Skip to main content
SpringerLink
Log in

On the decomposition of the damage variable in continuum damage mechanics

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Refinements and generalizations of the decomposition of the damage variable are presented within the framework of continuum damage mechanics. It is assumed that damage in a solid is due mainly to cracks and voids. The classical decomposition of the damage variable into a damage part due to cracks and another damage part due to voids is examined and extended consistently and mathematically. This is further elaborated upon by considering a solid with three types of defects: cracks, voids, and a third defect that is unspecified. Initially, the decomposition issues are carried out in one dimension using scalars. But this is generalized subsequently for the general case of three-dimensional deformation and damage using tensors. Finally, the special case of plane stress is illustrated as an example. It is shown that in the case of plane stress, two explicit decomposition equations are obtained along with a third implicit coupling equation that relates the various “crack” and “void” damage tensor components.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

\(\phi \) :

Scalar cross-sectional damage variable

\(\phi ^{c}\) :

Scalar cross-sectional damage variable due to cracks

\(\phi ^{v}\) :

Scalar cross-sectional damage variable due to voids

\(\phi ^{w}\) :

Scalar cross-sectional damage variable due to a third unspecified defect type

\(\varphi _0 \) :

Scalar cross-sectional damage variable due to a single defect type

\(\phi ^{*}\) :

New scalar cross-sectional damage variable

\(\ell \) :

Scalar elastic stiffness degradation damage variable

A :

Cross-sectional area in the deformed/damaged configuration

\({\bar{A}}\) :

Cross-sectional area in the effective/undamaged configuration

\(A^{c}\) :

Cross-sectional area of cracks

\(A^{v}\) :

Cross-sectional area of voids

\(\sigma \) :

Cauchy stress

\({\bar{\sigma }} \) :

Effective Cauchy stress

E :

Elastic modulus in the deformed/damaged configuration

\({\bar{E}}\) :

Effective elastic modulus (in the effective/undamaged configuration)

n :

Number of defect types, with \(n=1, 2, 3, 4, 5 ...\)

[M]:

Matrix representation of fourth-rank damage effect tensor (based on cross-sectional area)

[\(M^{c}\)]:

Matrix representation of fourth-rank damage effect tensor (based on cross-sectional area) due to cracks

[\(M^{v}\)]:

Matrix representation of fourth-rank damage effect tensor (based on cross-sectional area) due voids

[\({M^{w}}\)]:

Matrix representation of fourth-rank damage effect tensor (based on cross-sectional area) due to a third unspecified defect typerth-rank identity tensor

I :

Fourth-rank identity tensor

\(\{\sigma \}\) :

Vector representation of the second-rank stress tensor

\(\{{\bar{\sigma }}\}\) :

Vector representation of the second-rank effective stress tensor

e :

Exponential function

x :

Unknown or unspecified variable

y :

Unknown or unspecified variable

f :

Unknown or unspecified function

[A]:

Exponent matrix that represents a fourth-rank tensor

\(\Delta \) :

Expression related to the second-rank damage tensor

\(\Delta ^{c}\) :

Expression related to the second-rank damage tensor due to cracks

\(\Delta ^{v}\) :

Expression related to the second-rank damage tensor due to voids

\(\phi _{11} , \phi _{22} , \phi _{12} \) :

Components of the second-rank damage tensor

\(\phi _{11}^c , \phi _{22}^c , \phi _{12}^c \) :

Components of the second-rank damage tensor due to cracks

\(\phi _{11}^v , \phi _{22}^v , \phi _{12}^v \) :

Components of the second-rank damage tensor due to voids

References

  1. Basaran, C., Nie, S.: An irreversible thermodynamic theory for damage mechanics of solids. Int. J. Damage Mech. 13(3), 205–224 (2004)

    Article  Google Scholar 

  2. Basaran, C., Yan, C.Y.: A thermodynamic framework for damage mechanics of solder joints. Trans. ASME J. Electron. Pack. 120, 379–384 (1998)

    Article  Google Scholar 

  3. Celentano, D.J., Tapia, P.E., Chaboche, J.-L.: Experimental and numerical characterization of damage evolution in steels. In: Buscaglia G., Dari E., Zamonsky, O. (eds.) Mecanica Computacional, vol. XXIII. Bariloche, Argentina (2004)

  4. Chaboche, J.L.: Continuum damage mechanics: a tool to describe phenomena before crack initiation. Nucl. Eng. Des. 64, 233–247 (1981)

    Article  Google Scholar 

  5. Chaboche, J.L.: Continuum damage mechanics: present state and future trends. In: International Seminar on Modern Local Approach of Fracture, Moret-sur-Loing, France (1986)

  6. Chaboche, J.L.: Continuum damage mechanics: Part I: general concepts. J. Appl. Mech. ASME 55, 59–64 (1988a)

    Article  Google Scholar 

  7. Chaboche, J.L.: Continuum damage mechanics: Part II: damage growth, crack initiation, and crack growth. J. Appl. Mech. ASME 55, 65–72 (1988b)

    Article  Google Scholar 

  8. Chow, C., Wang, J.: An anisotropic theory of elasticity for continuum damage mechanics. Int. J. Fract. 33, 3–16 (1987)

    Article  Google Scholar 

  9. Doghri, I.: Mechanics of Deformable Solids: Linear and Nonlinear, Analytical and Computational Aspects. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  10. Hansen, N.R., Schreyer, H.L.: A thermodynamically consistent framework for theories of elastoplasticity coupled with damage. Int. J. Solids Struct. 31(3), 359–389 (1994)

    Article  MATH  Google Scholar 

  11. Kachanov, L.: On the creep fracture time. Izv. Akad. Nauk. USSR Otd Tech. 8, 26–31 (1958). (in Russian)

    Google Scholar 

  12. Kattan, P.I., Voyiadjis, G.Z.: A coupled theory of damage mechanics and finite strain elasto-plasticity—part I: damage and elastic deformations. Int. J. Eng. Sci. 28(5), 421–435 (1990)

    Article  MATH  Google Scholar 

  13. Kattan, P.I., Voyiadjis, G.Z.: A plasticity-damage theory for large deformation of solids—part II: applications to finite simple shear. Int. J. Eng. Sci. 31(1), 183–199 (1993)

    Article  MATH  Google Scholar 

  14. Kattan, P.I., Voyiadjis, G.Z.: Decomposition of damage tensor in continuum damage mechanics. J. Eng. Mech. ASCE 127(9), 940–944 (2001a)

    Article  Google Scholar 

  15. Kattan, P.I., Voyiadjis, G.Z.: Damage Mechanics with Finite Elements: Practical Applications with Computer Tools. Springer, Berlin (2001b)

    Google Scholar 

  16. Krajcinovic, D.: Damage Mechanics, p. 776. North Holland, Amsterdam (1996)

    Google Scholar 

  17. Ladeveze, P., Poss, M., Proslier, L.: "Damage and Fracture of Tridirectional Composites", in progress in science and engineering of composites. Proc. Fourth Int. Conf. Compos. Mater. Jpn. Soc. Compos. Mater. 1, 649–658 (1982)

    Google Scholar 

  18. Ladeveze, P., Lemaitre, J.: Damage Effective Stress in Quasi-Unilateral Conditions. In: The 16th International Cogress of Theoretical and Applied Mechanics, Lyngby, Denmark (1984)

  19. Lee, H., Peng, K., Wang, J.: An anisotropic damage criterion for deformation instability and its application to forming limit analysis of metal plates. Eng. Fract. Mech. 21, 1031–1054 (1985)

    Article  Google Scholar 

  20. Lemaitre, J.: How to use damage mechanics. Nucl. Eng. Des. 80, 233–245 (1984)

    Article  Google Scholar 

  21. Lemaitre, J.: A continuous damage mechanics model for ductile fracture. J. Eng. Mater. Technol. 107, 83–89 (1985)

    Article  Google Scholar 

  22. Lemaitre, J.: Local approach of fracture. Eng. Fract. Mech. 25(5/6), 523–537 (1986)

    Article  Google Scholar 

  23. Lemaitre, J.: A Course on Damage Mechanics. Springer, New York (1992)

    Book  MATH  Google Scholar 

  24. Lemaitre, J., Chaboche, J.L.: Mecanique de Materiaux Solides. Dunod, Paris (1985)

    Google Scholar 

  25. Lemaitre, J., Dufailly, J.: Damage measurements. Eng. Fract. Mech. 28(5/6), 643–661 (1987)

    Article  Google Scholar 

  26. Lubineau, G.: A pyramidal modeling scheme for laminates—identification of transverse cracking. Int. J. Damage Mech. 19(4), 499–518 (2010)

    Article  MathSciNet  Google Scholar 

  27. Lubineau, G., Ladeveze, P.: Construction of a micromechanics-based intralaminar mesomodel, and illustrations in ABAQUS/standard. Comput. Mater. Sci. 43(1), 137–145 (2008)

    Article  Google Scholar 

  28. Luccioni, B., Oller, S.: A directional damage model. Comput. Methods Appl. Mech. Eng. 192, 1119–1145 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nichols, J.M. Abell, A.B.: Implementing the Degrading Effective Stiffness of Masonry in a Finite Element Model. In: North American Masonry Conference, Clemson, South Carolina, USA (2003)

  30. Nichols, J.M., Totoev, Y.Z.: Experimental Investigation of the Damage Mechanics of Masonry Under Dynamic In-plane Loads. North American Masonry Conference, Austin, Texas, USA (1999)

  31. Rabotnov, Y.: Creep rupture. In: Hetenyi, M., Vincenti, W.G. (eds.) Proceedings, Twelfth International Congress of Applied Mechanics, Stanford, pp. 342–349. Springer, Berlin (1968)

    Google Scholar 

  32. Rice, J.R.: Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 (1971)

    Article  MATH  Google Scholar 

  33. Sidoroff, F.: Description of anisotropic damage application in elasticity. IUTAM Colloqium on Physical Nonlinearities in Structural Analysis, pp. 237–244. Springer, Berlin (1981)

    Chapter  Google Scholar 

  34. Voyiadjis, G.Z.: Degradation of elastic modulus in elastoplastic coupling with finite strains. Int. J. Plast 4, 335–353 (1988)

    Article  Google Scholar 

  35. Voyiadjis, G.Z., Kattan, P.I.: A coupled theory of damage mechanics and finite strain elasto-plasticity—part II: damage and finite strain plasticity. Int. J. Eng. Sci. 28(6), 505–524 (1990)

    Article  MATH  Google Scholar 

  36. Voyiadjis, G.Z., Kattan, P.I.: A plasticity-damage theory for large deformation of solids—part I: theoretical formulation. Int. J. Eng. Sci. 30(9), 1089–1108 (1992)

    Article  MATH  Google Scholar 

  37. Voyiadjis, G.Z., Kattan, P.I.: Damage Mechanics. Taylor and Francis (CRC Press), Boca Raton (2005)

    Book  MATH  Google Scholar 

  38. Voyiadjis, G.Z., Kattan, P.I.: Advances in Damage Mechanics: Metals and Metal Matrix Composites with an Introduction to Fabric Tensors, 2nd edn. Elsevier, New York (2006a)

    MATH  Google Scholar 

  39. Voyiadjis, G.Z., Kattan, P.I.: A new fabric-based damage tensor. J. Mech. Behav. Mater. 17(1), 31–56 (2006b)

    Article  Google Scholar 

  40. Voyiadjis, G.Z., Kattan, P.I.: Damage mechanics with fabric tensors. Mech. Adv. Mater. Struct. 13(4), 285–301 (2006c)

    Article  Google Scholar 

  41. Voyiadjis, G.Z., Kattan, P.I.: A comparative study of damage variables in continuum damage mechanics. Int. J. Damage Mech. 18(4), 315–340 (2009)

    Article  Google Scholar 

  42. Voyiadjis, G.Z., Kattan, P.I.: A new class of damage variables in continuum damage mechanics. ASME J. Mater. Technol. 134(2), 10 (2012)

    Google Scholar 

  43. Voyiadjis, G.Z., Kattan, P.I.: Mechanics of damage processes in series and in parallel: a conceptual framework’. Acta Mech. 223(9), 1863–1878 (2012)

    Article  MATH  Google Scholar 

  44. Voyiadjis, G.Z., Kattan, P.I.: Mechanics of damage, healing, damageability, and integrity of materials: a conceptual framework. Int. J. Damage Mech. (accepted for publication) (2017)

  45. Voyiadjis, G.Z., Kattan, P.I.: Elasticity of damaged graphene: a damage mechanics approach. Int. J. Damage Mech. (accepted for publication) (2017)

  46. Voyiadjis, G.Z., Kattan, P.I.: Decomposition of elastic stiffness degradation in continuum damage mechanics. ASME J. Mater. Technol. (2017)

  47. Voyiadjis, G.Z., Kattan, P.I.: Introducing damage mechanics templates for the systematic and consistent formulation of holistic material damage models. Acta Mech. (2017)

Download references

Author information

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA, 70803, USA

    George Z. Voyiadjis

  2. Petra Books, Amman, PO Box 1392, Jordan

    Peter I. Kattan

Authors
  1. George Z. Voyiadjis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter I. Kattan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Z. Voyiadjis.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Voyiadjis, G.Z., Kattan, P.I. On the decomposition of the damage variable in continuum damage mechanics. Acta Mech 228, 2499–2517 (2017). https://doi.org/10.1007/s00707-017-1836-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-017-1836-1

Search

Navigation