SpringerLink
Log in

Simulation of Crack Propagation in Heterogeneous Materials by a Fracture Phase Field

  • Chapter
  • First Online:
New Achievements in Mechanics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 205))

  • 142 Accesses

Abstract

We examine a method to compute the unknown fracture toughness of a heterogeneous material by the means of a time dependent boundary condition. This boundary condition imposes a steadily propagating crack on an exactly defined heterogeneous body, from which it is possible to derive the corresponding fracture toughness of the underlying material. The goal is to identify toughening mechanisms and to compute the toughness parameter for the particular material definition, without restricting the model to randomness or small contrast.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alnaes, M.S., Logg, A., Ølgaard, K.B., Rognes, M.E., Wells, G.N.: Unified form language: A domain-specific language for weak formulations of partial differential equations. ACM Trans. Math. Softw. 40 (2014). https://doi.org/10.1145/2566630

  2. Ambati, M., Gerasimov, T., De Lorenzis, L.: Phase-field modeling of ductile fracture. Comput. Mech. 55, 1017–1040 (2015)

    Article  MathSciNet  Google Scholar 

  3. Ambati, M., Gerasimov, T., De Lorenzis, L.: A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput. Mech. 55, 383–405 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bonamy, D., Ponson, L., Prades, S., Bouchaud, E., Guillot, C.: Scaling exponents for fracture surfaces in homogeneous glass and glassy ceramics. Phys. Rev. Lett. 97(13), 135504 (2006). https://doi.org/10.1103/PhysRevLett.97.135504

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  6. Bourdin, B., Francfort, G.A., Marigo, J.J.: The variational approach to fracture. J. Elast. 91(1–3), 5–148 (2008). https://doi.org/10.1007/s10659-007-9107-3

    Article  MathSciNet  Google Scholar 

  7. Braun, M.: Configurational forces induced by finite-element discretization. Proc. Estonian Acad. Sci. Phys. Math 46(1/2), 24–31 (1997)

    Google Scholar 

  8. Fischer, F.D., Predan, J., Müller, R., Kolednik, O.: On problems with the determination of the fracture resistance for materials with spatial variations of the Young’s modulus. Int. J. Fract. 190(1–2), 23–38 (2014). https://doi.org/10.1007/s10704-014-9972-2

    Article  Google Scholar 

  9. Geers, M.G., Kouznetsova, V.G., Brekelmans, W.: Multi-scale computational homogenization: Trends and challenges. J. Comput. Appl. Math. 234(7), 2175–2182 (2010)

    Article  Google Scholar 

  10. Gross, D., Seelig, T.: Bruchmechanik. Springer, Berlin (2016). https://doi.org/10.1007/978-3-662-46737-4

  11. Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics, vol. 137. Springer Science & Business Media (1999)

    Google Scholar 

  12. Gurtin, M.E.: The nature of configurational forces. In: Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids: A Collection of Reprints of 14 Seminal Papers, pp. 281–314. Springer (1999)

    Google Scholar 

  13. Hansen, C.D., Johnson, C.R. (eds.): The Visualization Handbook. Elsevier-Butterworth Heinemann, Amsterdam, Boston (2005)

    Google Scholar 

  14. Heider, Y.: A review on phase-field modeling of hydraulic fracturing. Eng. Fract. Mech. 253, 107881 (2021)

    Article  Google Scholar 

  15. Hossain, M., Hsueh, C.J., Bourdin, B., Bhattacharya, K.: Effective toughness of heterogeneous media. J. Mech. Phys. Solids 71, 15–32 (2014). https://doi.org/10.1016/j.jmps.2014.06.002, linkinghub.elsevier.com/retrieve/pii/S0022509614001215

  16. Hsueh, C., Avellar, L., Bourdin, B., Ravichandran, G., Bhattacharya, K.: Stress fluctuation, crack renucleation and toughening in layered materials. J. Mech. Phys. Solids 120, 68–78 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  17. Kienzler, R., Herrmann, G.: Mechanics in Material Space: With Applications to Defect and Fracture Mechanics. Springer Science & Business Media (2000)

    Google Scholar 

  18. Kolednik, O., Predan, J., Shan, G., Simha, N., Fischer, F.: On the fracture behavior of inhomogeneous materials-a case study for elastically inhomogeneous bimaterials. Int. J. Solids Struct. 42(2), 605–620 (2005). https://doi.org/10.1016/j.ijsolstr.2004.06.064, linkinghub.elsevier.com/retrieve/pii/S0020768304003415

  19. Kuhn, C.: Numerical and analytical investigation of a phase field model for fracture. No. Bd. 6 in Forschungsbericht/Technische Universität Kaiserslautern, Lehrstuhl für Technische Mechanik. Technical University, Lehrstuhl für Technical Mechanik, Kaiserslautern (2013)

    Google Scholar 

  20. Kuhn, C., Müller, R.: A continuum phase field model for fracture. Eng. Fract. Mech. 77(18), 3625–3634 (2010). https://doi.org/10.1016/j.engfracmech.2010.08.009, linkinghub.elsevier.com/retrieve/pii/S0013794410003668

  21. Kuhn, C., Noll, T., Müller, R.: On phase field modeling of ductile fracture. GAMM-Mitteilungen 39(1), 35–54 (2016)

    Article  MathSciNet  Google Scholar 

  22. Kuhn, C., Schlüter, A., Müller, R.: On degradation functions in phase field fracture models. Comput. Mater. Sci. 108, 374–384 (2015)

    Article  Google Scholar 

  23. Loehnert, S., Mueller-Hoeppe, D., Wriggers, P.: 3d corrected XFEM approach and extension to finite deformation theory. Int. J. Numer. Methods Eng. 86(4–5), 431–452 (2011)

    Article  MathSciNet  Google Scholar 

  24. Logg, A., Wells, G.N., Hake, J.: DOLFIN: a C++/Python finite element library. In: Logg, K.M.A., Wells, G.N. (eds.) Automated Solution of Differential Equations by the Finite Element Method, Lecture Notes in Computational Science and Engineering, vol. 84. Springer (2012)

    Google Scholar 

  25. Maugin, G.A.: Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics. CRC Press (2016)

    Google Scholar 

  26. Maugin, G.A.: Material Inhomogeneities in Elasticity. CRC Press (2020)

    Google Scholar 

  27. Miehe, C.: Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int. J. Numer. Methods Eng. 55(11), 1285–1322 (2002)

    Article  MathSciNet  Google Scholar 

  28. Miehe, C., Schröder, J., Becker, M.: Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro-and macro-scales of periodic composites and their interaction. Comput. Methods Appl. Mech. Eng. 191(44), 4971–5005 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  29. Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Comput. Methods Appl. Mech. Eng. 171(3–4), 387–418 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  30. Milton, G.W.: The Theory of Composites. No. 6 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, New York (2002)

    Google Scholar 

  31. Moës, N., Belytschko, T.: Extended finite element method for cohesive crack growth. Eng. Fract. Mech. 69(7), 813–833 (2002)

    Article  Google Scholar 

  32. Mueller, R., Kolling, S., Gross, D.: On configurational forces in the context of the finite element method. Int. J. Numer. Methods Eng. 53(7), 1557–1574 (2002)

    Article  MathSciNet  Google Scholar 

  33. Mueller, R., Maugin, G.: On material forces and finite element discretizations. Comput. Mech. 29, 52–60 (2002)

    Article  MathSciNet  Google Scholar 

  34. Nemat-Nasser, S., Hori, M.: Micromechanics: overall properties of heterogeneous materials. No. v. 37 in North-Holland Series in Applied Mathematics and Mechanics. North-Holland, Amsterdam, New York (1993)

    Google Scholar 

  35. Ponson, L., Bonamy, D.: Crack propagation in brittle heterogeneous solids: Material disorder and crack dynamics. Int. J. Fract. 162(1–2), 21–31 (2010). https://doi.org/10.1007/s10704-010-9481-x

  36. Ramanathan, S., Ertaş, D., Fisher, D.S.: Quasistatic crack propagation in heterogeneous media. Phys. Rev. Lett. 79(5), 873–876 (1997). https://doi.org/10.1103/PhysRevLett.79.873

  37. Schlüter, A., Kuhn, C., Müller, R.: Phase field approximation of dynamic brittle fracture: phase field approximation of dynamic brittle fracture. PAMM 14(1), 143–144 (2014). https://doi.org/10.1002/pamm.201410059

    Article  Google Scholar 

  38. Schneider, M.: A review of nonlinear FFT-based computational homogenization methods. Acta Mech. 232(6), 2051–2100 (2021)

    Article  MathSciNet  Google Scholar 

  39. Shao, Y., Zhao, H.P., Feng, X.Q., Gao, H.: Discontinuous crack-bridging model for fracture toughness analysis of nacre. J. Mech. Phys. Solids 60(8), 1400–1419 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  40. Steinmann, P.: Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting. Int. J. Solids Struct. 37(48-50), 7371–7391 (2000)

    Google Scholar 

  41. Steinmann, P., Ackermann, D., Barth, F.: Application of material forces to hyperelastostatic fracture mechanics. II. Computational setting. Int. J. Solids Struct. 38(32-33), 5509–5526 (2001)

    Google Scholar 

  42. Wu, J.Y., Nguyen, V.P., Nguyen, C.T., Sutula, D., Sinaie, S., Bordas, S.P.: Phase-field modeling of fracture. Adv. Appl. Mech. 53, 1–183 (2020)

    Article  Google Scholar 

  43. Yuan, Z., Fish, J.: Toward realization of computational homogenization in practice. Int. J. Numer. Methods Eng. 73(3), 361–380 (2008)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Mechanics, TU Darmstadt, Franziska-Braun-Straße 7, 64287, Darmstadt, Germany

    Matthias Bohnen, Ralf Müller & Dietmar Gross

Authors
  1. Matthias Bohnen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ralf Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dietmar Gross
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ralf Müller .

Editor information

Editors and Affiliations

  1. Institute of Mechanics, Technical University of Berlin, Berlin, Berlin, Germany

    Wolfgang H. Müller

  2. Department of Mechanical Engineering, South Westphalia University of Applied Sciences, Soest, Germany

    Alfons Noe

  3. Faculty of Mechanical Engineering, Paderborn University, Paderborn, Germany

    Ferdinand Ferber

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bohnen, M., Müller, R., Gross, D. (2024). Simulation of Crack Propagation in Heterogeneous Materials by a Fracture Phase Field. In: Müller, W.H., Noe, A., Ferber, F. (eds) New Achievements in Mechanics. Advanced Structured Materials, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-031-56132-0_10

Download citation

Publish with us

Policies and ethics

Search

Navigation