Representation of Materials Constitutive Responses in Finite Element-Based Design Codes



Finite element analysis codes developed originally for engineering structural analysis and design have been adopted by many investigators for materials science studies, and for development of computational material models on the continuum scale. The variety of modeling tools, solution paths, and utilities for constructing new material models make the commercial finite element codes an attractive environment for material model development. This chapter reviews several commonly used continuum mechanics codes, with emphasis on capabilities for representing important classes of material behaviors. A detailed discussion is presented of modeling anisotropic and heterogeneous material structures using representative volume elements and repeating unit cells, with particular emphasis on metallic and intermetallic engineering materials. The presentation includes numerical representations of microscopic and macroscopic material behaviors, and recent efforts to link the responses at these length scales. Numerical and phenomenological aspects of the development of material constitutive models are discussed.


Slip System Representative Volume Element Crystal Plasticity Finite Element Code Finite Element Method Modeling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Abdelaziz Y, Hamouine A (2008) A survey of the extended finite element. Comput Struct 86: 1141–1151Google Scholar
  2. Acharya A, Roy A, Sawant A (2006) Continuum theory and methods for coarse-grained, mesoscopic plasticity. Scripta Mater 54:705–710Google Scholar
  3. Acharya A, Roy A (2006) Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: part I. JMech Phys Solids 54:1687–1710MATHMathSciNetGoogle Scholar
  4. Arsenlis A, Parks DM (2002) Modeling the evolution of crystallographic dislocation density in crystal plasticity. J Mech Phys Solids 50:1979–2009MATHGoogle Scholar
  5. Arsenlis A, Tang M (2003) Simulations on the growth of dislocation density during state 0 deformation in BCC metals. Modell Simul Mater Sci Eng 11:251–264Google Scholar
  6. Asaro RJ, Needleman A (1985) Texture development and strain hardening in rate dependent polycrystals. Acta Metall 33:923–953Google Scholar
  7. Balasubramanian S, Anand L (1996) Single crystal and polycrystal elasto-viscoplasticity: application to earing in cup drawing of FCC materials. Comput Mech 17:209–225Google Scholar
  8. Balasubramanian S, Anand L (2002) Plasticity of initially textured hexagonal polycrystals at high homologous temperatures: application to titanium. Acta Mater 50:133–148Google Scholar
  9. Bansal Y, Pindera M-J (2006) Finite-volume direct averaging micromechanics of heterogeneous materials with elastic-plastic phases. Int J Plast 22:775–825MATHGoogle Scholar
  10. Barbe F, Decker L, Jeulin D, Cailletaud G (2001a) Intergranular and intragranular behavior of polycrystalline aggregates. Part 1: FE model. Int J Plast 17:513–536MATHGoogle Scholar
  11. Barbe F, Forest S, Cailletaud G (2001b) Intergranular and intragranular behavior of polycrystalline aggregates. Part 2: results. Int J Plast 17:537–563MATHGoogle Scholar
  12. Beaudoin AJ, Mathur KK, Dawson PR, Johnson GC (1993) Three-dimensional deformation process simulation with explicit use of polycrystal plasticity models. Int J Plast 9:833–860MATHGoogle Scholar
  13. Beaudoin AJ, Dawson PR, Mathur KK, Kocks UF, Korzekwa DA (1994) Application of polycrystal plasticity to sheet forming. Comput Methods Appl Mech Eng 117:49–70MATHGoogle Scholar
  14. Beaudoin AJ, Dawson PR, Mathur KK, Kocks UF (1995) A hybrid finite element formulation for polycrystal plasticity with consideration of macrostructural and microstructural linking. Int J Plast 11:501–521MATHGoogle Scholar
  15. Beaudoin AJ, Acharya A, Chen SR, Korzekwa DA, Stout MG (2000) Consideration of grain-size effect and kinetics in the plastic deformation of metal polycrystals. Acta Mater 48:3409–3423Google Scholar
  16. Becker R (1991) Analysis of texture evolution in channel die compression-I. Effects of grain interaction. Acta Metall Mater 39:1211–1230Google Scholar
  17. Becker R, Panchanadeeswaran S (1995) Effects of grain interactions on deformation and local texture in polycrystals. Acta Metall Mater 43:2701–2719Google Scholar
  18. Becker R (2002) Developments and trends in continuum plasticity. J Comput Aided Mater Des 9:145–163Google Scholar
  19. Bergan PG, Horrigmoe G, Bråkeland B, Søreide TH (1978) Solution techniques for non-linear finite element problems. Int J Numer Methods Eng 12:1677–1696MATHGoogle Scholar
  20. Bhattacharya AK (1991) A composite model to predict plastic flow of a superalloy based on its constituent properties. Scripta Metall Mater 25:1663–1667Google Scholar
  21. Bonora N, Ruggiero A (2006) Micromechanincal modeling of composites with mechanical interface– Part II: damage mechanics assessment. Compos Sci Technol 66:323–332Google Scholar
  22. Brockman RA (2003) Analysis of elastic-plastic deformation in TiAl polycrystals. Int J Plast 19:1749–1772MATHGoogle Scholar
  23. Bronkhorst CA, Kalidindi SR, Anand L (1992) Polycrystalline plasticity and the evolution of crystallographic texture in FCC metals. Philos Trans R Soc Lond A 341:443–477Google Scholar
  24. Busso EP, Meissonnier FT, O’Dowd NP (2000) Gradient-dependent deformation of two-phase single crystals. J Mech Phys Solids 48:2333–2361MATHGoogle Scholar
  25. Cailletaud G, Diard O, Feyel F, Forest S (2003) Computational crystal plasticity: from single crystal to homogenized polycrystals. Technische Mechanik 23:130–145Google Scholar
  26. Carter BJ, Wawrzynek PA, Ingraffea AR (2000) Automated 3D crack growth simulation. Int J Numer Methods Eng 47:229–253MATHGoogle Scholar
  27. Chanwandi R, Timbrell C (2007) Simulation of 3-D non-planar crack propagation. Proc NAFEMS World Congress, Vancouver, British ColumbiaGoogle Scholar
  28. Cheong KS, Busso EP, Arsenlis A (2005) A study of microstructural length scale effects on the behaviour of FCC polycrystals using strain gradient concepts. Int J Plast 21:1797–1814MATHGoogle Scholar
  29. Choi YS, Parthasarathy TA, Dimiduk DM, Uchic MD (2005) Numerical study of the flow responses and the geometric constraint effects in Ni-base two-phase single crystals using strain gradient plasticity. Mater Sci Eng A397:69–83Google Scholar
  30. Christman T, Needleman A, Suresh S (1989) An experimental and numerical study of deformation in metal-ceramic composites. Acta Metall 37:3029–3050Google Scholar
  31. Devincre B, Kubin L, Hoc T (2006) Physical analysis of crystal plasticity by DD simulations. Scripta Mater 54:741–746Google Scholar
  32. Devincre B, Hoc T, Kubin L (2008) Dislocation mean free paths and straining hardening of crystals. Science 320:1745–1748Google Scholar
  33. Dimiduk DM, Koslowski M, LeSar R (2006) Preface to the viewpoint set on: statistical mechanics and coarse graining of dislocation behavior for continuum plasticity. Scripta Mater 54:701–704Google Scholar
  34. Du Z-Z, Zok FW (1998) Limit stress conditions for weakly bonded fiber composites subject to transverse biaxial tensile loading. Int J Solids Struct 35:2821–2842MATHGoogle Scholar
  35. Dunne F, Petrinik N (2005) Introduction to computational plasticity. Oxford University Press, NewYorkMATHGoogle Scholar
  36. El-Azab (2006) Statistical mechanics of dislocation systems. Scripta Mater 54:723–727Google Scholar
  37. El-Azab, Deng J, Tang M (2007) Statistical characterization of dislocation ensembles. Philos Mag 87:1201–1223Google Scholar
  38. Evers LP, Parks DM, Brekelmans WAM, Geers MGD (2002) Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation. J Mech Phys Solids 50:2403–2424MATHGoogle Scholar
  39. Fivel M, Tabourot E, Rauch E, Canova G (1998) Identification through mesoscopic simulations of macroscopic parameters of physically based constitutive equations for the plastic behaviour of fcc single crystals. J Phys IV France 8:Pr8.151–Pr8.158Google Scholar
  40. Franciosi P, Zaoui A (1982) Multislip in FCC crystals: a theoretical approach compared with experimental data. Acta Metall 30:1627–1637Google Scholar
  41. Frank G, Brockman RA (2001) A viscoelastic-viscoplastic constitutive model for glassy polymers. Int J Solids Struct 38:5149–5164MATHGoogle Scholar
  42. Frénois S, Munier E, Feaugas X, Pilvin P (2001) A polycrystalline model for stress-strain behaviour of tantalum at 300K. J Phys IV France 11:Pr5.302–Pr5.308Google Scholar
  43. Ganghoffer JF, Hazotte A, Denis S, Simon A (1991) Finite element calculation of internal mismatch stresses in a single crystal nickel base superalloy. Scripta Metall Mater 25:2491–2496Google Scholar
  44. Ganguly P, Poole WJ (2005) Rearrangement of local stress and strain fields due to damage initiation in a model composite system. Comput Mater Sci 34:107–122Google Scholar
  45. Ghosh S, Dakshinamurthy V, Hu C, Bai J (2008) Multi-scale characterization and modeling of ductile failure in cast aluminum alloys. Int J Comput Meth Eng Sci Mech 9:1–18Google Scholar
  46. Giner E, Sukumar N, Denia FD, Fuenmayor FJ (2008) Extended finite element method for fretting fatigue crack propagation. Int J Solids Struct 45:5675–5687MATHGoogle Scholar
  47. Glatzel U, Feller-Kniepmeier M (1989) Calculations of internal stresses in the γ∕γ microstructure of a nickel-base superalloy with high volume fraction of γ-phase. Scripta Metall 23: 1839–1844Google Scholar
  48. Gurtin ME, Anand L (2005), The decomposition FeFp, material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous, Int J Plast 21:1686–1719MATHGoogle Scholar
  49. Harder J (1999) A crystallographic model for the study of local deformation processes in polycrystals. Int J Plast 15:605–624MATHGoogle Scholar
  50. Harren SV, Asaro SV (1989) Nonuniform deformations in polycrystals and aspects of the validity of the Taylor model. J Mech Phys Solids 37:191–232MATHGoogle Scholar
  51. Hasija V, Ghosh S, Mills MJ, Joseph DS (2003) Deformation and creep modeling in polycrystalline Ti-6Al alloys. Acta Mater 51:4533–4549Google Scholar
  52. Hibbitt HD Karlsson BI (1979) Analysis of pipe whip, Paper 79-PVP-122, ASME Pressure Vessels and Piping Conference, San Francisco, CaliforniaGoogle Scholar
  53. Hughes TJR, Winget J (1980) Finite rotation effects in numerical integration of rate constitutive equations arising in large deformation analysis, Int J Numer Methods Eng 15:1862–1867MATHMathSciNetGoogle Scholar
  54. Ismar H, Schroter F, Streicher F (2001) Effects of interfacial debonding on the transverse loading behaviour of continuous fibre-reinforced metal matrix composites. Comput Struct 79: 1713–1722Google Scholar
  55. Jäger P, Steinmann P, Kuhl E (2008), Modeling three-dimensional crack propagation– a comparison of crack path tracking strategies, Int J Numer Methods Eng 76:1328–1352MATHGoogle Scholar
  56. Jin K-K, Oh J-H, Ha S-K (2007) Effect of fiber arrangement on residual thermal stress distributions in a unidirectional composite. J Compos Mater 41:591–611Google Scholar
  57. Kalidindi SR, Bronkhorst CA, Anand L (1992) Crystallographic texture evolution in bulk deformation processing of FCC metals. J Mech Phys Solids 40:537–569Google Scholar
  58. Kalidindi SR, Anand L (1994) Macroscopic shape change and evolution of crystallographic texture in pre-textured FCC metals. J Mech Phys Solids 42:459–490Google Scholar
  59. Kirchner E (2001) Modeling single crystals: time integration, tangent operators, sensitivity analysis and shape optimization. Int J Plast 17:907–942MATHGoogle Scholar
  60. Kocks UF (1976) Laws for work-hardening and low-temperature creep. J Eng Mater Technol 98:76–85Google Scholar
  61. Kocks UF (2000) Kinematics and kinetics of plasticity. In: Kocks UF, Tomé CN, Wenk H-R (eds) Texture and anisotropy. Cambridge University Press, CambridgeGoogle Scholar
  62. Kocks UF, Argon AS, Ashby MF (1975) Thermodynamics and kinetics of slip. Prog Mater Sci 19:1–288Google Scholar
  63. Kok S, Beaudoin AJ, Tortorelli DA (2002) On the development of stage IV hardening using a model based on the mechanical threshold. Acta Mater 50:1653–1667Google Scholar
  64. Kovač M, Cizelj L (2005) Modeling elasto-plastic behavior of polycrystalline grain structure of steels at mesoscopic level. Nucl Eng Des 235:1939–1950Google Scholar
  65. Kröner E (1960) Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Arch Rational Mech Anal 4:273–334MATHMathSciNetGoogle Scholar
  66. Kumar RS, Wang A-J, McDowell DL (2006) Effects of microstructure variability on intrinsic fatigue resistance of nickel-base superalloys– a computational micromechanics approach. Int J Fract 137:173–210MATHGoogle Scholar
  67. Kuttner T, Wahi RP (1998) Modelling of internal stress distribution and deformation behaviour in the precipitation hardened superalloy SC16. Mater Sci Eng A242:259–267Google Scholar
  68. Lee EH (1969) Elastic plastic deformation at finite strain. Trans ASME J Appl Mech 36:1–6MATHGoogle Scholar
  69. Lewis AC, Suh C, Stukowski M, Geltmacher AB, Spanos G, Rajan K (2006) Quantitative analysis and feature recongnition in 3-D microstructural data sets. JOM 58 (12):52–56Google Scholar
  70. Lewis AC, Jordan KA, Geltmacher AB (2008) Determination of critical microstructural features in an austenitic stainless steel using image-based finite element modeling. Metall Mater Trans A 39:1109–1117Google Scholar
  71. Li C, Ellyin F (1998) A micro-macro correlation analysis for metal matrix composites undergoing multiaxial damage. Int J Solids Struct 35:637–649MATHGoogle Scholar
  72. Li S, Anchana W (2004) Unit cells for micromechanical analyses of particle-reinforced composites. Mech Mater 36:543–572Google Scholar
  73. Luciano R, Bisegna P (1998) Bounds on the overall properties of composites with debonded frictionless interfaces. Mech Mater 28:23–32Google Scholar
  74. Ma A, Dye D, Reed RC (2008) A model for the creep deformation behavior of single-crystal superalloy CMSX-4. Acta Mater 56:1657–1670Google Scholar
  75. MacLachlan DW, Knowles DM (2002) The effect of material behavior on the analysis of single crystal turbine blades: Part II– component analysis. Fatigue Fract Eng Mater Struct 25: 399–409Google Scholar
  76. Madec R, Devincre B, Kubin LP (2002) From dislocation junctions to forest hardening. Phys Rev Lett 89:255508-1–25508-4Google Scholar
  77. Madec R, Devincre B, Kubin LP, Hoc T, Rodney D (2003) The role of collinear interaction in dislocation-induced hardening. Science 301:1879–1882Google Scholar
  78. Marin EB, Dawson PR (1998) On modeling the elasto-viscoplastic response of metals using polycrystal plasticity. Comput Methods Appl Mech Eng 165:1–21MATHGoogle Scholar
  79. Mathur KK, Dawson PR (1989) On modeling the development of crystallographic texture in bulk forming processes. Int J Plast 5:67–94Google Scholar
  80. McHugh PE, Connolly P (1994) Modelling the thermo-mechanical behaviour of an Al alloy-SiCp composite. Effects of particle shape and microscale fracture. Comput Mater Sci 3:199–206Google Scholar
  81. Meissonnier FT, Busso EP, O’Dowd NP (2001) Finite element implementation of a generalized non-local rate-dependent crystallographic formulation for finite strains. Int J Plast 17:601–640MATHGoogle Scholar
  82. Metzger DR, Duan X, Jain M, Wilkinson DS, Mishra R, Kim S, Sachdev AK (2006) The influence of particle distribution and volume fraction on the post-necking behaviour of aluminum alloys. Mech Mater 38:1026–1038Google Scholar
  83. Mika DP, Dawson PR (1999) Polycrystal plasticity modeling of intracrystalline boundary textures. Acta Mater 47:1355–1369Google Scholar
  84. Müller L, Glatzel U, Feller-Kniepmeier M (1993) Calculation of the internal stresses and strains in the microstructure of a single crystal nickel-base superalloy during creep. Acta Metall Mater 41:3401–3411Google Scholar
  85. Murray NGD, Dunand DC (2004) Effect of thermal history on the superplastic expansion of argon-filled pores in titanium: part II modeling of kinetics. Acta Mater 52:2279–2291Google Scholar
  86. Nadgorny E (1988) Dislocation dynamics and mechanical properties of crystals. Prog Mater Sci 31:1–530Google Scholar
  87. Nakamachi E, Tam NN, Morimoto H (2007) Multi-scale finite element analysis of sheet metals by using SEM-EBSD measured crystallographic RVE models. Int J Plast 23:450–489MATHGoogle Scholar
  88. Nemat-Nasser S, Hori M (1993) Micromechanics: overall properties of heterogeneous materials. North-Holland, AmsterdamMATHGoogle Scholar
  89. Niordson CF, Tvergaard V (2002) Nonlocal plasticity effects on fibre debonding in a whisker-reinforced metal. Eur J Mech A-Solid 21:239–248MATHGoogle Scholar
  90. Nouailhas D, Cailletaud G (1996) Finite element analysis of the mechanical behavior of two-phase single-crystal superalloys. Scripta Mater 34:565–571Google Scholar
  91. Nouailhas D, Lhuillier S (1997) On the micro-macro modeling of γ∕γ single crystal behavior. Comput Mater Sci 9:177–187Google Scholar
  92. Pollock TM, Argon AS (1992) Creep resistance of CMSX-3 nickel base superalloy single crystals. Acta Metall Mater 40:1–30Google Scholar
  93. Ponthot JP (2002), Unified stress update algorithms for the numerical simulation of large deformation elastic-plastic and elasto-viscoplastic processes, Int J Plast 18:91–126MATHGoogle Scholar
  94. Potirniche GP, Hearndon JL, Horstemeyer MF, Ling XW (2006) Lattice orientation effects on void growth and coalescence in fcc single crystals. Int J Plast 22:921–942MATHGoogle Scholar
  95. Prabu SB, Karunamoorthy L, Kandasami GS (2004) A finite element analysis study of micromechanical interfacial characteristics of metal matrix composites. J Mater Proc Tech 153–154:992–997Google Scholar
  96. Preußner J, Rudnik Y, Völkl R, Glatzel U (2005) Finite-element modeling of anisotropic single-crystal superalloy creep deformation based on dislocation densities of individual slip systems. Z Metallkd 96:595–601Google Scholar
  97. Ramberg W, Osgood WR (1943) Description of stress-strain curves by three parameters (Technical Note No. 902). National Advisory Committee for Aeronautics, WashingtonDCGoogle Scholar
  98. Rickman JM, Vinals J, LeSar R (2005) Unified framework for dislocation-based defect energetics. Philos Mag 85:917–929Google Scholar
  99. Rickman JM, LeSar R (2006) Issues in the coarse-graining of dislocation energetic and dynamics. Scripta Mater 54:735–739Google Scholar
  100. Roy A, Puri S, Acharya A (2007) Phenomenological mesoscopic field dislocation mechanics, lower-order gradient plasticity, and transport of mean excess dislocation density. Modell Simul Mater Sci Eng 15:S167–S180Google Scholar
  101. Sansour C, Karšaj I, Sorić J (2008) On a numerical implementation of a formulation of anisotropic continuum elastoplasticity at finite strains. J Comput Phys 227:7643–7663MATHMathSciNetGoogle Scholar
  102. Sarma GB, Radhakrishnan B, Zacharia T (1998) Finite element simulations of cold deformation at the mesoscale. Comput Mater Sci 12:105–123Google Scholar
  103. Segurado J, LLorca J (2005) A computational micromechanics study of the effect of interface decohesion on the mechanical behavior of composites. Acta Mater 53:4931–4942Google Scholar
  104. Soussou JE, Moavenzadeh F, Gradowczyk MH (1970) Application of Prony series to linear viscoelasticity. J Rheol 14:573–584Google Scholar
  105. Swaminathan S, Ghosh S (2006) Statistically equivalent representative volume elements for unidirectional composite microstructures: part I– without damage. J Compos Mater 40:583–604Google Scholar
  106. Tabourot L, Dumoulin S, Balland P (2001) An attempt for a unified description from dislocation dynamics to metallic plastic behaviour. J Phys IV France 11:Pr5.111–Pr5.118Google Scholar
  107. Tabourot L, Fivel M, Rauch E (1997) Generalized constitutive laws for fcc single crystals. Mater Sci Eng A 234–236:639–642Google Scholar
  108. Taylor GI (1938a) Plastic strain in metals. J Inst Metals 62:307–324Google Scholar
  109. Taylor GI (1938b) Analysis of plastic strain in a cubic crystal, S. Timoshenko 60th Anniversary Volume. Macmillan, NewYorkGoogle Scholar
  110. Tirtom I, Güden M, Yildiz H (2008) Simulation of the strain rate sensitive flow behavior of SiC-particulate reinforce aluminum metal matrix composites. Comp Mater Sci 42:570–578Google Scholar
  111. Tvergaard V (1990) Analysis of tensile properties for a whisker-reinforced metal-matrix composite. Acta Metall Mater 38:185–194Google Scholar
  112. Wilkins M (1964) Calculation of elastoplastic flows. In: Alder B (ed) Methods in computational physics, vol 3. Academic Press, New York, pp 211–263Google Scholar
  113. Xie CL, Ghosh S, Groeber M (2004) Modeling cyclic deformation of HSLA steels using crystal plasticity. J Eng Mater Tech 126:339–352Google Scholar
  114. Zaiser M, Hochrainer T (2006) Some steps towards a continuum representation of 3D dislocation systems. Scripta Mater 54:717–721Google Scholar
  115. Zeghadi A, N’guyen F, Forest S, Gourgues A-F, Bouaziz O (2007a) Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure– part 1: anisotropic elastic behaviour. Philos Mag 87:1401–1424Google Scholar
  116. Zeghadi A, Forest S, Gourgues A-F, Bouaziz O (2007b) Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure– Part 2: crystal plasticity. Philos Mag 87:1425–1446Google Scholar
  117. Zhao Z, Kuchnicki S, Radovitzky R, Cuitiňo (2007) Influence of in-grain mesh resolution on the prediction of deformation textures in fcc polycrystals by crystal plasticity FEM. Acta Mater 55: 2361–2373Google Scholar
  118. Zong BY, Zhang F, Wang G, Zuo L (2007) Strengthening mechanism of load sharing of particulate reinforcements in a metal matrix composite. J Mater Sci 42:4215–4226Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Universal Energy SystemsDaytonUSA

Personalised recommendations