This study aims to validate a cavity shape optimization approach for improving the debonding resistance of dental restorations by carrying out fracture tests on restored model teeth with standard and optimized cavity designs.
The bio-mimetic stress-induced material transformation (SMT) optimization method was incorporated into the finite element (FE) program ABAQUS as a user material (UMAT) subroutine. The method uses stress minimization to optimize the cavity shape of a MOD restoration in an artificial premolar with special reference to the tooth–restoration interface under occlusal loads. The mechanical performance of the optimized design was first verified through FE analysis and then compared with that of the conventional design using fracture tests on model teeth.
The SMT optimization process indicated a T-shape cavity as a more favorable design for the MOD restoration in the artificial premolar. Compared with the conventional parallel wall, or undercut design, the T-shape cavity was shown numerically to reduce the interfacial stresses by up to 69%, and experimentally to increase the mean debonding resistance of the model teeth by 23% ( p < 0.05).
Cavity shape optimization can help increase the debonding resistance of restored teeth by reducing the interfacial stresses between tooth and restoration under occlusal loads.
Secondary caries is caries that occur at the degraded margins of an existing dental restoration. Recent clinical surveys indicated that secondary caries was the most common reason for the failure and replacement of restorations , followed by fracture of the tooth. Therefore, maintaining the integrity of the tooth–restoration interface is essential in ensuring the longevity of dental restorations .
Much work has been done to explore the possible factors that lead to marginal deterioration of dental restorations. Among these, tooth–restoration interfacial stresses, produced by either shrinkage of the dental composite during light-curing or the occlusal loads during mastication, have been shown to compromise the marginal integrity of composite restorations by causing interfacial debonding . Using appropriate techniques and material properties, for example, an appropriate adhesive layer thickness or a more flexible restorative material, can help reduce shrinkage stress . Geometrical configuration of the cavity will also affect the magnitude of the interfacial stresses . Using both numerical and analytical methods, Hubsch et al. studied the interfacial shrinkage stresses of restorations with different cavity shapes. They showed that a drawn-out bevel could help reduce the stress singularity at the surface margin of the restoration. De Vree et al. compared the interfacial stresses in amalgam restorations with conventional and modified cavities under occlusal loads. Using the finite element (FE) method, they showed that the modified design resulted in fewer stress concentrations and lower stress values. In Ref. , fracture tests on human teeth with different amalgam restoration designs were conducted. The experimental results showed a relationship between the fracture strength and cavity dimensions.
Recently, modern shape optimization techniques have been applied to the design of dental restorations, including cavity preparations, bridges and implants . Combined with the powerful stress analysis capability of the FE method, shape optimization techniques can improve significantly the mechanical performance of structures with complex geometries and non-linear material properties. Based on the adaptive growth process of biological structures , Shi et al. have developed shape optimization methods in the form of stress-induced material transformation (SMT) and stress-induced volume transformation (SVT) and incorporated them into the commercial FE package ABAQUS via its user material (UMAT) subroutine. In , the shape of a MOD restoration in a premolar was optimized using the above techniques and the resulting diamond-shape cavity was shown to have greatly reduced interfacial stresses under external load.
The work presented in this paper extends the theoretical work of Shi et al. on cavity shape optimization, the aim being to validate the approach using destructive mechanical tests. As a first attempt, preclinical artificial teeth were used in the experiments to minimize the variations in tooth anatomy and material properties. The SMT method was thus applied to the optimization of a MOD cavity in a model maxillary premolar to minimize the stresses at the tooth–restoration interface. FE simulations and mechanical tests were then conducted to verify the effectiveness of the new cavity shape in increasing the resistance of the restored tooth against interfacial debonding.
Materials and methods
All the FE analyses presented in this study, including the shape optimization, were performed using the commercial FE package ABAQUS .
Shape optimization using SMT
To reduce the interfacial stresses between the tooth material and the restoration, the stress-induced material transformation (SMT) method was utilized to optimize the cavity shape . SMT is one of the simplest structural topology optimization methods that mimic the biological mineralization process in living bone: material is deposited in loaded areas while non-load-bearing material is gradually removed. This process can be simulated in a FE analysis by making Young’s modulus ( E ) of the material a function of the mechanical stress/strain, which will be non-uniform over the volume of structure being optimized. The following equation shows the changes in E between two time steps or increments:
where E i and σ i are Young’s modulus and stress at the i th increment, respectively; σ ref is a reference or threshold stress above which changes to Young’s modulus will occur; and k is a parameter that controls the rate of changes of Young’s modulus. A more detailed description of the SMT method has been presented elsewhere .
The above optimization method has been incorporated into the finite element software ABAQUS via a user material (UMAT) subroutine . During the optimization process, the UMAT subroutine is called upon repeatedly until the spatial distribution of Young’s modulus reaches a steady state.
Shape optimization of MOD cavity in a maxillary premolar
The Lava Scan ST Scanner (3M ESPE, USA) was used to obtain the geometric model of the artificial maxillary premolar. A 2-D FE model was then constructed in ABAQUS from the 3D geometric model; see Fig. 1 (a) . By St. Venant’s principle , assuming that the tooth is well supported by the surrounding bone, the exclusion of its root from the model should have little effect on the stress distribution around the restoration. Therefore, only the crown of the tooth was modeled.
The constraints on the tooth from the bones were simplified by fixing all degrees of freedom at the base of the tooth model. A standard MOD restoration was then constructed, as shown in Fig. 1 (a). The dimensions of the restoration are given in Table 1 . In total, the model contained 8051 plane-strain elements (CPE3 and CPE4 ) and 8148 nodes.
|W||D||W 1||W 2||D 1||D 2|
|Test samples (mean, SD)||1.91, 0.11||4.51, 0.23||2.86, 0.20||1.89, 0.10||3.35, 0.20||4.51, 0.19|
A compressive load was applied to the tooth through a semi-cylindrical steel loading head with an outer radius of 3 mm ( Fig. 1 (a)). A uniform pressure of 15.15 MPa was applied on the top surface of the loading head, giving a total force of 400 N, which represented the maximum occlusal force typically reported in the literature. Polymerization shrinkage of the composite was not considered.
There were three components with different materials in the FE model ( Fig. 1 (a)): the artificial tooth, composite restoration and steel loading head. All materials were assumed to be linearly elastic. The composite restoration was assumed to be fully cured without the presence of residual stresses. The mechanical properties of the three materials are shown in Table 2 . Young’s moduli of the artificial tooth and composite restoration were determined using 3-point bend tests on small beam specimens of the materials. That of the steel loading head was based on typical values from the literature. Poisson’s ratios of all the materials were assumed to be 0.3.
|Young’s modulus (GPa)||6||12||210|
Shape optimization process
The shape optimization process began with the initial model as shown in Fig. 1 (a). The first step of the process involved loading the tooth with the compressive force to create stresses. Areas of stress concentration would appear around the tooth–restoration interface and these were the areas that required reinforcement through material transformation. Thus, in the second step, the SMT process based on Eq. (1) was executed via the user material (UMAT) subroutine to transform the artificial tooth material around the restoration with overtly high stresses to the stronger restorative material which had a higher Young’s modulus. The optimized shape of the restoration was therefore expected to be bigger than the original standard shape. The SMT process took place in small increments, with the stresses, σ i , updated at the end of each increment until convergence or a steady state was achieved. The parameters in Eq. (1) were set as: σ ref = 11 MPa and k = 50. The maximum principal stress was chosen to be the stress to cause material changes since debonding was considered to be caused predominantly by a tensile stress.
Verification of optimized design by means of FEA
To verify the potential effectiveness of the optimized design in improving debonding resistance, a 2-D FE stress analysis based on the model in Fig. 1 (a) was carried out to evaluate the interfacial stresses of the restored tooth for both optimized and non-optimized cavities. The loading and boundary conditions were the same as those shown in Fig. 1 (a), and the material properties were also the same as those shown in Table 2 . A tied contact interaction was defined between the tooth and restoration surfaces to facilitate extraction of the interfacial stresses that were normal and parallel to the interface.
Verification of optimized design by means of experiment
Even though FE analysis could provide very useful numerical results for evaluating the effectiveness of the optimized design, experimental proof was still necessary. Therefore, mechanical tests were performed to compare the resistance against debonding of restored artificial model teeth with the optimized cavity against those with the standard design.
Artificial maxillary premolars were used in this seminal study to minimize the variations in geometry and material properties, as would be found with natural human teeth. 50 specimens, separated into 3 groups, were tested: 12 intact teeth (Group I), 20 teeth with a standard restoration (Group S) and 18 teeth with an optimized restoration (Group O). Group I was used as a control group.
The standard and optimized cavities were prepared by the same operator following standard clinical procedures with a high-speed handpiece and dental cutting burs. The dimensions of the cavities for both the numerical model and actual specimens are shown in Table 1 . Ethyl-2-cyanoacrylate-based superglue, used as a bonding agent between the restoration and artificial tooth, was applied thinly on the cavity walls. The cavities were then filled with a dental composite (Z100, 3M ESPE, USA) incrementally to minimize shrinkage stress and to ensure complete polymerization of the composite . A blue curing light (3M ESPE, USA) was used to cure the dental composite.
The restored samples were potted into Teflon mounting rings using an orthodontic resin (DENTSPLY International Inc., USA), as shown in Fig. 2 (a) . To ensure all the specimens have the same position and orientation relative to the load axis, an index was made using a vinyl polysiloxane impression material (3M ESPE, USA) to accurately locate the specimen within the mounting ring ( Fig. 2 (a)). The positioning index was fixed to the ring via two diametrically opposite positioners that fit into the two corresponding holes of the ring. The resin was then poured through a cutout of the index to fill the ring up to the level of the cementum–enamel junction, so as to be consistent with the boundary condition used in the FE analysis ( Fig. 1 (a)).