Skip to main content
SpringerLink
Log in

Ductile Failure Modeling: Stress Dependence, Non-locality and Damage to Fracture Transition

  • Reference work entry
  • First Online:
Handbook of Damage Mechanics

Abstract

In this chapter, some recent developments and proposals for improvement of material models at the constitutive level to deal with ductile damage at large plastic strains are addressed. Numerical tests are carried out to test their performance on shear-dominated stress states where their main differences lie. Subsequently, aspects of the use of nonlocal models for the regularization of the numerical values associated with damage models, namely, discretization dependency, are reviewed. Different approaches on the choice of the regulation variable or variables are tested at different stress states characterized by different values of triaxiality and third invariant of the deviatoric stress tensor. Finally, a simple strategy on how to handle the transition from damage to fracture by means of the extended finite element method is described.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 649.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Y. Abdelaziz, A. Hamouine, A survey of the extended finite element. Comput. Struct. 86(11–12), 1141–1151 (2008)

    Article  Google Scholar 

  • J. Alfaiate, G. Wells, L. Sluys, On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture. Eng. Fract. Mech. 69(6), 661–686 (2002)

    Article  Google Scholar 

  • F.X.C. Andrade, Non-local modelling of ductile damage: formulation and numerical issues. PhD Thesis, Faculty of Engineering, University of Porto, Porto, Portugal, 2011

    Google Scholar 

  • F.X.C. Andrade, F.M. Andrade Pires, J.M.A. Cesar de Sa, L. Malcher, Nonlocal integral formulation for a plasticity-induced damage model. Comput. Methods Mater. Sci. 9(1), 49–54 (2009)

    Google Scholar 

  • F. Andrade, J. Cesar de Sa, F. Andrade Pires, A ductile damage nonlocal model of integral-type at finite strains: formulation and numerical issues. Int. J. Damage Mech. 20, 515–557 (2011b)

    Article  Google Scholar 

  • P. Areias, N. Van Goethem, E. Pires, A damage model for ductile crack initiation and propagation. Comput. Mech. 47, 641–656 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • F. Armero, K. Garikipati, An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids. Int. J. Solids Struct. 33(20–22), 2863–2885 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • A.G. Atkins, Possible explanation for unexpected departures in hydrostatic tension-fracture strain relations. Metal Sci. 15, 81–83 (1981)

    Article  Google Scholar 

  • Y. Bai, Effect of loading history on necking and fracture. PhD Thesis, Massachusetts Institute of Technology, 2008

    Google Scholar 

  • Y. Bai, T. Wierzbicki, A new model of metal plasticity and fracture with pressure and Lode dependence. Int. J. Plast. 24, 1071–1096 (2008)

    Article  MATH  Google Scholar 

  • Y. Bao, Prediction of ductile crack formation in uncracked bodies. PhD Thesis, Massachusetts Institute of Technology, 2003

    Google Scholar 

  • Y. Bao, T. Wierzbicki, On fracture locus in the equivalent strain and stress triaxiality space. Int. J. Mech. Sci. 46(81), 81–98 (2004)

    Article  Google Scholar 

  • I. Barsoum, J. Faleskog, Rupture in combined tension and shear: experiments. Int. J. Solids Struct. 44, 1768–1786 (2007a)

    Article  MATH  Google Scholar 

  • I. Barsoum, J. Faleskog, Rupture in combined tension and shear: micromechanics. Int. J. Solids Struct. 44, 5481–5498 (2007b)

    Article  MATH  Google Scholar 

  • S.R. Beissel, G.R. Johnson, C.H. Popelar, An element-failure algorithm for dynamic crack propagation in general directions. Eng. Fract. Mech. 61(3–4), 407–425 (1998)

    Article  Google Scholar 

  • T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601–620 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • T. Belytschko, J. Fish, B.E. Engelmann, A finite element with embedded localization zones. Comput. Methods Appl. Mech. Eng. 70(1), 59–89 (1988)

    Article  MATH  Google Scholar 

  • T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures (Wiley, Chichester, 2000)

    MATH  Google Scholar 

  • E. Benvenuti, A regularized XFEM framework for embedded cohesive interfaces. Comput. Methods Appl. Mech. Eng. 197, 4367–4607 (2008)

    Article  MATH  Google Scholar 

  • E. Benvenuti, A. Tralli, Iterative LCP solver for non-local loading unloading conditions. Int. J. Numer. Methods Eng. 58, 2343–2370 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • G. Borino, P. Fuschi, C. Polizzotto, A thermodynamic approach to nonlocal plasticity and related variational principles. J. Appl. Mech. 66, 952–963 (1999)

    Article  Google Scholar 

  • P.O. Bouchard, F. Bay, Y. Chastel, I. Tovena, Crack propagation modelling using an advanced remeshing technique. Comput. Methods Appl. Mech. Eng. 189(3), 723–742 (2000)

    Article  MATH  Google Scholar 

  • P.W. Bridgman, Studies in Large Plastic Flow and Fracture (McGraw-Hill Book, New-York, 1952)

    MATH  Google Scholar 

  • M. Brünig, S. Berger, H. Obrecht, Numerical simulation of the localization behavior of hydrostatic-stress-sensitive metals. Int. J. Mech. Sci. 42, 2147–2166 (2008)

    Article  Google Scholar 

  • M. Brunig, O. Chyra, D. Albrecht, L. Driemeier, M. Alves, A ductile damage criterion at various stress triaxialities. Int. J. Plast. 24, 1731–1755 (2008)

    Article  Google Scholar 

  • F. Cazes, M. Coret, A. Combescure, A. Gravouil, A thermodynamic method for the construction of a cohesive law from a non local damage model. Int. J. Solids Struct. 46, 1476–1490 (2009)

    Article  MATH  Google Scholar 

  • F. Cazes, A. Simatos, M. Coret, A. Combescure, A cohesive zone model which is energetically equivalent to a gradient-enhanced coupled damage-plasticity model. Eur. J. Mech. A/Solids 29, 976–998 (2010)

    Article  MathSciNet  Google Scholar 

  • J.M.A. Cesar de Sa, P.M.A. Areias, C. Zheng, Damage modelling in metal forming problems using an implicit non-local gradient model. Comput. Methods Appl. Mech. Eng. 195, 6646–6660 (2006)

    Article  MATH  Google Scholar 

  • J.M.A. Cesar de Sa, F.M. Andrade Pires, F.X.C. Andrade, Local and nonlocal modeling of ductile damage, in Advanced Computational Materials Modelling: From Classical to Multi-Scale Techniques, ed. by M. Vaz Jr., E.A. De Souza Neto, P.A. Muñoz-Rojas (Wiley-VCH, Weinheim, 2010)

    Google Scholar 

  • J.L. Chaboche, M. Boudifa, K.A. Saanouni, CDM approach of ductile damage with plastic compressibility. Int. J. Fract. 137, 51–75 (2006)

    Article  MATH  Google Scholar 

  • T.P. Chang, Z.P. Bazant, Instability of nonlocal continuum and strain averaging. J. Eng. Mech. ASCE 110, 1441–1450 (1984)

    Article  Google Scholar 

  • J. Chessa, H. Wang, T. Belytschko, On the construction of blending elements for local partition of unity enriched finite elements. Int. J. Numer. Methods Eng. 57, 1015–1038 (2003)

    Article  MATH  Google Scholar 

  • M.G. Cockcroft, D.J. Latham, Ductility and workability of metals. J. Inst. Metals 96, 33–39 (1968)

    Google Scholar 

  • J.A. Cottrell, T. Hughes, Y. Bazilevs, Isogeometric Analysis – Toward Integration of CAD and FEA (Wiley, Chichester, 2009)

    Book  Google Scholar 

  • J. Datsko, Material Properties and Manufacturing Process (Wiley, New York, 1966)

    Google Scholar 

  • C. de Boor, A Practical Guide to Splines (Springer, New York, 1978)

    Book  MATH  Google Scholar 

  • R. De Borst, H. Mühlhaus, Gradient-dependent plasticity: formulation and algorithmic aspects. Int. J. Numer. Methods Eng. 35, 521–539 (1992)

    Article  MATH  Google Scholar 

  • E.A. De Souza Neto, D. Peric, D.R.J. Owen, Computational Methods for Plasticity: Theory and Applications (Wiley, Chichester, 2008)

    Book  Google Scholar 

  • J.H.P. De Vree, W.A.M. Brekelmans, M.A.J. van Gils, Comparison of nonlocal approaches in continuum damage mechanics. Comput. Struct. 4, 581–588 (1995)

    Article  Google Scholar 

  • L. Driemeier, M. Brünig, G. Micheli, M. Alves, Experiments on stress-triaxiality dependence of material behavior of aluminum alloys. Mech. Mater. 42(2), 207–217 (2010)

    Article  Google Scholar 

  • D. Edelen, N. Laws, On the thermodynamics of systems with nonlocality. Arch. Ration. Mech. Anal. 43, 24–35 (1971)

    MathSciNet  MATH  Google Scholar 

  • D. Edelen, A. Green, N. Laws, Nonlocal continuum mechanics. Arch. Ration. Mech. Anal. 43, 36–44 (1971)

    MathSciNet  MATH  Google Scholar 

  • K. Enakousta, J.B. Leblond, G. Perrin, Numerical implementation and assessment of a phenomenological nonlocal model of ductile rupture. Comput. Methods Appl. Mech. Eng. 196, 1946–1957 (2007)

    Article  Google Scholar 

  • R. Engelen, M. Geers, R. Ubachs, Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. Int. J. Plast. 19(4), 403–433 (2003)

    Article  MATH  Google Scholar 

  • M. Fagerstrom, R. Larsson, A thermo-mechanical cohesive zone formulation for ductile fracture. J. Mech. Phys. Solids 56(10), 3037–3058 (2008)

    Article  MathSciNet  Google Scholar 

  • M. Feucht, Ein gradientenabhangiges Gursonmodell zur Beschreibung duktiler Schadigung mit Entfestigung. PhD Thesis, Technische Universitat Darmstadt, 1999

    Google Scholar 

  • A.M. Freudenthal, The Inelastic Behaviour of Engineering Materials and Structures (Wiley, New York, 1950)

    Google Scholar 

  • T.-P. Fries, A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Methods Eng. 75, 503–532 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • T.-P. Fries, T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84, 253–304 (2010)

    MathSciNet  MATH  Google Scholar 

  • X. Gao, J. Kim, Modeling of ductile fracture: significance of void coalescence. Int. J. Solids Struct. 43, 6277–6293 (2006)

    Article  MATH  Google Scholar 

  • X. Gao, T. Wang, J. Kim, On ductile fracture initiation toughness: effects of void volume fraction, void shape and void distribution. Int. J. Solids Struct. 42, 5097–5117 (2005)

    Article  MATH  Google Scholar 

  • X. Gao, G. Zhang, C. Roe, A study on the effect of the stress state on ductile fracture. Int. J. Damage Mech. 19, 75–94 (2009)

    Google Scholar 

  • X. Gao, T. Zhang, J. Zhou, S.M. Graham, M. Hayden, C. Roe, On stress-state dependent plasticity modeling: significance of the hydrostatic stress, the third invariant of stress deviator and the non-associated flow rule. Int. J. Plast. 27(2), 217–231 (2011)

    Article  Google Scholar 

  • A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth – Part I. Yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99, 2–15 (1977)

    Article  Google Scholar 

  • P. Hakansson, M. Wallin, M. Ristinmaa, Thermomechanical response of non-local porous material. Int. J. Plast. 22, 2066–2090 (2006)

    Article  MATH  Google Scholar 

  • J.W. Hancock, A.C. Mackenzie, On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. J. Mech. Phys. Solids 24, 147–160 (1976)

    Article  Google Scholar 

  • A. Huespe, A. Needleman, J. Oliver, P.J. Sanchez, A finite strain, finite band method for modeling ductile fracture. Int. J. Plast. 28(1), 53–69 (2012)

    Article  Google Scholar 

  • M. Jirasek, Nonlocal models for damage and fracture: comparison of approaches. Int. J. Solids Struct. 35, 4133–4145 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • M. Jirasek, Comparative study on finite elements with embedded discontinuities. Comput. Methods Appl. Mech. Eng. 188(1–3), 307–330 (2000)

    Article  MATH  Google Scholar 

  • M. Jirásek, Nonlocal damage mechanics. Revue Européene de Génie Civil 11, 993–1021 (2007)

    Article  Google Scholar 

  • M. Jirásek, S. Rolshoven, Comparison of integral-type nonlocal plasticity models for strain-softening materials. Int. J. Eng. Sci. 41, 1553–1602 (2003)

    Article  MATH  Google Scholar 

  • M. Jirasek, T. Zimmermann, Analysis of rotating crack model. J. Eng. Mech., ASCE 124, 842–851 (1998)

    Article  Google Scholar 

  • G.R. Johnson, W.H. Cook, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech. 21(1), 31–48 (1985)

    Article  Google Scholar 

  • L.M. Kachanov, Time of the rupture process under creep condition. Izv. Akad. Nauk. SSSR, Otd. Tekhn. Nauk 8, 26–31 (1958)

    Google Scholar 

  • J. Kim, X. Gao, T.S. Srivatsan, Modeling of crack growth in ductile solids: a three-dimensional analysis. Int. J. Solids Struct. 40, 7357–7374 (2003)

    Article  MATH  Google Scholar 

  • J. Kim, X. Gao, T.S. Srivatsan, Modeling of void growth in ductile solids: effects of stress triaxiality and initial porosity. Eng. Fract. Mech. 71, 379–400 (2004)

    Article  Google Scholar 

  • J. Kim, G. Zhang, X. Gao, Modeling of ductile fracture: application of the mechanism-based concepts. Int. J. Solids Struct. 44, 1844–1862 (2007)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  • J. Lemaitre, Coupled elasto-plasticity and damage constitutive equations. Comput. Methods Appl. Mech. Eng. 51, 31–49 (1985b)

    Article  MATH  Google Scholar 

  • J. Lemaître, Local Approach of fracture. Eng. Fract. Mech. 25, 523–537 (1986)

    Article  Google Scholar 

  • J. Lemaitre, A Course on Damage Mechanics (Springer, New York, 1996)

    Book  MATH  Google Scholar 

  • J. Lemaitre, R. Desmorat, Engineering Damage Mechanics (Springer, Berlin, 2005)

    Google Scholar 

  • F.A. McClintock, A criterion for ductile fracture by growth of holes. J. Appl. Mech. 35, 363–371 (1968)

    Article  Google Scholar 

  • J. Mediavilla, Continuous and discontinuous modeling of ductile fracture. PhD Thesis, Technische Universiteit Eindhoven, 2005

    Google Scholar 

  • J. Mediavilla, R.H.J. Peerlings, M.G.D. Geers, A robust and consistent remeshing-transfer operator for ductile fracture simulations. Comput. Struct. 84(8–9), 604–623 (2006)

    Article  Google Scholar 

  • G. Mirone, D. Corallo, A local viewpoint for evaluating the influence of stress triaxiality and Lode angle on ductile failure and hardening. Int. J. Plast. 26(3), 348–371 (2010)

    Article  MATH  Google Scholar 

  • M.S. Mirza, D.C. Barton, P. Church, The effect of stress triaxiality and strain rate on the fracture characteristics of ductile metals. J. Mater. Sci. 31, 453–461 (1996)

    Article  Google Scholar 

  • N. Möes, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)

    Article  Google Scholar 

  • N. Möes, C. Stolz, P.-E. Bernard, N. Chevaugeon, A level set based model for damage growth: the thick level set approach. Int. J. Numer. Methods Eng. 86(3), 358–380 (2011)

    Article  Google Scholar 

  • D.M. Norris, J.E. Reaugh, B. Moran, D.F. Quiñones, A plastic-strain, mean-stress criterion for ductile fracture. J. Eng. Mater. Technol., Trans ASME 100, 279–286 (1978)

    Article  Google Scholar 

  • M. Ortiz, Y. Leroy, A. Needleman, A finite element method for localized failure analysis. Comput. Methods Appl. Mech. Eng. 61(2), 189–214 (1987)

    Article  MATH  Google Scholar 

  • M. Oyane, S. Shima, T. Tabata, Considerations of basic equations and their application in the forming of metal powders and porous metals. J. Mech. Tech. 1, 325–341 (1978)

    Article  Google Scholar 

  • R. Peerlings, R. De Borst, W.A.M. Brekelmans, J.H.P. De Vree, Gradient-enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering. 39, 1512–1533 (1996)

    Google Scholar 

  • L. Piegl, Fundamental Developments of Computer Aided Geometric Design (Academic, San Diego, 1993)

    Google Scholar 

  • G. Pijaudier-Cabot, Z.P. Bažant, Nonlocal damage theory. J. Eng. Mech. 113(10), 1512–1533 (1987)

    Article  Google Scholar 

  • C. Polizzotto, A nonlocal strain gradient plasticity theory for finite deformations. Int. J. Plast. 25(7), 1280–1300 (2009)

    Article  MATH  Google Scholar 

  • C. Polizzotto, G. Borino, P. Fuschi, A thermodynamic consistent formulation of nonlocal and gradient plasticity. Mech. Res. Commun. 25(1), 75–82 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Y.N. Rabotnov, On the equations of state for creep, in Progress in Applied Mechanics, Prager Anniversary Volume, New York: MacMillan, pp 307–315 (1963)

    Google Scholar 

  • F. Reusch, B. Svendsen, D. Klingbeil, A non-local extension of Gurson based ductile damage modeling. Comput. Mater. Sci. 26, 219–229 (2003a)

    Article  Google Scholar 

  • F. Reusch, B. Svendsen, D. Klingbeil, Local and non-local Gurson based ductile damage and failure modelling at large deformation. Eur. J. Mech. A/Solids 22, 779–792 (2003b)

    Article  MATH  Google Scholar 

  • J.R. Rice, D.M. Tracey, On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17, 201–217 (1969)

    Article  Google Scholar 

  • K. Saanouni, On the numerical prediction of the ductile fracture in metal forming. Eng. Fract. Mech. 75(11), 3545–3559 (2008)

    Article  Google Scholar 

  • M.K. Samal, M. Seidenfuss, E. Roos, B.K. Dutta, H.S. Kushwaha, Finite element formulation of a new nonlocal damage model. Finite Elem. Anal. Des. 44, 358–371 (2008)

    Article  Google Scholar 

  • P.J. Sanchez, A.E. Huespe, J. Oliver, On some topics for the numerical simulation of ductile fracture. Int. J. Plast. 24(6), 1008–1038 (2008)

    Article  MATH  Google Scholar 

  • M. Seabra, P. Sustaric, J. Cesar de Sa, T. Rodic, Damage driven crack initiation and propagation in ductile metals using XFEM. Comput. Mech. (2012). doi:10.1007/s00466-012-0804-9

    Google Scholar 

  • J.C. Simo, T.J.R. Hughes, Computational Inelasticity (Springer, New York, 1998)

    MATH  Google Scholar 

  • J.C. Simo, J. Oliver, F. Armero, An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput. Mech. 12, 277–296 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • A. Simone, G. Wells, L. Sluys, From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Comput. Methods Appl. Mech. Eng. 192, 4581–4607 (2003)

    Article  MATH  Google Scholar 

  • J.-H. Song, H. Wang, T. Belytschko, A comparative study on finite element methods for dynamic fracture. Comput. Mech. 42(2), 239–250 (2008)

    Article  MATH  Google Scholar 

  • L. Strömberg, M. Ristinmaa, FE-formulation of a nonlocal plasticity theory. Comput. Methods Appl. Mech. Eng. 136, 127–144 (1996)

    Article  MATH  Google Scholar 

  • W. Tai, B.X. Yang, A new damage mechanics criterion for ductile fracture. Eng. Fract. Mech. 27, 371–378 (1987)

    Article  Google Scholar 

  • X. Teng, Numerical prediction of slant fracture with continuum damage mechanics. Eng. Fract. Mech. 75, 2020–2041 (2008)

    Article  Google Scholar 

  • V. Tvergaard, A. Needleman, Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 32, 157–169 (1984)

    Article  Google Scholar 

  • V. Tvergaard, A. Needleman, Effects of nonlocal damage in porous plastic solids. Int. J. Solids Struct. 32(8/9), 1063–1077 (1995)

    Article  MATH  Google Scholar 

  • M. Vaz, D.R.J. Owen, Aspects of ductile fracture and adaptive mesh re-finement in damaged elasto-plastic materials. Int. J. Numer. Methods Eng. 50(1), 29–54 (2001)

    Article  MATH  Google Scholar 

  • G. Voyiadjis, G. Pekmezi, B. Deliktas, Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening. Int. J. Plast. 26, 1335–1356 (2010)

    Article  MATH  Google Scholar 

  • L. Xue, Ductile Fracture Modeling – Theory, Experimental Investigation and Numerical Verification (Massachusetts Institute of Technology, 2007)

    Google Scholar 

  • L. Xue, Constitutive modeling of void shearing effect in ductile fracture of porous materials. Eng. Fract. Mech. 75, 3343–3366 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, s/n, 4200-465, Porto, Portugal

    J. M. A. Cesar de Sa, F. M. A. Pires & M. R. R. Seabra

  2. DYNAmore GmbH, Industriestr. 2, D-70565, Stuttgart, Germany

    F. X. C. Andrade

  3. Department of Mechanical Engineering, University of Brasilia, Campus Darcy Ribeiro, EdifÚcio SG-9, 70910900, Asa Norte - Brasília, DF, Brazil

    L. Malcher

Authors
  1. J. M. A. Cesar de Sa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. M. A. Pires
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. X. C. Andrade
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. L. Malcher
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. M. R. R. Seabra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. M. A. Cesar de Sa .

Editor information

Editors and Affiliations

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

    George Z. Voyiadjis

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this entry

Cite this entry

de Sa, J.M.A.C., Pires, F.M.A., Andrade, F.X.C., Malcher, L., Seabra, M.R.R. (2015). Ductile Failure Modeling: Stress Dependence, Non-locality and Damage to Fracture Transition. In: Voyiadjis, G. (eds) Handbook of Damage Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5589-9_39

Download citation

Publish with us

Policies and ethics

Search

Navigation