Abstract
The mathematical modeling of a stress-strain state of the viscoelastic periodontal membrane is carried out. Internal and external surfaces of the periodontal ligament are described by a symmetrical two-sheeted hyperboloid. Tooth root is assumed to be a rigid body. Displacements of points on the internal surface of the periodontal ligament coincide with the displacements of the corresponding points of the external surface of the tooth root. The relationships between the displacements and strains for periodontal ligaments are formulated as an assumption that the periodontal tissue approaches to incompressible materials. Viscoelastic constitutive law with a fractional exponential kernel for periodontal ligament was used. The equations of motion for the periodontal ligament relative to translational displacements and rotation angles of its points are derived. In the particular case the vertical translational motion of the tooth root, as well as corresponding displacements are analyzed. Constants of the fractional viscoelastic function were assessed on the basis of the experimental data about the behavior of the periodontal ligament. The obtained results can be used to determine a load for orthodontic tooth movement corresponding to the optimal stresses, as well as to simulate bone remodeling on the basis of changes of stresses and strains in the periodontal ligament during orthodontic movement.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Masella, R.S., Meister, M.: Current concepts in the biology of orthodontic tooth movement. Am. J. Orthod. Dentofac. Orthop. 129, 458–468 (2006)
Wise, G.E., King, G.J.: Mechanisms of tooth eruption and orthodontic tooth movement. J. Dent. Res. 87, 414–434 (2008)
Fill, T.S., Toogood, R.W., Major, P.W., Carey, J.P.: Analytically determined mechanical properties of, and models for the periodontal ligament: Critical review of literature. J. Biomech. 45, 9–16 (2012)
Komatsu, K.: Mechanical strength and viscoelastic response of the periodontal ligament in relation to structure. J. Dent. Biomech. 1, 1–18 (2010)
Qian, L., Todo, M., Morita, Y., Matsushita, Y., Koyano, K.: Deformation analysis of the periodontium considering the viscoelasticity of the periodontal ligament. Dent. Mater. 25, 1285–1292 (2009)
Wood, S.A., Strait, D.S., Dumont, E.R., Ross, C.F., Grosse, I.R.: The effects of modeling simplifications on craniofacial finite element models: The alveoli (tooth sockets) and periodontal ligaments. J. Biomech. 44, 1831–1838 (2011)
Ferrari, M., Sorrentino, R., Zarone, F., Apicella, D., Aversa, R., Apicella, A.: Non-linear viscoelastic finite element analysis of the effect of the length of glass fiber posts on the biomechanical behaviour of directly restored incisors and surrounding alveolar bone. Dent. Mater. J. 27, 485–498 (2008)
Natali, A.N., Pavan, P.G., Scarpa, C.: Numerical analysis of tooth mobility: Formulation of a non-linear constitutive law for the periodontal ligament. Dent. Mater. 20, 623–629 (2004)
Toms, S.R., Eberhardt, A.W.: A nonlinear finite element analysis of the periodontal ligament under orthodontic tooth loading. Am. J. Orthod. Dentofac. Orthop. 123, 657–665 (2003)
Bergomi, M., Cugnoni, J., Galli, M., Botsis, J., Belser, U.C., Wiskott, H.W.A.: Hydro-mechanical coupling in the periodontal ligament: A porohyperelastic finite element model. J. Biomech. 44, 34–38 (2011)
Naveh, G.R.S., Chattah, N.L.-T., Zaslansky, P., Shahar, R., Weiner, S.: Tooth-PDL-bone complex: Response to compressive loads encountered during mastication - a review. Arch. Oral Biol. 57, 1575–1584 (2012)
Yoshida, N., Koga, Y., Peng, Ch.-L., Tanaka, E., Kobayashi, K.: In vivo measurement of the elastic modulus of the human periodontal ligament. Med. Eng. Phys. 23, 567–572 (2001)
Cronau, M., Ihlow, D., Kubein-Meesenburg, D., Fanghanel, J., Dathe, H., Nagerl, H.: Biomechanical features of the periodontium: An experimental pilot study in vivo. Am. J. Orthod. Dentofac. Orthop. 129, 599.e13–599.e21 (2006)
Fill, T.S., Carey, J.P., Toogood, R.W., Major, P.W.: Experimentally determined mechanical properties of, and models for, the periodontal ligament: Critical review of current literature. J. Dent. Biomech. 2, 1–11 (2011)
Uchaikin, V.: Fractional Derivatives for Physicists and Engineers, vols. I–II. Springer/Higher Education Press, Berlin/Beijing (2013)
Koeller, R.C.: A theory relating creep and relaxation for linear materials with memory. J. Appl. Mech. 77, 031008-1–031008-9 (2010)
West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, NewYork (2003)
Rogosin, S., Mainardi, F.: George William Scott Blair - the pioneer of factional calculus in rheology. Commun. Appl. Ind. Math. 6(1), e481 (2014)
Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press/World Scientific, London/Singapore (2010)
Rossikhin, Yu.A., Shitikova, M.V.: Nonlinear dynamic response of a fractionally damped suspension bridge subjected to small external force. Int. J. Mech. 7, 155–163 (2013)
Rossikhin, Yu.A., Shitikova, M.V., Popov, I.I.: Dynamic response of a hereditarily elastic beam with Rabotnov’s kernel impacted by an elastic rod. In: Proceedings of the 2014 International Conference on Mathematical Models and Methods in Applied Sciences, pp. 25–31. Saint Petersburg State Polytechnic University, Saint-Petersburg (2014)
Sibatov, R.T., Svetukhin, V.V., Uchaikin, V.V., Morozova, E. V.: Fractional model of electron diffusion in dye-sensitized nanocrystalline solar cells. In: Proceedings of the 2014 International Conference on Mathematical Models and Methods in Applied Sciences, pp. 118–121. Saint Petersburg State Polytechnic University, Saint-Petersburg (2014)
Rabotnov, Yu.N.: Elements of Hereditary Solid Mechanics. Mir Publishers, Moscow (1980)
Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299–307 (1984)
Rossikhin, Yu.A., Shitikova, M.V.: Centennial jubilee of academician Rabotnov and contemporary handling of his fractional operator. Fract. Calc. Appl. Anal. 17, 674–683 (2014)
Rossikhin, Yu.A., Shitikova, M.V.: Two approaches for studying the impact response of viscoelastic engineering systems: An overview. Comput. Math. Appl. 66, 755–773 (2013)
Gorenflo, R., Kilbas, A., Mainardi, F., Rogosin, S.: Mittag-Leffler Functions, Related Topics and Applications. Springer, New York (2014)
Hohmann, A., Kober, C., Young, Ph., Dorow, Ch., Geiger, M., Boryor, A., Sander, F.M., Sander, Ch., Sander, F.G.: Influence of different modeling strategies for the periodontal ligament on finite element simulation results. Am. J. Orthod. Dentofac. Orthop. 139, 775–783 (2011)
Provatidis, C.G.: An analytical model for stress analysis of a tooth in translation. Int. J. Eng. Sci. 39, 1361–1381 (2001)
Van Schepdael, A., Geris, L., Van der Sloten, J.: Analytical determination of stress patterns in the periodontal ligament during orthodontic tooth movement. Med. Eng. Phys. 35, 403–410 (2013)
Rabotnov, Yu.N.: Equilibrium of an elastic medium with after-effect. Fract. Calc. Appl. Anal. 17, 684–696 (2014)
Tanne, K., Nagataki, T., Innoue, Y., Sakuda, M., Burstone, C.J.: Patterns of initial tooth displacement associated with various root lengths and alveolar bone heights. Am. J. Orthod. Dentofac Orthop. 100, 66–71 (1991)
Slomka, N., Vardimon, A.D., Gefen, A., Pilo, R., Bourauel, C., Brosh, T.: Time-related PDL: Viscoelastic response during initial orthodontic tooth movement of a tooth with functioning interproximal contact – a mathematical model. J. Biomech. 41, 1871–1877 (2008)
Rossikhin, Yu.A.: Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Appl. Mech. Rev. 63, 010701-1–010701-12 (2010)
Acknowledgements
The research is supported by the FP7 IRSES Marie Curie grant TAMER No 610547. The authors are thankful to professor Francesco Mainardi and to professor Ivan Argatov for valuable discussions of the results of the paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bosiakov, S., Rogosin, S. (2015). Analytical Modeling of the Viscoelastic Behavior of Periodontal Ligament with Using Rabotnov’s Fractional Exponential Function. In: Mastorakis, N., Bulucea, A., Tsekouras, G. (eds) Computational Problems in Science and Engineering. Lecture Notes in Electrical Engineering, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-319-15765-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-15765-8_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15764-1
Online ISBN: 978-3-319-15765-8
eBook Packages: EngineeringEngineering (R0)