Mean-field/FMBEM homogenization of 3-D particulate composites

Downloads

Authors

  • J. Ptaszny Department of Computational Mechanics and Engineering, Silesian University of Technology, Poland

Abstract

An approach to homogenization of particulate composite materials is proposed. The mean-field assumption for averaging over phases is combined with numerical calculations of strain-concentration tensors, thus making it independent from the analytical Eshelby solution for ellipsoidal inclusions. The fast multipole boundary element method (FMBEM) is applied to 3-D elasticity and two-phase composites. As opposed to the finite element method (FEM), this method allows for easy modeling of large structures without the need to discretize volumes. Single-inhomogeneity problems are solved, and the calculated strain concentration tensors are used in the averaging formula under the assumption of the Mori–Tanaka approach. An interpolative scheme involving the inverse Mori–Tanaka assumption, known from the literature, is also applied to increase the accuracy of the approximation for higher volume fractions of particles. Examples include composites with spherical and cubic particles, and hybrid materials with auxetic components. The results are consistent with analytical solutions and RVE/FEM models.

Keywords:

3-D particulate composites, linear elasticity, mean-field homogenization, strain concentration tensor, fast multipole boundary element method

References


  1. J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proceedings of the Royal Society A, 241, 1226, 376–396, 1957, https://doi.org/10.1098/rspa.1957.0133

  2. T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metallurgica, 21, 571–574, 1973, https://doi.org/10.1016/0001-6160(73)90064-3

  3. Y. Benveniste, A new approach to the application of Mori–Tanaka’s theory in composite materials, Mechanics of Materials, 6, 147–157, 1987, https://doi.org/10.1016/0167-6636(87)90005-6

  4. T. Chen, G.J. Dvorak, Y. Benveniste, Mori–Tanaka estimates of the elastic moduli of certain composite materials, Journal of Applied Mechanics, 59, 3, 539–546, 1992, https://doi.org/10.1115/1.2893757

  5. T. Mura, Micromechanics of Defects in Solids, Kluwer Academic Publishers, Dordrecht, 1987, https://doi.org/10.1007/978-94-009-3489-4

  6. S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, Amsterdam, 1999.

  7. O. Pierard, C. Friebel, I. Doghri, Mean-field homogenization of multi-phase thermo-elastic composites: a general framework and its validation, Composites Science and Technology, 64, 10–11, 1587–1603, 2004, https://doi.org/10.1016/j.compscitech.2003.11.009

  8. J. Qu, M. Cherkaoui, Fundamentals of Micromechanics of Solids, John Wiley & Sons, New Jersey, 2006, https://doi.org/10.1002/9780470117835

  9. T.I. Zohdi, P. Wriggers, An Introduction to Computational Micromechanics, Springer-Verlag, Berlin–Heidelberg, 2008, https://doi.org/10.1007/978-3-540-32360-0

  10. R.D. Bradshaw, F.T. Fisher, L.C. Brinson, Fiber waviness in nanotube-reinforced polymer composites – II: modeling via numerical approximation of the dilute strain concentration tensor, Composites Science and Technology, 63, 11, 1705–1722, 2003, https://doi.org/10.1016/S0266-3538(03)00070-8

  11. L. Brassart, I. Doghri, L. Delannay, Homogenization of elasto-plastic composites coupled with a nonlinear finite element analysis of the equivalent inclusion problem, International Journal of Solids and Structures, 47, 5, 716–729, 2010, https://doi.org/10.1016/j.ijsolstr.2009.11.013

  12. G. Srinivasulu, R. Velmurugan, S. Jayasankar, A hybrid method for computing the effective properties of composites containing arbitrarily shaped inclusions, Computers & Structures, 150, 63–70, 2015, https://doi.org/10.1016/j.compstruc.2014.12.010

  13. P. Sadowski, K. Kowalczyk-Gajewska, S. Stupkiewicz, Consistent treatment and automation of the incremental Mori–Tanaka scheme for elasto-plastic composites, Computational Mechanics, 60, 493–511, 2017, https://doi.org/10.1007/s00466-017-1418-z

  14. W. Ogierman, Hybrid Mori–Tanaka/Finite Element Method in homogenization of composite materials with various reinforcement shape and orientation, International Journal for Multiscale Computational Engineering, 17, 3, 281–295, 2019, https://doi.org/10.1615/IntJMultCompEng.2019028827

  15. W. Ogierman, A new model for time-efficient analysis of nonlinear composites with arbitrary orientation distribution of fibres, Composite Structures, 273, 114310, 2021, https://doi.org/10.1016/j.compstruct.2021.114310

  16. W. Ogierman, A data-driven model based on the numerical solution of the equivalent inclusion problem for the analysis of nonlinear short-fibre composites, Composites Science and Technology, 250, 110516, 2024, https://doi.org/10.1016/j.compscitech.2024.110516

  17. O. Eroshkin, I. Tsukrov, On micromechanical modeling of particulate composites with inclusions of various shapes, International Journal of Solids and Structures, 42, 2, 2005, https://doi.org/10.1016/j.ijsolstr.2004.06.045

  18. A. Trofimov, B. Drach, I. Sevostianov, Effective elastic properties of composites with particles of polyhedral shapes, International Journal of Solids and Structures, 120, 157–170, 2017, https://doi.org/10.1016/j.ijsolstr.2017.04.037

  19. A. Markov, A. Trofimov, I. Sevostianov, A unified methodology for calculation of compliance and stiffness contribution tensors of inhomogeneities of arbitrary 2D and 3D shapes embedded in isotropic matrix – open access software, International Journal of Engineering Science, 157, 103390, 2020, https://doi.org/10.1016/j.ijengsci.2020.103390

  20. Q.H. Qin, Micromechanics–BE solution for properties of piezoelectric materials with defects, Engineering Analysis with Boundary Elements, 28, 7, 809–814, 2004, https://doi.org/10.1016/j.enganabound.2003.12.006

  21. Q.H. Qin, Material properties of piezoelectric composites by BEM and homogenization method, Composite Structures, 66, 1–4, 295–299, 2004, https://doi.org/10.1016/j.compstruct.2004.04.051

  22. G. Dziatkiewicz, Analysis of effective properties of piezocomposites by the subregion BEM–Mori–Tanaka approach, Mechanics and Control, 30, 4, 194–202, 2011.

  23. P. Fedeliński, R. Górski, G. Dziatkiewicz, J. Ptaszny, Computer modelling and analysis of effective properties of composites, Computer Methods in Materials Science, 11, 1, 3–8, 2011, https://doi.org/10.7494/cmms.2011.1.0304

  24. T. Czyż, G. Dziatkiewicz, P. Fedeliński [ed.], R. Górski, J. Ptaszny, Advanced Computer Modelling in Micromechanics, Silesian University of Technology Press, Gliwice, 2013.

  25. J. Ptaszny, A fast multipole BEM with higher-order elements for 3-D composite materials, Computers & Mathematics with Applications, 82, 148–160, 2021, https://doi.org/10.1016/j.camwa.2020.10.024

  26. G. Lielens, P. Pirotte, A. Couniot, F. Dupret, R. Keunings, Prediction of thermo-mechanical properties for compression moulded composites, Composites Part A: Applied Science and Manufacturing, 29, 1–2, 63–70, 1998,
     https://doi.org/10.1016/S1359-835X(97)00039-0

  27. I. Shufrin, E. Pasternak, A.V. Dyskin, Hybrid materials with negative Poisson’s ratio inclusions, International Journal of Engineering Science, 89, 100–120, 2015, https://doi.org/10.1016/j.ijengsci.2014.12.006

  28. K. Long, X. Du, S. Xu, Y.M. Xie, Maximizing the effective Young’s modulus of a composite material by exploiting the Poisson effect, Composite Structures, 153, 593–600, 2016, https://doi.org/10.1016/j.compstruct.2016.06.061

  29. J. Schjèdt-Thomsen, R. Pyrz, The Mori–Tanaka stiffness tensor: diagonal symmetry, complex fibre orientations and non-dilute volume fractions, Mechanics of Materials, 33, 10, 531–544, 2001, https://doi.org/10.1016/S0167-6636(01)00072-2

  30. C.A. Brebbia, J. Dominguez, Boundary Elements an Introductory Course, McGraw-Hill, New York, 1992.

  31. T. Burczyński, Boundary Element Method in Mechanics, Scientific and Technical Publishing WNT, Warsaw, 1995 (in Polish).

  32. X.W. Gao, G. Davies, Boundary Element Programming in Mechanics, Cambridge University Press, Cambridge, 2002.

  33. G. Beer, I. Smith, C. Duenser, The Boundary Element Method with Programming for Engineers and Scientists, Springer–Verlag, Wien–New York, 2008, https://doi.org/10.1007/978-3-211-71576-5

  34. Z. Yao, F. Kong, X. Zheng, Simulation of 2D elastic bodies with randomly distributed circular inclusions using the BEM, Electronic Journal of Boundary Elements, 1, 2, 270–282, 2003, https://doi.org/10.14713/ejbe.v1i2.761

  35. L. Greengard, V. Rokhlin, A fast algorithm for particle simulations, Journal of Computational Physics, 73, 2, 325–348, 1987, https://doi.org/10.1016/0021-9991(87)90140-9

  36. Y.J. Liu, Fast Multipole Boundary Element Method Theory and Applications in Engineering, Cambridge University Press, 2009, https://doi.org/10.1017/CBO9780511605345

  37. J. Ptaszny, M. Hatłas, Evaluation of the FMBEM efficiency in the analysis of porous structures, Engineering Computations, 35, 2, 843–866, 2018, https://doi.org/10.1108/EC-12-2016-0436

  38. L.H. Dai, Z.P. Huang, R. Wang, Explicit expressions for bounds for the effective moduli of multi-phased composites by the generalized self-consistent method, Composites Science and Technology, 59, 11, 1691–1699, 1999, https://doi.org/10.1016/S0266-3538(99)00031-7

  39. S. Torquato, Random Heterogeneous Materials, Springer-Verlag, New York, 2002, https://doi.org/10.1007/978-1-4757-6355-3

  40. A. Gillman, G. Amadio, K. Matouš, T.L. Jackson, Third-order thermo-mechanical properties for packs of platonic solids using statistical micromechanics, Proceedings of The Royal Society A, 471, 20150060, 2015, https://doi.org/10.1098/rspa.2015.0060