A Variational Approach for the Mixed Problem in the Elastostatics of Bodies with Dipolar Structure

  • Marin Marin
  • Vicenţiu Rădulescu
Open Access


In this study, we address the mixed initial boundary value problem in the elastostatics of dipolar bodies. Using the equilibrium equations, we build the operator of dipolar elasticity and prove that this operator is positively defined even in the general case of an elastic inhomogeneous and anisotropic dipolar solid. This helps us to prove the existence of a generalized solution for first boundary value problem and also the uniqueness of the solution. Moreover, relying on this property of the operator of dipolar elasticity to be positively defined, we can apply the known variational method proposed by Mikhlin.


Dipolar bodies first boundary value problem operator of dipolar elasticity generalized solution variational method 

Mathematics Subject Classification

Primary 70G75 Secondary 53A45 58E30 74A20 



The authors are grateful to the anonymous reviewer for the careful reading of the initial version of this paper and for numerous suggestions that improved the present work. V.D. Rădulescu acknowledges the support through the Project MTM2017-85449-P of the DGISPI (Spain).


  1. 1.
    Eringen, A.C.: Theory of thermo-microstretch elastic solids. Int. J. Engng. Sci. 28, 1291–1301 (1990)CrossRefGoogle Scholar
  2. 2.
    Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999)CrossRefGoogle Scholar
  3. 3.
    Iesan, D., Pompei, A.: Equilibrium theory of microstretch elastic solids. Int. J. Eng. Sci. 33, 399–410 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Iesan, D., Quintanilla, R.: Thermal stresses in microstretch bodies. Int. J. Eng. Sci. 43, 885–907 (2005)CrossRefGoogle Scholar
  5. 5.
    Marin, M.: Weak solutions in elasticity of dipolar porous materials. Math. Probl. Eng. 2008(158908), 1–8 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Marin, M.: An approach of a heat-flux dependent theory for micropolar porous media. Meccanica 51, 1127–1133 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ciarletta, M.: On the bending of microstretch elastic plates. Int. J. Eng. Sci. 37, 1309–1318 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Marin, M., Florea, O.: On temporal behaviour of solutions in thermoelasticity of porous micropolar bodies. An. St. Univ. Ovidius Constanta-Seria Mathematics 22(1), 169–188 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Marin, M., Oechsner, A.: An initial boundary value problem for modeling a piezoelectric dipolar body. Continuum Mech. Thermodyn. 30, 267–278 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Straughan, B.: Heat Waves. Applied Mathematical Sciences. Springer, New York (2011)CrossRefGoogle Scholar
  11. 11.
    Marin, M.: Lagrange identity method for microstretch thermoelastic materials. J. Math. Anal. Appl. 363, 275–286 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78 (1964)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Rational Mech. Anal. 17, 113–147 (1964)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fried, E., Gurtin, M.E.: Thermomechanics of the interface between a body and its environment. Contin. Mech. Thermodyn. 19, 253–271 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Brun, L.: Méthodes énergetiques dans les systèmes linéaires. J. Mécanique 8, 167–192 (1969)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Knops, R.J., Payne, L.E.: On uniqueness and continuous data dependence in dynamical problems of linear thermoelasticity. Int. J. Solids Struct. 6, 1173–1184 (1970)CrossRefGoogle Scholar
  17. 17.
    Levine, H.A.: On a theorem of Knops and Payne in dynamical thermoelasticity. Arch. Rational Mech. Anal. 38, 290–307 (1970)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rionero, S., Chirita, S.: The Lagrange identity method in linear thermoelasticity. Int. J. Eng. Sci. 25, 935–947 (1987)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wilkes, N.S.: Continuous dependence and instability in linear thermoelasticity. SIAM J. Appl. Math. 11, 292–299 (1980)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Green, A.E., Laws, N.: On the entropy production inequality. Arch. Rational Mech. Anal. 45, 47–59 (1972)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mikhlin, S.G.: The Problem of the Minimum of a Quadratic Functional, Holden-Day Series in Mathematical Physics. Holden-Day Inc, San Francisco, Calif.-London-Amsterdam (1965)Google Scholar
  22. 22.
    Iesan, D.: Existence theorems in micropolar elastostatics. Int. J. Eng. Sci. 9, 59–78 (1971)CrossRefGoogle Scholar
  23. 23.
    Fichera, G.: Linear Elliptic Differential Systems and Eigenvalue Problems. Lectures Notes in Mathematics, Springer, Baltimore, MD (1965)Google Scholar

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© The Author(s) 2018

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Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceTransilvania University of BraşovBrasovRomania
  2. 2.Faculty of Applied MathematicsAGH University of Science and TechnologyKrakówPoland
  3. 3.Department of MathematicsUniversity of CraiovaCraiovaRomania
  4. 4.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

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