M Khanala, Z Chen, Y Zheng
core, endodontic restorations, post, simulation, stress analysis
M Khanala, Z Chen, Y Zheng. Finite Element Analysis Of Mechanical Behaviour Of Dental Materials. The Internet Journal of Bioengineering. 2005 Volume 1 Number 2.
This work investigates the stress distribution of endodontically treated dental post and core of a central maxillary incisor. Three different cases are simulated through a 3D finite element analysis method — a.) Post and core both are made from titanium and are a single unit, b.) Post and core are made from titanium but are two different parts and c.) Post is made from titanium and core with epoxy resin. Different loading conditions and model shapes are considered to understand the stress distributions. The post experiences slightly higher stress when the resin core (case c) has been used but the dentin is less stressed than the other cases. The root anatomy seems to affect the stress distributions.
Post and core are often used to restore the endodontically treated tooth. Generally there are two types of posts – cast and prefabricated posts. The cast posts are made according to the shape and size of the root canal. They are customized to the profile of the residual structure of the tooth. Mostly cast posts are metallic but recently Genovese et al.  have proposed the customized composite post system made of Targis/ Vectris (TV) for endodontically treated tooth.
The customized posts are suitable for large and irregular canals where the intimate contact between canal and post can be obtained. The cast metal post systems are mould according to the root cavity . They conserve more radicular tooth structure. The major disadvantage of the cast posts is that, they may cause root fracture because of very high stiffness of metallic alloys. Some authors [1,3] have reported that, generally, the cast posts are fabricated with short length. This is probably a major reason that they are not much retentive and have greater chance of splitting the tooth [4,5].
The prefabricated posts which have standard dimensions are made of a variety of metallic and nonmetallic materials. These posts possess high strength and good retention. The drawback of such posts is the loss of tooth profile (if the prefabricated posts are larger than the available tooth structure) and higher chance of corrosion for stainless steel posts. The prefabricated composite posts provides better mechanical and aesthetic properties and reproduce better natural load transmission mechanisms because their stiffness is very close to dentine . The use of prefabricated composite posts may become critical if restoration of a wide flared root cavity is entailed because a lot of cement has to be used to fix it . And the cement causes inhomogeneties in mechanical behavior. The thickness of the cement used to fix the post influences the stress transmission to the root .
Basically there are three different shapes of the prefabricated posts – parallel, tapered, and parallel and tapered combination. The parallel posts provide good retention and produce uniform stress distribution along the post length . The disadvantage of such posts is that the presence of a sharp angle of the post at the apex causes the stress concentration . The parallel posts are suitable for a shorter root length.
The tapered posts conform to the natural root shape and canal configuration. Some of the disadvantages of the tapered posts are – wedging effect, stress concentration at the coronal portion of the root and low retentive strength . The parallel and tapered combination posts preserve the dentin at the apex and achieve sufficient retention due to the parallel design.
The ideal post system should have the same material for post and core and also the post's modulus should match with the dentin's modulus. The dissimilar alloys may create galvanic action, which leads to the corrosion of the less noble alloy . The post should stabilize the core and not weaken the root . The mismatch between stiffness of intraradicular devices and dental tissues will cause large stress concentration at the restorative materials/tooth interfaces . Therefore, the use of customized post systems made of materials whose elastic properties are very close to those of the natural tooth is preferable.
The inhomogeneous properties of the tooth material and the irregularity of the tooth contours cause a very complicated stress patterns inside the tooth . Many literatures [1,2,3,9,10] are available which use the finite element method to study the stress generation inside the endodontically treated tooth under different loading conditions. The stress distribution in the endodontically treated tooth has been analyzed with [2,5,10,11,12] and without [3,13,14,15,16] considering periodontical ligament.
Genovese et al.  proved that maximum stress values in restored tooth are rather insensitive to post types and materials. Whereas Asmussen et al.  concluded that post related factors (modulus of elasticity, diameter and length) influence the stress fields in the restored tooth. Fujihara et al.  have observed a reduction in peak tensile and shear stresses in functionally graded dental posts as compared to the stainless steel post. The direction of the functional load has greater effect than the post design on maximum stresses and displacements . The horizontal loading generates higher stresses than the vertical loading. In terms of strength, core material has greater importance than post materials and size .
Pegoretti et al.  have reported that the gold cast post and core produce the greatest stress concentration at the post dentin interface. They have observed the more critical values of the von Mises stresses inside the post and at the interface during the vertical loading and on the post surface of the labial side during oblique loading. Similarly, Lanza et al.  have also observed that the stiffer systems work against the natural function of the tooth and create zones of tension and shear both in the dentine, and at the interfaces of the luting cement and the post during oblique loading. The tooth restored with metallic alloys and carbon fiber posts possess quite high stresses in the post itself . Greater deflection and higher stresses were generated with horizontal loading than with vertical loading . Also, the loading direction had a much greater effect than dowel design on maximum stress and displacement . The von Mises and tensile stresses influence the risk of root fracture .
It is known from the literatures that the post design causes the variation in stress distributions. Also, different designs of the tooth are considered in different literatures. The researchers are interpreted their results based on their model. In this case, it is necessary to know, whether the interpretation based on one model is applicable to another model or not. With reference to this, the research aims to investigate the effects of tooth anatomies (model shapes) on stress distributions. The objective of the research is also to evaluate and compare the stress distributions during different loading conditions (bruxism, mastication and horizontal loadings) under three different post-core combinations – a.) post and core both are made from titanium and are a single unit, b.) post and core are made from titanium but are two different parts and c.) post is made from titanium and core with the composite resin.
Finite Element Modeling
It is a known fact that there is no uniformity on exact design of geometrically complicated tooth in different individuals. By considering this fact, two different three dimensional geometrical tooth models have been considered as shown in Figure 1. This is required to examine the effects of geometries in stress distributions. The geometrical dimensions of the tooth are taken as an average from the literatures [1,17] and are shown in Figure 1a. The model consists of post, core, crown, cement, dentin and ligament. Since the modeled tooth consists of a prefabricated post, a varying thickness of cement is also considered. For simplicity, the thickness of the cement is assumed to vary linearly from 0.2 to 0.1 mm from top to bottom. Due to the higher specific strength, titanium is replacing steel in endodontic post restorations . In this model also posts are modeled with titanium. The respective material properties are assigned to the modeled tooth. Both models use the same material properties as shown in Table 1.
These data are taken from the literatures. The aim of analyzing two different models is to find out the effects of root anatomies on stress distributions. The basic difference between these two models lies on the apical design of the root. In model 1 the end of the root is like half hemisphere whereas in model 2 the apical design is tapered, more natural. The three different cases analyzed are:
a.) Post and core both are made from titanium and are a single unit,
b.) Post and core are made from titanium but are two different parts and
c.) Post is made from titanium and core with the composite resin.
The geometrically complicated crown and core are modeled with AutoCAD and imported to Abaqus 6.5  whereas cement, post, dentin and ligament are modeled with Abaqus 6.5. Model 1 has 414731 elements. Due to complicated geometries of the models, free meshing technique  has been adopted and tetrahedral elements have been adopted to discrete the parts. It is suggested that the 4 node first order linear tetrahedral solid elements, C3D4, which are used for the stress analysis, should be used with fine meshes to obtain the accurate results because the constant stress tetrahedral elements exhibit slow convergence . Therefore, fine meshes are assigned to the elements. The simulations are performed on Pentium 4 personal computer of 3.0 GHz CPU speed and 1 GB memory. Due to the large number of elements, the average calculation time is 2 hours 17 minutes for model 1. To compare the effects of root shapes on the stress generation, model 1 and model 2 are remeshed with 106116 and 122206 elements, respectively, without considering the ligaments. All other parameters are same for both the models. In these cases, the average simulation time for model 1 is 6 minutes and model 2 is 17 minutes.
According to , the materials used for the modeling tooth parts are assumed to be linearly elastic, isotropic and homogeneous. The different adjacent parts are tied together with the surface to surface contact (TIE constraint in Abaqus) to simulate the tooth as a single unit. The master – slave type contact conditions are defined at the interfaces without friction. As a boundary condition no translational and rotational displacements have been allowed for the outer nodes on the bottom part (below the apical part of the post) of the model (figure 1b).
Anderson  has recorded the maximum vertical loads between 70 and 150N during chewing and swallowing a variety of foods. Similarly, maximum biting forces of 100 – 200N have been measured for incisor [20,21]. The three different cases are differentiated by different loading conditions and the loading intensities. In this research, the bruxism, mastication and two impacts have been simulated with - 100 N load acting on vertical direction, 100 N load acting on 55 ° oblique with reference to the horizontal axis and 16 N load from buccal and labial directions (Figure 1b). The applied loads have been chosen and interpreted from the literatures [1,9,10]. The loads are impact type loads, i.e. the load is applied suddenly to the loading area. Table 2 shows the magnitudes of loads during different loading conditions. Four different loads are considered to evaluate the effects of loading on the stress distributions. The impacts from the horizontal axis could not be exactly horizontal to the tooth surface because of its geometry. However, they can be assumed as the horizontal impact loadings. All the forces have been applied on the respective portion of the geometry as distributed pressure. For analyzing the stress distributions in the restored tooth, three dimensional static finite element analysis simulations have been performed.
Generally, two different stresses –von Mises stress [1,3] and maximum principal stress  are used for the stress analysis of the restored tooth. The von Mises stress is calculated by combining stresses in three dimensions, with the result compared to the tensile strength of the material loaded in one dimension. The von Mises stress is also evaluated at two different cross-sections along a horizontal axis, Figure 1b, to study the propagation and distribution of stresses during loadings. The abscissa of the plot starts from the left hand outside of the ligament. In this paper, the stress analysis graphs are referred with the mid cross-section of the models.
Post and Core as a Single Unit
Figure 2 shows the stress distribution on vertically loaded tooth, post and core (titanium), crown (ceramic) and dentin. The maximum stress distribution inside the post is observed at the interface of the post and cement on the loading side. In the dentin, the outer part in contact with ligament and where the apical part of the post resides experience the higher stress. The stress in the dentin is lower as compared to the maximum stress on the tooth.
Figure 3 shows the stress distributions on tooth, core-post, crown and dentin during oblique loading. Though the loading forces are equal on bruxism and mastication, the stresses are higher on the mastication. The maximum stresses are generated on the post at the interface of post and cement at the lower part. This stress patterns can be explained with the bending of the post inside the rigid structure. The crown is highly stressed at the place of loading. In the dentin, the stresses are higher at the interface of dentin and ligament.
Two different horizontal loading cases – loading from buccal and labial sides – are evaluated to observe the effect of loading directions on the restored tooth. The horizontal loads bend the system towards the opposite side. Figure 4 shows the stress distribution on the tooth and post-core during horizontal loading from buccal and labial sides.
Figure 5 shows the von Mises stress distribution along the mid cross-section of the model during two different types of horizontal loadings. Three different cases are evaluated for each loading condition. The stresses in the ligament are very small and sudden ascends on the curves shows the increase in stresses at the interfaces of ligament-dentin and dentin-cement-post. The outer surface of the post has maximum stresses. This is because of the change in stiffness of the parts at the interfaces and tie constrain defined at the contact between the parts.
The maximum peak stress of 15.11 MPa is observed at the post during the buccal side loading along the mid cross-section whereas the maximum peak stress is 9.75 MPa during the labial side loading. In both cases the maximum stresses are noticed on the opposite side of the loading and they are observed in the case where post and core are integrated.
Post and Core as Two Separate Units
Figure 6 shows the stress distribution on restored tooth and, post and core during vertical loading. For the vertical loading, as compared to case a, case b shows relatively small stresses though the difference between the stresses is lower. This is further verified by the stress distributions obtained from mastication and horizontal loadings. Since the results are similar, the stress patterns of these two loadings are not shown here. This suggests that if the core and post are to be used from the same material, then it is better to separate them.
Resin Core and Titanium Post
Figure 7 shows the stress distribution on resin core restored tooth, post and core, and dentin during oblique loading. Smaller stress is generated on the resin core because of its low modulus of elasticity.
Stress Comparison among Vertical, Oblique and Horizontal Loadings
Table 3 shows the maximum von Mises stress in dentin for the different cases and loading conditions. The stresses on the dentin are almost equal during different cases of bruxism and horizontal loading from labial side whereas during mastication and loading from buccal side the resin core restored tooth yields less stresses as compared to others.
Figure 8 shows the von Mises stress distribution along the mid cross-section of the restored tooth during oblique loading. The interface of the post-cement generates higher stresses. The stress patterns obtained in vertical and horizontal loadings are also similar to each other in respective loading conditions for integrated, separate and resin core – titanium post systems.
Effect of Model Shapes
Two different models of the tooth (without ligament) are simulated to investigate the effect of root shape on stress distributions. In this paper, only the stress distributions under vertical loading conditions are compared between these two models. Figure 9 shows the stress distributions on the restored tooth for two different models. Figure 10 shows the comparison of first principal stress between two different models measured at the bottom and mid cross-sections (Figure 1).
For the post and core as a single unit, the maximum stress distribution (during vertical loading) inside the post at the interface of the post and cement on the loading side is justifiable because the applied load on the crown causes the bending of the core-post. The bending of the core-post is restricted by the rigid structure of the dentin. The modulus mismatch between the crown and core causes the maximum stress at the inner corner of the crown (Figure 2). This observation agrees with Genovese et al. .
During horizontal loadings, due to the bending of the core and post, the higher stresses have been developed on the post at the interface between post and cement. This was reported by Pegoretti et al. . The significant bending of the restored tooth may be considered as a beam clamped into the cortical bone cavity. Though the applied load is very less (16 N) as compared to the other two cases (100 N), comparatively high stresses are generated on the restored tooth (Figure 4), which is similar to the observation . This proves that the loading direction and planes affect the stress distributions on the restored tooth. Typically, the horizontal loading from the buccal direction may not always be realistic. But this could happen while, for example, opening the plastic packages by tooth, holding something by tooth for a moment etc.
In the figure 5, two stress peaks are observed with almost the same intensity. It is interesting to note different stress patterns and peak stresses on these two types of horizontal loadings though the applied loads are equal. This is because of the asymmetrical model and different loading planes. In fact, due to the geometry of the tooth model, in both cases there are no perfect horizontal loading. The horizontal loading plane is 85 ° on the labial direction and 75 ° on the buccal direction with respect to abscissa. The applied pressure is normal to these planes. Hence, the loading is not perfectly horizontal on both the conditions, which causes the difference in magnitudes of the stresses.
Compared to case a (integrated post-core, Figure 2), case b (separate post-core, Figure 6) shows relatively small stresses though the difference between the stresses is lower. In this research, the titanium core has been used which may not be a common practice because of its high stiffness compared to the resin made cores. However, this explores one possibility of using post and core as separate parts to reduce the stresses if both are to be made from the same materials. Recently, it has been proposed that the ideal post-core system should have high stiffness at the coronal part and low stiffness at the apical part . In such case, the titanium core, whose stiffness (116 GPa) is comparable with the stiffness of the crown (120 GPa), can be a suitable option for the core. The same explanation of case a is valid to explain the generation of maximum stress on the post at post-cement interface.
In the resin core restored tooth (Figure 7), post has larger stresses as compared to other two cases. The applied load has been transferred to the post through the resin core. The dentin is less stressed as compared to the above two cases. In this case, the maximum stress on the dentin is 47.65 MPa whereas the maximum stresses on the other two cases are higher than 90 MPa. This suggests that the maximum of the applied stress is taken by the post and as a result the dentin is less stressed. Hence, it can be inferred that the modulus of the tooth parts affects the stress factors .
For different types of loading conditions, the maximum stresses are observed at the post – cement interfaces. The post is stiffer than other parts of the restored tooth, hence, it is natural to observe the maximum stresses on the post because it will carry larger load fractions.
During oblique loading, the interface of the post-cement generates higher stress concentrations (Figure 8) similar to . This is due to the contact conditions and stiffness difference of the post and cement, which yield the sudden fall in stress magnitudes. The stresses are higher on the post at the cement-post interface on the opposite side of the loading. The titanium post and core produces higher stresses than the resin core-titanium post system. The soft cores absorb the stresses and transmit to the post.
Compare model 1 and model 2 in Figure 9 to observe the effects of model shapes on stress distributions. In the figure, model 2 yields smaller stresses on the restored tooth as compared to model 1, though the stress patterns are almost similar. Model 2 is much similar to the natural tooth than the model 1. Therefore, it can be noted that the more realistic shape generates less stresses on the post – core of the restored tooth. The natural tooth shape generates favorable stresses for the post – core system. In other words, the tapered root design (outer design) yields less stresses in comparison with the half hemispherical root design. Figure 10 shows that when the restored tooth is loaded vertically, the load bends the tooth as a result, the compressive and tensile stresses are generated on the tooth.
The paper conducted a parametric study of the stress distribution of endodontically treated dental post and core of a central maxillary incisor. The stress distributions during different types of loading under three different cases (post-core integral with same material, post-core separate with same materials and post-core separate with different materials) have been evaluated. It has been found that post cement interface yields higher stresses during loadings. The model predicted stress distributions are helpful to define the critical stress levels. Failure is expected to start once the equivalent stress reaches a critical value. The separate post and core with same material showed relatively lower stress than with the integrated post and core. The core made from resin transferred less stress to the dentin.
Different researchers have used different models of the tooth. From this research, it has been observed that the model shapes influence the distribution of the stresses on the restored tooth though the stress patterns are similar. Hence, it has been realized that one should consider the shape of the restored tooth and its components while comparing and validating the stress patterns for different model shapes.
In future, the model will be analyzed with different contact types (among adjoining parts) to understand the effects of contact on the stress distributions and propose the contact type which mimics the natural tooth system. In reality ligament is not linear elastic (which was assumed in this simulation). Hence, the model will be simulated by assigning the visco-elastic property to the ligament. The post will be evaluated with reference to the composite material, which is designed in our lab.
The authors thank Atlantic Innovation Foundation, Canada for financial support and Dr Robert J Rogers for his assistance during simulations.
Dr. Ying Zheng Department of Chemical Engineering University of New Brunswick 15 Dineen Drive, P.O. Box 4400 Fredericton, NB Canada, E3B 5A3 Phone: 506-447-3329, Fax: 506-453-3591, E-mail: email@example.com; firstname.lastname@example.org