Skip to main content
SpringerLink
Log in

Gradient plasticity crack tip characterization by means of the extended finite element method

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

Abstract

Strain gradient plasticity theories are being widely used for fracture assessment, as they provide a richer description of crack tip fields by incorporating the influence of geometrically necessary dislocations. Characterizing the behavior at the small scales involved in crack tip deformation requires, however, the use of a very refined mesh within microns to the crack. In this work a novel and efficient gradient-enhanced numerical framework is developed by means of the extended finite element method (X-FEM). A mechanism-based gradient plasticity model is employed and the approximation of the displacement field is enriched with the stress singularity of the gradient-dominated solution. Results reveal that the proposed numerical methodology largely outperforms the standard finite element approach. The present work could have important implications on the use of microstructurally-motivated models in large scale applications. The non-linear X-FEM code developed in MATLAB can be downloaded from www.empaneda.com/codes.

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

Similar content being viewed by others

References

  1. Dahlberg CFO, Faleskog J (2013) An improved strain gradient plasticity formulation with energetic interfaces: theory and a fully implicit finite element formulation. Comput Mech 51:641–659

    Article  MathSciNet  MATH  Google Scholar 

  2. Bayerschen E, Böhlke T (2016) Power-law defect energy in a single-crystal gradient plasticity framework: a computational study. Comput Mech 58:13–27

    Article  MathSciNet  MATH  Google Scholar 

  3. Martínez-Pañeda E, Niordson CF, Bardella L (2016) A finite element framework for distortion gradient plasticity with applications to bending of thin foils. Int J Solids Struct 96:288–299

    Article  Google Scholar 

  4. Aifantis EC (1984) On the microstructural origin of certain inelastic models. J Eng Mater Technol 106:326–330

    Article  Google Scholar 

  5. Fleck NA, Hutchinson JW (2001) A reformulation of strain gradient plasticity. J Mech Phys Solids 49:2245–2271

    Article  MATH  Google Scholar 

  6. Gao H, Huang Y, Nix WD, Hutchinson JW (1999) Mechanism-based strain gradient plasticity I. Theory. J Mech Phys Solids 47:1239–1263

    Article  MathSciNet  MATH  Google Scholar 

  7. Huang Y, Qu S, Hwang KC, Li M, Gao H (2004) A conventional theory of mechanism-based strain gradient plasticity. Int J Plast 20:753–782

    Article  MATH  Google Scholar 

  8. Hao S, Liu WK, Moran B, Vernerey F, Olson GB (2004) Multi-scale constitutive model and computational framework for the design of ultra-high strength, high toughness steels. Comput Methods Appl Mech Eng 193:1865–1908

    Article  MATH  Google Scholar 

  9. Vernerey F, Liu WK, Moran B (2007) Multi-scale micromorphic theory for hierarchical materials. J Mech Phys Solids 55:2603–2651

    Article  MathSciNet  MATH  Google Scholar 

  10. McDowell DL (2008) Viscoplasticity of heterogeneous metallic materials. Mater Sci Eng R 62:67–123

    Article  Google Scholar 

  11. McVeigh C, Liu WK (2008) Linking microstructure and properties through a predictive multiresolution continuum. Comput Methods Appl Mech Eng 197:3268–3290

    Article  MATH  Google Scholar 

  12. O’Keeffe SC, Tang S, Kopacz AM, Smith J, Rowenhorst DJ, Spanos G, Liu WK, Olson GB (2015) Multiscale ductile fracture integrating tomographic characterization and 3-D simulation. Acta Mater 82:503–510

    Article  Google Scholar 

  13. Korn D, Elssner G, Cannon RM, Rühle M (2002) Fracture properties of interfacially doped \(\text{ Nb }\)–\(\text{ Al }_2\text{ O }_3\) bicrystals: I, Fracture characteristics. Acta Mater 50:3881–3901

    Article  Google Scholar 

  14. Komaragiri U, Agnew SR, Gangloff RP, Begley MR (2008) The role of macroscopic hardening and individual length-scales on crack tip stress elevation from phenomenological strain gradient plasticity. J Mech Phys Solids 56:3527–3540

    Article  MATH  Google Scholar 

  15. Tvergaard V, Niordson CF (2008) Size effects at a crack-tip interacting with a number of voids. Philos Mag 88:3827–3840

    Article  Google Scholar 

  16. Mikkelsen LP, Goutianos S (2009) Suppressed plastic deformation at blunt crack-tips due to strain gradient effects. Int J Solids Struct 46:4430–4436

    Article  MATH  Google Scholar 

  17. Jiang H, Huang Y, Zhuang Z, Hwang KC (2001) Fracture in mechanism-based strain gradient plasticity. J Mech Phys Solids 49:979–993

    Article  MATH  Google Scholar 

  18. Pan X, Yuan H (2011) Computational assessment of cracks under strain-gradient plasticity. Int J Fract 167:235–248

    Article  MATH  Google Scholar 

  19. Pan X, Yuan H (2011) Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories. Eng Fract Mech 78:452–461

    Article  Google Scholar 

  20. Martínez-Pañeda E, Betegón C (2015) Modeling damage and fracture within strain gradient plasticity. Int J Solids Struct 59:208–215

    Article  Google Scholar 

  21. Martínez-Pañeda E, Niordson CF (2016) On fracture in strain gradient plasticity. Int J Plast 80:154–167

    Article  Google Scholar 

  22. Martínez-Pañeda E, Niordson CF, Gangloff RP (2016) Strain gradient plasticity-based modeling of hydrogen environment assisted cracking. Acta Mater 117:321–332

    Article  Google Scholar 

  23. Martínez-Pañeda E, del Busto S, Niordson CF, Betegón C (2016) Strain gradient plasticity modeling of hydrogen diffusion to the crack tip. Int J Hydrog Energy 41:10265–10274

    Article  Google Scholar 

  24. Shi M, Huang Y, Jiang H, Hwang KC, Li M (2001) The boundary-layer effect on the crack tip field in mechanism-based strain gradient plasticity. Int J Fract 112:23–41

    Article  Google Scholar 

  25. Hutchinson JW (1968) Singular behavior at the end of a tensile crack tip in a hardening material. J Mech Phys Solids 16:13–31

    Article  MATH  Google Scholar 

  26. Rice JR, Rosengren GF (1968) Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 16:1–12

    Article  MATH  Google Scholar 

  27. Hwang KC, Jiang H, Huang Y, Gao H (2003) Finite deformation analysis of mechanism-based strain gradient plasticity: torsion and crack tip field. Int J Plast 19:235–251

    Article  MATH  Google Scholar 

  28. Niordson CF, Hutchinson JW (2003) On lower order strain gradient plasticity theories. Eur J Mech A Solids 22:771–778

    Article  MATH  Google Scholar 

  29. Qu S, Huang Y, Jiang H, Liu C, Wu PD, Hwang KC (2004) Fracture analysis in the conventional theory of mechanism-based strain gradient (CMSG) plasticity. Int J Fract 129:199–220

    Article  MATH  Google Scholar 

  30. Elguedj T, Gravouil A, Combescure A (2006) Appropriate extended functions for X-FEM simulation of plastic fracture mechanics. Comput Methods Appl Mech Eng 195:501–515

    Article  MATH  Google Scholar 

  31. Duflot M, Bordas S (2008) A posteriori error estimation for extended finite elements by an extended global recovery. Int J Numer Methods Eng 76:1123–1138

    Article  MathSciNet  MATH  Google Scholar 

  32. Duflot M (2006) A study of the representation of cracks with level sets. Int J Numer Methods Eng 70:1261–1302

    Article  MathSciNet  MATH  Google Scholar 

  33. Moumnassi M, Belouettar S, Béchet E, Bordas S, Quoirin D, Potier-Ferry M (2011) Finite element analysis on implicitly defined domains: an accurate representation based on arbitrary parametric surfaces. Comput Methods Appl Mech Eng 200:774–796

    Article  MathSciNet  MATH  Google Scholar 

  34. Fries TP, Baydoun M (2012) Crack propagation with the extended finite element method and a hybrid explicit–implicit crack description. Int J Numer Methods Eng 89:1527–1558

    Article  MathSciNet  MATH  Google Scholar 

  35. Gracie R, Wang H, Belytschko T (2008) Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods. Int J Numer Methods Eng 74:1645–1669

    Article  MathSciNet  MATH  Google Scholar 

  36. Fries TP (2008) A corrected XFEM approximation without problems in blending elements. Int J Numer Methods Eng 75:503–532

    Article  MathSciNet  MATH  Google Scholar 

  37. Ventura G, Gracie R, Belytschko T (2008) Fast integration and weight function blending in the extended finite element method. Int J Numer Methods Eng 77:1–29

    Article  MathSciNet  MATH  Google Scholar 

  38. Xiao QZ, Karihaloo BL (2007) Implementation of hybrid crack element on a general finite element mesh and in combination with XFEM. Comput Methods Appl Mech Eng 196:1864–1873

  39. Réthoré J, Roux S, Hild F (2010) Hybrid analytical and extended finite element method (HAX-FEM): a new enrichment procedure for cracked solids. Int J Numer Methods Eng 81:269–285

  40. Natarajan S, Song C (2013) Representation of singular fields without asymptotic enrichment in the extended finite element method. Int J Numer Methods Eng 96:813–841

    Article  MathSciNet  MATH  Google Scholar 

  41. Legay A, Wang HW, Belytschko T (2005) Strong and weak arbitrary discontinuities in spectral finite elements. Int J Numer Methods Eng 64:991–1008

  42. Laborde P, Pommier J, Renard Y, Salaün M (2005) High-order extended finite element method for cracked domains. Int J Numer Methods Eng 64:354–381

    Article  MATH  Google Scholar 

  43. Natarajan S, Bordas S, Mahapatra DR (2009) Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping. Int J Numer Methods Eng 80:103–134

    Article  MathSciNet  MATH  Google Scholar 

  44. Ventura G (2006) On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method. Int J Numer Methods Eng 66:767–795

    Article  MathSciNet  MATH  Google Scholar 

  45. Mousavi SE, Sukumar N (2010) Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method. Comput Methods Appl Mech Eng 199:3237–3249

    Article  MathSciNet  MATH  Google Scholar 

  46. Bordas S, Natarajan S, Kerfriden P, Augarde C, Mahapatra DR, Rabczuk T, Pont SD (2011) On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM). Int J Numer Methods Eng 86:637–666

    Article  MATH  Google Scholar 

  47. Xiao QZ, Karihaloo BL (2006) Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery. Int J Numer Methods Eng 66:1378–1410

    Article  MATH  Google Scholar 

  48. Chin EB, Lasserre JB, Sukumar N (2016) Modeling crack discontinuities without element-partitioning in the extended finite element method. Int J Numer Methods Eng. doi:10.1002/nme.5436

  49. Bower AF (2009) Applied mechanics of solids. CRC Press, Boca Raton

    Google Scholar 

  50. Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46:5109–5115

    Article  Google Scholar 

  51. González-Albuixech VF, Giner E, Taracón JE, Fuenmayor FJ, Gravouil A (2013) Convergence of domain integrals for stress intensity factor extraction in 2-D curved cracks problems with the extended finite element method. Int J Numer Methods Eng 94:740–757

    Article  MathSciNet  MATH  Google Scholar 

  52. Robinson J (1987) Some new distortion measures for quadrilaterals. Finite Elem Anal Des 3:183–197

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge financial support from the European Research Council (ERC Starting Grant Agreement No. 279578). E. Martínez-Pañeda also acknowledges financial support from the Ministry of Economy and Competitiveness of Spain through Grant MAT2014-58738-C3-1, and the University of Oviedo through Grant UNOV-13-PF.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800, Kgs. Lyngby, Denmark

    E. Martínez-Pañeda

  2. Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai, 600036, India

    S. Natarajan

  3. Faculté des Sciences, de la Technologie et de la Communication, University of Luxembourg, Luxembourg-Kirchberg, 1359, Luxembourg

    S. Bordas

  4. School of Engineering, Cardiff University, Cardiff, CF24 3AA, Wales, UK

    S. Bordas

Authors
  1. E. Martínez-Pañeda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Natarajan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Bordas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Martínez-Pañeda.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 194 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martínez-Pañeda, E., Natarajan, S. & Bordas, S. Gradient plasticity crack tip characterization by means of the extended finite element method. Comput Mech 59, 831–842 (2017). https://doi.org/10.1007/s00466-017-1375-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-017-1375-6

Keywords

Search

Navigation