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Numerical Implementation

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Strain Gradient Plasticity-Based Modeling of Damage and Fracture

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

While solving analytically (or semi-analytically) simple problems, such as pure bending or shear of an infinite layer, has been particularly useful to compare and benchmark SGP theories, quantitative assessment of gradient effects in engineering applications requires the use of numerical methods. A wide range of ad hoc numerical solutions have been proposed for each gradient plasticity model, ranging from the relatively easy to implement lower order theories to the more complicated gradient plasticity formulations falling within the mathematical framework of Cosserat-Koiter-Mindlin theories of higher order elasticity. Verification of each numerical implementation is performed by solving different boundary value problems and comparing the output with numerical results from other authors.

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References

  1. 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. doi:10.1016/j.ijplas.2003.08.002

    Article  MATH  Google Scholar 

  2. 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. doi:10.1016/S0749-6419(01)00039-0

    Article  MATH  Google Scholar 

  3. Martínez-Pañeda E, Betegón C (2015) Modeling damage and fracture within strain-gradient plasticity. Int J Solids Struct 59:208–215. doi:10.1016/j.ijsolstr.2015.02.010

    Article  Google Scholar 

  4. Shi M, Huang Y, Jiang H, Hwang KC, Li M (2001) The boundary-layer effect on the crack tip field in mechanismbased strain gradient plasticity. Int J Fract 23–41. doi:10.1023/A:1013548131004

  5. Qu S (2004) A conventional theory of mechanism-based strain gradient plasticity. Ph.D. thesis. University of Illinois at Urbana-Champaign

    Google Scholar 

  6. Kok S, Beaudoin AJ, Tortorelli DA (2002) A polycrystal plasticity model based on the mechanical threshold. Int J Plast 18:715–741. doi:10.1016/S0749-6419(01)00051-1

    Article  MATH  Google Scholar 

  7. Gao H, Huang Y, Nix WD, Hutchinson JW (1999) Mechanism-based strain gradient plasticity-I. Theory. J Mech Phys Solids 47:128–152. doi:10.1016/S0022-5096(98)00103-3

  8. Qu S, Huang Y, Jiang H, Liu C (2004) Fracture analysis in the conventional theory of mechanism-based strain gradient (CMSG) plasticity. Int J Fract 199–220. doi:10.1023/B:FRAC.0000047786.40200.f8

  9. Jiang H, Huang Y, Zhuang Z, Hwang KC (2001) Fracture in mechanism-based strain gradient plasticity. J Mech Phys Solids 49:979–993. doi:10.1016/S0022-5096(00)00070-3

    Article  MATH  Google Scholar 

  10. Niordson CF, Hutchinson JW (2003) On lower order strain gradient plasticity theories. Eur J Mech A Solids 22:771–778. doi:10.1016/S0997-7538(03)00069-X

    Article  MATH  Google Scholar 

  11. Komaragiri U, Agnew S, Gangloff RP, Begley M (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. doi:10.1016/j.jmps.2008.08.007

    Article  MATH  Google Scholar 

  12. Tvergaard V, Niordson CF (2008) Size effects at a crack-tip interacting with a number of voids. Philo Mag 88:3827–3840. doi:10.1080/14786430802225540

    Article  Google Scholar 

  13. Mikkelsen LP, Goutianos S (2009) Suppressed plastic deformation at blunt crack-tips due to strain gradient effects. Int J Solids Struct 46:4430–4436. doi:10.1016/j.ijsolstr.2009.09.001

    Article  MATH  Google Scholar 

  14. Martínez-Pañeda E, Niordson CF (2016) On fracture in finite strain gradient plasticity. Int J Plast 80:154–167. doi:10.1016/j.ijplas.2015.09.009

    Article  Google Scholar 

  15. Pan X, Yuan H (2010) Computational assessment of cracks under strain-gradient plasticity. Int J Fract 167:235–248. doi:10.1007/s10704-010-9548-8

    Article  MATH  Google Scholar 

  16. 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. doi:10.1016/j.engfracmech.2010.08.024

    Article  Google Scholar 

  17. 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. doi:10.1016/j.cma.2005.02.007

    Article  MATH  Google Scholar 

  18. Hutchinson JW (1968) Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids. 16:13–31. doi:10.1016/0022-5096(68)90014-8

  19. 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. doi:10.1016/0022-5096(68)90013-6

  20. 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. doi:10.1002/nme.2332

    Article  MathSciNet  MATH  Google Scholar 

  21. Duflot M (2006) A study of the representation of cracks with level sets. Int J Numer Methods Eng 70:1261–1302. doi:10.1002/nme.1915

    Article  MathSciNet  MATH  Google Scholar 

  22. Fries TP (2008) A corrected XFEM approximation without problems in blending elements. Int J Numer Methods Eng 75:503–532. doi:10.1002/nme.2259

    Article  MathSciNet  MATH  Google Scholar 

  23. Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46:5109–5115. doi:10.1016/S1359-6454(98)00153-0

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

    Article  MathSciNet  Google Scholar 

  25. Fleck NA, Hutchinson JW (2001) A reformulation of strain gradient plasticity. J Mech Phys Solids 49:2245–2271. doi:10.1016/S0022-5096(01)00049-7

    Article  MATH  Google Scholar 

  26. Niordson CF, Hutchinson JW (2003) Non-uniform plastic deformation of micron scale objects. Int J Numer Methods Eng 56:961–975. doi:10.1002/nme.593

    Article  MATH  Google Scholar 

  27. Gudmundson P (2004) A unified treatment of strain gradient plasticity. J Mech Phys Solids 52:1379–1406. doi:10.1016/j.jmps.2003.11.002

    Article  MathSciNet  MATH  Google Scholar 

  28. Gurtin ME, Anand L (2005) A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: small deformations. J Mech Phys Solids 53:1624–1649. doi:10.1016/j.jmps.2004.12.008

  29. Nielsen KL, Niordson CF (2013) A 2D finite element implementation of the Fleck-Willis strain-gradient flow theory. Eur J Mech A Solids 41:134–142. doi:10.1016/j.euromechsol.2013.03.002

    Article  MathSciNet  Google Scholar 

  30. Nielsen KL, Niordson CF (2014) A numerical basis for strain-gradient plasticity theory: Rate-independent and ratedependent formulations. J Mech Phys Solids 63:113–127. doi:10.1016/j.jmps.2013.09.018

    Article  MathSciNet  MATH  Google Scholar 

  31. Fleck NA, Willis JR (2009) A mathematical basis for straingradient plasticity theory–Part I: scalar plastic multiplier. J Mech Phys Solids 57:161–177. doi:10.1016/j.jmps.2008.09.010

    Article  MathSciNet  MATH  Google Scholar 

  32. Fleck NA, Willis JR (2009) A mathematical basis for straingradient plasticity theory. Part II: tensorial plastic multiplier. J Mech Phys Solids 57:1045–1057. doi:10.1016/j.jmps.2009.03.007

  33. Fredriksson P, Gudmundson P (2005) Size-dependent yield strength of thin films. Int J Plast 21:1834–1854. doi:10.1016/j.ijplas.2004.09.005

    Article  MATH  Google Scholar 

  34. Niordson CF, Legarth BN (2010) Strain gradient effects on cyclic plasticity. J Mech Phys Solids 58:542–557. doi:10.1016/j.jmps.2010.01.007

    Article  MathSciNet  MATH  Google Scholar 

  35. Idiart MI, Deshpande VS, Fleck NA, Willis JR (2009) Size effects in the bending of thin foils. Int J Eng Sci 47:1251–1264. doi:10.1016/j.ijengsci.2009.06.002

    Article  MathSciNet  MATH  Google Scholar 

  36. Poh LH, Peerlings RHJ (2016) The plastic rotation effect in an isotropic gradient plasticity model for applications at the meso scale. Int J Solids Struct 78–79:57–69. doi:10.1016/j.ijsolstr.2015.09.017

    Article  Google Scholar 

  37. Ostien J, Garikipati K (2008) A discontinuous Galerkin method for an incompatibility-based strain gradient plasticity theory. In: IUTAM symposium on theoretical, computational and modelling aspects of inelastic media. Springer, pp 217–226. doi:10.1007/978-1-4020-9090-5_20

  38. Gurtin ME (2004) A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J Mech Phys Solids 52(2004):2545–2568. doi:10.1016/j.jmps.04.010

    Article  MathSciNet  MATH  Google Scholar 

  39. Bardella L, Panteghini A (2015) Modelling the torsion of thin metal wires by distortion gradient plasticity. J Mech Phys Solids 78:467–492. doi:10.1016/j.jmps.2015.03.003

    Article  MathSciNet  MATH  Google Scholar 

  40. 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. doi:10.1016/j.ijsolstr.2016.06.001

    Article  Google Scholar 

  41. Niordson CF, Hutchinson JW (2011) Basic strain gradient plasticity theories with application to constrained film deformation. J Mech Mater Struct 6:395–416. doi:10.2140/jomms.2011.6.395

    Article  Google Scholar 

  42. Bardella L (2010) Size effects in phenomenological strain gradient plasticity constitutively involving the plastic spin. Int J Eng Sci 48:550–568. doi:10.1016/j.ijengsci.2010.01.003

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Construction and Manufacturing Engineering, University of Oviedo, Gijón, Spain

    Emilio Martínez Pañeda

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  1. Emilio Martínez Pañeda
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Correspondence to Emilio Martínez Pañeda .

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Martínez Pañeda, E. (2018). Numerical Implementation. In: Strain Gradient Plasticity-Based Modeling of Damage and Fracture. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-63384-8_3

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  • DOI: https://doi.org/10.1007/978-3-319-63384-8_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-63383-1

  • Online ISBN: 978-3-319-63384-8

  • eBook Packages: EngineeringEngineering (R0)

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