Abstract
An attempt is presented to provide for a global and holistic approach to the study of continuum damage mechanics. For this purpose, the authors classify material damage models into three categories. Furthermore, the concept of ready-made model templates for damage mechanics is proposed. In this regard, the authors provide details for fourteen basic damage mechanics templates. Each template comes with a schematic diagram along with a set of equations that govern the respective material damage template. The user can then use the chosen template and fill in the details to obtain the constitutive equations. The obtained material damage model is guaranteed to be systematic and consistent provided that proper use is made of the chosen template. Furthermore, three additional sections are added to provide details on how to generate advanced, special, and more complex damage mechanics templates that go beyond the rule of mixtures and for specific types of materials. The details of 14 complete and comprehensive damage mechanics templates for both metals and composite materials are presented in this work. Finally, using one chosen basic template, it is shown how to generate five additional and more complex templates.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- C :
-
Damaged material configuration
- \(\overline{{C}}\) :
-
Effective fictitious undamaged material configuration
- \(C^{m}\) :
-
Damaged material configuration of the matrix
- \(\overline{{C}}^{m}\) :
-
Effective fictitious damaged material configuration of the matrix
- \(C^{f}\) :
-
Damaged material configuration of the fibers
- \(\overline{{C}}^{f}\) :
-
Effective fictitious damaged material configuration of the fibers
- \(C^{c}\) :
-
Damaged material configuration when cracks are present only
- \(\overline{{C}}^{c}\) :
-
Effective fictitious damaged material configuration when cracks are present and voids are removed
- \(C^{v}\) :
-
Damaged material configuration when voids are present only
- \(\overline{{C}}^{v}\) :
-
Effective fictitious damaged material configuration when voids are present and cracks are removed
- \(C^{mc}\) :
-
Damaged material configuration of the matrix when cracks are present only
- \(\overline{{C}}^{mc}\) :
-
Effective fictitious damaged material configuration of the matrix when cracks are present and voids are removed
- \(C^{mv}\) :
-
Damaged material configuration of the matrix when voids are present only
- \(\overline{{C}}^{mv}\) :
-
Effective fictitious damaged material configuration of the matrix when voids are present and cracks are removed
- \(C^{fc}\) :
-
Damaged material configuration of the fibers when cracks are present only
- \(\overline{{C}}^{fc}\) :
-
Effective fictitious damaged material configuration of the fibers when cracks are present and voids are removed
- \(C^{fv}\) :
-
Damaged material configuration of the fibers when voids are present only
- \(\overline{{C}}^{fv}\) :
-
Effective fictitious damaged material configuration of the fibers when voids are present and cracks are removed
- \(\overline{{C}}^{d}\) :
-
Effective fictitious configuration with respect to de-bonding, i.e., interfacial damage
- M :
-
Fourth-rank damage effect tensor
- \(M^{m}\) :
-
Fourth-rank matrix damage effect tensor
- \(M^{f}\) :
-
Fourth-rank fiber damage effect tensor
- \(M^{c}\) :
-
Fourth-rank damage effect tensor when cracks are present only
- \(M^{v}\) :
-
Fourth-rank damage effect tensor when voids are present only
- \(M^{mc}\) :
-
Fourth-rank matrix damage effect tensor of the matrix when cracks are present only
- \(M^{mv}\) :
-
Fourth-rank matrix damage effect tensor of the matrix when voids are present only
- \(M^{fc}\) :
-
Fourth-rank matrix damage effect tensor of the fibers when cracks are present only
- \(M^{fv}\) :
-
Fourth-rank matrix damage effect tensor of the fibers when voids are present only
- \(M^{d}\) :
-
De-bonding (i.e., interfacial damage) fourth-rank damage effect tensor
- \(A^{m}\) :
-
Matrix strain concentration factor/tensor
- \(A^{f}\) :
-
Fiber strain concentration factor/tensor
- \(\overline{{A}}^{m}\) :
-
Effective matrix strain concentration factor/tensor
- \(\overline{{A}}^{f}\) :
-
Effective fiber strain concentration factor/tensor
- \(A^{c}\) :
-
Strain concentration factor/tensor when cracks are present only
- \(A^{v}\) :
-
Strain concentration factor/tensor when voids are present only
- \(\overline{{A}}^{c}\) :
-
Effective strain concentration factor/tensor when cracks are present only and voids are removed
- \(\overline{{A}}^{v}\) :
-
Effective strain concentration factor/tensor when voids are present only and cracks are removed
- \(B^{m}\) :
-
Matrix stress concentration factor/tensor
- \(B^{f}\) :
-
Fiber stress concentration factor/tensor
- \(\overline{{B}}^{m}\) :
-
Effective matrix stress concentration factor/tensor
- \(\overline{{B}}^{f}\) :
-
Effective fiber stress concentration factor/tensor
- \(B^{c}\) :
-
Stress concentration factor/tensor when cracks are present only
- \(B^{v}\) :
-
Stress concentration factor/tensor when voids are present only
- \(\overline{{B}}^{c}\) :
-
Effective stress concentration factor/tensor when cracks are present only and voids are removed
- \(\overline{{B}}^{v}\) :
-
Effective stress concentration factor/tensor when voids are present only and cracks are removed
- \(\phi \) :
-
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
- A :
-
Cross-sectional area in the deformed/damaged configuration
- \(\overline{{A}}\) :
-
Cross-sectional area in the effective/undamaged configuration
- \(\sigma \) :
-
Cauchy stress
- \(\sigma ^{m}\) :
-
Matrix Cauchy stress
- \(\sigma ^{f}\) :
-
Fiber Cauchy stress
- \(\overline{{\sigma }}\) :
-
Effective Cauchy stress
- \(\overline{{\sigma }}^{m}\) :
-
Effective matrix Cauchy stress
- \(\overline{{\sigma }}^{f}\) :
-
Effective fiber Cauchy stress
- E :
-
Elastic modulus in the deformed/damaged configuration
- \(E^{m}\) :
-
Matrix elastic modulus in the deformed/damaged configuration
- \(E^{f}\) :
-
Fiber elastic modulus in the deformed/damaged configuration
- \(\overline{{E}}\) :
-
Effective elastic modulus (in the fictitious/undamaged configuration)
- \(\overline{{E}}^{m}\) :
-
Matrix effective elastic modulus (in the fictitious/undamaged configuration)
- \(\overline{{E}}^{f}\) :
-
Fiber effective elastic modulus (in the fictitious/undamaged configuration)
- G :
-
Shear modulus in the deformed/damaged configuration
- \(G^{m}\) :
-
Matrix shear modulus in the deformed/damaged configuration
- \(G^{f}\) :
-
Fiber shear modulus in the deformed/damaged configuration
- \(\upsilon \) :
-
Poisson’s ratio in the deformed/damaged configuration
- \(\upsilon ^{m}\) :
-
Matrix Poisson’s ratio in the deformed/damaged configuration
- \(\upsilon ^{f}\) :
-
Fiber Poisson’s ratio in the deformed/damaged configuration
- S :
-
Eshelby tensor
- \(c^{m}\) :
-
Matrix volume fraction
- \(c^{f}\) :
-
Fiber volume fraction
- \(\overline{c}^{m}\) :
-
Effective matrix volume fraction
- \(\overline{{c}}^{f}\) :
-
Effective fiber volume fraction
- \(c^{c}\) :
-
Crack volume fraction
- \(c^{v}\) :
-
Void volume fraction
- \(\overline{{c}}^{c}\) :
-
Effective crack volume fraction
- \(\overline{{c}}^{v}\) :
-
Effective void volume fraction
- \(V^{m}\) :
-
Matrix volume fraction
- \(V^{f}\) :
-
Fiber volume fraction
- \(\eta \) :
-
Stress-portioning factor
- \(\eta ^{m}\) :
-
Stress-portioning factor
- \(\eta ^{f}\) :
-
Shear-portioning factor
- s :
-
Deviatoric stress tensor
- f :
-
Yield function
- \(\alpha \) :
-
Backstress tensor
- \(\varepsilon ^{e}\) :
-
Elastic strain tensor
- \(\varepsilon ^{p}\) :
-
Plastic strain tensor
- c :
-
Scalar parameter related to kinematic hardening
- \(\lambda \) :
-
Scalar parameter related to flow rule
- \(I_2 \) :
-
Second-rank identity tensor
- \(I_4 \) :
-
Fourth-rank identity tensor
- \(\delta \) :
-
Kronecker delta
References
Allen, D.H., Harris, C.E.: A thermomechanical constitutive theory for elastic composites with distributed damage—I: theoretical formulation. Int. J. Solids Struct. 23, 1301–1318 (1987)
Arnold, S. M., Kruch, S.: Differential continuum damage mechanics models for creep and fatigue of unidirectional metal matrix composites. Technical Memorandum, 105213, NASA ( 1991)
Arnold, S.M., Kruch, S.: A differential cdm model for fatigue of unidirectional metal matrix composites. Technical Memorandum, 105726, NASA (1991)
Barbero, E.J.: Finite Element Analysis of Composite Materials. CRC Press, Boca Raton (2007)
Basaran, C., Lin, M., Ye, H.: A thermodynamic model for electrical current induced damage. Int. J. Solids Struct. 40, 7315–7327 (2003)
Basaran, C., Nie, S.: An irreversible thermodynamic theory for damage mechanics of solids. Int. J. Damage Mech 13, 205–224 (2004)
Basaran, C., Yan, C.Y.: A thermodynamic framework for damage mechanics of solder joints. Trans. ASME J. Electron. Packag. 120, 379–384 (1998)
Berryman, J.G.: Explicit Schemes for Estimating Elastic Properties of Multiphase Composites. Stanford Exploration Project, Report 79, November 16, pp. 1–259 (1997)
Chaboche, J.L.: Continuum damage mechanics: a tool to describe phenomena before crack initiation. Nucl. Eng. Des. 64, 233–247 (1981)
Chaboche, J.L.: Continuum Damage Mechanics: Present State and Future Trends. International Seminar on Modern Local Approach of Fracture. Moret-sur-Loing, France (1986)
Chaboche, J.L.: Continuum damage mechanics: part I: general concepts. J. Appl. Mech. ASME 55, 59–64 (1988)
Chaboche, J.L.: Continuum damage mechanics: part II: damage growth, crack initiation, and crack growth. J. Appl. Mech. ASME 55, 65–72 (1988)
Chateau, C., Gelebart, L., Bornert, M., Crepin, J., Caldemaison, D.: Modeling of damage in unidirectional ceramic matrix composites and multi-scale experimental validation on third generation sic/sic mini-composites. J. Mech. Phys. Solids 63, 298–319 (2014)
Chiang, Y.-C.: Mechanics of matrix cracking in bonded composites. J. Mech. 23, 95–106 (2007)
Darabi, M.K.: Abu Al-Rub, R.K., Little, D.N.: A continuum damage mechanics framework for modeling micro-damage healing. Int. J. Solids Struct. 49, 492–513 (2012)
Doghri, I.: Mechanics of Deformable Solids: Linear and Nonlinear. Analytical and Computational Aspects. Springer, Berlin (2000)
Dvorak, G.J., Bahei-El-Din, Y.A.: A bimodal plasticity theory of fibrous composite materials. Acta Mech. 69, 219–244 (1987)
Gavazzi, A.C., Lagoudas, D.C.: On the numerical evaluation of Eshelby’s tensor and its applications to elasto-plastic fibrous composites. Comput. Mech. 7, 13–19 (1990)
Gent, A.N., Wang, C.: Matrix Cracking Initiated by Fiber Breaks in Model Composites. Office of Naval Research, Technical Report No. 28, 62 pages (1991)
Hansen, N.R., Schreyer, H.L.: A thermodynamically consistent framework for theories of elastoplasticity coupled with damage. Int. J. Solids Struct. 31, 359–389 (1994)
Hutchinson, J.W., Jensen, H.M.: Models of fiber debonding and pullout in brittle composites with friction. Mech. Mater. 9, 139–163 (1990)
Hyer, M.: Stress Analysis of Fiber-Reinforced Composite Materials. McGraw-Hill, New York (1998)
Kachanov, L.: On the creep fracture time. Izv Akad Nauk USSR Otd Tech. 8, 26–31 (1958). (in Russian)
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, 421–435 (1990)
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, 183–199 (1993)
Kattan, P.I., Voyiadjis, G.Z.: Micromechanical modeling of damage in uniaxially loaded unidirectional fiber-reinforced composite lamina. Int. J. Solids Struct. 30, 19–36 (1993)
Kattan, P.I., Voyiadjis, G.Z.: Overall damage and elasto-plastic deformation in fibrous metal matrix composites. Int. J. Plast 9, 931–949 (1993)
Kattan, P.I., Voyiadjis, G.Z.: Damage-plasticity in a uniaxially loaded composite lamina: overall analysis. Int. J. Solids Struct. 33, 555–576 (1993)
Kattan, P.I., Voyiadjis, G.Z.: Decomposition of damage tensor in continuum damage mechanics. J. Eng. Mech. ASCE 127, 940–944 (2001)
Kattan, P.I., Voyiadjis, G.Z.: Damage Mechanics with Finite Elements: Practical Applications with Computer Tools. Springer, Berlin (2001)
Krajcinovic, D.: Damage Mechanics. North Holland, New York (1996)
Kuster, G.T., Toksoz, M.N.: Velocity and attenuation of seismic waves in two-phase media, part I: theoretical formulations. Geophysics 39, 587–606 (1974)
Ladeveze, P., Lemaitre, J.: Damage effective stress in quasi-unilateral conditions. In: The 16th International Congress of Theoretical and Applied Mechanics. Lyngby, Denmark (1984)
Ladeveze, P., Poss, M., Proslier, L.: Damage and fracture of tridirectional composites. In: Progress in Science and Engineering of Composites. Proceedings of the Fourth International Conference on Composite Materials, Japan Society for Composite Materials. Vol. 1, pp. 649–658 (1982)
Lagoudas, D.C., Gavazzi, A.C., Nigam, H.: Elasto-plastic behavior of metal matrix composites based on incremental plasticity and the Mori-Tanaka averaging scheme. Comput. Mech. 8, 193–203 (1991)
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)
Lemaitre, J.: A continuous damage mechanics model for ductile fracture. J. Eng. Mater. Technol. 107, 83–89 (1985)
Lemaitre, J.: Local approach of fracture. Eng. Fract. Mech. 25, 523–537 (1986)
Lemaitre, J.: A Course on Damage Mechanics. Springer, New York (1992)
Lemaitre, J., Chaboche, J.L.: Mecanique de Materiaux Solides. Dunod, Paris (1985)
Lemaitre, J., Dufailly, J.: Damage measurements. Eng. Fract. Mech. 28, 643–661 (1987)
Li, S., Basaran, C.: A computational damage mechanics model for thermomigration. Mech. Mater. 41, 271–278 (2009)
Li, S., Ghosh, S.: Modeling interfacial debonding and matrix cracking in fiber-reinforced composites by the extended voronoi cell FEM. Finite Elem. Anal. Des. 43, 397–410 (2007)
Li, S., Reid, S.R., Soden, P.D.: A continuum damage model for transverse matrix cracking in laminated fiber-reinforced composites. Philos. Trans. R. Soc. A 356, 2379–2412 (1998)
Liu, Y.J., Xu, N.: Modeling of interface cracks in fiber-reinforced composites with the presence of interphases using the boundary element method. Mech. Mater. 32, 769–783 (2000)
Lubineau, G.: A pyramidal modeling scheme for laminates—identification of transverse cracking. Int. J. Damage Mech 19, 499–518 (2010)
Lubineau, G., Ladeveze, P.: Construction of a micromechanics-based intralaminar mesomodel, and illustrations in ABAQUS/standard. Comput. Mater. Sci. 43, 137–145 (2008)
Lubliner, J.: Plasticity Theory. Dover Books on Engineering, Mineola (2008)
Milton, G.W.: The Theory of Composites (Cambridge Monographs on Applied and Computational Mathematics). Cambridge University Press, Cambridge (2002)
Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574 (1973)
Pollayi, H., Wu, W.: Modeling matrix cracking in composite rotor blades with VABS framework. Compos. Struct. 110, 62–76 (2014)
Pupurs, A., Varna, J.: FEM modeling of fiber/matrix debond growth in tensio-tension cyclic loading of unidirectional composites. Int. J. Damage Mech. 22, 1144–1160 (2013)
Rabotnov, Y.: Creep rupture. In: Hetenyi, M., Vincenti, W.G. (eds.) Proceedings, Twelfth International Congress of Applied Mechanics, pp. 342–349. Stanford, 1968. Springer, Berlin (1969)
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)
Sidoroff, F.: Description of anisotropic damage application in elasticity. In: IUTAM Colloqium on Physical Nonlinearities in Structural Analysis, pp. 237–244. Springer, Berlin (1981)
Swolfs, Y., McMeeking, R.M., Verpoest, I., Gorbatikh, L.: Matrix cracks around fiber breaks and their effect on stress redistribution and failure development in unidirectional composites. Compos. Sci. Technol. (2015). doi:10.1016/j.compscitech.2015.01.002
Voyiadjis, G.Z.: Degradation of elastic modulus in elastoplastic coupling with finite strains. Int. J. Plast. 4, 335–353 (1988)
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, 505–524 (1990)
Voyiadjis, G.Z., Kattan, P.I.: A plasticity-damage theory for large deformation of solids—part I: theoretical formulation. Int. J. Eng. Sci. 30, 1089–1108 (1992)
Voyiadjis, G.Z., Kattan, P.I.: Damage of fiber-reinforced composite materials with micromechanical characterization. Int. J. Solids Struct. 30, 2757–2778 (1993)
Voyiadjis, G.Z., Kattan, P.I.: Local approach to damage in elasto-plastic metal matrix composites. Int. J. Damage Mech 2, 92–114 (1993)
Voyiadjis, G.Z., Kattan, P.I.: Micromechanical characterization of damage-plasticity in metal matrix composites. In: Voyiadjis, G.Z. (ed.) Studies in Applied Mechanics. Damage in Composite Materials, vol. 34, pp. 67–102. Elsevier, Amsterdam, Netherlands (1993)
Voyiadjis, G.Z., Kattan, P.I.: Evolution of a damage tensor for metal matrix composites. In: MECAMAT 93: International Seminar on Micromechanics of Materials, vol. 84, pp. 406–417. Moret-sur-Loing, France (1993)
Voyiadjis, G.Z., Kattan, P.I.: Equivalence of the overall and local approaches to damage in metal matrix composites. Int. J. Plast 14, 273–288 (1998)
Voyiadjis, G.Z., Kattan, P.I.: Damage Mechanics. Taylor and Francis (CRC Press), Boca Raton (2005)
Voyiadjis, G.Z., Kattan, P.I.: Mechanics of Composite Materials with MATLAB. Springer, Berlin (2005)
Voyiadjis, G.Z., Kattan, P.I.: Advances in Damage Mechanics: Metals and Metal Matrix Composites with an Introduction to Fabric Tensors, 2nd edn. Elsevier, Amsterdam (2006)
Voyiadjis, G.Z., Kattan, P.I.: A comparative study of damage variables in continuum damage mechanics. Int. J. Damage Mech. 18, 315–340 (2009)
Voyiadjis, G.Z., Kattan, P.I.: Mechanics of damage processes in series and in parallel: a conceptual framework. Acta Mech. 223, 1863–1878 (2012)
Voyiadjis, G.Z., Kattan, P.I.: Mechanics of damage, healing, damageability, and integrity of materials: a conceptual framework. Int. J. Damage Mech., p. 55. (2016a). doi:10.1177/1056789516635730
Voyiadjis, G.Z., Kattan, P.I.: Elasticity of damaged graphene: a damage mechanics approach. Int. J. Damage Mech., p. 50. (2016b). doi:10.1177/1056789516656747
Voyiadjis, G.Z., Park, T.: Elasto-plastic stress and strain concentration tensors for damaged fibrous composites. Stud. Appl. Mech. 44, 81–106 (1996). doi:10.1016/S0922-5382(96)80006-7
Voyiadjis, G.Z., Park, T.: Local and interfacial damage analysis of metal matrix composites using the finite element method. J. Eng. Fracture Mech. 56, 483–511 (1997)
Voyiadjis, G.Z., Shojaei, A., Li, G.: A thermodynamic consistent damage and healing model for self-healing materials. Int. J. Plast 27, 1025–1044 (2011)
Voyiadjis, G.Z., Shojaei, A., Li, G., Kattan, P.I.: A theory of anisotropic healing and damage mechanics of materials. Proc. R. Soc. A Math. Phys. Eng. Sci. 468, 163–183 (2012)
Voyiadjis, G.Z., Shojaei, A., Li, G., Kattan, P.I.: Continuum Damage–Healing mechanics with introduction to new healing variables. Int. J. Damage Mech 21, 391–414 (2012)
Voyiadjis, G.Z., Venson, A.R., Kattan, P.I.: Experimental determination of damage parameters in uniaxially-loaded metal matrix composites using the overall approach. Int. J. Plast 11, 895–926 (1995)
Weng, G.: Some elastic properties of reinforced solids with special reference to isotropic ones containing spherical inclusions. Int. J. Eng. Sci. 22, 845–856 (1984)
Yao, W., Basaran, C.: Computational damage mechanics of electromigration and thermomigration. J. Appl. Phys. 114, 103708 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Voyiadjis, G.Z., Kattan, P.I. Introducing damage mechanics templates for the systematic and consistent formulation of holistic material damage models. Acta Mech 228, 951–990 (2017). https://doi.org/10.1007/s00707-016-1747-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-016-1747-6