Skip to main content
SpringerLink
Log in

Modeling of Nonlocal Damage Using the Phase Field Method

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

  • 16k Accesses

Abstract

In this chapter, the application of the phase field method (PFM) into continuum damage mechanics is discussed. It is shown that the effect of the damage gradient can be deduced using the PFM which yields a nonlocal damage model. This is derived for isotropic damage using a scalar variable. The derivation is in the elastic region and the damage rate equation shows the evolution of damage for brittle materials. However, this theory may be coupled with a plasticity model. The framework of the phase field method is discussed in a simple scalar form. After a brief review of isotropic damage, the order parameter is related to the damage variable and a free energy functional in damaged materials is derived which is capable in capturing the evolution of nonlocal damage through the Allen–Cahn equation. It is shown that there is no need to follow the conventional normality rule – which is common in previously proposed models – using this variational approach. Specific length scale due to damage is proposed and the general state of stress with scalar damage variable is discussed. Details of three different finite difference schemes are discussed and the application and regularization capabilities of the model are demonstrated by a 1D numerical example.

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

Similar content being viewed by others

References

  • R.K. Abu Al-Rub, G.Z. Voyiadjis, On the coupling of anisotropic damage and plasticity models for ductile materials. Int. J. Solids Struct. 40, 2611–2643 (2003)

    Article  MATH  Google Scholar 

  • S.M. Allen, J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)

    Article  Google Scholar 

  • H. Amor, J.-J. Marigo, C. Maurini, Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J. Mech. Phys. Solids 57, 1209–1229 (2009)

    Article  MATH  Google Scholar 

  • I. Aranson, V. Kalatsky, V. Vinokur, Continuum field description of crack propagation. Phys. Rev. Lett. 85, 118–121 (2000)

    Article  Google Scholar 

  • P.W. Bates, S. Brown, J. Han, Numerical analysis for a nonlocal Allen-Cahn equation. Num. Anal. Model. 6, 33–49 (2009)

    MathSciNet  MATH  Google Scholar 

  • W. Boettinger, J. Warren, C. Beckermann, A. Karma, Phase-field simulation of solidification 1. Annu. Rev. Mater. Res. 32, 163–194 (2002)

    Article  Google Scholar 

  • M.J. Borden, C.V. Verhoosel, M.A. Scott, T.J. Hughes, C.M. Landis, A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217, 77–95 (2012)

    Article  MathSciNet  Google Scholar 

  • G. Boussinot, Y. Le Bouar, A. Finel, Phase-field simulations with inhomogeneous elasticity: comparison with an atomic-scale method and application to superalloys. Acta Mater. 58, 4170–4181 (2010)

    Article  Google Scholar 

  • J. Cahn, S. Allen, A microscopic theory for domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics. Le J. de Phys. Colloques 38, 7–7 (1977)

    Google Scholar 

  • J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258 (1958)

    Article  Google Scholar 

  • P.R. Cha, D.H. Yeon, J.K. Yoon, A phase field model for isothermal solidification of multicomponent alloys. Acta Mater. 49, 3295–3307 (2001)

    Article  Google Scholar 

  • L.Q. Chen, Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140 (2002)

    Article  Google Scholar 

  • L. Chen, J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 108, 147–158 (1998)

    Article  MATH  Google Scholar 

  • J.W. Choi, H.G. Lee, D. Jeong, J. Kim, An unconditionally gradient stable numerical method for solving the Allen–Cahn equation. Phys.Statist. Mech. Appl. 388, 1791–1803 (2009)

    Article  MathSciNet  Google Scholar 

  • R. Courant, K. Friedrichs, H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  • M. Del Pino, M. Kowalczyk, F. Pacard, J. Wei, Multiple-end solutions to the Allen–Cahn equation in R2. J. Funct. Anal. 258, 458–503 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • R.J. Dorgan, G.Z. Voyiadjis, Nonlocal coupled damage-plasticity model incorporating functional forms of hardening state variables. AIAA J. 45, 337–346 (2007)

    Article  Google Scholar 

  • K. Elder, N. Provatas, Phase-Field Methods in Materials Science and Engineering, 1st edn. (Wiley-VCH, Weinheim, 2010)

    Google Scholar 

  • D. Fan, L.Q. Chen, Computer simulation of grain growth using a continuum field model. Acta Mater. 45, 611–622 (1997)

    Article  Google Scholar 

  • X. Feng, A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • W. Feng, P. Yu, S. Hu, Z. Liu, Q. Du, L. Chen, Spectral implementation of an adaptive moving mesh method for phase-field equations. J. Comput. Phys. 220, 498–510 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Furukawa, K. Nakajima, Advances in Crystal Growth Research. (Elsevier Science, Amsterdam, The Netherlands, 2001)

    Google Scholar 

  • A. Gaubert, Y. Le Bouar, A. Finel, Coupling Phase Field and Visco-Plasticity to Study Rafting in Ni-Base Superalloys. (Philosophical Magazine 90, 2010), pp. 375–404

    Google Scholar 

  • V. Ginzburg, L.D. Landau, On the theory of superconductivity, Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950). Translation in Collected papers of L.D.Landau. (Pergamon, Oxford, 1965)

    Google Scholar 

  • L. Gránásy, T. Pusztai, J.A. Warren, Modelling polycrystalline solidification using phase field theory. J. Phys. Condens. Matter 16, R1205 (2004)

    Article  Google Scholar 

  • X. Guo, S.Q. Shi, X. Ma, Elastoplastic phase field model for microstructure evolution. Appl. Phys. Lett. 87, 221910–221910–221913 (2005)

    Google Scholar 

  • X. Guo, S. Shi, Q. Zhang, X. Ma, An elastoplastic phase-field model for the evolution of hydride precipitation in zirconium. Part I: smooth specimen. J. Nuclear Mater. 378, 110–119 (2008)

    Article  Google Scholar 

  • M.E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Phys. Nonlinear Phenomena 92, 178–192 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • S. Hu, L. Chen, A phase-field model for evolving microstructures with strong elastic inhomogeneity. Acta Mater. 49, 1879–1890 (2001)

    Article  Google Scholar 

  • S. Hu, M. Baskes, M. Stan, Phase-field modeling of microvoid evolution under elastic-plastic deformation. Appl. Phys. Lett. 90, 081921–081921–081923 (2007)

    Google Scholar 

  • A. Hunter, M. Koslowski, Direct calculations of material parameters for gradient plasticity. J. Mech. Phys. Solids 56, 3181–3190 (2008)

    Article  MATH  Google Scholar 

  • L.M. Kachanov, On the creep fracture time. Izv Akad. Nauk USSR Otd. Tekh 26–31 (1958)

    Google Scholar 

  • A. Karma, Phase-field formulation for quantitative modeling of alloy solidification. Phys. Rev. Lett. 87, 115701 (2001)

    Article  Google Scholar 

  • A. Karma, D.A. Kessler, H. Levine, Phase-field model of mode III dynamic fracture. Phys. Rev. Lett. 87, 45501 (2001)

    Article  Google Scholar 

  • A.K. Kassam, L.N. Trefethen, Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 1214–1233 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • A.G. Khachaturyan, Theory of Structural Transformations in Solids (Wiley, New York, 1983)

    Google Scholar 

  • M. Koslowski, A.M. Cuitino, M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals. J. Mech. Phys. Solids 50, 2597–2635 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • C. Kuhn, R. Müller, A continuum phase field model for fracture. Eng. Fract. Mech. 77, 3625–3634 (2010)

    Article  Google Scholar 

  • J. Lemaitre, R. Desmorat, Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures (Springer, New York, 2005)

    Google Scholar 

  • V.I. Levitas, I.B. Ozsoy, Micromechanical modeling of stress-induced phase transformations. part 1. thermodynamics and kinetics of coupled interface propagation and reorientation. Int. J. Plast. 25, 239–280 (2009a)

    Article  MATH  Google Scholar 

  • V.I. Levitas, I.B. Ozsoy, Micromechanical modeling of stress-induced phase transformations. Part 2. computational algorithms and examples. Int. J. Plast. 25, 546–583 (2009b)

    Article  MATH  Google Scholar 

  • V.I. Levitas, D.L. Preston, Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. I. Austenite↔ martensite. Phys. Rev. B 66, 134206 (2002)

    Article  Google Scholar 

  • C. Miehe, M. Hofacker, F. Welschinger, A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. 199, 2765–2778 (2010a)

    Article  MathSciNet  MATH  Google Scholar 

  • C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J. Numer. Methods Eng. 83, 1273–1311 (2010b)

    Article  MathSciNet  MATH  Google Scholar 

  • N. Moelans, A quantitative and thermodynamically consistent phase-field interpolation function for multi-phase systems. Acta Mater. 59, 1077–1086 (2011)

    Article  Google Scholar 

  • N. Moelans, B. Blanpain, P. Wollants, An introduction to phase-field modeling of microstructure evolution. Calphad 32, 268–294 (2008)

    Article  Google Scholar 

  • E. Nauman, N.P. Balsara, Phase equilibria and the Landau–Ginzburg functional. Fluid Phase Equilib. 45, 229–250 (1989)

    Article  Google Scholar 

  • S. Nemat-Nasser, Decomposition of strain measures and their rates in finite deformation elastoplasticity. Int. J. Solids Struct. 15, 155–166 (1979)

    Article  MATH  Google Scholar 

  • S. Nemat-Nasser, On Finite Plastic Flow of Crystalline Solids and Geomaterials. J. Appl. Mech. 50(4b), 1114 (1983)

    Google Scholar 

  • N. Ofori-Opoku, N. Provatas, A quantitative multi-phase field model of polycrystalline alloy solidification. Acta Mater. 58, 2155–2164 (2010)

    Article  Google Scholar 

  • M. Ohno, K. Matsuura, Quantitative phase-field modeling for two-phase solidification process involving diffusion in the solid. Acta Mater. 58, 5749–5758 (2010)

    Article  Google Scholar 

  • A. Onuki, Ginzburg-Landau approach to elastic effects in the phase separation of solids. J. Phys. Soc. Jap. 58, 3065–3068 (1989)

    Article  Google Scholar 

  • K. Pham, H. Amor, J.J. Marigo, C. Maurini, Gradient damage models and their use to approximate brittle fracture. Int. J. Damage Mech. 20, 618–652 (2011)

    Article  Google Scholar 

  • D. Rodney, Y. Le Bouar, A. Finel, Phase field methods and dislocations. Acta Mater. 51, 17–30 (2003)

    Article  Google Scholar 

  • T. Sadowski, S. Samborski, Z. Librant, Damage growth in porous ceramics. Key Eng. Mater. 290, 86–93 (2005)

    Article  Google Scholar 

  • D. Salac, W. Lu, Controlled nanocrack patterns for nanowires. J. Comput. Theor. Nanosci. 3, 263–268 (2006)

    Google Scholar 

  • S. Samborski, T. Sadowski, On the method of damage assessment in porous ceramics, in Conference Proceedings of 11th Conference on Fracture, Turin, 2005

    Google Scholar 

  • R. Sankarasubramanian, Microstructural evolution in elastically-stressed solids: a phase-field simulation. Def. Sci. J. 61, 383–393 (2011)

    Article  Google Scholar 

  • J.P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity (Oxford University Press, New York, 2006)

    Google Scholar 

  • J. Shen, X. Yang, Numerical approximations of allen-cahn and cahn-hilliard equations. Discrete Contin. Dyn. Syst 28, 1669–1691 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • F. Sidoroff, Description of Anisotropic Damage Application to Elasticity (Springer, Berlin, 1981), pp. 237–244

    Google Scholar 

  • R. Spatschek, D. Pilipenko, C. Müller-Gugenberger, E.A. Brener, Phase field modeling of fracture and composite materials. Phys. Rev. Lett. 96, 015502 (2006)

    Article  Google Scholar 

  • R. Spatschek, C. Müller-Gugenberger, E. Brener, B. Nestler, Phase field modeling of fracture and stress-induced phase transitions. Phys. Rev. E. 75, 066111 (2007)

    Article  MathSciNet  Google Scholar 

  • I. Steinbach, M. Apel, Multi phase field model for solid state transformation with elastic strain. Phys. Nonlinear Phenomena 217, 153–160 (2006)

    Article  MATH  Google Scholar 

  • I. Steinbach, F. Pezzolla, B. Nestler, M. Seeßelberg, R. Prieler, G. Schmitz, J. Rezende, A phase field concept for multiphase systems. Phys. Nonlinear Phenomena 94, 135–147 (1996)

    Article  MATH  Google Scholar 

  • T. Uehara, T. Tsujino, N. Ohno, Elasto-plastic simulation of stress evolution during grain growth using a phase field model. J. Cryst. Growth 300, 530–537 (2007)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • G.Z. Voyiadjis, B. Deliktas, A coupled anisotropic damage model for the inelastic response of composite materials. Comput. Methods Appl. Mech. Eng. 183, 159–199 (2000)

    Article  MATH  Google Scholar 

  • G.Z. Voyiadjis, R.J. Dorgan, Framework using functional forms of hardening internal state variables in modeling elasto-plastic-damage behavior. Int. J. Plast. 23, 1826–1859 (2007)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • G.Z. Voyiadjis, P.I. Kattan, A plasticity-damage theory for large deformation of solids–I. Theoretical formulation. Int. J. Eng. Sci. 30, 1089–1108 (1992)

    Article  MATH  Google Scholar 

  • G.Z. Voyiadjis, P.I. Kattan, Advances in Damage Mechanics: Metals and Metal Matrix Composites (Elsevier, Oxford, ISBN 0-08-043601-3, 1999), p. 542

    Google Scholar 

  • G.Z. Voyiadjis, P.I. Kattan, Advances in Damage Mechanics: Metals and Metal Matrix Composites with an Introduction to Fabric Tensors. (2nd edn.) (Elsevier, Oxford, London, ISBN: 0-08-044688-4, 2006), p. 742

    Google Scholar 

  • G.Z. Voyiadjis, P.I. Kattan, A new class of damage variables in continuum damage mechanics. J. Eng. Mater. Technol. 134 (2012)

    Google Scholar 

  • G.Z. Voyiadjis, N. Mozaffari, Nonlocal damage model using the phase field method: theory and applications. Int. J. Solids Struct. 50, 3136–3151 (2013)

    Article  Google Scholar 

  • G.Z. Voyiadjis, R.K. Abu Al-Rub, A.N. Palazotto, Thermodynamic framework for coupling of non-local viscoplasticity and non-local anisotropic viscodamage for dynamic localization problems using gradient theory. Int. J. Plast. 20, 981–1038 (2004)

    Article  MATH  Google Scholar 

  • G.Z. Voyiadjis, B. Deliktas, A.N. Palazotto, Thermodynamically consistent coupled viscoplastic damage model for perforation and penetration in metal matrix composite materials. Compos. Part B 40, 427–433 (2009)

    Article  Google Scholar 

  • S.-L. Wang, R. Sekerka, A. Wheeler, B. Murray, S. Coriell, R. Braun, G. McFadden, Thermodynamically-consistent phase-field models for solidification. Phys. Nonlinear Phenomena 69, 189–200 (1993)

    Article  MATH  Google Scholar 

  • Y. Wang, Y. Jin, A. Cuitino, A. Khachaturyan, Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations. Acta Mater. 49, 1847–1857 (2001)

    Article  Google Scholar 

  • Y.U. Wang, Y.M. Jin, A.G. Khachaturyan, Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid. J. Appl. Phys. 92, 1351–1360 (2002)

    Article  Google Scholar 

  • J.A. Warren, R. Kobayashi, A.E. Lobkovsky, W. Craig Carter, Extending phase field models of solidification to polycrystalline materials. Acta Mater. 51, 6035–6058 (2003)

    Article  Google Scholar 

  • A. Yamanaka, T. Takaki, Y. Tomita, Elastoplastic phase-field simulation of self-and plastic accommodations in Cubic–> tetragonal martensitic transformation. Mater. Sci. Eng. A 491, 378–384 (2008)

    Article  Google Scholar 

  • P. Yu, S. Hu, L. Chen, Q. Du, An iterative-perturbation scheme for treating inhomogeneous elasticity in phase-field models. J. Comput. Phys. 208, 34–50 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • N. Zhou, C. Shen, M. Mills, Y. Wang, Contributions from elastic inhomogeneity and from plasticity to [gamma]’rafting in single-crystal Ni-Al. Acta Mater. 56, 6156–6173 (2008)

    Article  Google Scholar 

  • J. Zhu, L.Q. Chen, J. Shen, Morphological evolution during phase separation and coarsening with strong inhomogeneous elasticity. Model. Simul. Mater. Sci. Eng. 9, 499 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Louisiana State University, 3508B Patrick F. Taylor Hall, 70803, Baton Rouge, LA, USA

    George Z. Voyiadjis & Navid Mozaffari

Authors
  1. George Z. Voyiadjis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Navid Mozaffari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Z. Voyiadjis .

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

Voyiadjis, G.Z., Mozaffari, N. (2015). Modeling of Nonlocal Damage Using the Phase Field Method. In: Voyiadjis, G. (eds) Handbook of Damage Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5589-9_47

Download citation

Publish with us

Policies and ethics

Search

Navigation