In a bibliographical review about finite element modellings in dentistry, FEM was described as a prevalent and useful technique in the evaluation of stress distributions in biomechanic models . During the constitution of FEM models, it was suggested that increase in the number of elements and nodes resulted in a more detailed and realistic structure . In the present study, 183,528 nodes and 863,441 elements were used in model construction, which was quite satisfactory when compared with some previous three-dimensional FEM studies [20, 21].
All elements tested in the present FEM study were assumed to be isotropic, homogeneous, and linearly elastic as in the literature [14, 15]. Due to the differences between the finite element models and the actual situation owing to material properties and boundary conditions, the results of the FEM studies should be deciphered with caution. For example, the PDL is a non-linear, viscoelastic, and anisotropic material actually . However, it occupies only a little volume in total skull models in FEM studies and its elastic properties are incomparably minor to dense structures as cortical bone and teeth. Therefore, insignificant assumptions like in the current example may form negligible effects on the outcomes derived from FEM analysis.
Previous studies have shown that the computed tomographic images are not reliable for generating detailed 3D models of teeth [13, 22, 23]. Dental age, tooth wear, metabolic calcium content, individual volumetric changes of dental sub-structures, and difference in hardness calibration techniques used to determine elastic properties of the tissues were shown among the factors that avoid the researchers to maintain a specific data for dentin, pulp, and enamel . Therefore, a single elastic modulus and Poisson ratio data that was derived from previous FEM studies was assigned to teeth in the current study which mainly focused on the effects of unilateral corticotomy of the bone model [13, 25].
Every element in nature has an elastic limit under pressure above which the deformations stop being elastic and irreversible deformation occurs. Von Mises stress is a measure of distortion energy density at a particular point in a system which is useful in ascertaining failure in ductile materials . Von Mises stress allows researchers to determine the elastic limit for any material easily; therefore, it is commonly used in computer engineering-based diagnostic experiments [9, 14, 15]. On the other hand, maximum and minimum principle stress values, defining the highest tension and highest compression, respectively, can be obtained by suitably rotating an element with no shear stress . In other words, the principle stress is the normal stress that an element will ever see under specified applied loads, like hammering a nail in one direction without causing shear forces. Principle stress could be the right criterion for the FEM studies focusing on mini-screw anchorage and orthodontic tooth movement. Based on the concerns mentioned above, von Mises criterion was chosen in the present study to investigate multi-directional 3D realistic stress occurred during corticotomy applied maxillary expansion.
Rapid maxillary expansion was pointed out to be among the most promising methods in the treatment of transversal maxillary deficiency [27,28,29]. It was stated that the main resistance regions to the midpalatal suture opening during maxillary expansion both in clefted and healthy subjects were zygomatic buttress areas in the lateral sides and the pterygoid junctions in the posterior [29, 30]. Therefore, some relieving corticotomies should be applied over these surrounding tissues for decreasing pain during maxillary expansion in adult patients .
Two corticotomies, which were performed vertically between the central incisors to the anterior nasal spine and horizontally from aperture piriformis to tuber maxilla lying under the zygomatic buttress, separated the side with transversal deficiency from the maxillary body in the present study. The corticotomy in the pterygoid junction were also added to decrease the unwanted stress on this region during RME application. Therefore, the stress relieving corticotomies recommended in the literature were simulated in the present study [15, 30, 31].
The results of the current study expressed that stress values above the corticotomy line were decreased creating a more stressed area below this line. In other words, the force generated by the RME remained in the corticotomy applied maxillary part. The stress levels of the non-corticotomy side were lower below the zygomatic buttress and higher above this region. Therefore, the over-stressed maxillary half splitted by the corticotomy could be moved easily in the transverse direction whereas the other part resisted to the RME force. Our findings were harmonious with a previous study which pointed out stress changes depending on the extent of surgical approaches .
Similar to the findings above, the stress levels calculated on the crowns and roots of the teeth on the corticotomy side were lower when compared with the teeth of the other side. The stress values decrease from crowns to the roots on both sides, which might be interpreted as buccal tipping of the posterior teeth well matching with the classic literature [1, 32].