Skip to main content
SpringerLink
Log in

An enriched finite volume formulation for the simulation of ductile material failure under shock loading

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A method is proposed to model failure under shock loading. Finite Volume schemes are known to be efficient to take into account the density variations in the shock regions. The proposed method, called eXtended Finite Volume (XFV), is able to model failure in a finite volume framework. The material degradation is modeled using a cohesive zone model to dissipate the amount of energy required to create new surfaces. The XFV method allows to locate the cohesive surface inside elements and to introduce a displacement jump inside the cracked cells without remeshing, in an explicit dynamics finite volume framework. Attention is paid to the mass lumping and scheme stability. The XFV method is validated to simulate a plate impact experiment. It shows the ability to reproduce spall patterns and free surface velocities. However, numerical stability issues still need to be fixed before being able to compare the simulation with experimental data.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33

Similar content being viewed by others

Notes

  1. \(\sigma _{N0}\) depictes the cohesive traction projected on the cohesive surface, as a force per unit surface.

  2. Cases A and E are limit cases respectively approaching a linear cohesive law and a Dugdale model. However, in the implementation of the cohesive model in Hesione code, it is not possible to choose \(\delta _1 = 0 \) or \(\delta _1 = \delta _{Nc}\). These values have therefore been approximated to get \(\delta _1 \approx 0 \) and \( \delta _1 \approx \delta _{Nc} \).

  3. Once again, cases A and E are limit cases corresponding respectively to a linear cohesive law and a Dugdale model.

References

  1. Roy G (2003) Vers une modélisation approfondie de l’endommagement ductile dynamique: investigation expérimentale d’une nuance de tantale et développements théoriques. PhD thesis, Poitiers

  2. Besson J (2010) Continuum models of ductile fracture: a review. Int J Damage Mech 19(1):3–52

    Google Scholar 

  3. Johnson JN (1981) Dynamic fracture and spallation in ductile solids. J Appl Phys 52(4):2812–2825

    Google Scholar 

  4. Czarnota C, Jacques N, Mercier S, Molinari A (2008) Modelling of dynamic ductile fracture and application to the simulation of plate impact tests on tantalum. J Mech Phys Solids 56(4):1624–1650

    Google Scholar 

  5. Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: part 1—yield criteria and flow rules for porous ductile media. J Eng Mater Technol Trans ASME 99(1):2–15

    Google Scholar 

  6. Tvergaard V, Needleman A (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 32(1):157–169

    Google Scholar 

  7. Bargellini R, Besson J, Lorentz E, Michel-Ponnelle S (2009) A non-local finite element based on volumetric strain gradient: application to ductile fracture. Comput Mater Sci 45(3):762–767

    Google Scholar 

  8. Ambati M, Gerasimov T, De Lorenzis L (2015) Phase-field modeling of ductile fracture. Comput Mech 55(5):1017–1040

    MathSciNet  MATH  Google Scholar 

  9. Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104

    Google Scholar 

  10. Needleman A (1987) A continuum model for void nucleation by inclusion debonding. J Appl Mech 54(3):525–531

    MATH  Google Scholar 

  11. Gullerud AS, Gao X, Dodds RH Jr, Haj-Ali R (2000) Simulation of ductile crack growth using computational cells: numerical aspects. Eng Fract Mech 66(1):65–92

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Stershic AJ, Dolbow JE, Moës N (2017) The thick level-set model for dynamic fragmentation. Eng Fract Mech 172:39–60

    Google Scholar 

  14. Zhang H, Li L, An X, Ma G (2010) Numerical analysis of 2-d crack propagation problems using the numerical manifold method. Eng Anal Bound Elem 34(1):41–50

    MathSciNet  MATH  Google Scholar 

  15. Terada K, Asai M, Yamagishi M (2003) Finite cover method for linear and non-linear analyses of heterogeneous solids. Int J Numer Methods Eng 58(9):1321–1346

    MATH  Google Scholar 

  16. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150

    MathSciNet  MATH  Google Scholar 

  17. Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193(33):3523–3540

    MathSciNet  MATH  Google Scholar 

  18. Areias PM, Belytschko T (2006) A comment on the article “A finite element method for simulation of strong and weak discontinuities in solid mechanics” by a. hansbo and p. hansbo [comput. methods appl. mech. engrg. 193 (2004) 3523–3540]. Comput Methods Appl Mech Eng 195(9):1275–1276

    MATH  Google Scholar 

  19. Rozycki P, Moës N, Bechet E, Dubois C (2008) X-fem explicit dynamics for constant strain elements to alleviate mesh constraints on internal or external boundaries. Comput Methods Appl Mech Eng 197(5):349–363

    MATH  Google Scholar 

  20. Song J-H, Areias P, Belytschko T (2006) A method for dynamic crack and shear band propagation with phantom nodes. Int J Numer Methods Eng 67(6):868–893

    MATH  Google Scholar 

  21. Menouillard T, Rethore J, Combescure A, Bung H (2006) Efficient explicit time stepping for the extended finite element method (x-fem). Int J Numer Methods Eng 68(9):911–939

    MathSciNet  MATH  Google Scholar 

  22. Elguedj T, Gravouil A, Maigre H (2009) An explicit dynamics extended finite element method. Part 1: mass lumping for arbitrary enrichment functions. Comput Methods Appl Mech Eng 198(30):2297–2317

    MATH  Google Scholar 

  23. Menouillard T, Rethore J, Moes N, Combescure A, Bung H (2008) Mass lumping strategies for x-fem explicit dynamics: application to crack propagation. Int J Numer Methods Eng 74(3):447–474

    MathSciNet  MATH  Google Scholar 

  24. VonNeumann J, Richtmyer RD (1950) A method for the numerical calculation of hydrodynamic shocks. J Appl Phys 21(3):232–237

    MathSciNet  MATH  Google Scholar 

  25. Wilkins M L (1963) Calculation of elastic-plastic flow. Technical report, California Univ Livermore Radiation Lab

  26. Flament J, Perlat J-P (2011) Méthode de couplage euler-lagrange pour la dynamique rapide. In 10e colloque national en calcul des structures

  27. Longère P, Dragon A (2013) Description of shear failure in ductile metals via back stress concept linked to damage-microporosity softening. Eng Fract Mech 98:92–108

    Google Scholar 

  28. Desgraz JC, Lascaux P (1976) Stabilite de la discretisation des equations de l’hydrodynamique lagrangienne 2d. Computing Methods in Applied Sciences. Springer, Berlin, pp 510–529

    Google Scholar 

  29. Grüneisen E (1912) Theorie des festen zustandes einatomiger elemente. Ann Phys 344(12):257–306

    MATH  Google Scholar 

  30. McQueen RG, Marsh SP, Taylor JW, Fritz JN, Carter WJ (1970) The equation of state of solids from shock wave studies. In: High velocity impact phenomena, vol 293, pp 293–417

  31. Steinberg D, Cochran S, Guinan M (1980) A constitutive model for metals applicable at high-strain rate. J Appl Phys 51(3):1498–1504

    Google Scholar 

  32. Wilkins ML (1980) Use of artificial viscosity in multidimensional fluid dynamic calculations. J Comput Phys 36(3):281–303

    MathSciNet  MATH  Google Scholar 

  33. Tvergaard V, Hutchinson JW (1992) The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. J Mech Phys Solids 40(6):1377–1397

    MATH  Google Scholar 

  34. Scheider I (2009) Derivation of separation laws for cohesive models in the course of ductile fracture. Eng Fract Mech 76(10):1450–1459

    Google Scholar 

  35. Baranger J, Maitre J-F (1996) Connection between finite volume and mixed finite element methods. ESAIM Math Modell Numer Anal 30(4):445–465

    MathSciNet  MATH  Google Scholar 

  36. Chan RK-C (1975) A generalized arbitrary Lagrangian–Eulerian method for incompressible flows with sharp interfaces. J Comput Phys 17(3):311–331

    MATH  Google Scholar 

  37. Buy F, Llorca F (2002) Shock wave effects in copper: design of an experimental device for post recovery mechanical testing. In: AIP conference proceedings, vol 620. AIP, pp 319–322

  38. Courant R, Friedrichs K, Lewy H (1928) Über die partiellen differenzengleichungen der mathematischen physik. Math Ann 100(1):32–74

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CEA-DAM Ile de France, 91297, Arpajon Cedex, France

    Marie Gorecki, Guillaume Peillex & Laurianne Pillon

  2. GeM UMR CNRS 6183, Ecole Centrale de Nantes, 1 rue de la Noe, 44321, Nantes Cedex, France

    Marie Gorecki & Nicolas Moës

Authors
  1. Marie Gorecki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guillaume Peillex
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Laurianne Pillon
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nicolas Moës
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marie Gorecki.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorecki, M., Peillex, G., Pillon, L. et al. An enriched finite volume formulation for the simulation of ductile material failure under shock loading. Comput Mech 65, 1267–1288 (2020). https://doi.org/10.1007/s00466-020-01818-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-020-01818-0

Keywords

Search

Navigation