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Embedded unit cell homogenization model for localized non-periodic elasto-plastic zones

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Abstract

We extend the embedded unit cell (EUC) homogenization approach, to efficiently and accurately capture the multiscale solution of a solid with localized domains undergoing plastic yielding. The EUC approach is based on a mathematical homogenization formulation with non-periodic domains, in which the macroscopic and microscopic domain are concurrently coupled. The formulation consists of a theoretical derivation and the development of special boundary conditions representing the variations of the local displacement field across the unit cell boundaries. In particular, we introduce a restraining band surrounding the local domain in order to support the consistency of the solution in the transition layer between the micro and macro scales. The method is neither limited to a specific plasticity model nor to the number of localized features, thereby providing great flexibility in modeling. Several numerical examples illustrate that the proposed approach is accurate compared with direct finite element simulations, yet requires less computational cost.

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References

  1. Gibson RF (2011) Principles of composite material mechanics. CRC Press, Boca Raton

    Book  Google Scholar 

  2. Tadmor EB, Ortiz M, Phillips R (1996) Quasicontinuum analysis of defects in solids. Philos Mag A 73(6):1529–1563

    Article  Google Scholar 

  3. Weinan E (2011) Principles of multiscale modeling. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  4. Fish J (2006) Bridging the scales in nano engineering and science. J Nanoparticle Res 8(5):577–594

    Article  Google Scholar 

  5. Fish J (2010) Multiscale methods: bridging the scales in science and engineering. Oxford University Press on Demand

  6. Castañeda PP, Willis JR (1999) Variational second-order estimates for nonlinear composites Proc. R. Soc. London. Ser. A Math. Phys. Eng. Sci., vol. 455, no. 1985, pp. 1799–1811

  7. Fish J, Shek K, Pandheeradi M, Shephard MS (1997) Computational plasticity for composite structures based on mathematical homogenization: theory and practice. Comput Methods Appl Mech Eng 148(1):53–73

    Article  MathSciNet  MATH  Google Scholar 

  8. Doghri I, Friebel C (2005) Effective elasto-plastic properties of inclusion-reinforced composites. Study of shape, orientation and cyclic response. Mech Mater 37(1):45–68

    Article  Google Scholar 

  9. Doghri I, Brassart L, Adam L, Gérard J-S (2011) A second-moment incremental formulation for the mean-field homogenization of elasto-plastic composites. Int J Plast 27(3):352–371

    Article  MATH  Google Scholar 

  10. Fish J, Fan R (2008) Mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading. Int J Numer Methods Eng 76(7):1044–1064

    Article  MathSciNet  MATH  Google Scholar 

  11. Matouš K, Geubelle PH (2006) Multiscale modelling of particle debonding in reinforced elastomers subjected to finite deformations. Int J Numer Methods Eng 65(2):190–223

    Article  MathSciNet  MATH  Google Scholar 

  12. Agoras M, Castañeda PP (2011) Homogenization estimates for multi-scale nonlinear composites. Eur J Mech 30(6):828–843

    Article  MathSciNet  MATH  Google Scholar 

  13. Mosby M, Matouš K (2015) Hierarchically parallel coupled finite strain multiscale solver for modeling heterogeneous layers. Int J Numer Methods Eng 102(3–4):748–765

    Article  MathSciNet  MATH  Google Scholar 

  14. Hill R (1965) A self-consistent mechanics of composite materials. J Mech Phys Solids 13(4):213–222

    Article  Google Scholar 

  15. Mandel J (1966) Conditions de stabilité et postulat de Drucker. Rheology and soil mechanics/Rhéologie et mécanique des sols. Springer, Berlin, pp 58–68

    Google Scholar 

  16. Kaleel I, Petrolo M, Carrera E, Waas AM (2019) Computationally efficient concurrent multiscale framework for the nonlinear analysis of composite structures. AIAA J 57(9):4029–4041

    Article  Google Scholar 

  17. Talebi H, Silani M, Rabczuk T (2015) Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Adv Eng Softw 80:82–92

    Article  Google Scholar 

  18. Silani M, Ziaei-Rad S, Talebi H, Rabczuk T (2014) A semi-concurrent multiscale approach for modeling damage in nanocomposites. Theor Appl Fract Mech 74:30–38

    Article  Google Scholar 

  19. Lu G, Tadmor EB, Kaxiras E (2006) From electrons to finite elements: a concurrent multiscale approach for metals. Phys Rev B 73(2):24108

    Article  Google Scholar 

  20. Gracie R, Belytschko T (2011) An adaptive concurrent multiscale method for the dynamic simulation of dislocations. Int J Numer Methods Eng 86(4–5):575–597

    Article  MATH  Google Scholar 

  21. Fish J (2011) Multiscale modeling and simulation of composite materials and structures. Multiscale methods in computational mechanics. Springer, Netherlands, pp 215–231

    Chapter  MATH  Google Scholar 

  22. Zhang X, Oskay C (2017) Sparse and scalable eigenstrain-based reduced order homogenization models for polycrystal plasticity. Comput Methods Appl Mech Eng 326:241–269

    Article  MathSciNet  MATH  Google Scholar 

  23. Matouš K, Geers MGD, Kouznetsova VG, Gillman A (2017) A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J Comput Phys 330:192–220

    Article  MathSciNet  Google Scholar 

  24. Guo N, Zhao J (2014) A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media. Int J Numer Methods Eng 99(11):789–818

    Article  MathSciNet  MATH  Google Scholar 

  25. Bouvard JL, Ward DK, Hossain D, Nouranian S, Marin EB, Horstemeyer MF Review of hierarchical multiscale modeling to describe the mechanical behavior of amorphous polymers. J Eng Mater Technol, vol. 131, no. 4, 2009.

  26. Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234(7):2175–2182

    Article  MATH  Google Scholar 

  27. Chun J, Ahn S Bengio Y (2016) Hierarchical multiscale recurrent neural networks. arXiv Prepr. arXiv1609.01704

  28. Wu W, Yuan Z, Fish J (2010) Eigendeformation-based homogenization of concrete. Int J Multiscale Comput Eng 8(1):1–15

    Article  Google Scholar 

  29. Tabarraei A, Song J-H, Waisman H (2013) A two-scale strong discontinuity approach for evolution of shear bands under dynamic impact loads. Int J Multiscale Comput Eng 11(6):543–563

    Article  Google Scholar 

  30. V. G. Kouznetsova (2004) Computational homogenization for the multi-scale analysis of multi-phase materials

  31. Yuan Z, Fish J (2008) Toward realization of computational homogenization in practice. Int J Numer Methods Eng 73(3):361–380

    Article  MathSciNet  MATH  Google Scholar 

  32. Geers MGD, Coenen EWC, Kouznetsova VG (2007) Multi-scale computational homogenization of structured thin sheets. Model Simul Mater Sci Eng 15(4):S393

    Article  Google Scholar 

  33. Wu L, Noels L, Adam L, Doghri I (2013) A combined incremental-secant mean-field homogenization scheme with per-phase residual strains for elasto-plastic composites. Int J Plast 51:80–102

    Article  Google Scholar 

  34. Prasad NE, Srivatsan TS, Wanhill RJH, Malakondaiah G, Kutumbarao VV (2014) Fatigue behavior of aluminum-lithium alloys. Elsevier, Amsterdam, pp 341–379

    Book  Google Scholar 

  35. Geers M, Kouznetsova VG, Brekelmans WAM (2003) Multiscale first-order and second-order computational homogenization of microstructures towards continua. Int J Multiscale Comput Eng 1(4):371–386

    Article  Google Scholar 

  36. Kouznetsova V, Geers MGD, Brekelmans WAM (2002) Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 54(8):1235–1260

    Article  MATH  Google Scholar 

  37. de Souza Neto EA, Peric D, Owen DRJ (2011) Computational methods for plasticity: theory and applications. John Wiley & Sons, Hoboken

    Google Scholar 

  38. Ghosh S, Moorthy S (1995) Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi Cell finite element method. Comput Methods Appl Mech Eng 121(1–4):373–409

    Article  MATH  Google Scholar 

  39. Doghri I, Ouaar A (2003) Homogenization of two-phase elasto-plastic composite materials and structures: study of tangent operators, cyclic plasticity and numerical algorithms. Int J Solids Struct 40(7):1681–1712

    Article  MATH  Google Scholar 

  40. Kelly P (2013) Solid mechanics. Part II, Lecture notes, The University of Auckland

  41. Andrianopoulos NP, Manolopoulos VM (2014) Elastic strain energy density decomposition in failure of ductile materials under combined torsion-tension. Int J Mech Mater Eng 9(1):1–12

    Article  Google Scholar 

  42. Lemaitre J (2001) Handbook of materials behavior models, three-volume set: nonlinear models and properties. Elsevier, Amsterdam

    Google Scholar 

  43. Suganuma K (2018) Wide bandgap power semiconductor packaging: materials, components, and reliability. Woodhead Publishing, Cambridge

    Google Scholar 

  44. Belytschko T, Loehnert S, Song JH (2008) Multiscale aggregating discontinuities: a method for circumventing loss of material stability. Int J Numer Methods Eng 73(6):869–894. https://doi.org/10.1002/nme.2156

    Article  MathSciNet  MATH  Google Scholar 

  45. Song JH, Belytschko T (2009) Multiscale aggregating discontinuities method for micro-macro failure of composites. Compos Part B Eng 40(6):417–426. https://doi.org/10.1016/j.compositesb.2009.01.007

    Article  Google Scholar 

  46. Grigorovitch M, Gal E (2017) Homogenization of non-periodic zones in periodic domains using the embedded unit cell approach. Comput Struct 179:95–108

    Article  Google Scholar 

  47. M. Grigorovitch and E. Gal, “The local response in structures using the Embedded Unit Cell Approach,” Comput. Struct., vol. 157, 2015, doi: https://doi.org/10.1016/j.compstruc.2015.05.006.

  48. Song JH, Yoon YC (2014) Multiscale failure analysis with coarse-grained micro cracks and damage. Theor Appl Fract Mech 72(1):100–109. https://doi.org/10.1016/j.tafmec.2014.04.005

    Article  MathSciNet  Google Scholar 

  49. Belytschko T, Song JH (2010) Coarse-graining of multiscale crack propagation. Int. J. Numer Methods Eng 81(5):537–563. https://doi.org/10.1002/nme.2694

    Article  MATH  Google Scholar 

  50. “Abaqus.” [Online]. Available: https://www.3ds.com/products-services/simulia/products/abaqus/.

  51. Gal E, Yuan Z, Wu W, Fish J (2007) A multiscale design system for fatigue life prediction. Int. J. Multiscale Comput Eng. 5(6):435–446

    Article  Google Scholar 

  52. Fish J, Wagiman A (1993) Multiscale finite element method for a locally nonperiodic heterogeneous medium. Comput Mech 12(3):164–180. https://doi.org/10.1007/BF00371991

    Article  MathSciNet  MATH  Google Scholar 

  53. Allaire G, Briane M (1996) Multiscale convergence and reiterated homogenisation. Proc R Soc Edinburgh Sect A Math 126(2):297–342

    Article  MathSciNet  MATH  Google Scholar 

  54. Ryvkin M, Hadar O, Kucherov L (2017) Multiscale analysis of non-periodic stress state in composites with periodic microstructure. Int J Eng Sci 121:167–181

    Article  MathSciNet  MATH  Google Scholar 

  55. Geers MGD, Yvonnet J (2016) Multiscale modeling of microstructure–property relations. MRS Bull 41(8):610–616

    Article  Google Scholar 

  56. Sridhar A, Kouznetsova VG, Geers MGD (2018) A general multiscale framework for the emergent effective elastodynamics of metamaterials. J Mech Phys Solids 111:414–433

    Article  MathSciNet  MATH  Google Scholar 

  57. Geers MGD, Kouznetsova VG, Matouš K, Yvonnet J (2017) Homogenization methods and multiscale modeling: nonlinear problems. Encyclopaedia of scomputational mechanics. John & Wiley sons, Chichester, pp 1–34

    Google Scholar 

  58. Dolbow JE (2000) An extended finite element method with discontinuous enrichment for applied mechanics

  59. Samimi M, Van Dommelen JAW, Geers MGD (2009) An enriched cohesive zone model for delamination in brittle interfaces. Int J Numer Methods Eng 80(5):609–630

    Article  MATH  Google Scholar 

  60. Waseem A, Heuzé T, Stainier L, Geers MGD, Kouznetsova VG (2021) Two-scale analysis of transient diffusion problems through a homogenized enriched continuum. Eur J Mech 87:104212

    Article  MathSciNet  MATH  Google Scholar 

  61. Fish J, Yuan Z (2007) Multiscale enrichment based on partition of unity for nonperiodic fields and nonlinear problems. Comput Mech 40(2):249–259

    Article  MATH  Google Scholar 

  62. Oskay C (2013) Variational multiscale enrichment method with mixed boundary conditions for modeling diffusion and deformation problems. Comput Methods Appl Mech Eng 264:178–190

    Article  MathSciNet  MATH  Google Scholar 

  63. Gupta V, Duarte CA, Babuška I, Banerjee U (2013) A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics. Comput Methods Appl Mech Eng 266:23–39

    Article  MathSciNet  MATH  Google Scholar 

  64. Kim D, Pereira JP, Duarte CA (2010) Analysis of three-dimensional fracture mechanics problems: a two-scale approach using coarse-generalized FEM meshes. Int J Numer Methods Eng 81(3):335–365

    Article  MATH  Google Scholar 

  65. Pereira JP, Duarte CA, Guoy D, Jiao X (2009) hp-Generalized FEM and crack surface representation for non-planar 3-D cracks. Int J Numer Methods Eng 77(5):601–633

    Article  MathSciNet  MATH  Google Scholar 

  66. Hiriyur B, Tuminaro RS, Waisman H, Boman EG, Keyes DE (2012) A quasi-algebraic multigrid approach to fracture problems based on extended finite elements. SIAM J Sci Comput 34(2):A603–A626

    Article  MathSciNet  MATH  Google Scholar 

  67. Gerstenberger A, Tuminaro RS (2013) An algebraic multigrid approach to solve extended finite element method based fracture problems. Int J Numer Methods Eng 94(3):248–272

    Article  MathSciNet  MATH  Google Scholar 

  68. Dvorak GJ (1900) Transformation field analysis of inelastic composite materials. Proc R Soc Lond A 1992(437):311–327

    MATH  Google Scholar 

  69. Matouš K (2003) Damage evolution in particulate composite materials. Int J Solids Struct 40(6):1489–1503

    Article  MATH  Google Scholar 

  70. Fritzen F, Leuschner M, Hodapp M (2015) Nonlinear homogenization using model order reduction: two-scale simulations and novel developments using the pRBMOR on GPUs

  71. Covezzi F, de Miranda S, Fritzen F, Marfia S, Sacco E (2018) Comparison of reduced order homogenization techniques: pRBMOR, NUTFA and MxTFA. Meccanica 53(6):1291–1312

    Article  Google Scholar 

  72. Roussette S, Michel J-C, Suquet P (2009) Nonuniform transformation field analysis of elastic–viscoplastic composites. Compos Sci Technol 69(1):22–27

    Article  Google Scholar 

  73. Michel J-C, Suquet P (2016) A model-reduction approach in micromechanics of materials preserving the variational structure of constitutive relations. J Mech Phys Solids 90:254–285

    Article  MathSciNet  Google Scholar 

  74. Monaldo E, Marfia S (2021) Computational homogenization of 3D printed materials by a reduced order model. Int J Mech Sci 197:106332

    Article  Google Scholar 

  75. Covezzi F, de Miranda S, Marfia S, Sacco E (2016) A HOMOGENIZATION TECHNIQUE FOR ELASTO-PLASTIC COMPOSITES.”

  76. Oskay C, Fish J (2007) Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials. Comput Methods Appl Mech Eng 196(7):1216–1243

    Article  MathSciNet  MATH  Google Scholar 

  77. Yuan Z, Jiang T, Fish J, Morscher G (2014) Reduced-order multiscale-multiphysics model for heterogeneous materials. Int. J. Multiscale Comput Eng 12(1):45–64

    Article  Google Scholar 

  78. Zhuang X, Wang Q, Zhu H (2015) A 3D computational homogenization model for porous material and parameters identification. Comput Mater Sci 96:536–548

    Article  Google Scholar 

  79. Jeong S, Zhu F, Lim H, Kim Y, Yun GJ (2019) 3D stochastic computational homogenization model for carbon fiber reinforced CNT/epoxy composites with spatially random properties. Compos Struct 207:858–870. https://doi.org/10.1016/j.compstruct.2018.09.025

    Article  Google Scholar 

  80. Jain JR, Ghosh S (2009) Damage evolution in composites with a homogenization-based continuum damage mechanics model. Int J Damage Mech 18(6):533–568

    Article  Google Scholar 

  81. Duarte CA, Babuška I, Oden JT (2000) Generalized finite element methods for three-dimensional structural mechanics problems. Comput Struct 77(2):215–232

    Article  MathSciNet  Google Scholar 

  82. Galvanetto U, Aliabadi MHF (2009) Multiscale modeling in solid mechanics: computational approaches, 3rd edn. Imperial College Press, London

    Book  MATH  Google Scholar 

  83. Babuška I (1976) Homogenization and its application. Mathematical and computational problems. Numerical solution of partial differential equations–III. Academic Press, Cambridge, pp 89–116

    Chapter  Google Scholar 

  84. Bensoussan A, Lions J-L, Papanicolaou G (2011) Asymptotic analysis for periodic structures, vol 374. American Mathematical Soc, Providence, Rhode Island

    MATH  Google Scholar 

  85. Sanchez-Palencia E (1983) Homogenization method for the study of composite media. Asymptotic analysis II. Springer, Cham, pp 192–214

    Chapter  Google Scholar 

  86. Panasenko GP (2005) Multi-scale modelling for structures and composites. Springer, Cham

    MATH  Google Scholar 

  87. Grigorovitch M, Gal E (2015) The local response in structures using the Embedded Unit Cell Approach. Comput Struct 157:189–200

    Article  Google Scholar 

  88. Kelly P (2013) Solid mechanics. Lect. notes, Univ. Auckl, Auckland

    Google Scholar 

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No funding was received to assist with the preparation of this manuscript. No funding was received for conducting this study.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Ben Gurion University of the Negev, Beer Sheva, Israel

    Marina Grigorovitch & Erez Gal

  2. Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, 10027, USA

    Haim Waisman

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  1. Marina Grigorovitch
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  2. Erez Gal
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  3. Haim Waisman
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Marina Grigorovitch, Erez Gal and Haim Waisman contributed to the design and implementation of the research, to the analysis of the results and to the writing of the manuscript.

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Correspondence to Marina Grigorovitch.

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Grigorovitch, M., Gal, E. & Waisman, H. Embedded unit cell homogenization model for localized non-periodic elasto-plastic zones. Comput Mech 68, 1437–1456 (2021). https://doi.org/10.1007/s00466-021-02077-3

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