Skip to main content
SpringerLink
Log in

Estimation of Errors in Stress Distributions Computed in Finite Element Simulations of Polycrystals

  • Technical Article
  • Published:
Integrating Materials and Manufacturing Innovation Aims and scope Submit manuscript

Abstract

The accuracy of the stresses predicted from crystal plasticity-based finite element formulation depends on estimation and control of the errors associated with the discretization. In the current work, the errors in the stress distribution are estimated in virtual polycrystalline samples of α-phase titanium (hexagonal close-packed phase of Ti–6Al–4V). To estimate the error, the stress field, which does not possess inter-element continuity, is smoothed over a grain using an \(L_2\) projection, thereby providing continuous stress distributions with inter-element continuity. The differences between the continuous (smooth) and discontinuous (raw) stress fields are calculated at individual Gauss quadrature points and used to estimate errors for corresponding elements and grains. Error estimations are performed for a Voronoi-tessellated microstructure, an equiaxed microstructure, and two microstructures with varying grain sizes for tensile loading extending into the fully plastic regime (\(\approx \) 5% extension). Magnitudes of the errors are found to depend on microstructural characteristics, particularly the shape and size of grains. Samples having variations in grain size or having less spherical grains exhibited higher errors than samples with uniformly sized, equiaxed grains, with the size variations having a more pronounced effect. Errors correlate with proximity to grain boundaries at small (elastic) strains and with deformation-induced features (deformation bands) at large strains.

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.

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

Similar content being viewed by others

Notes

  1. Rate-dependent crystal plasticity admits plastic flow at any resolved shear stress, but with low rate sensitivity the slip system activity is very small unless the resolved shear stress is close to the slip system strength.

References

  1. Roters F, Eisenlohr P, Hantcherli L, Tjahjanto DD, Bieler TR, Raabe D (2010) Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications. Acta Mater 58(4):1152–1211

    Article  CAS  Google Scholar 

  2. Zienkiewicz OC, Taylor RL, Zhu JZ (2005) Chapter 13. The finite element method its basis and fundamentals, 6th edn. Elsevier Butterworth-Heinemann, Amsterdam

    Google Scholar 

  3. Oden JT, Moser R, Ghattas O (2010) Computer predictions with quantified uncertainty, part 1. SIAM News 43(9):1–3

    Google Scholar 

  4. Oden JT, Moser R, Ghattas O (2010) Computer predictions with quantified uncertainty, part 2. SIAM News 43(10):1–4

    Google Scholar 

  5. Buchheit TE, Wellman GW, Battaile CC (2005) Investigating the limits of polycrystal plasticity modeling. Int J Plast 21:221–249

    Article  Google Scholar 

  6. Poshadel AC, Gharghouri M, Dawson PR (2019) Sensitivity of crystal stress distributions to the definition of virtual two-phase microstructures. Metall Mater Trans A. https://doi.org/10.1007/s11661-018-5085-2

    Article  Google Scholar 

  7. Rizzi R, Jones RE, Templeton JA, Ostien JT, Boyce BL (2017) Plasticity models of material variability based on uncertainty quantification techniques. arXiv:1802.01487v1 [cond-mat.mtrl-sci]

  8. Zienkiewicz Olgierd C, Zhu Jian Z (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 24(2):337–357

    Article  Google Scholar 

  9. Dawson PR, Boyce DE (April 2015) FEpX—finite element polycrystals: theory, finite element formulation, numerical implementation and illustrative examples. ArXiv e-prints

  10. Kasemer M, Quey R, Dawson P (2017) The influence of mechanical constraints introduced by \(\beta \) annealed microstructures on the yield strength and ductility of Ti–6Al–4V. J Mech Phys Solids 103:179–198

    Article  CAS  Google Scholar 

  11. Kasemer M, Echlin MP, Stinville JC, Pollock TM, Dawson P (2017) On slip initiation in equiaxed \(\alpha \)/\(\beta \) Ti–6Al–4V. Acta Mater 136:288–302

    Article  CAS  Google Scholar 

  12. Chatterjee K, Echlin MP, Kasemer MP, Callahan PG, Pollock TM, Dawson PR (2018) Prediction of tensile stiffness and strength of Ti–6Al–4V using instantiated volume elements and crystal plasticity. Acta Mater 157:21–32

    Article  CAS  Google Scholar 

  13. Ainsworth M, Tinsley Oden J (1997) A posteriori error estimation in finite element analysis. Comput Methods Appl Mech Eng 142(1):1–88

    Article  Google Scholar 

  14. Becker R, Rannacher R (2001) An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer 10:1–102

    Article  Google Scholar 

  15. Grätsch T, Bathe K-J (2005) A posteriori error estimation techniques in practical finite element analysis. Comput Struct 83(4):235–265

    Article  Google Scholar 

  16. Lütjering G, Williams JC (2007) Titanium. Engineering materials and processes, 2nd edn. Springer, Berlin

    Google Scholar 

  17. Quey R, Dawson PR, Barbe F (2011) Large-scale 3D random polycrystals for the finite element method: generation, meshing and remeshing. Comput Method Appl Mech Eng 200(17):1729–1745

    Article  Google Scholar 

  18. Wielewski E, Boyce DE, Park J-S, Miller MP, Dawson PR (2017) A methodology to determine the elastic moduli of crystals by matching experimental and simulated lattice strain pole figures using discrete harmonics. Acta Mater 126:469–480

    Article  CAS  Google Scholar 

  19. Dawson PR, Boyce DE, Park J-S, Wielewski E, Miller MP (2018) Determining the strengths of HCP slip systems using harmonic analyses of lattice strain distributions. Acta Mater 144:92–106

    Article  CAS  Google Scholar 

  20. Maddali S, Ta’asan S, Suter RM (2016) Topology-faithful nonparametric estimation and tracking of bulk interface networks. Comput Mater Sci 125:328–340

    Article  Google Scholar 

  21. Resk H, Delannay L, Bernacki M, Coupez T, Log’e R (2009) Adaptive mesh refinement and automatic remeshing in crystal plasticity finite element simulations. Model Simul Mater Sci Eng 17:1–22

    Article  Google Scholar 

  22. Spearman C (1904) The proof and measurement of association between two things. Am J Psychol 15(1):72–101

    Article  Google Scholar 

  23. McDonald JH (2014) Handbook of biological statistics, 3rd edn. Sparky House Publishing, Baltimore, pp 209–212

    Google Scholar 

  24. Zwillinger D, Kokoska S (2000) CRC standard probability and statistics tables and formulae, chapter 14.7. Chapman & Hall, London

    Google Scholar 

  25. Skroch M, Owen SJ, Staten ML, Quadros RW, Hanks B, Clark B, Hensley T, Meyers RJ, Ernst C, Morris R, McBride C, Stimpson C (2019) Cubit geometry and mesh generation toolkit 15.4 user documentation. Sandia National Laboratories

  26. Knupp PM (2000) Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities, part I. Int J Numer Methods Eng 48(3):401–420

    Article  Google Scholar 

  27. Quey R, Renversade L (2018) Optimal polyhedral description of 3D polycrystals: method and application to statistical and synchrotron X-ray diffraction data. Comput Methods Appl Mech Eng 330:308–333

    Article  Google Scholar 

  28. Owen SJ, Brown JA, Ernst CD, Lim H, Long KN (2017) Hexahedral mesh generation for computational materials modeling. Procedia Eng 203:167–179

    Article  Google Scholar 

  29. Carson R, Dawson P (2019) Formulation and characterization of a continuous crystal lattice orientation finite element method (LOFEM) and its application to dislocation fields. J Mech Phys Solids 126:1–19

    Article  CAS  Google Scholar 

  30. Ahrens J, Geveci B, Law C, Hansen C, Johnson C (2005) ParaView: an end-user tool for large-data visualization. In: Hansen CD, Johnson CR (eds) The visualization handbook, vol 717. Elsevier, Amsterdam, pp 717–731

    Chapter  Google Scholar 

Download references

Acknowledgements

Principal support for this research was provided by the Office of Naval Research under Grant N00014-16-1-2982. This work was performed partially by Robert A. Carson under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Visualizations of simulation results were rendered using the ParaView software [30].

Author information

Authors and Affiliations

  1. Sibley School of Mechanical and Aerospace Engineering, Cornell University, 405 Upson Hall, Ithaca, NY, 14850, USA

    Kamalika Chatterjee, Robert A. Carson & Paul R. Dawson

Authors
  1. Kamalika Chatterjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert A. Carson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paul R. Dawson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul R. Dawson.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 255 KB)

Supplementary material 2 (pdf 7716 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chatterjee, K., Carson, R.A. & Dawson, P.R. Estimation of Errors in Stress Distributions Computed in Finite Element Simulations of Polycrystals. Integr Mater Manuf Innov 8, 476–494 (2019). https://doi.org/10.1007/s40192-019-00158-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40192-019-00158-z

Keywords

Search

Navigation