# Effect of Graded Mechanical Properties on Stress Distribution in Dental Restorations

Léa Guérandelle, Nathalie Brulat-Bouchard, Yannick Tillier

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Léa Guérandelle, Nathalie Brulat-Bouchard, Yannick Tillier. Effect of Graded Mechanical Properties on Stress Distribution in Dental Restorations. Multidisciplinary Biomechanics Journal, 2025, 50th congress of the.., 10.46298/mbj.16136. hal-05191688v2

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# Effect of Graded Mechanical Properties on Stress Distribution in Dental Restorations

Léa Guérandelle<sup>a*</sup>, Nathalie Brulat-Bouchard<sup>a, b</sup>, Yannick Tillier<sup>a</sup>

Received date: 05/04/2025

Accepted date: 27/06/2025

Publication date: 27/10/2025

**Keywords:** mechanical properties of teeth, finite element model, gradient

© 2025 The Authors
Licence CC-BY 4.0
Published by Société de Biomécanique

## 1. Introduction

Teeth are composed of several tissues with different mechanical properties, combining stiffness, strength and elasticity, which contribute to their longevity (Magne & Belser, 2022). Experimental studies have shown that enamel and dentin are graded materials, particularly in terms of Young's modulus. In enamel, Young's modulus decreases from approximately 100 GPa at the surface to 70 GPa at the dentin interface, while in dentin it decreases from approximately 20 GPa at the enamel junction to 15 GPa near the pulp (Andrejovská et al., 2023).

Restorative dentistry aims to reproduce the original tooth as closely as possible, which requires an understanding of these heterogeneous properties. The use of homogeneous materials in restorations can lead to stress concentration at the interfaces, increasing the risk of failure. Finite element modelling is widely used to assess mechanical behaviour, although most models assume isotropic and homogeneous tissues and restorative materials (Ouldyerou et al., 2023; Houdaifa et al., 2022). This study investigates how the incorporation of graded material properties affects stress distribution and whether it is interesting to develop functionally graded restoration materials.

## 2. Methods

A simplified molar tooth model was created using SolidWorks® 2023. It has four cusps, two roots, and

mesio-distal and bucco-lingual symmetry. The enamel, dentin, pulp and a class I restoration are also simplified. A 100 µm thick layer of adhesive was added at the tooth/restoration interface.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Different components and Young's modulus gradient of the tooth.</figcaption></figure>

Credit: Léa Guérandelle.

Each geometry was imported into Forge® NxT 4.0 software for 3D finite element simulation (~1.3 M elements). The Young's modulus gradient in enamel and

<footnote><sup>a</sup> Mines Paris, PSL University, Centre for material forming (CEMEF), UMR7635 CNRS, 06904 Sophia Antipolis, France</footnote>

<footnote><sup>b</sup> Université Côte d'Azur, UFR d'Odontologie, Université Nice Sophia Antipolis, Nice, France</footnote>

<footnote>* Corresponding author: lea.guerandelle@minesparis.psl.eu</footnote>
dentin is defined using a distance field, allowing a linear gradual change in Young's modulus (Fig. 1.). For the restoration part, when a gradient is applied, it depends on a single axis, allowing the creation of graded horizontal layers, like what a dentist might do in a direct restoration. Several types of behaviour were tested, covering the properties of the main dental materials available for indirect restoration (ceramics, composites resins). The tooth is fixed by its roots in a rigid, non-deformable part. A static load of 600 N is applied vertically with a spherical indenter (8 mm diameter). The load was applied through a food bowl to simulate chewing soft food. The bolus/tooth contact and the contact between adhesive layer, tooth and restoration are frictional (μ = 0.3) or sticking, while the bolus/indenter contact is sliding. All materials are assumed to be elastic and isotropic.

<table><caption>**Table 1:** Materials properties (*homogeneous, **min/max when gradient).**</caption><thead><tr><th>Material</th><th>Young's Modulus (GPa)</th><th>Poisson's ratio</th><th>Ref.</th></tr></thead><tbody><tr><td>Enamel</td><td>84.1*<br>70 to 100**</td><td>0.33</td><td>Ouldyerou et al., 2023</td></tr><tr><td>Dentin</td><td>18.6*<br>15 to 20**</td><td>0.31</td><td>Ouldyerou et al., 2023</td></tr><tr><td>Pulp</td><td>0.002</td><td>0.45</td><td>Houdaifa et al., 2022</td></tr><tr><td>Food bolus</td><td>0.1</td><td>0.3</td><td>Fouquet et al., 2025</td></tr><tr><td>Adhesive</td><td>2</td><td>0.3</td><td>/</td></tr><tr><td>Restoration</td><td>40*<br>10 to 40**</td><td>0.3</td><td>/</td></tr></tbody></table>

Credit: Léa Guérandelle.

The pulp and adhesive layer are homogeneous, while the enamel, dentin and restoration may be homogeneous or have a Young's modulus gradient. There are four configurations: (i) homogeneous tooth and homogeneous restoration, (ii) homogeneous tooth and heterogeneous restoration, (iii) heterogeneous tooth and homogeneous restoration and (iv) heterogeneous tooth and heterogeneous restoration. Relevant material properties are given in Table 1.

### 3. Results and discussion

The aim is to compare homogeneous and graded restorations on both homogeneous and graded teeth. Stress

is evaluated using the 1st principal stress, as enamel, dentin, and adhesive layers are more prone to tensile failure. In a homogeneous tooth, a homogeneous restoration causes stress concentrations at the occlusal surface of the enamel due to the lower modulus of the restoration and on the lower surface of the restoration due to the difference in modulus with the dentin. In a graded tooth, a homogeneous restoration induces a higher stress concentration on the enamel surface, but a lower one in the part of the restoration in contact with the dentin, because the difference in modulus is greater with enamel than with dentin. A graded restoration in a homogeneous tooth induces a stress concentration similar to the previous case because the difference is also smaller than in the purely homogeneous case. More stress is absorbed by the graded restoration. A graded restoration in a graded tooth is the case where the stress in the restoration is the lowest. This can be explained by the combination of the gradient in the restoration and the tooth, which limits the difference in modulus and increases the stress absorption capacity of the restoration.

### 4. Conclusions

This study shows the effect of the property gradient in the tooth and restoration on the amplitude and distribution of stress. There is a difference between homogeneous and graded teeth, regardless of the type of restoration. It seems appropriate to consider the graded properties of the tooth to obtain stress distributions that are closer to reality. Our results show that creating a restoration with graded properties leads to lower stress levels in both the tooth and the restoration. Integrating the gradient in the tooth into future finite element modelling, along with the development of restorative materials with graded properties, has the potential to significantly enhance current restorative dentistry. This approach would effectively limit the risk of damage to healthy tissue or the restoration itself. In a second phase, it could be interesting to also take into account the combined impact of the periodontal ligament and the bone (cortical and cancellous).

### Acknowledgements

The authors would like to thank Dr José R. Alves Z. and Patrice Lasne of Transvalor S.A. for providing the computer resources needed to create the gradient.
**Conflict of Interest Statement**

None.

**Contributor Roles**

LG: Conceptualization, Methodology, Writing original draft; NB: Writing-review & editing; YT: Writing-review & editing.

**Funding**

This work was supported by the CMADENT ANR 22-CE51-0017 of the French National Research Agency (ANR).

**References**

Magne, P., & Belser, U. (2022). *Dentisterie restauratrice biomimétique*. Quintessence publishing.

Andrejovská, J., Petruš, O., Medveď, D., Vojtko, M., Riznič, M., Kizek, P., & Dusza, J. (2023). Hardness and indentation modulus of human enamel and dentin. *Surface and Interface Analysis*, 55(4), 270–278.
https://doi.org/10.1002/sia.7187

Ouldyerou, A., Mehboob, H., Mehboob, A., Merdji, A., Aminallah, L., Mukdadi, O. M., Barsoum, I., & Junaedi, H. (2023). Biomechanical performance of resin composite on dental tissue restoration: A finite element analysis. *PLOS ONE*, 18(12), e0295582.
https://doi.org/10.1371/journal.pone.0295582

Houdaifa, R., Alzoubi, H., Jamous, I., Houdaifa, R., Alzoubi, H., & Jamous, I. (2022). Three-dimensional finite element analysis of worn molars with prosthetic crowns and onlays made of various materials. *Cureus*, 14(10).
https://doi.org/10.7759/cureus.30240

Fouquet, V., Larsen, N., Stchepinsky, A.-C., Vennat, E., Benoit, A., & Tapie, L. (2024). A parametrical finite element analysis for functionally graded material overlay restoration. *Journal of the Mechanical Behavior of Biomedical Materials*, 152, 106409.
https://doi.org/10.1016/j.jmbbm.2024.106409