Skip to main content
SpringerLink
Log in

On adaptive thermo-electro-elasticity within a Green–Naghdi type II or III theory

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

We develop a thermo-electro-mechanical continuum theory for a bone remodeling model in order to understand and predict the features of the remodeling process under the control of the strain for a normal living bone. Bone remodeling refers to the continual processes of growth, reinforcement and resorption which arise in living bone. Unlike other approaches to the subject, we follow the Green–Naghdi approach to thermodynamics that employs the concept of thermal displacement and an entropy equality instead of an entropy inequality. We study the bone remodeling process in the context of thermo-electro-elasticity and introduce new balance laws of momentum, energy and entropy. Then, we derive the local balance laws, the constitutive assumptions, the constitutive restrictions and finally focus on the case of transversely isotropic bodies. Last but not least, we prove that the mathematical model is well posed in the nonlinear case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alblas, J.B.: General theory of electro- and magneto-elasticity. In: Parkus, H. (ed.) Electromagnetic Interactions in Elastic Solids. CISM Course, Brussels (1979)

    Google Scholar 

  2. Bargmann, S., Favata, A., Podio-Guidugli, P.: A revised exposition of the Green Naghdi theory of heat propagation. J. Elas. 114(2), 143–154 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bassett, C.A.L., Becker, R.O.: Generation of electric potentials in bone in response to mechanical stress. Science 137, 1063–1064 (1962)

    Article  ADS  Google Scholar 

  4. Becker, R.O.: Search for evidence of axial current flow in peripheral nerves of salamander. Science 134, 101–102 (1961)

    Article  ADS  Google Scholar 

  5. Becker, R.O.: Stimulation of partial limb regeneration in rats. Nature 235, 109–111 (1972)

    Article  ADS  Google Scholar 

  6. Becker, R.O.: The significance of bioelectric potentials. Bioelectrochem. Bioenerg. 1, 187–199 (1974)

    Article  Google Scholar 

  7. Bhatti, M.M., Lu, D.Q.: Head-on collision between two hydroelastic solitary waves in shallow water. Qual. Theory Dyn. Syst. 17, 103–122 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cerrolaza, M., Duarte, V., Garzon-Alvarado, D.: Analysis of bone remodeling under piezoelectricity effects using boundary elements. J. Bionic Eng. 14, 659–671 (2017)

    Article  Google Scholar 

  9. Chirilă, A., Marin, M.: Diffusion in microstretch thermoelasticity with microtemperatures and microconcentrations. In: Flaut, C., Hošková-Mayerová, Š., Flaut, D. (eds.) Models and Theories in Social Systems. Studies in Systems, Decision and Control, pp. 149–164. Springer, Cham (2019)

    Chapter  Google Scholar 

  10. Chirilă, A., Marin, M.: The theory of generalized thermoelasticity with fractional order strain for dipolar materials with double porosity. J. Mater. Sci. 53(5), 3470–3482 (2018)

    Article  ADS  Google Scholar 

  11. Cowin, S.C., Hegedus, D.H.: Bone remodeling I: theory of adaptive elasticity. J. Elast. 6(3), 313–326 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fernandez, J.R., Kuttler, K.L.: An existence and uniqueness result for an elasto-piezoelectric problem with damage. Math. Models Methods Appl. Sci. 19(1), 31–50 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ghiba, I.D.: On the deformation of transversely isotropic porous elastic circular cylinder. Arch. Mech. 61(5), 407–421 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Giorgi, C., Montanaro, A.: Constitutive equations and wave propagation in Green-Naghdi type II and III thermoelectroelasticity. J. Therm. Stress. 39(9), 1051–1073 (2016)

    Article  Google Scholar 

  15. Giorgio, I., Andreaus, U., Madeo, A.: The influence of different loads on the remodeling process of a bone and bioresorbable material mixture with voids. Contin. Mech. Thermodyn. 28, 21–40 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Giorgio, I., Andreaus, U., Scerrato, D., Braidotti, P.: Modeling of a non-local stimulus for bone remodeling process under cyclic load: application to a dental implant using a bioresorbable porous material. Math. Mech. Solids 22(9), 1790–1805 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goodman, M.A., Cowin, S.C.: A continuum theory for granular materials. ARMA 44, 249–266 (1972)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. Ser. A 432, 171–194 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Greiner, W., Neise, L., Stöcker, H.: Thermodynamics and Statistical Mechanics. Springer, Berlin (1995)

    MATH  Google Scholar 

  20. Ieşan, D.: Thermoelastic Models of Continua. Springer, Berlin (2004)

    MATH  Google Scholar 

  21. Liu, I.S.: On entropy flux of transversely isotropic elastic bodies. J. Elas. 96, 97–104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Louna, Z., Goda, I., Ganghoffer, J.: Identification of a constitutive law for trabecular bone samples under remodeling in the framework of irreversible thermodynamics. Contin. Mech. Thermodyn. 30, 529–551 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lubarda, V.A., Chen, M.C.: On the elastic moduli and compliances of transversely isotropic and orthotropic materials. J. Mech. Mater. Struct. 3(1), 153–171 (2008)

    Article  Google Scholar 

  24. Marin, M., Ellahi, R., Chirilă, A.: On solutions of Saint-Venant’s problem for elastic dipolar bodies with voids. Carpathian J. Math. 33(2), 199–212 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Marin, M., Ochsner, A.: The effect of a dipolar structure on the Hölder stability in Green-Naghdi thermoelasticity. Contin. Mech. Thermodyn. 29(6), 1365–1374 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Marino, A.A., Becker, R.O.: Biological effects of extremely low frequency electrical and magnetic fields: a review. Physiol. Chem. Phys. 9, 131–147 (1977)

    Google Scholar 

  27. Scala, I., Spingarn, C., Rémond, Y., Madeo, A., George, D.: Mechanically-driven bone remodeling simulation: application to LIPUS treated rat calvarial defects. Math. Mech. Solids 22(10), 1976–1988 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Tiersten, H.F.: On the nonlinear equations of thermoelectroelasticity. Int. J. Eng. Sci. 9, 587–604 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, J.S., Batra, R.C.: A second-order theory for piezoelectric materials. J. Acoust. Soc. Am. 97, 280 (1995)

    Article  ADS  Google Scholar 

  30. Yang, Y.: An Introduction to the Theory of Piezoelectricity. Springer, Berlin (2005)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewers for useful comments that led to a better presentation of the results.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Transilvania University of Braşov, str. Iuliu Maniu 50, 500091, Braşov, Romania

    Adina Chirilă & Marin Marin

  2. Department of Mathematics Tullio Levi-Civita, University of Padua, Via Trieste 63, 35121, Padua, Italy

    Adriano Montanaro

Authors
  1. Adina Chirilă
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marin Marin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Adriano Montanaro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adina Chirilă.

Additional information

Communicated by Andreas Öchsner.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chirilă, A., Marin, M. & Montanaro, A. On adaptive thermo-electro-elasticity within a Green–Naghdi type II or III theory. Continuum Mech. Thermodyn. 31, 1453–1475 (2019). https://doi.org/10.1007/s00161-019-00766-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-019-00766-2

Keywords

Search

Navigation