A generalization of the Saint-Venant’s principle for an elastic body with dipolar structure

  • Marin MarinEmail author
  • Andreas Öchsner
  • Eduard M. Craciun
Original Article


This study is concerned with the linear elasticity theory for bodies with a dipolar structure. In this context, we approach transient elastic processes and the steady state in a cylinder consisting of such kind of body which is only subjected to some boundary restrictions at a plane end. We will show that at a certain distance \(d=d(t)\), which can be calculated, from the loaded plan, the deformation of the body vanishes. For the points of the cylinder located at a distance less than d, we will use an appropriate measure to assess the decreasing of the deformation relative to the distance from the loaded plane end. The fact that the measure, that assess the deformation, decays with respect to the distance at the loaded end is the essence of the principle of Saint-Venant.


Dipolar bodies Thermoelastostatics Saint-Venant’s principle Spatial decay 



  1. 1.
    Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VIa/2. Springer, New York (1972)Google Scholar
  2. 2.
    Edelstein, W.S.: A spatial decay estimate for the heat equation. Z. Angew. Math. Phys. 20, 900–905 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Knowles, J.K.: On Saint-Venant’s principle in the two-dimensional linear theory of elasticity. Arch. Ration. Mech. Anal. 21, 1–22 (1966)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Toupin, R.A.: Saint-Venant’s principle. Arch. Ration. Mech. Anal. 18, 83–96 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Horgan, C.O.: Recent developments concerning Saint-Venant’s principle: an update. Appl. Mech. Rev. 42(11), 295–303 (1989)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Karp, D., Durban, D.: Saint-Venant’s principle in dynamics of structure. Appl. Mech. Rev. 64(2), 020801-020801-20 (2011)ADSCrossRefGoogle Scholar
  7. 7.
    Knowles, J.K.: On the spatial decay of solutions of the heat equation. Z. Angew. Math. Phys. 22, 1050–1056 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ericksen, J.L.: Uniformity in shells. Arch. Ration. Mech. Anal. 37, 73–84 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chirita, S.: On the spatial decay estimates in certain time-dependent problems of continuum mechanics. Arch. Mech. 47, 755–771 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Horgan, C.O., Payne, L.E., Wheeler, L.T.: Spatial decay estimates in transient heat conduction. Q. Appl. Math. 42, 119–127 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nunziato, J.W.: On the spatial decay of solutions in the nonlinear theory of heat conduction. J. Math. Anal. Appl. 48, 687–698 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chirita, S.: Spatial decay estimates for solutions describing harmonic vibrations in a thermoelastic cylinder. J. Therm. Stresses 18, 421–436 (1995)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Iesan, D., Quintanilla, R.: Decay estimates and energy bounds for porous elastic cylinders. Z. Angew. Math. Phys. 46, 268–281 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Flavin, J.N., Knops, R.J.: Some spatial decay estimates in continuum dynamics. J. Elast. 17, 249–264 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Flavin, J.N., Knops, R.J., Payne, L.E.: Energy bounds in dynamical problems for a semi-infinite elastic beam. In: Eason, G., Ogden, R.W. (eds.) Elasticity: Mathematical Methods and Applications, pp. 101–111. Ellis-Horwood, Chichester (1990)Google Scholar
  16. 16.
    Chirita, S., Quintanilla, R.: On Saint-Venant’s principle in linear elastodynamics. J. Elast. 42, 201–215 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chirita, S.: On Saint-Venant’s principle in dynamic viscoelasticity. Q. Appl. Math. 55(1), 139–149 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Eringen, A.C.: Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci. 28, 1291–1301 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Eringen, A.C.: Microcontinuum Field Theory. Foundations and Solids, vol. I. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  20. 20.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113–147 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fried, E., Gurtin, M.E.: Thermomechanics of the interface between a body and its environment. Contin. Mech. Therm. 19(5), 253–271 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Iesan, D., Quintanilla, R.: On Saint-Venant’s principle for microstretch elastic bodies. Int. J. Eng. Sci. 35(14), 1277–1290 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Othman, M.I.A., Marin, M.: Effect of thermal loading due to laser pulse on thermoelastic porous medium under G-N theory. Results Phys. 7, 3863–3872 (2017)ADSCrossRefGoogle Scholar
  25. 25.
    Marin, M., Öchsner, A.: The effect of a dipolar structure on the Holder stability in Green–Naghdi thermoelasticity. Contin. Mech. Therm. 30(2), 267–278 (2018)CrossRefzbMATHGoogle Scholar
  26. 26.
    Marin, M., Craciun, E.M.: Uniqueness results for a boundary value problem in dipolar thermoelasticity to model composite materials. Compos. B Eng. 126, 27–37 (2017)CrossRefGoogle Scholar
  27. 27.
    Hassan, M., Marin, M., Ellahi, R., Alamri, S.Z.: Exploration of convective heat transfer and flow characteristics synthesis by Cu-Ag/Water hybrid-nanofluids. Heat Transf. Res. 49(18), 1837–1848 (2018)CrossRefGoogle Scholar
  28. 28.
    Marin, M., Nicaise, S.: Existence and stability results for thermoelastic dipolar bodies with double porosity. Contin. Mech. Therm. 28(6), 1645–1657 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Marin, M., Ellahi, R., Chirila, A.: On solutions of Saint-Venant’s problem for elastic dipolar bodies with voids. Carpathian J. Math. 33(2), 219–232 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Mehrabadi, M.M., Cowin, S.C., Horgan, C.O.: Strain energy density bounds for linear anisotropic elastic materials. J. Elast. 30, 191–196 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Marin, M., Öchsner, A.: Complements of Higher Mathematics. Springer, New York (2018)CrossRefzbMATHGoogle Scholar
  32. 32.
    Marin, M., Öchsner, A., Radulescu, V.: A polynomial way to control the decay of solutions for dipolar bodies. Contin. Mech. Therm. 31(1), 331–340 (2019)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Marin, M., Öchsner, A.: Propagation of a straight crack in dipolar elastic bodies. Contin. Mech. Therm. 30(4), 775–782 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Marin, M., Öchsner, A.: An initial boundary value problem for modeling a piezoelectric dipolar body. Contin. Mech. Therm. 30(2), 267–278 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Ellahi, R., Hassan, M., Zeeshan, A.: A study of heat transfer in power law nanofluid. Therm. Sci. 20(6), 2015–2026 (2016)CrossRefGoogle Scholar
  36. 36.
    Bhatti, M.M., Zeeshan, A., Ellahi, R., Ijaz, N.: Heat and mass transfer of two-phase flow with electric double layer effects induced due to peristaltic propulsion in the presence of transverse magnetic field. J. Mol. Liq. 230, 237–246 (2017)CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceTransilvania University of BrasovBrasovRomania
  2. 2.Faculty of Mechanical EngineeringEsslingen University of Applied SciencesEsslingenGermany
  3. 3.Faculty of Mechanical, Industrial and Maritime EngineeringOvidius University of ConstantaConstantaRomania

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