Computational Mechanics

, Volume 56, Issue 4, pp 613–630 | Cite as

Modeling and simulation of cooling-induced residual stresses in heated particulate mixture depositions in additive manufacturing

  • T. I. ZohdiEmail author
Original Paper


One key aspect of many additive manufacturing processes is the deposition of heated mixtures of particulate materials onto surfaces, which then bond and cool, leading to complex microstructures and possible residual stresses. The overall objective of this work is to construct a straightforward computational approach that researchers in the field can easily implement and use as a numerically-efficient simulation and design tool. Specifically because multifield coupling is present, a recursive, staggered, temporally-adaptive, finite difference time domain scheme is developed to resolve the internal microstructural thermal and mechanical fields, accounting for the simultaneous elasto-plasticity and damage. The time step adaptation allows the numerical scheme to iteratively resolve the changing physical fields by refining the time-steps during phases of the process when the system is undergoing large changes on a relatively small time-scale and can also enlarge the time-steps when the processes are relatively slow. The spatial discretization grids are uniform and dense. The deposited microstructure is embedded into spatial discretization. The regular grid allows one to generate a matrix-free iterative formulation which is amenable to rapid computation and minimal memory requirements, making it ideal for laptop computation. Numerical examples are provided to illustrate the approach. This formulation is useful for material scientists who seek ways to deposit such materials while simultaneously avoiding inadvertent excessive residual stresses.


Particulates Multiphysics Residual stresses 


  1. 1.
    Avci B, Wriggers P (2012) A DEM-FEM coupling approach for the direct numerical simulation of 3D particulate flows. J Appl Mech 79:010901-1–010901-7CrossRefGoogle Scholar
  2. 2.
    Akisanya AR, Cocks ACF, Fleck NA (1997) The yield behavior of metal powders. Int J Mech Sci 39:1315–1324CrossRefGoogle Scholar
  3. 3.
    Ames WF (1977) Numerical methods for partial differential equations, 2nd edn. Academic Press, New YorkzbMATHGoogle Scholar
  4. 4.
    Anand L, Gu C (2000) Granular materials: constitutive equations and shear localization. J Mech Phys Solids 48:1701–1733zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Axelsson O (1994) Iterative solution methods. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  6. 6.
    Bianco M, Bilardi G, Pesavento F, Pucci G, Schrefler BA (2003) A frontal solver tuned for fully coupled non-linear hygro-thermo-mechanical problems. Int J Numer Methods Eng 57:18011818CrossRefGoogle Scholar
  7. 7.
    Bolintineanu DS, Grest GS, Lechman JB, Pierce F, Plimpton SJ, Schunk PR (2014) Particle dynamics modeling methods for colloid suspensions. Comput Part Mech 1(3):321–356CrossRefGoogle Scholar
  8. 8.
    Brown S, Abou-Chedid G (1994) Yield behavior of metal powder assemblages. J Mech Phys Solids 42:383–398CrossRefGoogle Scholar
  9. 9.
    Campello EMB, Zohdi TI (2014) A computational framework for simulation of the delivery of substances into cells. Int J Numer Methods Biomed Eng 30(11):1132–1152MathSciNetCrossRefGoogle Scholar
  10. 10.
    Campello EMB, Zohdi TI (2014) Design evaluation of a particle bombardment system to deliver substances into cells. Comput Mech Eng Sci 98(2):221–245MathSciNetGoogle Scholar
  11. 11.
    Cante J, Davalos C, Hernandez JA, Oliver J, Jonsen P, Gustafsson G, Haggblad HA (2014) PFEM-based modeling of industrial granular flows. Comput Part Mech 1(1):47–70CrossRefGoogle Scholar
  12. 12.
    Carbonell JM, Onate E, Suarez B (2010) Modeling of ground excavation with the particle finite element method. J Eng Mech, ASCE 136:455–463CrossRefGoogle Scholar
  13. 13.
    Deckard C (1986) Method and apparatus for producing parts by selective sintering US Patent 4,863,538Google Scholar
  14. 14.
    Domas F (1997) Eigenschaft profile und Anwendungsübersicht von EPE und EPP. Technical report of the BASF CompanyGoogle Scholar
  15. 15.
    Donev A, Cisse I, Sachs D, Variano EA, Stillinger F, Connelly R, Torquato S, Chaikin P (2004a) Improving the density of jammed disordered packings using ellipsoids. Science 303:990–993Google Scholar
  16. 16.
    Donev A, Stillinger FH, Chaikin PM, Torquato S (2004b) Unusually dense crystal ellipsoid packings. Phys Rev Lett 92:255506CrossRefGoogle Scholar
  17. 17.
    Donev A, Torquato S, Stillinger F (2005a) Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles-I. Algorithmic details. J Comput Phys 202:737zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Donev A, Torquato S, Stillinger F (2005b) Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles-II. Application to ellipses and ellipsoids. J Comput Phys 202:765zbMATHMathSciNetGoogle Scholar
  19. 19.
    Donev A, Torquato S, Stillinger FH (2005c) Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings. Phys Rev E 71:011105MathSciNetCrossRefGoogle Scholar
  20. 20.
    Duran J (1997) Sands, powders and grains. An introduction to the physics of granular matter. Springer, New YorkGoogle Scholar
  21. 21.
    Dwivedi G, Wentz T, Sampath S, Nakamura T (2010) Assessing process and coating reliability through monitoring of process and design relevant coating properties. J Thermal Spray Technol 19:695–712CrossRefGoogle Scholar
  22. 22.
    Fleck NA (1995) On the cold compaction of powders. J Mech Phys Solids 43:1409–1431zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Fuller SB, Wilhelm EJ, Jacobson JM (2002) Ink-jet printed nanoparticle microelectromechanical systems. J Microelectromech Syst 11:54–60CrossRefGoogle Scholar
  24. 24.
    Gamota D, Brazis P, Kalyanasundaram K, Zhang J (2004) Printed organic and molecular electronics. Kluwer Academic Publishers, New YorkCrossRefGoogle Scholar
  25. 25.
    Gethin DT, Lewis RW, Ransing RS (2003) A discrete deformable element approach for the compaction of powder systems. Model Simul Mater Sci Eng 11(1):101–114CrossRefGoogle Scholar
  26. 26.
    Ghosh S (2011) Micromechanical analysis and multi-scale modeling using the Voronoi cell finite element method. CRC Press/Taylor & Francis, Boca RatonzbMATHCrossRefGoogle Scholar
  27. 27.
    Ghosh S, Dimiduk D (2011) Computational methods for microstructure-property relations. Springer, New YorkCrossRefGoogle Scholar
  28. 28.
    Gu C, Kim M, Anand L (2001) Constitutive equations for metal powders: application to powder forming processes. Int J Plast 17:147–209zbMATHCrossRefGoogle Scholar
  29. 29.
    Haruta M (2002) Catalysis of gold nanoparticles deposited on metal oxides. Cattech 6(3):102–115CrossRefGoogle Scholar
  30. 30.
    Hashin Z, Shtrikman S (1962) On some variational principles in anisotropic and nonhomogeneous elasticity. J Mech Phys Solids 10:335–342MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hashin Z (1983) Analysis of composite materials: a survey. ASME J Appl Mech 50:481–505zbMATHCrossRefGoogle Scholar
  32. 32.
    Housholder R (1979) Molding process. US Patent 4,247,508Google Scholar
  33. 33.
    Hull C (1984) Apparatus for production of three-dimensional objects by stereolithography. US Patent 4,575,330Google Scholar
  34. 34.
    Jikov VV, Kozlov SM, Olenik OA (1994) Homogenization of differential operators and integral functionals. Springer, New YorkCrossRefGoogle Scholar
  35. 35.
    Kachanov LM (1986) Introduction to continuum damage mechanics. Martinus Nijoff, DordrichtzbMATHCrossRefGoogle Scholar
  36. 36.
    Kachanov M (1993) Elastic solids with many cracks and related problems. Advance applied mechanics, vol 30. Academic Press, New YorkGoogle Scholar
  37. 37.
    Kachanov M, Tsukrov I, Shafiro B (1994) Effective moduli of solids with cavities of various shapes. Appl Mech Rev 47:S151–S174CrossRefGoogle Scholar
  38. 38.
    Kachanov M, Sevostianov I (2005) On the quantitative characterization of microstructures and effective properties. Int J Solids Struct 42:309–336zbMATHCrossRefGoogle Scholar
  39. 39.
    Kansaal A, Torquato S, Stillinger F (2002) Diversity of order & densities in jammed hard-particle packings. Phys Rev E 66:041109CrossRefGoogle Scholar
  40. 40.
    Martin P (2009) Handbook of deposition technologies for films and coatings, 3rd edn. Elsevier, OxfordGoogle Scholar
  41. 41.
    Martin P (2011) Introduction to surface engineering and functionally engineered materials. Scrivener and Elsevier, HobokenCrossRefGoogle Scholar
  42. 42.
    Labra C, Onate E (2009) High-density sphere packing for discrete element method simulations. Commun Numer Methods Eng 25(7):837–849zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Leonardi A, Wittel FK, Mendoza M, Herrmann HJ (2014) Coupled DEM-LBM method for the free-surface simulation of heterogeneous suspensions. Comput Part Mech 1(1):3–13CrossRefGoogle Scholar
  44. 44.
    Lewis RW, Gethin DT, Yang XSS, Rowe RC (2005) A combined finite-discrete element method for simulating pharmaceutical powder tableting. Int J Numer Methods Eng 62:853869CrossRefGoogle Scholar
  45. 45.
    Lewis RW, Schrefler BA, Simoni L (1992) Coupling versus uncoupling in soil consolidation. Int J Numer Anal Methods Geomech 15:533–548CrossRefGoogle Scholar
  46. 46.
    Lewis RW, Schrefler BA (1998) The finite element method in the static and dynamic deformation and consolidation of porous media, 2nd edn. Wiley press, New YorkzbMATHGoogle Scholar
  47. 47.
    Liu Y, Nakamura T, Dwivedi G, Valarezo A, Sampath S (2008) Anelastic behavior of plasma sprayed zirconia coatings. J Am Ceram Soc 91:4036–4043CrossRefGoogle Scholar
  48. 48.
    Liu Y, Nakamura T, Srinivasan V, Vaidya A, Gouldstone A, Sampath S (2007) Nonlinear elastic properties of plasma sprayed zirconia coatings and associated relationships to processing conditions. Acta Mater 55:4667–4678CrossRefGoogle Scholar
  49. 49.
    Maxwell JC (1867) On the dynamical theory of gases. Philos Trans Soc Lond 157:49CrossRefGoogle Scholar
  50. 50.
    Maxwell JC (1873) A treatise on electricity and magnetism, 3rd edn. Clarendon Press, OxfordGoogle Scholar
  51. 51.
    Mura T (1993) Micromechanics of defects in solids, 2nd edn. Kluwer Academic Publishers, DordrechtGoogle Scholar
  52. 52.
    Nakamura T, Liu Y (2007) Determination of nonlinear properties of thermal sprayed ceramic coatings via inverse analysis. Int J Solids Struct 44:1990–2009zbMATHCrossRefGoogle Scholar
  53. 53.
    Nakamura T, Qian G, Berndt CC (2000) Effects of pores on mechanical properties of plasma sprayed ceramic coatings. J Am Ceram Soc 83:578–584CrossRefGoogle Scholar
  54. 54.
    Nakanishi H, Bishop KJM, Kowalczyk B, Nitzan A, Weiss EA, Tretiakov KV, Apodaca MM, Klajn R, Stoddart JF, Grzybowski BA (2009) Photoconductance and inverse photoconductance in thin films of functionalized metal nanoparticles. Nature 460:371–375CrossRefGoogle Scholar
  55. 55.
    Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous solids, 2nd edn. Elsevier, AmsterdamGoogle Scholar
  56. 56.
    Onate E, Idelsohn SR, Celigueta MA, Rossi R (2008) Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows. Comput Methods Appl Mech Eng 197(19–20):1777–1800zbMATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    Onate E, Celigueta MA, Idelsohn SR, Salazar F, Surez B (2011) Possibilities of the particle finite element method for fluid-soil-structure interaction problems. Comput Mech 48:307–318zbMATHMathSciNetCrossRefGoogle Scholar
  58. 58.
    Pöschel T, Schwager T (2004) Computational granular dynamics. Springer, New YorkGoogle Scholar
  59. 59.
    Qian G, Nakamura T, Berndt CC (1998) Effects of thermal gradient and residual stresses on thermal barrier coating fracture. Mech Mater 27:91–110CrossRefGoogle Scholar
  60. 60.
    Ransing RS, Lewis RW, Gethin DT (2004) Using a deformable discrete-element technique to model the compaction behaviour of mixed ductile and brittle particulate systems. Philos Trans R Soc Ser A 362(1822):1867–1884zbMATHCrossRefGoogle Scholar
  61. 61.
    Rayleigh JW (1892) On the influence of obstacles arranged in rectangular order upon properties of a medium. Philos Mag 32:481–491CrossRefGoogle Scholar
  62. 62.
    Rojek J, Labra C, Su O, Onate E (2012) Comparative study of different discrete element models and evaluation of equivalent micromechanical parameters. Int J Solids Struct 49:1497–1517. doi: 10.1016/j.ijsolstr.2012.02.032 CrossRefGoogle Scholar
  63. 63.
    Rojek J (2014) Discrete element thermomechanical modelling of rock cutting with valuation of tool wear. Comput Part Mech 1(1):71–84CrossRefGoogle Scholar
  64. 64.
    Onate E, Celigueta MA, Latorre S, Casas G, Rossi R, Rojek J (2014) Lagrangian analysis of multiscale particulate flows with the particle finite element method. Comput Part Mech 1(1):85–102CrossRefGoogle Scholar
  65. 65.
    Schrefler BA (1985) A partitioned solution procedure for geothermal reservoir analysis. Commun Appl Numer Methods 1:53–56zbMATHCrossRefGoogle Scholar
  66. 66.
    Sevostianov I, Gorbatikh L, Kachanov M (2001) Recovery of information of porous/microcracked materials from the effective elastic/conductive properties. Mater Sci Eng A 318:1–14Google Scholar
  67. 67.
    Sevostianov I, Kachanov M (2008) Connections between elastic and conductive properties of heterogeneous materials. Adv Appl Mech 42:69–253CrossRefGoogle Scholar
  68. 68.
    Sevostianov I, Kachanov M (2000) Modeling of the anisotropic elastic properties of plasma-sprayed coatings in relation to their microstructure. Acta Mater 48(6):1361–1370CrossRefGoogle Scholar
  69. 69.
    Sevostianov I, Kachanov M (2001) Thermal conductivity of plasma sprayed coatings in relation to their microstructure. J Therm Spray Technol 9(4):478–482CrossRefGoogle Scholar
  70. 70.
    Sevostianov I, Kachanov M (2001) Plasma-sprayed ceramic coatings: anisotropic elastic and conductive properties in relation to the microstructure; cross-property correlations, with I. Sevostianov. Mater Sci Eng-A 297:235–243Google Scholar
  71. 71.
    Tatzel H (1996) Grundlagen der Verarbeitungstechnik von EPP-Bewährte und neue Verfahren. Technical report of the BASF CompanyGoogle Scholar
  72. 72.
    Torquato S (2001) Random heterogeneous materials: microstructure and macroscopic properties. Springer, New YorkGoogle Scholar
  73. 73.
    Turska E, Schrefler BA (1994) On consistency, stability and convergence of staggered solution procedures. Rend Mat Acc Lincei, Rome, S. 9, 5:265–271Google Scholar
  74. 74.
    Wang X, Schrefler BA (1998) A multifrontal parallel algorithm for coupled thermo-hydro-mechanical analysis of deforming porous media. Int J Numer Methods Eng 43:1069–1083zbMATHCrossRefGoogle Scholar
  75. 75.
    Widom B (1966) Random sequential addition of hard spheres to a volume. J Chem Phys 44:3888–3894CrossRefGoogle Scholar
  76. 76.
    Young DM (1950) Iterative methods for solving partial difference equations of elliptic type. Doctoral thesis. Harvard UniversityGoogle Scholar
  77. 77.
    Zienkiewicz OC (1984) Coupled problems & their numerical solution. In: Lewis RW, Bettes P, Hinton E (eds) Numerical methods in coupled systems. Wiley, Chichester, pp 35–38Google Scholar
  78. 78.
    Zienkiewicz OC, Paul DK, Chan AHC (1988) Unconditionally stable staggered solution procedure for soil-pore fluid interaction problems. Int J Numer Methods Eng 26:1039–1055zbMATHCrossRefGoogle Scholar
  79. 79.
    Zohdi TI (2002) An adaptive-recursive staggering strategy for simulating multifield coupled processes in microheterogeneous solids. Int J Numer Methods Eng 53:1511–1532zbMATHMathSciNetCrossRefGoogle Scholar
  80. 80.
    Zohdi TI (2003) On the compaction of cohesive hyperelastic granules at finite strains. Proc R Soc 454(2034):1395–1401CrossRefGoogle Scholar
  81. 81.
    Zohdi TI (2003) Genetic design of solids possessing a random-particulate microstructure. Philos Trans R Soc 361(1806):1021–1043zbMATHMathSciNetCrossRefGoogle Scholar
  82. 82.
    Zohdi TI (2004) Modeling and simulation of a class of coupled thermo-chemo-mechanical processes in multiphase solids. Comput Methods Appl Mech Eng 193(6–8):679–699zbMATHCrossRefGoogle Scholar
  83. 83.
    Zohdi TI (2006) Computation of the coupled thermo-optical scattering properties of random particulate systems. Comput Methods Appl Mech Eng 195:5813–5830zbMATHCrossRefGoogle Scholar
  84. 84.
    Zohdi TI, Wriggers P (2008) Introduction to computational micromechanics. Springer, New York Second ReprintingzbMATHGoogle Scholar
  85. 85.
    Zohdi TI (2010) Simulation of coupled microscale multiphysical-fields in particulate-doped dielectrics with staggered adaptive FDTD. Comput Methods Appl Mech Eng 199:79–101MathSciNetCrossRefGoogle Scholar
  86. 86.
    Zohdi TI (2012) Dynamics of charged particulate systems. Modeling, theory and computation. Springer, New YorkzbMATHCrossRefGoogle Scholar
  87. 87.
    Zohdi TI (2012) Electromagnetic properties of multiphase dielectrics. A primer on modeling, theory and computation. Springer, New YorkCrossRefGoogle Scholar
  88. 88.
    Zohdi TI (2013a) Numerical simulation of charged particulate cluster-droplet impact on electrified surfaces. J Comput Phys 233:509–526MathSciNetCrossRefGoogle Scholar
  89. 89.
    Zohdi TI (2013b) Rapid simulation of laser processing of discrete particulate materials. Arch Comput Methods Eng. doi: 10.1007/s11831-013-9092-6 pp 1–17
  90. 90.
    Zohdi TI (2014) A direct particle-based computational framework for electrically-enhanced thermo-mechanical sintering of powdered materials. Math Mech Solids. doi: 10.1007/s11831-013-9092-6 pp 1–21
  91. 91.
    Zohdi T (2014) Embedded electromagnetically sensitive particle motion in functionalized fluids. Comput Part Mech 1(1):27–45CrossRefGoogle Scholar
  92. 92.
    Zohdi TI (2014) Additive particle deposition and selective laser processing-a computational manufacturing framework. Comput Mech 54:171–191zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of CaliforniaBerkeleyUSA

Personalised recommendations