Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 101–132 | Cite as

Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates

  • Patrick FranciosiEmail author
  • Mario Spagnuolo
  • Oguz Umut Salman
Original Article


Composites comprising included phases in a continuous matrix constitute a huge class of meta-materials, whose effective properties, whether they be mechanical, physical or coupled, can be selectively optimized by using appropriate phase arrangements and architectures. An important subclass is represented by “network-reinforced matrices,” say those materials in which one or more of the embedded phases are co-continuous with the matrix in one or more directions. In this article, we present a method to study effective properties of simple such structures from which more complex ones can be accessible. Effective properties are shown, in the framework of linear elasticity, estimable by using the global mean Green operator for the entire embedded fiber network which is by definition through sample spanning. This network operator is obtained from one of infinite planar alignments of infinite fibers, which the network can be seen as an interpenetrated set of, with the fiber interactions being fully accounted for in the alignments. The mean operator of such alignments is given in exact closed form for isotropic elastic-like or dielectric-like matrices. We first exemplify how these operators relevantly provide, from classic homogenization frameworks, effective properties in the case of 1D fiber bundles embedded in an isotropic elastic-like medium. It is also shown that using infinite patterns with fully interacting elements over their whole influence range at any element concentration suppresses the dilute approximation limit of these frameworks. We finally present a construction method for a global operator of fiber networks described as interpenetrated such bundles.


Composites Fiber networks Green operator Effective properties Phase co-continuity 


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Mario Spagnuolo has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665850.


  1. 1.
    Christensen, R.M., Walls, F.M.: Effective stiffness of randomly oriented fiber compositions. J. Compos. Mater. 6, 518–531 (1972)ADSGoogle Scholar
  2. 2.
    Boucher, S.: On the effective moduli of isotropic two-phase elastic composites. J. Compos. Mater. 8, 82–89 (1974)ADSGoogle Scholar
  3. 3.
    Christensen, R.M.: Mechanics of Composite Materials. Wiley, Hoboken (1979)Google Scholar
  4. 4.
    Christensen, R.M.: Isotropic properties of platelet reinforced media. J. Eng. Mater. Technol. 101(3), 299–303 (1979)MathSciNetGoogle Scholar
  5. 5.
    Franciosi, P., El Omri, A.: Effective properties of fiber and platelet systems and related phase arrangements in n-phase heterogeneous media. Mech. Res. Commun. 38, 38–44 (2011)zbMATHGoogle Scholar
  6. 6.
    Franciosi, P.: Laminate system schemes for effective property estimates of architectured composites with co(dis)continuous phases. Mech. Res. Commun. 45, 70–76 (2012)Google Scholar
  7. 7.
    Veenstra, H., Verkooijen, P.C.J., van Lent, B.J.J., van Dam, J., de Boer, A.P., Nijhof, A.H.J.: On the mechanical properties of co-continuous polymer blends: experimental and modelling. Polymer 41(5), 1817–1826 (2000)Google Scholar
  8. 8.
    Kinney, J.H., Stölken, J.S., Smith, T.S., Ryaby, J.T., Lane, N.E.: An orientation distribution function for trabecular bone. Bone 36(2), 193–201 (2005)Google Scholar
  9. 9.
    Clyne, T.W., Markaki, A.E., Tan, J.C.: Mechanical and magnetic properties of metal fibre networks with and without a polymeric matrix. Compos. Sci. Technol. 65, 2492–2499 (2005)Google Scholar
  10. 10.
    Agoras, M., Lopez-Pamies, O., Ponte, Castaneda P.: Onset of macroscopic instabilities in fiber-reinforced elastomers at finite strain. J. Mech. Phys. Solids 57, 1828–1850 (2009)ADSMathSciNetzbMATHGoogle Scholar
  11. 11.
    Broedersz, C.P., Mao, X., Lubensky, T.C., MacKintosh, F.C.: Criticality and isostaticity in fibre networks. Nat. Phys. 12(7), 983–988 (2011)Google Scholar
  12. 12.
    Cuomo, M., Dell’Isola, F., Greco, L., Rizzi, N.L.: First versus second gradient energies for planar sheets with two families of inextensible fibres: investigation on deformation boundary layers, discontinuities and geometrical instabilities. Compos. B Eng. 115, 423–448 (2017)Google Scholar
  13. 13.
    Placidi, L., Barchiesi, E., Turco, E., Rizzi, N.L.: A review on 2D models for the description of pantographic fabrics. Zeitschrift für angewandte Mathematik und Physik 67(5), 121 (2016)ADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    Della Corte, A., Dell’Isola, F., Esposito, R., Pulvirenti, M.: Equilibria of a clamped Euler beam (Elastica) with distributed load: large deformations. Math. Models Methods Appl. Sci. 27(8), 1391–1421 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ivan, G., Della Corte, A., Dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)Google Scholar
  16. 16.
    Scerrato, D., Zhurba Eremeeva, I.A., Lekszyki, T., Rizzi, N.L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheet. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und mechanic 96(11), 1268–1279 (2016)MathSciNetGoogle Scholar
  17. 17.
    Dell’Isola, F., Lekszycki, T., Pawlikowski, M., Grygoruk, R., Greco, L.: Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Zeitschrift für angewandte Mathematik und Physik 66(6), 3473–3498 (2015)ADSMathSciNetzbMATHGoogle Scholar
  18. 18.
    Dell’Isola, F., Cuomo, M., Greco, L., Della Corte, A.: Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J. Eng. Math. 103(1), 127–157 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dell’Isola, F., Giorgio, I., Andreaus, U.: Elastic pantographic 2D lattices: a numerical analysis on static response and wave propagation. Proc. Est. Acad. Sci. 64(3), 219–225 (2015)Google Scholar
  20. 20.
    Dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. R. Soc. A 472(2185), 20150790 (2016)ADSGoogle Scholar
  21. 21.
    Rahali, Y., Giorgio, I., Ganghoffer, J.F., Dell’Isola, F.: Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Andreaus, U., Sawczuk, A.: Deflection of elastic-plastic frames at finite spread of yielding zones. Comput. Methods Appl. Mech. Eng. 39(1), 21–35 (1983)ADSzbMATHGoogle Scholar
  23. 23.
    Andreaus, U., D’Asdia, P.: Displacement analysis in elastic-plastic frames at plastic collapse. Comput. Methods Appl. Mech. Eng. 42(1), 19–35 (1984)ADSzbMATHGoogle Scholar
  24. 24.
    Andreaus, U., D’Asdia, P.: Incremental analysis of elastic-plastic frames at finite spread of yielding zones. Eng. Fract. Mech. 21(4), 827–839 (1985)Google Scholar
  25. 25.
    Andreaus, U., D’Asdia, P.: An incremental procedure for deformation analysis of elastic-plastic frames. Int. J. Numer. Methods Eng. 26(4), 769–784 (1988)zbMATHGoogle Scholar
  26. 26.
    König, J.A.: An iterative method of evaluation of elastic-plastic deflections of hyperstatic framed structures. Ingenieer-Archiv 55(3), 202–212 (1985)Google Scholar
  27. 27.
    Rangelov, T.V., Manolis, G.D., Dineva, P.S.: Elastodynamic fundamental solutions for certain families of 2d inhomogeneous anisotropic domains: basic derivations. Eur. J. Mech. A/Solids 24, 820–836 (2005)ADSMathSciNetzbMATHGoogle Scholar
  28. 28.
    Manolis, G.D., Makra, K., Dineva, P.S., Rangelov, T.V.: Seismic motions in a non-homogeneous soil deposit with tunnels by a hybrid computational technique. Earthq. Struct. 5(2), 161–205 (2013)Google Scholar
  29. 29.
    Chen, Z., Jeffrey, R.G., Pandurangan, V.: The far-field deformation caused by a hydraulic fracture in an inhomogeneous elastic half-space. Int. J. Solids Struct. 130, 220–231 (2018)Google Scholar
  30. 30.
    Willis, J.R., Acton, J.R.: Overall elastic moduli of a dilute suspension of spheres. Q. J. Mech. Appl. Mech. 29, 163–177 (1976)zbMATHGoogle Scholar
  31. 31.
    Kröner, E.: Modified Green functions in the theory of heterogeneous and/or anisotropic linearly elastic media. In: Weng, G.J., Taya, M., Abe, H. (eds.) Micromechanics and Inhomogeneity, pp. 197–211. Springer, New York (1990)Google Scholar
  32. 32.
    Lebensohn, R.A., Tome, C.N.: Anisotropic approach for the simulation of plastic deformation and texture development of polycrystals. Acta Metall. 41, 2611–2624 (1993)Google Scholar
  33. 33.
    El Omri, A., Fennan, A., Sidoroff, F., Hihi, A.: Elastic-plastic homogenization for layered composites. Eur. J. Mech. A/Solids 19, 585–601 (2000)zbMATHGoogle Scholar
  34. 34.
    Ponte Castaneda, P., Willis, J.R.: The effect of spatial distribution on the effective behaviour of composite materials and cracked media. J. Mech. Phys. Solids 43(12), 1919–1951 (1995)ADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Bornert, M., Stolz, C., Zaoui, A.: Morphologically representative pattern-based bounding in elasticity. J. Mech. Phys. Solids 44(3), 307–331 (1996)ADSMathSciNetzbMATHGoogle Scholar
  36. 36.
    Franciosi, P.: The boundary-due terms in the Green operator of inclusion patterns from distant to contact and to connected situations using Radon transforms: illustration for spheroid alignments in isotropic media. Int. J. Solids Struct. 47(2), 304–319 (2010)zbMATHGoogle Scholar
  37. 37.
    Franciosi, P., Barboura, S., Charles, Y.: Analytical mean Green operators/Eshelby tensors for patterns of coaxial finite long or flat cylinders in isotropic matrices. Int. J. Solids Struct. 66(1), 1–19 (2015)Google Scholar
  38. 38.
    Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A 421, 379–396 (1957)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Berveiller, M., Fassi-Ferhi, O., Hihi, A.: The problem of two plastic and heterogeneous inclusions in an anisotropic medium. Int. J. Eng. Sci. 25(6), 691–709 (1987)zbMATHGoogle Scholar
  40. 40.
    Kouris, D., Tsuchida, E.: On the elastic interaction between two fibres in a continuous fibre composite under thermal loading. Mech. Mater. 12, 131–146 (1991)Google Scholar
  41. 41.
    Anttreter, T., Fisher, F.D.: The stress state around two spatially arranged ellipsoidal inclusions. A case study for high-speed tool steel. Comput. Mater. Sci. 7, 247–252 (1996)Google Scholar
  42. 42.
    Franciosi, P., Lormand, G.: Using the Radon transform to solve inclusion problems in elasticity. Int. J. Solids Struct. 41(3/4), 585–606 (2004)zbMATHGoogle Scholar
  43. 43.
    Gel’fand, I.M., Graev, M.I., Vilenkin, NYa.: Generalized Functions, Integral Geometry and Representation Theory, vol. 5. Academic Press, New York (1966)zbMATHGoogle Scholar
  44. 44.
    Natterer, F.: The Mathematics of Computerized Tomography. Wiley, Stuttgart (1986)zbMATHGoogle Scholar
  45. 45.
    Ramm, A.G., Katsevitch, A.I.: Radon Transform and Local Tomography. CRC Press, Boca Raton (1996). ISBN 0849394929Google Scholar
  46. 46.
    Helgason, S.: The Radon Transform, Progress in Mathematics, vol. 5. Birkhausser, Boston (1980)zbMATHGoogle Scholar
  47. 47.
    Willis, J.R.: Interfacial stresses induced by arbitrary loading of dissimilar elastic half-spaces joined over a circular region. IMA J. Appl. Math. 7, 179–197 (1971)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Wang, C.-Y.: Elastic fields produced by a point source in solids of general anisotropy. J. Eng. Math. 32, 41–52 (1997)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Pan, E., Tonon, F.: Three dimensional Green’s functions in anisotropic piezoelectric solids. Int. J. Solids Struct. 37, 943–958 (2000)zbMATHGoogle Scholar
  50. 50.
    Meisner, M.J., Kouris, D.A.: Interaction of two elliptic inclusions. Int. J. Solids Struct. 32, 451–466 (1995)zbMATHGoogle Scholar
  51. 51.
    Ju, J.W., Sun, L.Z.: A novel formulation for the exterior point Eshelby’s tensor of an ellipsoidal inclusion. J. Appl. Mech. 66, 570–574 (1999)ADSGoogle Scholar
  52. 52.
    Nakasone, Y., Nishiyama, H., Nojiri, Y.: Numerical equivalent inclusion method: a new computational method for analyzing stress fields in and around inclusions of various shapes. Mater. Sci. Eng. A 285, 229–238 (2000)Google Scholar
  53. 53.
    Kushch, V.I., Shmegera, S.V., Buryachenko, V.A.: Interacting elliptic inclusions by the method of complex potentials. Int. J. Solids Struct. 42, 5491–5512 (2005)zbMATHGoogle Scholar
  54. 54.
    Zheng, Q.S., Zhao, Z.H., Du, D.X.: Irreducible structure, symmetry and average of Eshelby’s tensor field in isotropic elasticity. J. Mech. Phys. Solids 54(2), 368–383 (2006)ADSMathSciNetzbMATHGoogle Scholar
  55. 55.
    Franciosi, P.: Mean and axial Green and Eshelby tensors for an inclusion with finite cylindrical shape. Mech. Res. Commun. 59, 26–36 (2014)Google Scholar
  56. 56.
    Franciosi, P.: On the modified Green operator integral for polygonal, polyhedral and other non-ellipsoidal inclusions. Int. J. Solids Struct. 42(11/12), 3509–3531 (2005)zbMATHGoogle Scholar
  57. 57.
    Franciosi, P.: A Decomposition method for obtaining global mean Green operators of inclusions patterns. Application to parallel infinite beams in at least transversally isotropic media. Int. J. Solids Struct. (2018). Google Scholar
  58. 58.
    Hasegawa, H., Lee, G., Mura, T.: The stress field caused by a circular-cylindrical inclusion. J. Appl. Mech. 59, 107–114 (1992)zbMATHGoogle Scholar
  59. 59.
    Wu, L., Du, S.Y.: The elastic field caused by a circular cylindrical inclusion—part II: inside the region \(x1^{2}+x2^{2} >{a}^{2}\). J. Appl. Mech. 62, 585–589 (1995)ADSGoogle Scholar
  60. 60.
    Mura, T.: Micro-mechanics of Defects in Solids, 2nd edn. Martinus Nijhoff, Dordrecht (1987)zbMATHGoogle Scholar
  61. 61.
    Cherkaev, A.: Variational Methods for Structural Optimization, Applied Mathematical Sciences. Springer, Berlin (2000)zbMATHGoogle Scholar
  62. 62.
    Buryachenko, V.A.: Multiparticle effective field and related methods in micromechanics of composite materials. Appl. Mech. Rev. 54, 1–47 (2001)ADSGoogle Scholar
  63. 63.
    Buryachenko, V.A.: Micromechanics of Heterogeneous Materials. Springer, Berlin (2007)zbMATHGoogle Scholar
  64. 64.
    Buryachenko, V.A., Kushch, V.I., Roy, A.: Effective thermoelastic properties of ramdom structure composites reinforced by the clusters of deterministic structures (application to clay composites). Acta Mech. 192(1–4), 135–167 (2007)zbMATHGoogle Scholar
  65. 65.
    Franciosi, P., Charles, Y.: Mean Green operator and Eshelby tensor for hemispherical inclusions and bi-material spherical inclusions in infinite spaces. Mech. Res. Commun. 75, 57–66 (2016)Google Scholar
  66. 66.
    Franciosi, P.: Transversally isotropic Magneto-electro-elastic composites with co-(dis)continuous phases. Int. J. Solids Struct. 50, 1013–1031 (2013)Google Scholar
  67. 67.
    Walpole, L.J.: Elastic behavior of composites materials. Adv. Appl. Mech. 21, 169–242 (1981)zbMATHGoogle Scholar
  68. 68.
    Franciosi, P., Lebail, H.: Anisotropy features of phase and particle spatial pair distributions in various matrix/inclusions structures. Acta Mater. 52, 3161–3172 (2004)Google Scholar
  69. 69.
    Eremeyev, V.A., Dell’Isola, F., Boutin, C., Steigmann, D.: Linear pantographic sheets: existence and uniqueness of weak solutions. J. Elast. 1, 1–22 (2017)zbMATHGoogle Scholar
  70. 70.
    Altenbach, H., Eremeyev, V.A.: On the elastic plates and shells with residual surface stresses. Proc. IUTAM 21, 25–32 (2017)zbMATHGoogle Scholar
  71. 71.
    Andreaus, U., Chiaia, B., Placidi, L.: Soft-impact dynamics of deformable bodies. Contin. Mech. Thermodyn. 25(2–4), 375,398 (2013)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheet: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LSPM UPR 3407 CNRS, University of Paris13, USPCVilletaneuseFrance

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