Computational Mechanics

, Volume 57, Issue 5, pp 733–753 | Cite as

Stabilized tetrahedral elements for crystal plasticity finite element analysis overcoming volumetric locking

  • Jiahao Cheng
  • Ahmad Shahba
  • Somnath Ghosh
Original Paper


Image-based CPFE modeling involves computer generation of virtual polycrystalline microstructures from experimental data, followed by discretization into finite element meshes. Discretization is commonly accomplished using three-dimensional four-node tetrahedral or TET4 elements, which conform to the complex geometries. It has been commonly observed that TET4 elements suffer from severe volumetric locking when simulating deformation of incompressible or nearly incompressible materials. This paper develops and examines three locking-free stabilized finite element formulations in the context of crystal plasticity finite element analysis. They include a node-based uniform strain (NUS) element, a locally integrated B-bar (LIB) based element and a F-bar patch (FP) based element. All three formulations are based on the partitioning of TET4 element meshes and integrating over patches to obtain favorable incompressibility constraint ratios without adding large degrees of freedom. The results show that NUS formulation introduces unstable spurious energy modes, while the LIB and FP elements stabilize the solutions and are preferred for reliable CPFE analysis. The FP element is found to be computationally efficient over the LIB element.


Tetrahedral elements Locking-free stabilization Crystal plasticity FEM Finite deformation 



This work has been supported by the National Science Foundation, Mechanics and Structure of Materials Program through grant No. CMMI-1100818 (Program Manager: Dr. T. Siegmund), by the Army Research Office through grant No. W911NF-12-1-0376 (Program Manager: Dr. A. Rubinstein) and by the Air Force Office of Scientific through a grant FA9550-13-1-0062, (Program Manager: Dr. David Stargel). Computing support by the Homewood High Performance Compute Cluster (HHPC) is gratefully acknowledged.


  1. 1.
    Deka D, Joseph DS, Ghosh S, Mills MJ (2006) Crystal plasticity modeling of deformation and creep in polycrystalline Ti-6242. Metall Trans A 37A(5):1371–1388CrossRefGoogle Scholar
  2. 2.
    Hasija V, Ghosh S, Mills MJ, Joseph DS (2003) Modeling deformation and creep in Ti-6Al alloys with experimental validation. Acta Mater 51:4533–4549CrossRefGoogle Scholar
  3. 3.
    Venkataramani G, Kirane K, Ghosh S (2008) Microstructural parameters affecting creep induced load shedding in Ti-6242 by a size dependent crystal plasticity FE model. Int J Plast 24:428–454CrossRefzbMATHGoogle Scholar
  4. 4.
    Anahid M, Samal MK, Ghosh S (2011) Dwell fatigue crack nucleation model based on crystal plasticity finite element simulations of polycrystalline Titanium alloys. J Mech Phys Solids 59(10):2157–2176CrossRefzbMATHGoogle Scholar
  5. 5.
    Ghosh S, Anahid M (2013) Homogenized constitutive and fatigue nucleation models from crystal plasticity fe simulations of ti alloys, part 1: macroscopic anisotropic yield function. Int J Plast 47:182–201CrossRefGoogle Scholar
  6. 6.
    Ghosh S, Chakraborty P (2013) Microstructure and load sensitive fatigue crack nucleation in ti-6242 using accelerated crystal plasticity fem simulations. Int J Fatigue 48:231–246CrossRefGoogle Scholar
  7. 7.
    Keshavarz S, Ghosh S (2013) Multi-scale crystal plasticity fem approach to modeling nickel based superalloys. Acta Mater 61:6549–6561CrossRefGoogle Scholar
  8. 8.
    Simmetrix (2014)
  9. 9.
    Matou K, Maniatty AM (2004) Finite element formulation for modelling large deformations in elasto-viscoplastic polycrystals. Int J Numer Methods Eng 60:2313–2333MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dohrmann CR, Heinstein MW, Jung J, Key SW, Witkowski WR (2000) Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes. Int J Numer Meth Eng 47:1549–1568CrossRefzbMATHGoogle Scholar
  11. 11.
    Gee MW, Dohrmann CR, Key SW, Wall WA (2009) A uniform nodal strain tetrahedron with isochoric stabilization. Int J Numer Methods Eng 78:429–443MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    de Souza Neto EA, Andrade Pires FM, Owen DRJ (2005) F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I. Int J Numer Meth Eng 62:353–383CrossRefzbMATHGoogle Scholar
  13. 13.
    de Souza Neto EA, Peric D, Owen DRJ (2008) Computational methods for plasticity: theory and applications. Wiley, ChichesterCrossRefGoogle Scholar
  14. 14.
    Bonet J, Marriott H, Hassan O (2001) Stability and comparison of different linear tetrahedral formulations for nearly incompressible explicit dynamic applications. Int J Numer Methods Eng 50:119–133CrossRefzbMATHGoogle Scholar
  15. 15.
    Puso MA, Solberg J (2006) A stabilized nodally integrated tetrahedral. Int J Numer Meth Eng 67:841–867MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Laschet G, Caylak I, Benke S, Mahnken R Locally integrated node-based formulations for four-node tetrahedral meshes. Private communicationGoogle Scholar
  17. 17.
    Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY (2009) A node-based smoothed finite element method (ns-fem) for upper bound solutions to solid mechanics problems. Comput Struct 87:14–26CrossRefGoogle Scholar
  18. 18.
    Nguyen-Xuan H, Liu GR (2013) An edge-based smoothed finite element method softened with a bubble function (bes-fem) for solid mechanics. Comput Struct 128:14–30CrossRefGoogle Scholar
  19. 19.
    Liu GR, Nguyen-Thoi T, Lam KY (2009) An edge-based smoothed finite element method (es-fem) for static, free and forced vibration analyses of solids. J Sound Vib 320:1100–1130CrossRefGoogle Scholar
  20. 20.
    Nguyen-Thoi T, Liu GR, Lam KY, Zhang GY (2009) A face-based smoothed finite element method (fs-fem) for 3d linear and non-linear solid mechanics problems using 4-node tetrahedral elements. Int J Numer Meth Eng 78:324–353MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    de Souza Neto EA, Peric D, Dutko M, Owen DRJ (1996) Design of simple low order finite elements for large strain analysis of nearly incompressible solids. Int J Solids Struct 33:3277–3296MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wolff S, Bucher C (2011) A finite element method based on c0 continuous assumed gradients. Int J Numer Methods Eng 86:876–914MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mahnken R, Caylak I (2008) Stabilization of bi-mixed finite elements for tetrahedral with enhanced interpolation using volume and area bubble functions. Int J Numer Methods Eng 75:377–413MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mahnken R, Caylak I, Laschet G (2008) Two mixed finite element formulations with area bubble functions for tetrahedral elements. Comput Methods Appl Mech Eng 197:1147–1165MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Bathe KJ (1996) Finite element procedures. Prentice-Hall Inc., Englewood CliffszbMATHGoogle Scholar
  26. 26.
    Staroselsky A, Anand L (2003) A constitutive model for hcp materials deforming by slip and twinning: application to magnesium alloy az31b. Int J Plast 19:843–1864CrossRefzbMATHGoogle Scholar
  27. 27.
    Busso E, Meissonier F, ODowd N (2000) Gradient-dependent deformation of two-phase single crystals. J Mech Phys Solid 48(11):2333–2361CrossRefGoogle Scholar
  28. 28.
    Zambaldi C, Roters F, Raabe D, Glatzel U (2007) Modeling and experiments on the indentation deformation and recrystallization of a single-crystal nickel-base superalloy. Mater Sci Eng A 454–455:433–440CrossRefGoogle Scholar
  29. 29.
    Roters F, Eisenlohr P, Hantcherli L, Tjahjantoa 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):1152–1211CrossRefGoogle Scholar
  30. 30.
    Asaro RJ, Needleman A (1985) Texture development and strain hardening in rate dependent polycrystals. Acta Mater 33(6):923–953CrossRefGoogle Scholar
  31. 31.
    Kocks UF, Argon AS, Ashby MF (1975) Thermodynamics and kinetics of slip. Prog Mater Sci 19:141–145Google Scholar
  32. 32.
    Venkataramani G, Ghosh S, Mills MJ (2007) A size dependent crystal plasticity finite element model for creep and load-shedding in polycrystalline Titanium alloys. Acta Mater 55:3971–3986CrossRefGoogle Scholar
  33. 33.
    Sinha S, Ghosh S (2006) Modeling cyclic ratcheting based fatigue life of hsla steels using crystal plasticity fem simulations and experiments. Int J Fatigue 28:1690–1704CrossRefGoogle Scholar
  34. 34.
    Cheng J, Ghosh S (2015) A crystal plasticity fe model for deformation with twin nucleation in magnesium alloys. Int J Plast 67:148–170CrossRefGoogle Scholar
  35. 35.
    Ashby MF (1970) Deformation of plastically non-homogeneous materials. Philos Mag 21:399–424CrossRefGoogle Scholar
  36. 36.
    Ma A, Roters F, Raabe D (2006) A dislocation density based constitutive model for crystal plasticity fem including geometrically necessary dislocations. Acta Mater 54:2169–2179Google Scholar
  37. 37.
    Cao G, Fu L, Lin J, Zhang Y, Chen C (2000) The relationships of microstructure and properties of a fully lamellar tial alloy. Intermetallics 8:647–653CrossRefGoogle Scholar
  38. 38.
    Li JCM, Chou YT (2001) The role of dislocations in the flow stress grain size relationships. Metall Mater Trans 52:97–120Google Scholar
  39. 39.
    Nye JF (1953) Some geometrical relations in dislocated crystals. Acta Metall 1(2):153–162CrossRefGoogle Scholar
  40. 40.
    Arsenlis A, Parks DM (1998) Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Mater 47:1597–1611CrossRefGoogle Scholar
  41. 41.
    Crisfield MA (1997) Nonlinear finite element analysis of solids and structures, vol 1., EssentialsWiley, ChichesterzbMATHGoogle Scholar
  42. 42.
    Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall Inc., Eagle Wood CliffszbMATHGoogle Scholar
  43. 43.
    Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery (spr) and adaptive finite element refinement. Comput Method Appl Mech Eng 101:207–224MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Morawiec A (2004) Orientations and rotations: computations in crystallographic textures. Springer, BerlinCrossRefzbMATHGoogle Scholar
  45. 45.
    Bettles CJ, Gibson MA (2005) Material rate dependence and localized deformation in crystalline solids. J Miner Methods Mater Soc 57(5):46–49CrossRefGoogle Scholar
  46. 46.
    Kainer KU (2003) Magnesium alloys and their applications. Wiley, WeinheimCrossRefGoogle Scholar
  47. 47.
    Barnett LR (2007) Twinning and the ductility of magnesium alloys: Part I: tension twins. Mater Sci Eng A464:1–7CrossRefGoogle Scholar
  48. 48.
    Yu Q, Zhang J, Jiang Y (2011) Fatigue damage development in pure polycrystalline magnesium under cyclic tensioncompression loading. Mater Sci Eng: A 528(25–26):7816–7826CrossRefGoogle Scholar
  49. 49.
    Groeber MA, Jackson MA (2014) Dream. 3d: a digital representation environment for the analysis of microstructure in 3d. Integr Mater Manuf Innov 3:5CrossRefGoogle Scholar
  50. 50.
    Groeber M, Ghosh S, Uchic MD, Dimiduk DM (2008) A framework for automated analysis and simulation of 3d polycrystalline microstructures. Part 1: statistical characterization. Acta Mater 56(6):1257–1273CrossRefGoogle Scholar
  51. 51.
    Groeber M, Ghosh S, Uchic MD, Dimiduk DM (2008) A framework for automated analysis and simulation of 3d polycrystalline microstructures. part 2: Synthetic structure generation. Acta Mater 56(6):1274–1287CrossRefGoogle Scholar
  52. 52.
    Mishra R, Inal K (2013) Microstructure data. Unpublished workGoogle Scholar
  53. 53.
    Khan AS, Pandey A, Gnaupel-Herold T, Mishra RK (2011) Mechanical response and texture evolution of az31 alloy at large strains for different strain rates and temperatures. Int J Plast 27:688–706CrossRefzbMATHGoogle Scholar
  54. 54.
    Xiaoye S Li (2005) An overview of superlu: algorithms, implementation, and user interface. Trans Math Soft 31(3):302–325MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Meissonnier FT, Busso EP, O’Dowd NP (2001) Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains. Int J Plast 17:601–640CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Civil EngineeringJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations