Advertisement

An Incremental formulation for the linear analysis of viscoelastic beams: Relaxation differential approach using generalized variables

  • Claude Chazal
  • Rostand Mouto Pitti
  • Alaa Chateauneuf
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

Abstract

This paper is concerned with the development of a new incremental formulation in the time domain for linear, non-aging viscoelastic materials undergoing mechanical deformation. The formulation is derived from linear differential equations based on a discrete spectrum representation for the relaxation function. The incremental constitutive equations are then obtained by finite difference integration. Thus the difficulty of retaining the stress history in computer solutions is avoided. The influence of the whole past history on the behaviour at any time is given by a pseudo second order tensor. A complete general formulation of linear viscoelastic stress analysis is developed in terms of increments of midsurface strains and curvatures and corresponding stress resultants. The generality allowed by this approach has been established by finding incremental formulation through simple choices of the tensor relaxation components. This approach appears to open a wide horizon (to explore) of new incremental formulations according to particular relaxation components. Remarkable incremental constitutive laws, for which the above technique is applied, are given. This formulation is introduced in a finite element discretization in order to resolve complex boundary viscoelastic problems.

Keywords

Incremental formulation Viscoelasticity Discrete relaxation function Generalized variables 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    1- Bozza A. and Gentili G. (1995) Inversion and Quasi-Static Problems in Linear Viscoelasticity. Meccanica 30:321–335MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    2- Aleksey D. Drozdov and AlDorfmann (2004) A Constitutive Model in Finite Viscoelasticity of Particle-reinforced Rubbers. Meccanica 39:245–270MATHCrossRefGoogle Scholar
  3. 3.
    Kim K.S, Sung LEE H (2007) An incremental formulation for the prediction of two-dimensional fatigue crack growth with curved paths. Int J Num Methods Eng 72:697–721MATHCrossRefGoogle Scholar
  4. 4.
    Theocaris P.S (1964) Creep and relaxation contraction ratio of linear viscoelastic materials. J Mech Physics Solids 12:125–138CrossRefGoogle Scholar
  5. 5.
    Chazal C, Moutou Pitti R (2009) An incremental constitutive law for ageing viscoelastic materials: a three dimensional approach. C. R. Mecanique 337:30–33MATHCrossRefGoogle Scholar
  6. 6.
    Mouto Pitti R, Chazal C, Labesse F, Lapusta Y (2011) A generalization of Mv integral toGoogle Scholar
  7. 7.
    axisymmetric problems for viscoelastic materials, accepted for publication in Acta Mechanica, DOI10.1007/s00707-011-0460-8 Google Scholar
  8. 8.
    Moutou Pitti R, Chateauneuf A, Chazal C (2010) Fiabilité des structures en béton précontraint avec prise en compte du comportement viscoélastique Fiabilité des Matériaux et des Structures, 6èmes Journées Nationales de Fiabilité Toulouse, FranceGoogle Scholar
  9. 9.
    Krempl E, (1979) An experimental study of uniaxial room-temperature rate-sensitivity, creep and relaxation of AISI type 304 stainless steel. J Mech Physics Solids 27:363–375.CrossRefGoogle Scholar
  10. 10.
    Kujawski D, Kallianpur V, Krempl E (1980) An experimental study of uniaxial creep, cyclic Creep and relaxation of AISI type 304 stainless steel at room temperature. J Mech Physics Solids 28:129–148.CrossRefGoogle Scholar
  11. 11.
    Godunov S.K, Denisenko V.V, Kozin N.S, Kuzmina N.K (1975) Use of relaxation viscoelastic model in calculating uniaxial homogeneous strains and refining the interpolation equations for maxwellian viscosity. J Appl Mech Tech Physics 16:811–814.CrossRefGoogle Scholar
  12. 12.
    Duffrène L, Gy R, Burlet H, Piques R, Faivre A, Sekkat A, Perez J (1997) Generalized Maxwell model for the viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading. Rheologica Acta, 36:173–186.CrossRefGoogle Scholar
  13. 13.
    Boltzmann L (1878) Zur Theorie der elastischen Nachwirkung Sitzungsber, Mat Naturwiss. Kl. Kaiser. Akad. Wiss 70, 275.Google Scholar
  14. 14.
    Christensen R.M (1971) Theory of Viscoelasticity: an Introduction. Academic Press, New York. ISBN 0-12-174250-4Google Scholar
  15. 15.
    Mandel J (1978) Dissipativité normale et variables caches. Mech Res Commun 5:225–229.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Salençon J (1983) Viscoélasticité. Presse de l’école nationale des ponts et chaussées, Paris.Google Scholar
  17. 17.
    Chazal C, Dubois F (2001) A new incremental formulation in the time domain of crack initiation in an orthotropic linearly viscoelastic solid. Mech Time-Depend Mater 5:229–253.CrossRefGoogle Scholar
  18. 18.
    Andreas Jäger, Roman Lackner et al. (2007) Identification of viscoelastic properties by means of nanoindentation taking the real tip geometry into account. Meccanica 42:293–306MATHCrossRefGoogle Scholar
  19. 19.
    18- Herbert W. Müllner, Andreas Jäger et al. (2008) Experimental identification of viscoelastic properties of rubber compounds by means of torsional rhemetry. Meccanica 43:327–337MATHCrossRefGoogle Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2011

Authors and Affiliations

  • Claude Chazal
    • 1
  • Rostand Mouto Pitti
    • 2
  • Alaa Chateauneuf
    • 2
  1. 1.Heterogeneous Material Research Group, Civil Engineering and Durability TeamLimoges UniversityEgletonsFrance
  2. 2.Laboratoire de Mécanique et IngénieriesClermont Université, Université Blaise PascalClermont FerrandFrance

Personalised recommendations