ReviewNonlinear finite element analyses: Advances and challenges in dental applications
Introduction
The finite element method (FEM) involves a series of computational procedures to calculate the stress and strain in each element, which performs a model solution. Such a structural analysis allows the determination of stress and strain resulting from external force, pressure, thermal change, and other factors. This method is extremely useful for indicating mechanical aspects of biomaterials and human tissues that can hardly be measured in vivo. The results obtained can then be studied using visualization software within the FEM environment to view a variety of parameters, and to fully identify implications of the analysis.
Linear static models have been employed extensively in FEM studies. A constant elastic modulus that represents the linear stress–strain relationship of a material is input into a program. Linear analyses are valid if the structure exhibits a linear stress–strain relationship up to a stress level known as the proportional limit, and all the volumes are bonded as one unit. However, the validity of a linear static analysis may be questionable when the study objectives are to explore more realistic situations that are usually encountered in the intra-oral environment. The realistic situations will give rise to nonlinearities, which can be grouped into the following principal categories: (1) Material nonlinearities; (2) Changing interrelation of objects; and (3) Geometric nonlinearities. Material nonlinearities cause the stiffness of a structure to change with different load levels, and are expressed in a program as nonlinear stress–strain relationships. Many factors that influence stress–strain properties include the time- and path-dependent load history in elastoplastic response, environmental conditions such as temperature, and the amount of time that a load is applied in creep response. The dynamic behavior of the periodontal ligament (PDL) and oral soft tissues may well be simulated with this theory. Changing interrelation of objects is also a structural behavior that is commonly seen intra-orally in tooth-to-tooth and material-to-tissue contacts. Geometric nonlinearities are characterized by large deformations and/or rotations, and occasionally seen in dental materials such as the dental wires. Changing geometric configuration can cause the structure to respond nonlinearly. Often, the stiffness of a structure increases as the deflection increases.
Nonlinear analysis has become an increasingly powerful approach to predict stress and strain within structures in a realistic situation that cannot be solved by a linear static model. The key elements required for the design and appropriate utilization of this methodology should be fully clarified. This paper reviews the recent developments in the application of the nonlinear FEM in dentistry under the following headings: (1) nonlinear simulation of periodontal ligament property, (2) plastic and viscoelastic behaviors in materials, (3) tooth-to-tooth contact analysis, (4) contact analyses in implant structures and (5) interfacial stress in restorations.
Section snippets
Nonlinear simulation of periodontal ligament property
The force–displacement relationship of a tooth under applied force is represented not by a linearity but by a nonlinear curve.1 To assume the linear elasticity as a constant material property of the periodontal ligament may lead to an erroneous solution. If an intermediate constant is used as the elastic modulus of PDL, the tooth displacement would be too small under a relatively low bite force, and too high under a high load. The use of the linear static model is suitable only for expression
Plastic and viscoelastic behaviors in materials
Plasticity is characterized by an unrecoverable and path-dependent phenomenon, and begins when stress exceeds the material's yield point. Large deformation and large strain geometric nonlinearities are often associated with a plastic material response. Creep is the time-dependent material nonlinearity in which the material continues to deform under a constant load. If a displacement is imposed, the reaction force and stress will disappear over time, which in turn causes stress relaxation.
Tooth-to-tooth contact analysis
Sliding and friction phenomena critically affect the stress and strain created on the contact surfaces between teeth. The problems can be partially solved by performing contact analysis, which is highly nonlinear and difficult to solve due to the following major issues. First, the contact regions are unknown until a sequence of the problem has been solved. Depending upon the load, material, and environmental factors, the surfaces can move in and out of contact with each other in a highly
Contact analysis in implant structures
Contact and friction play important roles in the mechanical behavior of implant prostheses. The contact elements may be defined in two or more of the following components: implant bodies, threads, abutments, abutment screws, cylinders, and bone. The contact zones in the FEM models transfer only pressure and tangential frictional forces, while tension is not transferred. Since the screw joint integrity at the implant–abutment joint is essential for long-term success of implant restoration,
Interfacial stress in restorations
Stress analysis at the tooth–restoration complex has been performed to predict the failure risk at the interface as well as within the bonded tooth structures. In a linear static analysis, an interfacial surface between individual structures with different elastic properties shares the same node, representing the perfect bond. This conventional approach occasionally leads to erroneous interpretation in the FEM results.
Fig. 2 shows a simplified 2D FEM model of an endodontically treated
Conclusions
The nonlinear FE analysis has become an increasingly powerful approach to predict stress and strain within structures in a realistic situation that cannot be solved by conventional linear static models. The nonlinear simulation of the PDL properties enhances a precise estimation of the stress and strain with wide range of tooth movement. The determination of the elastic, plastic, and viscoelastic material properties of a target material often requires mechanical testing prior to FEM analyses.
Acknowledgement
This study was supported by KAKENHI 20592307 (to N.W.) and High-Tech Research Project 2005–2009 (to Iwate Medical University), both from JSPS/MEXT.
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