacement of 16pm reported by Hood is somewhere between the values obtained using an enamel modulus of 70GPa (16.3 pm) and 80 GPa (15.9 pm). Even in the intact tooth case, if it is assumed that the enamel modulus is 80GPa an modulus 15 GPa (case lo), the totad cuspal movement is 9.7,~m. Comparing this with the more accurate result predicted by the FZM when the enamel modulus was 40GPa gives an error of only 5%.

9. S. Rees, BP.p-Iv.Jacobsen

38

Table 5. Cuspal movement Case

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1.5

Den tine modulus (MPa)

(pm) for the intact tooth

(range in parentheses)

Enamel modulus (MPa)

40 000 50 000 60 000 70 000 80 000 40 000 50 000 60 000 IO 000 80 000 40 000 50 000 60 000 70 000 80 000

12000 12000 12000 12000 12000 15000 15000 15000 15000 15000 18 000 18 000 18000 18000 18000

Cuspal Movement (pm)

74 6.7 6.5 64 6.2 5% 5.6 5.4 5.3 5.2 5.1 4% 4.7 4.5 4.4

6.1 5.9 5.7 5.5 5.4. 5.1 4.9 4.7 4.6 45 44 4.2 4.1 4.0 3.9

It is also interesting to note that the dentine modulus of 15 GPa, suggested by the FEM as giving the most accurate prediction, is in the middle of the values obtained by laboratory testing. The enamel modulus value of X-8 suggested as representing a “real-life’ value is very similar to the range of values described by Craig et aL31 (Table 3). Furthermore, a similar study carried out by Sakaguchi et a1.36 claimed very good correlation between an in-vitro tooth loading system and a Tao-dimensional finite element model (Y> 0.95). The modulus values used in this investigation were 15.4 GPa for dentine and 46.9 GPa for enamel

Table 6. Cuspal movement Case

Dentine modulus (MPa)

entine

(pm) for the class I cavity (range in parentheses) -

Enamel modulus (MPa)

40 000 50 000 60 000 70 000 80 000 40 000 50 000 60 000 70 000 80 000 40 000 50 000 60 000 70 000 80 000

lll(.“lll_-_--l_ Pakztal

--

12 000 12 000 12000 12000 12000 15000 15000 15000 15000 15000 18000 18000 18000 18000 18000

not

Cuspal Movement (pm) Buecai

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

was

9.8 9.5 9.3 9.1 8.9 $“I 7.8 7.6 I..5 7.3 6.8 67 6.5 64 6.3

i9.2- 10.4‘)

j9.0-lO.Oj (8.8-9.7) (8G%5) (X.5-9.3) (7.5-8.6) (7.4-8.3) (7.2-8.1) (7G7.9) (7.O- 7.7) (6.4-74) (6.3-7-I) (6 I-69)

(60-68) (W-46) --.

abtal

11Ӥ 11.1

(Il~B-Piq (IQ”& i I-4)

4m 30.6

(1@6-I I-1)

206 20~1

(lO&l@9)

19.7

(10.2-10~7) ($1.2-9~7: (8%9”4j (8.7..9~2) (8+9.0) @la-8%) (7.8-8.3) (7”6&8,1) (7.4__7-9) (7”3_?~7) (7.&‘7.5)

19.3

104 9.4 9,2 8.9 8.8 8%

8.1 7.8 7,7 7.5 7.4

-.

21.3

17.5 _ 174 16.5 16.3

159 14.9 14.5 :i4,2 13.9 3;3 ‘y

The elastic moduli

consequently the modulus of dentine will have a greater influence over the response of a tooth to loading.

of enamel

17 18

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