Acta Mechanica

, 204:137 | Cite as

On the bending of viscoelastic plates made of polymer foams

  • Holm AltenbachEmail author
  • Victor A. Eremeyev


Considering the viscoelastic behavior of polymer foams a new plate theory based on the direct approach is introduced and applied to plates composed of functionally graded materials (FGM). The governing two-dimensional equations are formulated for a deformable surface, the viscoelastic stiffness parameters are identified assuming linear-viscoelastic material behavior. The material properties are changing in the thickness direction. Solving some problems of the global structural analysis it will be demonstrated that in some cases the results significantly differ from the results based on the Kirchhoff-type theory.


Foam Functionally Grade Material Relaxation Function Functionally Grade Material Polymer Foam 
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  1. 1.
    Altenbach H., Zhilin P.: A general theory of elastic simple shells (in Russian). Usp. Mek. 11, 107–14 (1988)MathSciNetGoogle Scholar
  2. 2.
    Altenbach H.: Eine direkt formulierte lineare Theorie für viskoelastische Platten und Schalen. Ing. Arch. 58, 215–228 (1988)zbMATHCrossRefGoogle Scholar
  3. 3.
    Altenbach H., Zhilin P.: The theory of simple elastic shells. In: Kienzler, R., Altenbach, H., Ott, I.(eds) Critical Review of the Theories of Plates and Shells and New Applications. Lect. Notes. Appl. Comp. Mech., vol. 16., pp. 1–12. Springer, Berlin (2004)Google Scholar
  4. 4.
    Altenbach H.: Determination of elastic moduli of anisotropic plates with nonhomogeneous material in thickness direction (in Russian). Mech. Solids 22, 135–141 (1987)Google Scholar
  5. 5.
    Altenbach H.: An alternative determination of transverse shear stiffnesses for sandwich and laminated plates. Int. J. Solids Struct. 37, 3503–3520 (2000)zbMATHCrossRefGoogle Scholar
  6. 6.
    Altenbach H.: On the determination of transverse shear stiffnesses of orthotropic plates. ZAMP 51, 629–649 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Altenbach, H., Eremeyev, V.A.: Direct approach based analysis of plates composed of functionally graded materials. Arch. Appl. Mech. doi: 10.1007/s00419-007-0192-3
  8. 8.
    Altenbach H., Eremeyev V.A.: Analysis of the viscoelastic behavior of plates made of functionally graded materials. ZAMM 88, 332–341 (2008)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Ashby M.F., Evans A.G., Fleck N.A., Gibson L.J., Hutchinson J.W., Wadley H.N.G.: Metal Foams: a Design Guide. Butterworth-Heinemann, Boston (2000)Google Scholar
  10. 10.
    Banhart J., Ashby M.F., Fleck N.A. (eds): Metal Foams and Porous Metal Structures. Verlag MIT Publishing, Bremen (1999)Google Scholar
  11. 11.
    Brinson H.F., Brinson C.L.: Polymer Engineering Science and Viscoelasticity. An Introduction. Springer, New York (2008)Google Scholar
  12. 12.
    Christensen R.M.: Theory of Viscoelasticity. An Introduction. Academic Press, New York (1971)Google Scholar
  13. 13.
    Collatz L.: Eigenwertaufgaben mit Technischen Anwendungen. Akademische Verlagsgesellschaft, Leipzig (1963)Google Scholar
  14. 14.
    Degischer, H.P., Kriszt B. (eds): Handbook of Cellular Metals. Wiley-VCH, Weinheim (2002)Google Scholar
  15. 15.
    Drozdov A.D.: Finite Elasticity and Viscoelasticity. World Scientific, Singapore (1996)zbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    Hartman Ph.: Ordinary Differential Equations. Wiley, New York (1964)zbMATHGoogle Scholar
  18. 18.
    Haupt P.: Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin (2002)zbMATHGoogle Scholar
  19. 19.
    Kraatz, A.: Berechnung des mechanischen Verhaltens von geschlossenzelligen Schaumstoffen unter Einbeziehung der Mikrostruktur. Diss., Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg (2007)Google Scholar
  20. 20.
    Lakes R.S.: Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987)CrossRefGoogle Scholar
  21. 21.
    Lakes R.S.: The time-dependent Poisson’s ratio of viscoelastic materials can increase or decrease. Cell. Polym. 11, 466–469 (1992)Google Scholar
  22. 22.
    Lakes R.S., Wineman A.: On Poisson’s ratio in linearly viscoelastic solids. J. Elast. 85, 45–63 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Landrock, A.H. (eds): Handbook of Plastic Foams. Types, Properties, Manufacture and Applications. Noes Publications,Park Ridge (1995)Google Scholar
  24. 24.
    Lee S.T., Ramesh N.S. (eds): Polymeric Foams. Mechanisms and Materials. CRC Press, Boca Raton (2004)Google Scholar
  25. 25.
    Mills N.: Polymer Foams Handbook. Engineering and Biomechanics Applications and Design Guide. Butterworth-Heinemann, Amsterdam (2007)Google Scholar
  26. 26.
    Mindlin R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Trans. ASME J. Appl. Mech. 18, 31–38 (1951)zbMATHGoogle Scholar
  27. 27.
    Naghdi P.M.: The theory of plates and shells. In: Flügge, S.(eds) Handbuch der Physik, Bd. VIa/2, pp. 425–640. Springer, Berlin (1972)Google Scholar
  28. 28.
    Rabotnov Yu N.: Elements of Hereditary Solid Mechanics. Mir Publishers, Moscow (1980)zbMATHGoogle Scholar
  29. 29.
    Reissner E.: On the theory of bending of elastic plates. J. Math. Phys. 23, 184–194 (1944)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Reissner E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, A69–A77 (1945)MathSciNetGoogle Scholar
  31. 31.
    Reissner E.: Reflection on the theory of elastic plates. Appl. Mech. Rev. 38, 1453–1464 (1985)CrossRefGoogle Scholar
  32. 32.
    Riande, E. (eds) et al.: Polymer Viscoelasticity: Stress and Strain in Practice. Marcel Dekker, New York (2000)Google Scholar
  33. 33.
    Rothert, H.: Direkte Theorie von Linien- und Flächentragwerken bei viskoelastischem Werkstoffverhalten. Techn.-Wiss. Mitteilungen des Instituts für Konstruktiven Ingenieurbau 73-2. Ruhr-Universität, Bochum (1973)Google Scholar
  34. 34.
    Shaw M.T., MacKnight W.J.: Introduction to Polymer Viscoelasticity, 3rd edn. Wiley, Hoboken (2005)Google Scholar
  35. 35.
    Stoer J., Bulirsch R.: Introduction to Numerical Analysis. Springer, New York (1980)Google Scholar
  36. 36.
    Timoshenko S.P.: On the correnction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. Ser. 6(41), 744–746 (1921)Google Scholar
  37. 37.
    Tschoegl N.W.: The Phenomenological Theory of Linear Viscoelastic Behavior. An Introduction. Springer, Berlin (1989)zbMATHGoogle Scholar
  38. 38.
    Zhilin P.A.: Applied Mechanics. Foundations of the Theory of Shells (in Russian). Petersburg State Polytechnical University, Saint Petersburg (2006)Google Scholar
  39. 39.
    Zhilin P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  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 DonRussian Federation

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