Abstract | | |
**Aim** : In this paper, the temperature and stress distributions in an exact 3D-model of a restored maxillary second premolar tooth are obtained with finite element approach. Objective : The carious teeth need to restore with appropriate restorative materials. There are too many restorative materials which can be used instead of tooth structures; since tooth structures are being replaced, the restorative materials should be similar to original structure as could as possible **.** Materials and Methods : In the present study, a Mesial Occlusal Distal (MOD) type of restoration is chosen and applied to a sound tooth model. Four cases of restoration are investigated: two cases in which base are used under restorative materials and two cases in which base is deleted. The restorative materials are amalgam and composite and glass-inomer is used as a base material. Modeling is done in the solid works ambient by means of an exact measuring of a typical human tooth dimensions. Tooth behavior under thermal load due to consuming hot liquids is analyzed by means of a three dimensional finite element method using ANSYS software. The highest values of tensile and compressive stresses are compared with tensile and compressive strength of the tooth and restorative materials and the value of shear stress on the tooth and restoration junctions is compared with the bond strength. Also, sound tooth under the same thermal load is analyzed and the results are compared with those obtained for restored models. **Results** : Temperature and stress distributions in the tooth are calculated for each case, with a special consideration in the vicinity of pulp and restoration region. Numerical results show that in two cases with amalgam, using the base material (Glass-ionomer) under the restorative material causes to decrease the maximum temperature in the restorative teeth **.** In the stress analysis, it is seen that the principal stress has its maximum values in composite restorations.
**Conclusion** : The maximum temperatures are found in the restoration case of amalgam without base. Besides, it is found that restoration has not any influence on the stress values at DEJ, such that for all cases, these values are close to sound tooth results.
**Keywords:** Finite element, maxillary second premolar, thermal and stress analysis
**How to cite this article:** Hashemipour MA, Mohammadpour A, Nassab SG. Transient thermal and stress analysis of maxillary second premolar tooth using an exact three-dimensional model. Indian J Dent Res 2010;21:158-64 |
**How to cite this URL:** Hashemipour MA, Mohammadpour A, Nassab SG. Transient thermal and stress analysis of maxillary second premolar tooth using an exact three-dimensional model. Indian J Dent Res [serial online] 2010 [cited 2020 Jul 13];21:158-64. Available from: http://www.ijdr.in/text.asp?2010/21/2/158/66624 |
Teeth always encounter different circumstances such as hot or cold liquids, acidic material and masticatory mechanical loads. These factors lead the teeth to corrosion. Carious teeth need to be restored with appropriate restorative material. There are many kinds of restorative materials available which can be used in restoration of tooth structures. Since tooth structures are being replaced the restorative material should be similar to original structure as much as possible. Besides, restorative material has to be investigated from the point thermal stresses caused by temperature changes in oral cavity.
Most studies in the field of thermal and stress analysis of restored teeth can be divided into two different groups. In the first type, the thermal behavior of restored teeth have investigated using 2D. ^{[1],[2]} In second type teeth is investigated by axisymmetric 3D models. ^{[3],[4]} It must be mentioned that the literature does not imply internal dimensions of tooth structures as exact as needed for engineering problems.
It is clear that thermal and stress analysis on an inaccurate model lead to unreliable results, especially in the calculation of stress field. This work deals with the stress and thermal analysis of restored tooth using an exact three-dimensional model to predict the temperature and stress fields while the tooth is heating in transient condition. The crucial influence of restorative material on the value of maximum thermal stress and also on the thermal behavior of restored teeth is thoroughly explored.
Materials and Methods | | |
In this study, a maxillary second premolar tooth is considered for analysis. The reasons for choosing this type of tooth are as follows:
- This tooth has a transitional shape and a more complex geometry in contrast to the other teeth.
- This tooth is one of the posterior teeth and needs to be analyzed for stress bearable restorative material.
The selected tooth is first embedded along its long axis in casting resin in a cylindrical molding cavity. After the resin is cured, the cylinder and the tooth embedded in it were milled using an automatic polishing machine with a rough degree. Once the first section is prepared, a digital photo of this section is recorded. All subsequent sections were photographed until the entire tooth milled away. A base line was super scribed on the peripheral surface parallel to long axis of cylinder as an indicator.
In this step, recorded photos are used to construct a 3D solid model. These photos need editing and aligning. This step is done using CorelDraw software. Then the recorded photos are converted to a series of profiles that specify the bounds of each tooth structure, clearly. These profiles are digitized using software named DigitizeIt and x; y coordinates of about 800 points at each axial section are recorded. Then the exact solid model is constructed in SolidWorks ambient. [Figure 1] shows the three dimensional profile view.
In [Table 1], the tooth external dimensions in mm which are measured during the modeling procedure are compared with those given in literatures ^{[5]} and a good consistency is found.
In this study a MOD restoration is developed. ^{[6]} Usually in this type of restoration, two different types of material are used; restorative material and base material. [Figure 2] shows the tooth with restoration cavity and dimensions.
Governing Equations | | |
The problem is being considered to find out the temperature and stress fields of a restored tooth caused by transient change of oral cavity temperature. Application of heat balance and Fourier's heat conduction law for homogeneous isotropic material leads to the following partied differential equation:
In this equation, α is the thermal diffusivity. Once the thermal boundary conditions on the tooth surface and the initial temperature distribution in the tooth at time t=0 have been specified, the transient temperature variation within the tooth can be determined.
In the numerical solution of Eq. 1, the following three different types of boundary conditions are imposed:
- Constant temperature on surfaces
- Convection heat transfer between surface boundaries and surrounding
- Zero displacement on fixed surfaces
As a separate analysis with using modifying Hook's law, the thermally induced elastic stresses in the tooth at any instant can be established to take into account the presence of thermal strain.
Finite Element Model | | |
Once the 3D model is constructed with Solidwork software, it is imported into ANSYS ambient. ^{[7],[8]} At the pulpo-dentinal junction (PDJ), heat transport is done by blood and lymph circulation. Because little is known about heat transport by means of blood and lymph circulation in the pulp, it is assumed that the pulp has an overall constant temperature equal to 37 ^{o} C. Thereby, the pulp solid body can be delete.d from the whole solid model with this assumption that its surface is isothermal.
Before meshing the solid model, mechanical and thermal properties of the tooth structures and restorative material are specified. The tooth was assumed to be isotropic, homogeneous and elastic ^{[3],[4]} and its physical properties are independent with temperature. ^{[9]}
[Table 2] shows mechanical and thermal properties for the tooth and restorative material. ^{[10],[11],[12],[13],[14],[15],[16],[17]}
The problem under consideration is a coupled analysis which consists of two fields: temperature and displacement. Therefore, four degrees of freedom are needed: UX, UY, UZ and TEMP. Solid98 is used to mesh the solid model which is a well suited element for modeling irregular meshes. Mesh sensitivity is done and it is clarified that a mesh with 13727 elements is an optimum mesh regarding the accuracy and amount of calculations. In addition, with the number of elements, curved boundaries can be generated without flawing surface smoothness. [Figure 3] shows the meshed solid model.
It is clear that the temperature of oral cavity varies by consuming hot or cold liquids. Fenner *et al*, ^{[3]} have conducted an experimental study to prescribe the variation of the oral cavity temperature with time in drinking hot liquids. In that work, it was concluded that the temperature variation on the buccal surface of the premolar is small, but at the palatal surface, the tooth temperature is affected strongly by the consumed liquid temperature.
Based on the measurement ^{[3]} the oral cavity temperature which is used in the present calculations is estimated.
Since there is a finite resistance to heat transfer between the hot liquid and the surface of tooth, the local tooth surface temperature T _{s} , is not equal to the oral cavity temperature, T _{o} . This resistance is measured in terms of the surface heat transfer coefficient, h, which turned out to be an essential parameter in the mathematical model. The value of this parameter is equal to 500 W/m ^{2 o} C in the present computation. ^{[18]}
Thermal load on the coronal part of the tooth is divided into two parts: constant temperature equal to 37 ^{o} C at the buccal aspect and some portions of mesial and distal surfaces bounded by restoration cavity boundaries and CEJ; heat transfer at occlusal, palatal surfaces and remained portions of mesial and distal. Also, restoration surfaces are subjected to the same heat transfer. On the outer surfaces of the root and the pulp chamber, the temperature was held constant at 37 ^{o} C. The tooth was assumed initially at a uniform temperature of 37 ^{o} C and to be stress-free at this temperature. A small surface at the bottom of the root was assumed to be rigidly fixed, and the remaining portion of the root was left since the periodontal ligament allows limited movement of the root.
The overall time for this analysis is considered 16 seconds in which the temperature of oral cavity temperature is varied according to the profile prescribed in [Figure 8]. An implicit analysis was done and a small time step of 0.08 second was specified in the time matching process.
Results | | |
The numerical solution of governing equations which are obtained by finite element method leads to determine the temperature and stress fields in the restored tooth. Some special points that are expected to be in critical state are defined below:
- The points on the dentino-enamel junction (DEJ)
- The points on the dentino-enamel-restoration junction (DERJ)
- The points at the vicinity of the pulp
- The points at the vicinity of the base
[Figure 4], [Figure 5], [Figure 6] show the location of these points.
The models of a sound tooth and four teeth with different restorative materials were evaluated, comparatively. These restoring models are as follows:
- Restoring with a MOD restoration by amalgam over a base
- Restoring with a MOD restoration by amalgam without base
- Restoring with a MOD restoration by composite resin over a base
- Restoring with a MOD restoration by composite resin
These cases are named case 1 to 4, briefly.
Moreover, the analysis is done for the sound tooth model under the same loading.
[Figure 7] shows the temperature variation versus time at points E1 and E2. Since, the restoration doesn't have a considerable influence on the thermal behavior of these points, only the results for case1 are shown.
It is seen that temperature increases in the first time period (about three seconds) with the maximum values of 52 ^{o} C and 53.5 ^{o} C for points E2 and E1, respectively, after which it falls. [Figure 7] shows that after spending 16 seconds, the temperature reaches almost close to its initial value. [Figure 8] and [Figure 9] show changes in temperature, with time, at points P1 and P2, respectively. These points are very critical because the pulp, is a very sensitive part of teeth. These figures illustrate the same trend of temperature variation at all four cases as in sound tooth. There is a sharp temperature increase during the first eight seconds after drinking hot liquid and then temperature decreases. The maximum values of temperature and their correlated times are summarized in [Table 3].
In two cases with amalgam, using the base material (Glass-ionomer) under the restorative material causes to decrease the maximum temperature especially at point p2. These temperature decreases for points p1 and p2 are 0.7% and 2.5% when amalgam is used and are 0.2% and 0.47% in the case of using composite.
The time variation of heat flow rate to the pulp in buccal and palatal cusps are observed respectively. The maximum heat flowed to the pulp horns is equal to 0.0451 watt at the buccal side that occurred at eight seconds after the initiation of heat load. The time lapse between the application of the heat transfer and a change of 0.1 ^{o} C at the points P1 and P2 is summarized in [Table 4]. This time can be used to determine the response of the patients to the heat load. In the amalgam restorations, using base has increased the time lapse about 72% and 178% at points P1 and P2, respectively. These values for composite restoration are 7% and 30%. The maximum time lapse is happened in the sound tooth. At the point P1, the time lapse for the sound tooth is 29.6% greater than those for the Case 1 and eight percent greater than those for Case 2. This parameter at P2 for the sound tooth is 129% greater than those for Case 1 and 32% greater than those for Case 2.
[Figure 10] and [Figure 11] show the temperature variation at the points B1 and B2 with time. As it was shown in [Figure 10], these points are very close to each other. The influence of using base on temperature gradient can be observed by comparing [Figure 10] and [Figure 11] with each other. There is a considerable temperature difference between these points in the case of using base.
As it was noted before, thermal stresses are generated in the tooth as a result of thermal loading.
In all four cases and also for a sound tooth, this stress is tensile. It is noted that the stress increases sharply during the first four seconds after which it falls. The maximum stress value is happened at point P1 in case 1. Case 2 has issued the nearest stresses to those for sound tooth.
[Figure 12] and [Figure 13] show the principal stress variations at points D1 and D2 with time. As it is seen, after a very short time (about two seconds), the tensile stresses reach to their maximum value which are about 3.5 MPa in all cases. Also, it is seen that in all four cases, the thermal stress is tensile, while in sound tooth, compression stress is also introduced at seven seconds after initiation of the heat load.
[Figure 14] , [Figure 15], [Figure 16] show the variation of principal stress with time for the points located on DERJ those are C1, C2 and C3. It is seen that the principal stress has its maximum values in composite restorations, such that the maximum stress is computed at point C1 with the value of 3.6 MPa.
Besides, the tensile and compressive stresses which were studied before, the shear thermal stress is also introduced as a result of temperature change in tooth. The value of this stress is very important especially at the points on tooth-restoration junctions. The shear stress on tooth-restoration junction may violate the bond strength and causes to fracture in these junctions. Three types of bonds are present: restorative material-enamel, restorative material-dentin and base-dentin. The values of shear stresses in these junctions are compared with the bond strength and it is clarified that the bond fracture process does not happen under this thermal load. [Table 5] shows shear stress values and related bond strength. ^{[19],[20]} In the computation, shear stress is calculated on surfaces parallel to XZ and YZ coordinate planes.
Concluding Remarks | | |
The present work comprises a comparative investigation on the temperature distribution and its changes in human teeth restored in four different various cases. It involves the computation of temperature and stress distributions on the restored maxillary second premolar in the time period of drinking hot liquid. The main conclusion can be summarized as follows:
- The maximum temperatures are found in the amalgam restoration without base (Case 2).
- It is found that base protects the pulp from sudden changes in temperature such that it causes a considerable increase in time lapse between application of thermal load and a change in pulp temperature.
- The nearest temperature profiles to sound tooth are found in Case 3.
- Stress analysis shows that using base does not influence stress at the vicinity of the pulp significantly.
- In the case of using base in amalgam restoration, the amount of stress decrease at the vicinity of the pulp is about 11%.
- Restoration has not any influence on the stress values at DEJ, such that for all cases, these values are close to sound tooth results.
- Shear stress, as a result of temperature changes in tooth, is very minimal than bond strengths. Consequently, the thermal loading which is under study may not cause fracture in the bonds.
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**Correspondence Address**: Maryam Alsadat Hashemipour Member of Kerman Oral and Dental Diseases Research Center Iran
**Source of Support:** None, **Conflict of Interest:** None
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**DOI:** 10.4103/0970-9290.66624
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11], [Figure 12], [Figure 13], [Figure 14], [Figure 15], [Figure 16]
[Table 1], [Table 2], [Table 3], [Table 4], [Table 5] |