Abstract
The purpose of this paper is to develop a homogeneous, couple-stress continuum model as a representation of 2D random fiber networks in the small deformation regime. The couple-stress substitution continuum is calibrated based on the response of a network model (window of analysis, WOA) subjected to prescribed kinematic boundary conditions applied on part of the WOA boundary, while the free surface boundary conditions are applied on complementary surfaces. Each fiber in the network is considered as a Timoshenko beam and the cross-links between fibers are modeled as welded joints in which the relative angles between the crossing beams remain constant during deformation, and hence they transmit moments along and between crossing fibers. The effective elastic constants of the couple-stress continuum are deduced by an equivalent strain energy method, and the characteristic lengths are identified from the resulting homogenized moduli. The competition between the affine (ADR) and non-affine (NADR) deformation regimes is shown to be quantified by the bending length, a scalar quantity that measures the relative importance of fiber bending in comparison with fiber stretch. The scaling laws of the effective moduli versus the bending length, network density and window size are determined in the affine and non-affine deformation regimes. The motivation of the adopted couple-stress substitution continuum is brought by comparing the identified effective non-classical properties with the mechanical properties predicted by FE simulations performed over the fully resolved fibrous network.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Fratzl, P.: Collagen: Structure and Mechanics (Springer, Max Planck Institute of Colloids and Interfaces, Department of Biomaterials, 14424 Potsdam, Germany, 2008)
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 (2015)
Placidi, L., Barchiesi, E., Turco, E., Rizzi, N.: A review on 2D models for the description of pantographic fabrics. Zeitschrift für angewandte Mathematik und Physik 67, 121 (2016)
Dell’Isola, F., Steigmann, D., Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67, 060804 (2016)
Steigmann, D., dell’Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mechanica Sinica 31(3), 373 (2015)
Cox, H.: The elasticity and strength of paper and other fibrous materials. J. Appl. Phys. 3, 72 (1952)
Petterson, D.R.: Mechanics of nonwoven fabrics. J. Ind. Eng. Chem. 51(8), 902 (1959)
Wu, X., Dzenis, Y.A.: Elasticity of planar fiber networks. J. Appl. Phys. 98, 093501 (2005)
MacKintosh, F., Kas, J., Jamney, P.: Elasticity of semiflexible biopolymer networks. Phys. Rev. Lett. 75, 4425 (1995)
Wilhelm, J., Frey, E.: Elasticity of stiff polymer networks. Phys. Rev. Lett. 91(10), 1 (2003)
Palmer, J., Boyc, M.: Constitutive modeling of the stress strain behavior of F-actin filament networks. Acta Biomaterialia 4(3), 597 (2008)
Lee, Y., Jasiuk, I.: Apparent elastic properties of random fiber networks. Comput. Mater. Sci. 79, 715 (1995)
Turco, E., Dell’Isola, F., Cazzani, A., Rizzi, N.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67, 85 (2016)
Alibert, J., Corte, A., Giorgio, I., Battista, A.: Extensional Elastica in large deformation as Gamma-limit of a discrete 1D mechanical system. Zeitschrift für angewandte Mathematik und Physik 68, 42 (2017)
DeMasi, A., Olla, S.: Quasi-static hydrodynamic limits. J. Stat. Phys. 61(5), 1037 (2015)
Berrehili, Y., Marigo, J.J.: The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers. Math. Mech. Complex Syst. 2(2), 181 (2014)
Picu, R.C.: Mechanics of random fiber networks-a review. Soft Matter. 7, 6768 (2012)
Shahsavari, A., Picu, R.: Size effect on mechanical behavior of random fiber networks. Int. J. Solids Struct. 50, 3332 (2013)
Astrom, J.A., Makinen, J.P., Alava, M.J., Timonen, J.: Elasticity of planar fiber networks. Phys. Rev. E 61, 5550 (2000)
Hatami-Marbini, H., Picu, R.C.: An eigenstrain formulation for the prediction of elastic moduli of defective fiber networks. Eur. J. Mech. A Solids 28, 305 (2009)
Hatami-Marbini, H., Picu, R.C.: Scaling of nonaffine deformation in random semiflexible fiber networks. Phys. Rev. E 77, 062103 (2008)
Cihan, T., Onck, P.R.: Size effects in two-dimensional Voronoi foams: a comparison between generalized continua and discrete models. J. Mech. Phys. Solids 56, 3541 (2008)
dell’Isola, F., Corte, A., Giorgio, I.: Higher-gradient continua: The legacy of Piola Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Solids 22(4), 852 (2016)
Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics ã la Hamilton Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375 (2013)
dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids 20(8), 887 (2014)
Aminpour, H., Rizzi, N.: On the modelling of carbon nano tubes as generalized continua. In: Altenbach, H., Forest, S. (eds.) Generalized Continua as Models for Classical and Advanced Materials, pp. 15–35. Springer, Berlin (2016)
Altenbach, H., Eremeyev, V.: On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math. Mech. Solids 3(3), 273 (2015)
Misra, A., Poorsolhjouy, P.: Grain- and macro-scale kinematics for granular micromechanics based small deformation micromorphic continuum model. Mech. Res. Commun. 21, 1 (2017)
Cosserat, E., Cosserat, F.: Theorie des Corps Deformables. Hermann, Paris (1909)
Grioli, G.: Elasticita asimmetrica. Annali di Matematica Pura ed Applicata 50(1), 389 (1960)
Rajagopal, E.S.: The existence of interfacial couples in infinitesimal elasticity. Annalen der Physik 461(3–4), 192 (1960)
Truesdell, C.A., Toupin, R.A.: The Classical Field Theories. Encyclopedia of Physics, III/1. Springer, Berlin (1960)
Aero, E.L., Kuvshinskii, E.V.: The main equations of the theory of elastic media with rotationally interacting particles. Fizika Tverdogo Tela 2, 1399 (1960)
Eringen, A.C.: Nonlinear Theory of Continuous Media. McGraw-Hill, New York (1962)
Mindlin, R.D., Tiersten, H.F.: Effects of couple stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415 (1962)
Koiter, W.T.: Effects of couple stresses in linear elasticity. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series B 67(1), 17 (1964)
Goda, I., Ganghoffer, J.F.: Identification of couple-stress moduli of vertebral trabecular bone based on the 3D internal architectures. J. Mech. Behav. Biomed. Mater. 51, 99 (2015)
Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Solids 3(3), 285 (2015)
Boutin, C., Dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets : asymptotic micro–macro models identification. Math. Mech. Complex Syst. 5(2), 127 (2017)
Giorgio, I.: Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures. Zeitschrift für angewandte Mathematik und Physik 67, 95 (2016)
Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J. Eng. Math. 103(1), 1 (2017)
Placidi, L., Andreaus, U., Corte, A., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Zeitschrift für angewandte Mathematik und Physik 66(6), 3699 (2015)
Miles, R.E.: Random polygons determined by random lines in a plane. Proc. Natl. Acad. Sci. USA 52(4), 901 (1964)
Kallmes, O., Corte, H.: The structure of paper. I. The statistical geometry of an ideal two dimensional fiber network. Tappi J. 43, 737 (1960)
Jasiuk, I., Ostoja-Starzewski, M.: Planar Cosserat elasticity of materials with holes and intrusions. Appl. Mech. Rev. 48(11), 11 (1995)
Liu, S., Su, W.: Effective couple-stress continuum model of cellular solids and size effects analysis. Int. J. Solids. Struct. 46(14–15), 2787 (2009)
Goda, I., Assidi, M., Ganghoffer, J.F.: A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech. Model. Mechanobiol. 13(1), 53 (2014)
Goda, I., Assidi, M., Belouettar, S., Ganghoffer, J.F.: A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. J. Mech. Behav. Biomed. Mater. 16, 87 (2012)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51 (1964)
Shahsavari, A., Picu, R.C.: Model selection for athermal cross-linked fiber networks. Phys. Rev. E 86, 011923 (2011)
Gere, J.M., Temoshenko, S.P.: Mechanics of Materials, pp. 02116–4324. PWS Publishing Company, 20 Park Plaza, Boston (1997)
Head, D., Levine, A., MacKintosh, F.: Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks. Phys. Rev. E 68(6), 061907 (2003)
Head, D.A., Levine, A.J., MacKintosh, F.C.: Deformation of cross-linked semiflexible polymer networks. Phys. Rev. Lett. 91(10), 108102 (2003)
Wilhelm, J., Frey, E.: Elasticity of stiff polymer networks. Phys. Rev. Lett. 91(10), 108103 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Francesco dell’Isola.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Berkache, K., Deogekar, S., Goda, I. et al. Identification of equivalent couple-stress continuum models for planar random fibrous media. Continuum Mech. Thermodyn. 31, 1035–1050 (2019). https://doi.org/10.1007/s00161-018-0710-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-018-0710-2