Numerical and experimental stress analysis of a crown-molar system.

Rusu-Casandra, Aurelia Liliana ; Iliescu, Nicolae ; Baciu, Florin 等

Abstract: The treatment of approximal carious lesions in molars and premolars is often difficult because of the significant lack of hard dental substance, thus special problems of retention, strength and pulp biology occur. The carious process can have particular developments spread on other dental surfaces. In these cases large preparations must be done resulting thus atypical cavities. In order to determine the most appropriate therapeutic schemes for the optimal retention elements, a comparative study, numerical and experimental, of the stress distribution in the crown-molar system was carried out.

Key words: stress analysis, FEM, photoelasticity, molar, crown

1. INTRODUCTION

The successful restoration of a tooth relies on the mechanical and biological capacity of the anatomical substitute to replace lost morphology and physiological functions. Treatment of the approximal carious lesions consists sometimes in large tooth preparations, resulting atypical cavities as: mesio-occlusal-distal (MOD), approximal-occlusal-vestibolar, cervico-approximal-occusal, etc. Crowns are often used for the restoration in these cases, due to their high strength and esthetic potential (Imanishi et al., 2003).

In order to achieve substantial improvement in clinical success, both numerical and experimental approaches have been used to analyze the stress distribution in a crown-molar system. The mathematical model of calculation obtained with the finite element method was validated using the photoelasticity technique that gave a fairly accurate picture of the stress variation. The study led to the correction of the restorative technique, increasing thus the stability and strength of the crown-molar system.

2. NUMERICAL CALCULUS

SOLIDWORKS software (***2011) was used to perform the finite element study. Two models, one of the molar and one of the crown, with the Poisson's ratio value applicable to photoelastie materials have been loaded similar to those of the photoelastic experiment. The finite element meshes were generated using tetrahedral elements (Huebner et al., 2001)

The contour plots of the principal stresses difference [[sigma].sub.1] - [[sigma].sub.2] for the models of the molar and crown using the finite element method are presented in Fig. 1 and Fig.2 respectively. In Fig.3 and Fig.4 are plotted the stress differences in the molar and in the crown respectively, at the interface crown-tooth.

3. PHOTOELASTIC INVESTIGATION

The stress state in the models of the tooth and the crown was experimentally investigated with the photoelastieity technique applied to plane stress. The photoelastic models were cut from an ARALDITE-D plate of 5.4mm thickness, cold molded, on which both contours of a crown and of a six-year molar with a MOD restoration were drawn increased using an X-ray.

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The experimental investigation took into consideration that the tooth and the crown have different modules of elasticity. Thus, in order to find out the stress state in the molar, the crown was simulated by a steel plate that has the value of the elastic modulus higher than the one of the photoelastic material. For the analysis of the stresses in the crown, the tooth was cut from a wooden board that has Young's modulus smaller than the modulus of elasticity of the photoelastic material.

Each of the two models was vertically loaded with the same force of 326,2N using a system of levers and weights, through a 35mm diameter plexiglass disc. The loaded models were examined in the polarized light of a circular polariscope, both in white and monochromatic light.

A strip made of same material as the two models was used to calibrate the material by means of the pure bendig method. The value of the stress photoelastic constant of the model was found to be [f.sub.[sigma]] = 2.57 MPa/fringe.

Figure 5 and Fig. 6 show the isochromatic patterns photographed for the two investigated models, the molar and the crown respectively. Isochromatics are the locus of the points in the model along which the difference in the first and second principal stress ([[sigma].sub.1] and [[sigma].sub.2]) remains the same, i.e.

[[sigma].sub.1] - [[sigma].sub.2] = N x [f.sub.[sigma]] (1)

where N represents the fringe order. Thus they are the lines which join the points with equal maximum shear stress magnitude (Paipetis, 1990). Taking into account the values of the fringe order N on the contours of the models (Fig.5 and Fig.6), in Fig.7 and Fig.8 are plotted the curves of the principal stress difference [[sigma].sub.1] - [[sigma].sub.2] for the tooth and crown respectively, at the interface crown-tooth.

4. CONCLUSIONS

The examination of the curves of variation of the principal stresses difference [[sigma].sub.1] - [[sigma].sub.2] on the contours of the two models and the comparison of the results of the finite element analysis (Fig.1 and Fig.2) with the photoelastic investigation (Fig.7 and Fig.8) led to the conclusions:

(a) Although the crown-molar assembly was loaded approximately symmetrical, due to the inner asymmetric cutting in the molar representing the dental pulp space and the shape of the crown, the loading of the tooth is not symmetrical. Thus the difference of the principal stresses has a maximum in the molar right side, at the base, in the connection area.

(b) According to the relationship (Rusu-Casandra, 2008)

[[sigma].sub.1] - [[sigma].sub.2] = 2 x [[tau].sub.max] (2)

where [[tau].sub.max] is the maximum shear stress, it can be noticed that in the above mentioned area high values of [[tau].sub.max] occur (Fig.3 and Fig.4). These stresses lead to the shearing of the interface crown-tooth creating microcraks in the layer of cement and finally to the mechanical loosening of the crown.

(c) In order to decrease the magnitude of the stresses on the contour of the molar it is recommended that during the tooth preparation the crown-molar contact area to be realized according to the occlusal geometry of the molar.

(d) No noticeable discrepancies occur between theoretical and experimental results, very small differences may be seen.

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The main conclusion of this study leads to the correction of the geometry of the tooth preparation and of the crown, thus as future work a new optimal geometry for the crown-molar assembly can be designed.

5. REFERENCES

Huebner, K.; Dewhirst, D.; Smith, D. & Byrom, T. (2001). The Finite Element Method for Engineers, Wiley-Interscience, ISBN 978-0471370789, Canada

Imanishi, A.; Nakamura, T. & Ohyama, T. (2003). 3-D Finite element analysis of all-ceramic posterior crowns. Journal of Oral Rehabilitation, Vol.30, No.8, pp. 818-822, ISSN 1365-2842

Paipetis, S. (1990). Photoelasticity in Engineering Practice, Routledge, ISBN 978-0853343639, United Kingdom

Rusu-Casandra, A. (2008). Elasticity in Engineering, Editura AGIR, ISBN 978-973-720-188-1, Bucuresti

*** (2010) Solidworks User Manual, Dassault Systemes SolidWorks Corp, Concord, MA, USA