logo CRAS
Comptes Rendus. Mécanique

Experimentally validated combined stiffness expression for finite domain containing multiple inclusions
Comptes Rendus. Mécanique, Tome 348 (2020) no. 2, pp. 113-135.

Traditional homogenization formulations for finite volume are extended to allow for the consideration of multiple inclusions and their spatial distributions along with their corresponding individual geometric attributes. For all inclusions present in different states, a combined homogenization formulation (dubbed generalized stiffness formulation, GSF) is posed for summing the five geometric attributes of volume fraction (VF), shape or aspect ratio (AR), orientation, location (position within the domain), and number of inclusions in the domain. For verification, the solutions are compared to two literature-reported calculations of simple cases of domains containing one inclusion of specific shape and orientation in which only one attribute is varied. Once verified for simple cases, GSF stiffness solutions were run for 5040 cases of domains containing multi inclusions with various configuration combinations. These solutions were compared against 36 identical cases of 3-dimensional (3D) domains printed using acrylonitrile butadiene styrene (ABS) and experimentally tested under mechanical compression. Additionally, finite element method (FEM) simulations are run of these test structures under compression. For these test cases, effective composite stiffness values are compared for numerical solutions, experimental tests, and FEM. Relative effects on stiffness of the five inclusion geometric attributes are assessed.

Reçu le : 2019-12-03
Révisé le : 2020-01-30
Accepté le : 2020-03-16
Publié le : 2020-06-24
DOI : https://doi.org/10.5802/crmeca.11
Mots clés: Composite homogenization, Stiffness formulation, Multiple inclusions
     author = {Ilige S. Hage and Ramsey F. Hamade},
     title = {Experimentally validated combined stiffness expression for finite domain containing multiple inclusions},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {113--135},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {348},
     number = {2},
     year = {2020},
     doi = {10.5802/crmeca.11},
     language = {en},
     url = {comptes-rendus.academie-sciences.fr/mecanique/item/CRMECA_2020__348_2_113_0/}
Ilige S. Hage; Ramsey F. Hamade. Experimentally validated combined stiffness expression for finite domain containing multiple inclusions. Comptes Rendus. Mécanique, Tome 348 (2020) no. 2, pp. 113-135. doi : 10.5802/crmeca.11. https://comptes-rendus.academie-sciences.fr/mecanique/item/CRMECA_2020__348_2_113_0/

[1] J. D. Eshelby The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. R. Soc. Lond. A, Volume 241 (1957), pp. 376-396 | MR 87326 | Zbl 0079.39606

[2] T. Mori; K. Tanaka Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall., Volume 21 (1973), pp. 571-574 | Article

[3] G. P. Tandon; G. J. Weng The effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites, Polym. Compos., Volume 5 (1984), pp. 327-333 | Article

[4] E. Marklund; J. Varna; R. C. Neagu; E. K. Gamstedt Stiffness of aligned wood fiber composites: effect of microstructure and phase properties, J. Compos. Mater., Volume 42 (2008), pp. 2377-2405 | Article

[5] V. Buryachenko Micromechanics of Heterogeneous Materials, Springer Science & Business Media, 2007 | Zbl 1133.74002

[6] A. L. Kalamkarov; I. V. Andrianov; V. V. Danishevsâ Asymptotic homogenization of composite materials and structures, Appl. Mech. Rev., Volume 62 (2009) (30802) | Article

[7] W. J. Parnell; M. B. Vu; Q. Grimal; S. Naili Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone, Biomech. Model. Mechanobiol., Volume 11 (2012), pp. 883-901 | Article

[8] H. M. Ma; X. L. Gao A new homogenization method based on a simplified strain gradient elasticity theory, Acta Mech., Volume 225 (2014), pp. 1075-1091 | Article | MR 3183435 | Zbl 1401.74236

[9] I. Hage; R. F. Hamade An experimentally validated combined stiffness formulation for a finite domain considering volume fraction, shape, orientation, and location of a single inclusion, Int. J. Appl. Mech., Volume 10 (2018) (1850011) | Article

[10] S. Li; G. Wang; R. A. Sauer The Eshelby tensors in a finite spherical domain—Part II: applications to homogenization, J. Appl. Mech., Volume 74 (2007), pp. 784-797 | Article

[11] J. W. Ju; L. Z. Sun A novel formulation for the exterior-point eshelby’s tensor of an ellipsoidal inclusion, J. Appl. Mech., Volume 66 (1999), pp. 570-574 | Article

[12] J. W. Ju; L. Z. Sun Effective elastoplastic behavior of metal matrix composites containing randomly located aligned spheroidal inhomogeneities. Part I: micromechanics-based formulation, Int. J. Solids Struct., Volume 38 (2001), pp. 183-201 | Article | Zbl 1085.74038

[13] L. Z. Sun; J. W. Ju Elastoplastic modeling of metal matrix composites containing randomly located and oriented spheroidal particles, J. Appl. Mech., Volume 71 (2004), pp. 774-785 | Article | Zbl 1111.74652

[14] H. Askari; H. M. Zbib; X. Sun Multiscale modeling of inclusions and precipitation hardening in metal matrix composites: application to advanced high-strength steels, J. Nanomech. Micromech., Volume 3 (2012), pp. 24-33 | Article

[15] T. Mura Micromechanics of Defects in Solids, Kluwer, 1987

[16] C. G. Koay On the six-dimensional orthogonal tensor representation of the rotation in three dimensions: a simplified approach, Mech. Mater., Volume 41 (2009), pp. 951-953 | Article

[17] W. L. Bond The mathematics of the physical properties of crystals, Bell Syst. Tech. J., Volume 22 (1943), pp. 1-72 | Article | MR 8404

[18] 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, Compos. Sci. Technol., Volume 59 (1999), pp. 1691-1699 | Article

[19] Y. Benveniste A new approach to the application of Mori–Tanaka’s theory in composite materials, Mech. Mater., Volume 6 (1987), pp. 147-157 | Article

[20] Z. Duan; X. Wang; Q. Gao; L. Zhang; B. Liu; I. Kim Highly active bifunctional cobalt-salen complexes for the synthesis of poly(ester-block-carbonate) copolymer via terpolymerization of carbon dioxide, propylene oxide, and norbornene anhydride isomer: roles of anhydride conformation consideration, J. Polym. Sci. A, Volume 52 (2014), pp. 789-795 | Article

[21] W. Voigt Theoretische studien über die elasticitätsverhältnisse der krystalle, Königliche Gesellschaft der Wissenschaften zu Göttingen, 1887

[22] Z. Hashin; S. Shtrikman A variational approach to the theory of the elastic behaviour of multiphase materials, J. Mech. Phys. Solids, Volume 11 (1963), pp. 127-140 | Article | MR 159459 | Zbl 0108.36902

[23] W. Xu; M. Jia; Z. Zhu; M. Liu; D. Lei; X. Gou n-Phase micromechanical framework for the conductivity and elastic modulus of particulate composites: design to microencapsulated phase change materials (MPCMs)-cementitious composites, Mater. Des., Volume 145 (2018), pp. 108-115 | Article

[24] Q. Chen; M. M. Nezhad; Q. Fisher; H. H. Zhu Multi-scale approach for modeling the transversely isotropic elastic properties of shale considering multi-inclusions and interfacial transition zone, Int. J. Rock Mech. Min. Sci., Volume 84 (2016), pp. 95-104 | Article

[25] W. Xu; F. Wu; Y. Jiao; M. Liu A general micromechanical framework of effective moduli for the design of nonspherical nano-and micro-particle reinforced composites with interface properties, Mater. Des., Volume 127 (2017), pp. 162-172 | Article

[26] W. Xu; B. Xu; F. Guo Elastic properties of particle-reinforced composites containing nonspherical particles of high packing density and interphase: DEM–FEM simulation and micromechanical theory, Comput. Methods Appl. Mech. Eng., Volume 326 (2017), pp. 122-143 | Article | MR 3709186 | Zbl 07186047