2.1

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.

2.2

Shape optimization of MOD cavity in a maxillary premolar

2.2.1

FE model

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.

Cavity shape optimization: (a) 2-D FE model, (b) results—Young’s modulus distribution indicating regions (red) that require reinforcement, and (c) optimized cavity shape with key positions (A–H) marked along the interface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

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.

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.

2.2.2

Material properties

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.

2.2.3

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.

2.3

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.

2.4

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.

2.4.1

Sample preparation

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)).

Compressive test: (a) mounting of a test specimen (A—resin, B—nylon mounting ring, C—model tooth specimen, D—positioning index, E—cutout for pouring resin and F—positioners) and (b) clamping jig and loading head.

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