Microstructural Statistics Informed Boundary Conditions for Statistically Equivalent Representative Volume Elements (SERVEs) of Polydispersed Elastic Composites

  • Somnath GhoshEmail author
  • Dhirendra V. Kubair
  • Craig Przybyla


The statistically equivalent RVE or P-SERVE have been introduced in Swaminathan et al. (J Compos Mater 40(7):583–604, 2006) and Ghosh (Micromechanical analysis and multi-scale modeling using the voronoi cell finite element method. CRC Press/Taylor & Francis, Boca Raton, 2011) as the smallest microstructural volume element in non-uniform microstructures that has effective material properties equivalent to those of the entire microstructure. An important consideration is the application of appropriate boundary conditions for optimal SERVE domains. The exterior statistics-based boundary conditions or ESBCs have been developed in Ghosh and Kubair (J Mech Phys Solids 96:1–24, 2016), Kubair and Ghosh (Int J Solids Struct 112:106–121, 2017), Kubair et al. (J Comput Mech 52(21):2919–2928, 2018), accounting for the statistics of fiber distributions and interactions in the domain exterior to the SERVE. The ESBC-based SERVEs have been validated for effective convergence in evaluating homogenized stiffnesses and optimal domains for micromechanical analysis. Validation is also conducted with an experimentally studied carbon-fiber epoxy-matrix polymer matrix composite (PMC). The performance of the SERVE with ESBCs is compared with other boundary conditions, as well as with the statistical volume elements (SVE). The tests clearly show the significant advantages of the ESBCs in terms of accuracy of the homogenized stiffness and efficiency.


Polymer matrix composite (PMC) Exterior statistics-based boundary conditions (ESBCs) Statistically equivalent RVE or SERVE Micromechanical analysis 



This work has been supported through a grant No. FA9550-12-1-0445 to the Center of Excellence on Integrated Materials Modeling (CEIMM) at Johns Hopkins University awarded by the AFOSR/RSL Computational Mathematics Program (Manager Dr. A. Sayir) and AFRL/RX (Monitors Drs. C. Woodward and C. Przybyla). These sponsorships are gratefully acknowledged. Computing support by the Homewood High-Performance Compute Cluster (HHPC) and Maryland Advanced Research Computing Center (MARCC) is gratefully acknowledged.


  1. 1.
    R. Hill, Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)CrossRefGoogle Scholar
  2. 2.
    Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11(2), 127–140 (1963)CrossRefGoogle Scholar
  3. 3.
    M. Stroeven, H. Askes, L.J. Sluys, Numerical determination of representative volumes for granular materials. Comput. Methods Appl. Mech. Eng. 193(30–32), 3221–3238 (2004)CrossRefGoogle Scholar
  4. 4.
    M. Thomas, N. Boyard, L. Perez, Y. Jarny, D. Delaunay, Representative volume element of anisotropic unidirectional carbon-epoxy composite with high-fibre volume fraction. Compos. Sci. Technol. 68(15–16), 3184–3192 (2008)CrossRefGoogle Scholar
  5. 5.
    C. Heinrich, M. Aldridge, A.S. Wineman, J. Kieffer, A.M. Waas, K. Shahwan, The influence of the representative volume element (RVE) size on the homogenized response of cured fiber composites. Model. Simul. Mater. Sci. Eng. 20(7), 075007 (2012)Google Scholar
  6. 6.
    R. Hill, The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15, 79–95 (1967)CrossRefGoogle Scholar
  7. 7.
    H.J. Böhm, A short introduction to continuum micromechanics, in Mechanics of Microstructured Materials: CISM Courses and Lectures, ed. by H.J. Böhm, vol. 464 (Springer, Wien, 2004), pp. 1–40Google Scholar
  8. 8.
    P.W. Chung, K.K. Tamma, R.R. Namburu, A finite element thermo-viscoelastic creep for heterogeneous structures with dissipative correctors. Finite Elem. Anal. Des. 36, 279–313 (2000)CrossRefGoogle Scholar
  9. 9.
    S. Ghosh, Micromechanical Analysis and Multi-scale Modeling Using the Voronoi Cell Finite Element Method (CRC Press/Taylor & Francis, Boca Raton, 2011)CrossRefGoogle Scholar
  10. 10.
    N. Willoughby, W.J. Parnell, A.L. Hazel, I.D. Abrahams, Homogenization methods to approximate the effective response of random fibre-reinforced composites. Int. J. Solids Struct. 49(13), 1421–1433 (2012)CrossRefGoogle Scholar
  11. 11.
    J. Fish, K. Shek, Multiscale analysis of composite materials and structures. Compos. Sci. Technol. 60, 2547–2556 (2000)CrossRefGoogle Scholar
  12. 12.
    S. Ghosh, K. Lee, S. Moorthy, Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method. Int. J. Solids Struct. 32(1), 27–62 (1995)CrossRefGoogle Scholar
  13. 13.
    S. Ghosh, K. Lee, S. Moorthy, Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and Voronoi cell finite element model. Comput. Methods Appl. Mech. Eng. 132(1–2), 63–116 (1996)CrossRefGoogle Scholar
  14. 14.
    J.M. Guedes, N. Kikuchi, Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput. Methods Appl. Mech. Eng. 83, 143–198 (1991)CrossRefGoogle Scholar
  15. 15.
    V. Kouznetsova, M.G.D. Geers, W.A.M. Brekelmans, Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Methods Eng. 54, 1235–1260 (2002)CrossRefGoogle Scholar
  16. 16.
    K. Terada, N. Kikuchi, Simulation of the multi-scale convergence in computational homogenization approaches. Int. J. Solids Struct. 37, 2285–2311 (2000)CrossRefGoogle Scholar
  17. 17.
    F. Feyel, J.H. Chaboche, FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput. Methods Appl. Mech. Eng. 183, 309–330 (2000)CrossRefGoogle Scholar
  18. 18.
    S. Swaminathan, S. Ghosh, N.J. Pagano, Statistically equivalent representative volume elements for unidirectional composite microstructures: part I-without damage. J. Compos. Mater. 40(7), 583–604 (2006)CrossRefGoogle Scholar
  19. 19.
    S. Swaminathan, N.J. Pagano, S. Ghosh, Statistically equivalent representative volume elements for unidirectional composite microstructures: part II-with interfacial debonding. J. Compos. Mater. 40(7), 605–621 (2006)CrossRefGoogle Scholar
  20. 20.
    M. Pinz, G. Weber, W. Lenthe, M. Uchic, T.M. Pollock, S. Ghosh, Microstructure and property based statistically equivalent RVEs for intragranular γ −γ′ microstructures of Ni-based superalloys. Acta Mater. 157(15), 245–258 (2018)CrossRefGoogle Scholar
  21. 21.
    A. Bagri, G. Weber, J.-C. Stinville, W. Lenthe, T. Pollock, C. Woodward, S. Ghosh, Microstructure and property-based statistically equivalent representative volume elements for polycrystalline Ni-based superalloys containing annealing twins. Metall. Mater. Trans. A 49(11), 5727–5744 (2018)CrossRefGoogle Scholar
  22. 22.
    X. Tu, A. Shahba, J. Shen, S. Ghosh, Microstructure and property based statistically equivalent RVEs for polycrystalline-polyphase aluminum alloys. Int. J. Plast. 115, 268–292 (2019)CrossRefGoogle Scholar
  23. 23.
    T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40(13–14), 3647–3679 (2003)CrossRefGoogle Scholar
  24. 24.
    P. Trovalusci, M. Ostoja-Starsewski, M.L. De Bellis, A. Murrali, Scale-dependent homogenization of random composites as micropolar continua. Eur. J. Mech. A Solids 49, 396–407 (2015)CrossRefGoogle Scholar
  25. 25.
    E. Recciaa, M.L. De Bellis, P. Trovalusci, R. Masiani, Sensitivity to material contrast in homogenization of random particle composites as micropolar continua. Compos. Part B 136, 39–45 (2018)CrossRefGoogle Scholar
  26. 26.
    R. Pyrz, Correlation of microstructure variability and local stress-field in 2-phase materials. Mater. Sci. Eng. A-Struct. Mater. Prop. Microstruct. Process. 177(1–2), 253–259 (1994)CrossRefGoogle Scholar
  27. 27.
    S. Torquato, Effective stiffness tensor of composite media-I. exact series expansions. J. Mech. Phys. Solids 45(9), 1421–1448 (1997)Google Scholar
  28. 28.
    S.E. Wilding, D.T. Fullwood, Clustering metrics for two-phase composites. Comput. Mater. Sci. 50(7), 2262–2272 (2011)CrossRefGoogle Scholar
  29. 29.
    E.-Y. Guo, N. Chawla, T. Jing, S. Torquato, Y. Jiao, Accurate modeling and reconstruction of three-dimensional percolating filamentary microstructures from two-dimensional micrographs via dilation-erosion method. Mater. Charact. 89, 33–42 (2014)CrossRefGoogle Scholar
  30. 30.
    Y. Jiao, F.H. Stillinger, S. Torquato, Modeling heterogeneous materials via two-point correlation functions: basic principles. Phys. Rev. E 76, 031110 (2007)CrossRefGoogle Scholar
  31. 31.
    Y. Jiao, F.H. Stillinger, S. Torquato, A superior descriptor of random textures and its predictive capacity. Proc. Nat. Acad. Sci. USA 106(42), 17634–17639 (2007)CrossRefGoogle Scholar
  32. 32.
    A Tewari, A.M Gokhale, J.E Spowart, D.B Miracle, Quantitative characterization of spatial clustering in three-dimensional microstructures using two-point correlation functions. Acta Mater. 52(2), 307–319 (2004)Google Scholar
  33. 33.
    D.T. Fullwood, S.R. Niezgoda, S.R. Kalidindi, Microstructure reconstructions from 2-point statistics using phase-recovery algorithms. Acta Mater. 56, 942–948 (2008)CrossRefGoogle Scholar
  34. 34.
    S.R. Niezgoda, D.T. Fullwood, S.R. Kalidindi, Delineation of the space of 2-point correlations in a composite material system. Acta Mater. 56(18), 5285–5292 (2008)CrossRefGoogle Scholar
  35. 35.
    D.V. Kubair, S. Ghosh, Exterior statistics based boundary conditions for establishing statistically equivalent representative volume elements of statistically nonhomogeneous elastic microstructures. Int. J. Solids Struct. 112, 106–121 (2017)CrossRefGoogle Scholar
  36. 36.
    T.I. Zohdi, P. Wriggers, An introduction to Computational Micromechanics (Springer-Verlag Berlin, Heidelberg, 2004). Google Scholar
  37. 37.
    M. Ostoja-Starzewski, Microstructural Randomness and Scaling in Mechanics of Materials (Chapman and Hall/CRC, Boca Raton, 2007)CrossRefGoogle Scholar
  38. 38.
    S. Ghosh, D.V. Kubair, Exterior statistics based boundary conditions for representative volume elements of elastic composites. J. Mech. Phys. Solids 96, 1–24 (2016)CrossRefGoogle Scholar
  39. 39.
    D. V. Kubair, M. Pinz, K. Kollins, C. Przybyla, S. Ghosh, Role of exterior statistics-based boundary conditions for property-based statistically equivalent RVEs of polydispersed elastic composites. J. Comput. Mech. 52(21), 2919–2928 (2018)Google Scholar
  40. 40.
    S. Torquato, Random Heterogeneous Materials; Microstructure and Macroscopic Properties (Springer, New York, 2002)CrossRefGoogle Scholar
  41. 41.
    T. Mura, Micromechanics of Defects in Solids, 2nd edn. (Kluwer Academic Publishers and Martinus Nijhoff, Dordrecht, 1987)CrossRefGoogle Scholar
  42. 42.
    J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A241, 376–396 (1957)Google Scholar
  43. 43.
    W. Lenthe, C. Pollock, Microstructural characterization of unidirectional composites. Private Communications (2014)Google Scholar
  44. 44.
    X. Yin, A. To, C. McVeigh, W.K. Liu, Statistical volume element method for predicting microstructure–constitutive property relations. Comput. Methods Appl. Mech. Eng. 197, 3516–3529 (2008)CrossRefGoogle Scholar
  45. 45.
    D.L. McDowell, S. Ghosh, S.R. Kalidindi, Representation and computational structure-property relations of random media. JOM, TMS 63(3), 45–51 (2011)CrossRefGoogle Scholar
  46. 46.
    S.W. Clay, P.M. Knoth, Experimental results of quasi-static testing for calibration and validation of composite progressive damage analysis methods. J. Compos. Mater. 10, 1333–1353 (2016)Google Scholar
  47. 47.
    HEXCEL, Composite materials and structures (2017)Google Scholar
  48. 48.
    C. Montgomery, N. Sottos, Experiments for properties of composites. Unpublished work. Private communication (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Somnath Ghosh
    • 1
    Email author
  • Dhirendra V. Kubair
    • 2
  • Craig Przybyla
    • 3
  1. 1.Departments of Civil, Mechanical Engineering and Materials Science & EngineeringJohns Hopkins UniversityBaltimoreUSA
  2. 2.Department of Civil EngineeringJohns Hopkins UniversityBaltimoreUSA
  3. 3.Air Force Research Laboratory/RXWright-Patterson Air Force BaseDaytonUSA

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