Skip to main content

Micromechanical model for viscoelastic materials undergoing damage

Abstract

We have derived a stress–strain relationship for viscoelastic materials undergoing damage using a granular micromechanics approach. This approach assumes the material to possess a granular meso-structure such that the material is treated as a discrete or a particulate system. By considering the particle kinematics in terms of Taylor series expansion, a continuum model of the discrete system is obtained. The material rate-dependence and damage are modeled by assuming appropriate inter-granular force–displacement relationships that satisfy thermodynamic constraints. The advantage of this micromechanical approach is that the resultant continuum model retains the discrete nature by incorporating the effect of nearest neighbor grain interactions through the inter-granular force–displacement relationship and orientation vector. As a result, the derived model has the ability to predict a number of material phenomena, such as loading-induced anisotropy, dilation or pressure sensitivity, and secondary creep, which often manifest due to material granularity.

This is a preview of subscription content, access via your institution.

References

  1. Nemat-Nasser S., Hori M.: Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, New York (1993)

    MATH  Google Scholar 

  2. Navier C.L.: Sur les lois de l’equilibre et du mouvement des corps solides elastiques. Memoire de l’Academie Royale de Sciences 7, 375–393 (1827)

    Google Scholar 

  3. Cauchy, A.-L.: Sur l’equilibre et le mouvement d’un systeme de points materiels sollicites par des forces d’attraction ou de repulsion mutuelle. Excercises de Mathematiques 3, 188–212 (1826–1830)

  4. Arndt M., Griebel M.: Derivation of higher order gradient continuum models from atomistic models for crystalline solids. Multiscale Model. Simulat. 4(2), 531–562 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  5. Blanc X., Le Bris C., Lions P.L.: From molecular models to continuum mechanics. Comptes Rendus De L Academie Des Sciences Serie I-Mathematique 332(10), 949–956 (2001)

    MathSciNet  ADS  MATH  Google Scholar 

  6. E W.N., Huang Z.Y.: A dynamic atomistic-continuum method for the simulation of crystalline materials. J. Computat. Phys. 182(1), 234–261 (2002). doi:10.1006/jcph.2002.7164

    MathSciNet  ADS  MATH  Article  Google Scholar 

  7. Misra A., Yang Y.: Micromechanical model for cohesive materials based upon pseudo-granular structure. Int. J. Solids Struct. 47(21), 2970–2981 (2010). doi:10.1016/j.ijsolstr.2010.07.002

    MATH  Article  Google Scholar 

  8. Chang C.S., Gao J.: 2nd-gradient constitutive theory for granular material with random packing structure. Int. J. Solids Struct. 32(16), 2279–2293 (1995)

    MATH  Article  Google Scholar 

  9. Alibert J.J., Seppecher P., Dell’Isola F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003). doi:10.1177/108128603029658

    MathSciNet  MATH  Article  Google Scholar 

  10. Seppecher P., Alibert J.-J., dell’Isola F.: Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys. Conf. Ser. 319(1), 012018 (2011)

    ADS  Article  Google Scholar 

  11. dell’Isola F., Vidoli S.: Continuum modelling of piezoelectromechanical truss beams: an application to vibration damping. Arch. Appl. Mech. 68(1), 1–19 (1998)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  12. Chang C.S., Askes H., Sluys L.J.: Higher-order strain/higher-order stress gradient models derived from a discrete microstructure, with application to fracture. Eng. Fract. Mech. 69(17), 1907–1924 (2002)

    Article  Google Scholar 

  13. Askes H., Metrikine A.V.: Higher-order continua derived from discrete media: continualisation aspects and boundary conditions. Int. J. Solids Struct. 42(1), 187–202 (2005). doi:10.1016/j.ijsolstr.2004.04.005

    MATH  Article  Google Scholar 

  14. Yang Y., Ching W.-Y., Misra A.: Higher-order continuum theory applied to fracture simulation of nano-scale intergranular glassy film. J. Nanomech. Micromech. 1(2), 60–71 (2011)

    Article  Google Scholar 

  15. Murrell J.N., Carter S., Farantos S.C., Huxley P., Varandas A.J.C.: Molecular Potential Energy Functions. Wiley, New York (1984)

    Google Scholar 

  16. Chang C.S., Misra A.: Packing structure and mechanical-properties of granulates. J. Eng. Mech. Asce 116(5), 1077–1093 (1990)

    Article  Google Scholar 

  17. Misra A., Chang C.S.: Effective elastic moduli of heterogeneous granular solids. Int. J. Solids Struct. 30(18), 2547–2566 (1993)

    MATH  Article  Google Scholar 

  18. Triantafyllidis N., Bardenhagen S.: On higher-order gradient continuum-theories in 1-D nonlinear elasticity—derivation from and comparison to the corresponding discrete models. J. Elast. 33(3), 259–293 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  19. dell’Isola F., Sciarra G., Vidoli S.: Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. Math. Phys. Eng. Sci. 465(2107), 2177–2196 (2009). doi:10.1098/rspa.2008.0530

    MathSciNet  ADS  MATH  Article  Google Scholar 

  20. dell’Isola, F., Seppecher, P.: Hypertractions and hyperstresses convey the same mechanical information. Continuum Mech. Thermodyn. 22, 163–176 (2010) by Prof. Podio Guidugli and Prof. Vianello and some related papers on higher gradient theories. Continuum Mech. Thermodyn. 23(5), 473–478 (2011) doi:10.1007/s00161-010-0176-3

  21. dell’Isola F., Seppecher P.: The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. Comptes Rendus De L Academie Des Sciences Serie Ii Fascicule B-Mecanique Physique Chimie Astronomie 321(8), 303–308 (1995)

    MATH  Google Scholar 

  22. dell’Isola F., Seppecher P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  23. Suiker A.S.J., Chang C.S.: Application of higher-order tensor theory for formulating enhanced continuum models. Acta Mech. 142(1–4), 223–234 (2000)

    MATH  Article  Google Scholar 

  24. dell’Isola, F., Seppecher, P., Madeo, A.: How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach “a la D’Alembert”. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) (2012). doi:10.1007/s00033-012-0197-9

  25. Yang, Y., Misra, A.: Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. Int. J. Solids Struct. (2012). doi:10.1016/j.ijsolstr.2012.05.024

  26. Johnson K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  27. Chang C.S., Misra A.: Theoretical and experimental-study of regular packings of granules. J. Eng. Mech. Asce 115(4), 704–720 (1989)

    Article  Google Scholar 

  28. Atanackovic T.M.: A modified Zener model of a viscoelastic body. Continuum Mech. Thermodyn. 14(2), 137–148 (2002). doi:10.1007/s001610100056

    MathSciNet  ADS  MATH  Article  Google Scholar 

  29. Rajagopal K.R., Srinivasa A.R., Wineman A.S.: On the shear and bending of a degrading polymer beam. Int. J. Plast. 23(9), 1618–1636 (2007). doi:10.1016/j.ijplas.2007.02.007

    MATH  Article  Google Scholar 

  30. Breuer S., Onat E.T.: On the determination of free energy in linear viscoelastic solids. Zeitschrift für Angewandte Mathematik und Physik 184–191(2), 15 (1964)

    Google Scholar 

  31. Day W.A.: Reversibility, recoverable work and free energy in linear viscoelasticity. Q. J. Mech. Appl. Math. 23, 1 (1970)

    MATH  Article  Google Scholar 

  32. Deseri L., Gentili G., Golden M.: An explicit formula for the minimum free energy in linear viscoelasticity. J. Elast. 54, 141–185 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  33. Deseri L., Golden J.M.: The minimum free energy for continuous spectrum materials. Siam J. Appl. Math. 67(3), 869–892 (2007). doi:10.1137/050639776

    MathSciNet  MATH  Article  Google Scholar 

  34. Gentili G.: Maximum recoverable work, minimum free energy and state space in linear viscoelasticity. Q. Appl. Math. 60(1), 153–182 (2002)

    MathSciNet  MATH  Google Scholar 

  35. Del Piero G., Deseri L.: On the concepts of state and free energy in linear viscoelasticity. Arch. Ration. Mech. 138, 1–35 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  36. Christensen R.M.: Theory of Viscoelasticity: An introduction. Academic Press, New York (1982)

    Google Scholar 

  37. Darabi M.K., Abu Al-Rub R.K., Masad E.A., Huang C.W., Little D.N.: A thermo-viscoelastic-viscoplastic-viscodamage constitutive model for asphaltic materials. Int. J. Solids Struct. 48(1), 191–207 (2011). doi:10.1016/j.ijsolstr.2010.09.019

    MATH  Article  Google Scholar 

  38. Zhou X.P., Yang H.Q., Zhang Y.X.: Rate dependent critical strain energy density factor of Huanglong limestone. Theor. Appl. Fract. Mech. 51(1), 57–61 (2009). doi:10.1016/j.tafmec.2009.01.001

    Article  Google Scholar 

  39. Kuhn M.R., Mitchell J.K.: New perspectives on soil-creep. J. Geotech. Eng. Asce 119(3), 507–524 (1993)

    Article  Google Scholar 

  40. Lade P.V., Liggio C.D., Nam J.: Strain rate, creep, and stress drop-creep experiments on crushed coral sand. J. Geotech. Geoenviron. Eng. 135(7), 941–953 (2009). doi:10.1061/(Asce)Gt.1943-5606.0000067

    Article  Google Scholar 

  41. dell’Isola F., Guarascio M., Hutter K.: A variational approach for the deformation of a saturated porous solid: a second-gradient theory extending Terzaghi’s effective stress principle. Arch. Appl. Mech. 70, 323–337 (2000)

    ADS  MATH  Article  Google Scholar 

  42. Dell’Isola F., Hutter K.: What are the dominant thermomechanical processes in the basal sediment layer of large ice sheets?. Proc. R. Soc. Lond. Ser. Math. Phys. Eng. Sci. 454(1972), 1169–1195 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  43. dell’Isola F., Rosa L., Wozniak C.: A micro-structured continuum modelling compacting fluid-saturated grounds: the effects of pore-size scale parameter. Acta Mech. 127(1–4), 165–182 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  44. Sciarra G., dell’Isola F., Hutter K.: A solid-fluid mixture model allowing for solid dilatation under external pressure. Continuum Mech. Thermodyn. 13(5), 287–306 (2001)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  45. dell’Isola F., Kosinski W.: Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers. Arch. Appl. Mech. 45, 333–359 (1993)

    MathSciNet  MATH  Google Scholar 

  46. dell’Isola F., Romano A.: On the derivation of thermomechanical balance-equations for continuous systems with a nonmaterial interface. Int. J. Eng. Sci. 25(11–12), 1459–1468 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  47. dell’Isola F., Romano A.: A phenomenological approach to phase-transition in classical field-theory. Int. J. Eng. Sci. 25(11–12), 1469–1475 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  48. Misra A., Marangos O.: Effect of contact viscosity and roughness on interface stiffness and wave propagation. In: Thompson, D.O., Chimenti, D.E. (eds) Review of Progress in Quantitative Nondestructive Evaluation, vol. 28A, pp. 105–112. AIP, New York (2009)

    Google Scholar 

  49. Misra A., Huang S.P.: Micromechanical stress-displacement model for rough interfaces: effect of asperity contact orientation on closure and shear behavior. Int. J. Solids Struct. 49(1), 111–120 (2012). doi:10.1016/j.ijsolstr.2011.09.013

    Article  Google Scholar 

  50. Kanatani K.I.: Distribution of directional-data and fabric tensors. Int. J. Eng. Sci. 22(2), 149–164 (1984)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anil Misra.

Additional information

Communicated by Francesco dell'Isola and Samuel Forest.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Misra, A., Singh, V. Micromechanical model for viscoelastic materials undergoing damage. Continuum Mech. Thermodyn. 25, 343–358 (2013). https://doi.org/10.1007/s00161-012-0262-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-012-0262-9

Keywords

  • Rate-dependence
  • Damage
  • Granular materials
  • Micromechanics
  • Discrete-continuum