Archive of Applied Mechanics

, Volume 80, Issue 3, pp 217–227 | Cite as

Acceleration waves and ellipticity in thermoelastic micropolar media

  • Holm Altenbach
  • Victor A. Eremeyev
  • Leonid P. Lebedev
  • Leonardo A. Rendón


Acceleration waves in nonlinear thermoelastic micropolar media are considered. We establish the kinematic and dynamic compatibility relations for a singular surface of order 2 in the media. An analogy to the Fresnel–Hadamard–Duhem theorem and an expression for the acoustic tensor are derived. The condition for acceleration wave’s propagation is formulated as an algebraic spectral problem. It is shown that the condition coincides with the strong ellipticity of equilibrium equations. As an example, a quadratic form for the specific free energy is considered and the solutions of the corresponding spectral problem are presented.


Acceleration waves Micropolar continuum Cosserat continuum Nonlinear thermoelasticity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Antman S.S.: Nonlinear Problems of Elasticity, 2nd edn. Springer Science Media, New York (2005)Google Scholar
  2. 2.
    Ashby M.F., Evans A.G., Fleck N.A., Gibson L.J., Hutchinson J.W., Wadley H.N.G.: Metal Foams: a Design Guid. Butterworth-Heinemann, Boston (2000)Google Scholar
  3. 3.
    Chen P.J.: Growth of acceleration waves in isotropic elastic materials. J. Acoust. Soc. Am. 43, 982–987 (1968)CrossRefGoogle Scholar
  4. 4.
    Chen P.J.: One dimensional acceleration waves in inhomogeneous elastic non-conductors. Acta Mech. 17, 17–24 (1973)MATHCrossRefGoogle Scholar
  5. 5.
    Cielecka I., Woźniak M., Woźniak C.: Elastodynamic behaviour of honeycomb cellular media. J. Elast. 60, 1–17 (2000)MATHCrossRefGoogle Scholar
  6. 6.
    Cosserat E., Cosserat F.: Théorie des corps déformables. Herman et Flis, Paris (1909)Google Scholar
  7. 7.
    Diebels S.: A micropolar theory of porous media: constitutive modelling. Transp. Porous Media 34, 193–208 (1999)CrossRefGoogle Scholar
  8. 8.
    Diebels S., Steeb H.: Stress and couple stress in foams. Comput. Mater. Sci. 28, 714–722 (2003)CrossRefGoogle Scholar
  9. 9.
    Dietsche A., Steinmann P., Willam K.: Micropolar elastoplasticity and its role in localization. Int. J. Plast. 9, 813–831 (1993)MATHCrossRefGoogle Scholar
  10. 10.
    Ehlers W., Ramm E., Diebels S., d’Addetta G.D.A.: From particle ensembles to Cosserat continua: Homogenization of contact forces towards stresses and couple stresses. Int. J. Solids Struct. 40, 6681–6702 (2003)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Eremeyev V.A.: Acceleration waves in micropolar elastic media. Dokl. Phys. 50(4), 204–206 (2005)CrossRefGoogle Scholar
  12. 12.
    Eremeyev V.A.: Nonlinear micropolar shells: theory and applications. In: Pietraszkiewicz, W., Szymczak, C. (eds) Shell Structures: Theory and Applications, pp. 11–18. Taylor and Francis, London (2005)Google Scholar
  13. 13.
    Eremeyev V.A., Zubov L.M.: On the stability of elastic bodies with couple stresses (in Russ.). Mech. Solids 3, 181–190 (1994)Google Scholar
  14. 14.
    Eremeyev V.A., Zubov L.M.: On constitutive inequalities in nonlinear theory of elastic shells. ZAMM 87(2), 94–101 (2007)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Eremeyev V.A., Lebedev L.P., Rendón L.A.: On the propagation of acceleration waves in thermoelastic micropolar media. Revista Colombiana de Matemáticas 41, 397–406 (2007)Google Scholar
  16. 16.
    Ericksen J.L., Truesdell C.: Exact tbeory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1(1), 295–323 (1958)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Eringen A.C.: Microcontinuum Field Theory. I. Foundations and Solids. Springer, New York (1999)Google Scholar
  18. 18.
    Eringen A.C.: Microcontinuum Field Theory. II. Fluent Media. Springer, New York (2001)Google Scholar
  19. 19.
    Eringen A.C., Maugin G.A.: Electrodynamics of Continua. Springer, New York (1990)Google Scholar
  20. 20.
    Eringen A.C., Suhubi E.S.: Elastodynamics, vol. 1. Academic Press, New York (1974)MATHGoogle Scholar
  21. 21.
    Erofeev V.I.: Wave Processes in Solids with Microstructure. World Scientific, Singapore (2003)Google Scholar
  22. 22.
    Fu Y.B., Scott N.H.: Acceleration wave propagation in an inhomogeneous heat conducting elastic rod of slowly varying cross section. J. Therm. Stresses 15, 253–264 (1988)MathSciNetGoogle Scholar
  23. 23.
    Gibson L.J., Ashby M.F.: Cellular Solids: Structure and Properties, 2nd edn. Cambridge Solid State Science Series. Cambridge University Press, Cambridge (1997)Google Scholar
  24. 24.
    Kafadar C.B., Eringen A.C.: Micropolar media–I. The classical theory. Int. J. Eng. Sci. 9, 271–305 (1971)CrossRefGoogle Scholar
  25. 25.
    Kafadar C.B., Eringen A.C.: Polar field theories. In: Eringen, A.C. (eds) Continuum Physics, vol. IV, pp. 1–75. Academic Press, New York (1976)Google Scholar
  26. 26.
    Knowles J.K., Sternberg E.: On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J. Elast. 10, 255–293 (1980)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Koiter W.T.: Couple–stresses in the theory of elasticity. Pt I–II. Proc. Koninkl Neterland Akad Wetensh B 67, 17–44 (1964)MATHGoogle Scholar
  28. 28.
    Lakes R.S.: Experimental microelasticity of two porous solids. Int. J. Solids Struct. 22, 55–63 (1986)CrossRefGoogle Scholar
  29. 29.
    Lakes R.S.: Experimental micro mechanics methods for conventional and negative Poisson’s ratio cellular solids as Cosserat continua. Trans. ASME J. Eng. Mater. Technol. 113, 148–155 (1991)CrossRefGoogle Scholar
  30. 30.
    Lurie A.I.: Nonlinear Theory of Elasticity. North-Holland, Amsterdam (1990)MATHGoogle Scholar
  31. 31.
    Lurie A.I.: Theory of Elasticity. Foundations of Engineering Mechanics. Springer, Berlin (2005)Google Scholar
  32. 32.
    Maugin G.A.: Acceleration waves in simple and linear viscoelastic micropolar materials. Int. J. Eng. Sci. 12, 143–157 (1974)MATHCrossRefGoogle Scholar
  33. 33.
    Maugin G.A.: Continuum Mechanics of Electromagnetic Solids. Elsevier, Oxford (1988)MATHGoogle Scholar
  34. 34.
    Maugin G.A.: On the structure of the theory of polar elasticity. Philos. Trans. R. Soc. Lond. A 356, 1367–1395 (1998)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Maugin G.A.: Nonlinear Waves on Elastic Crystals. Oxford University Press, Oxford (1999)Google Scholar
  36. 36.
    Naghdi P.: The theory of plates and shells. In: Flügge, S. (eds) Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Heidelberg (1972)Google Scholar
  37. 37.
    Neff P., Forest S.: A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elast. 87, 239–276 (2007)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Nikitin E., Zubov L.M.: Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress. J. Elast. 51, 1–22 (1998)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Noll W.: A new mathematical theory of simple materials. Arch. Rational Mech. Anal. 48, 1–50 (1972)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Nowacki W.: Theory of Asymmetric Elasticity. Pergamon-Press, Oxford (1986)MATHGoogle Scholar
  41. 41.
    Ogden R.W.: Growth and decay of acceleration waves in incompresible elastic solids. Q. J. Mech. Appl. Math. 37, 451–464 (1974)CrossRefGoogle Scholar
  42. 42.
    Pal’mov V.A.: Fundamental equations of the theory of asymmetric elasticity. J. Appl. Mech. Math. 28(3), 496–505 (1964)CrossRefMathSciNetGoogle Scholar
  43. 43.
    Park H.C., Lakes R.S.: Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent. J. Biomech. 19, 385–397 (1986)CrossRefGoogle Scholar
  44. 44.
    Pietraszkiewicz W., Eremeyev V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46(3–4), 774–787 (2009)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Rosakis P.: Ellipticity and deformation with discontinuous gradients in finite elastostatics. Arch. Ration. Mech. Anal. 109, 1–37 (1990)MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Rubin M.B.: Cosserat Theories: Shells, Rods and Points. Kluwer, Dordrecht (2000)MATHGoogle Scholar
  47. 47.
    Scott N.H.: Acceleration waves in constrained elastic materials. Arch. Ration. Mech. Anal. 58, 57–75 (1975)MATHCrossRefGoogle Scholar
  48. 48.
    Toupin R.A.: Theories of elasticity with couple–stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964)MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Truesdell C.: A First Course in Rational Continuum Mechanics. Academic Press, New York (1977)MATHGoogle Scholar
  50. 50.
    Truesdell C.: Rational Thermodynamics, 2nd edn. Springer, New York (1984)Google Scholar
  51. 51.
    Truesdell C., Noll W.: The nonlinear field theories of mechanics. In: Flügge, S. (eds) Handbuch der Physik, vol. III/3, pp. 1–602. Springer, Berlin (1965)Google Scholar
  52. 52.
    Vassilev, V., Djondjorov, P.: Acceleration waves in the von Kármán plate theory. In: Integral Methods in Science and Engineering, CRC Research Notes in Mathematics 418, vol. 83, pp. 131–136. Chapman & Hall, Boca Raton (2000)Google Scholar
  53. 53.
    Zee L., Sternberg E.: Ordinary and strong ellipticity in the equilibrium theory of incompressible hyperelastic solids. Arch. Ration. Mech. Anal. 83, 53–90 (1983)MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Zhilin P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976)CrossRefMathSciNetGoogle Scholar
  55. 55.
    Zubov L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin (1997)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Holm Altenbach
    • 1
  • Victor A. Eremeyev
    • 2
  • Leonid P. Lebedev
    • 3
  • Leonardo A. Rendón
    • 3
  1. 1.Lehrstuhl für Technische Mechanik, Zentrum für IngenieurwissenschaftenMartin-Luther-Universität Halle-WittenbergHalle (Saale)Germany
  2. 2.South Scientific Center of RASci and South Federal UniversityRostov on DonRussia
  3. 3.Universidad Nacional de ColombiaBogotáColombia

Personalised recommendations