Research Papers

Effect of Network Architecture on the Mechanical Behavior of Random Fiber Networks

[+] Author and Article Information
M. R. Islam

Department of Mechanical,
Aerospace and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180

R. C. Picu

Department of Mechanical,
Aerospace and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180
e-mail: picuc@rpi.edu

1Corresponding author.

Manuscript received April 5, 2018; final manuscript received May 7, 2018; published online June 4, 2018. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 85(8), 081011 (Jun 04, 2018) (8 pages) Paper No: JAM-18-1192; doi: 10.1115/1.4040245 History: Received April 05, 2018; Revised May 07, 2018

Fiber-based materials are prevalent around us. While microscopically these systems resemble a discrete assembly of randomly interconnected fibers, the network architecture varies from one system to another. To identify the role of the network architecture, we study here cellular and fibrous random networks in tension and compression, and in the context of large strain elasticity. We observe that, compared to cellular networks of same global parameter set, fibrous networks exhibit in tension reduced strain stiffening, reduced fiber alignment, and reduced Poisson's contraction in uniaxial tension. These effects are due to the larger number of kinematic constraints in the form of cross-links per fiber in the fibrous case. The dependence of the small strain modulus on network density is cubic in the fibrous case and quadratic in the cellular case. This difference persists when the number of cross-links per fiber in the fibrous case is rendered equal to that of the cellular case, which indicates that the different scaling is due to the higher structural disorder of the fibrous networks. The behavior of the two network types in compression is similar, although softening induced by fiber buckling and strain localization is less pronounced in the fibrous case. The contribution of transient interfiber contacts is weak in tension and important in compression.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Janmey, P. A. , 1991, “ Mechanical Properties of Cytoskeletal Polymers,” Curr. Opin. Cell Biol., 3(1), pp. 4–11. [CrossRef] [PubMed]
Pritchard, R. H. , Huang, Y. Y. S. , and Terentjev, E. M. , 2014, “ Mechanics of Biological Networks: From the Cell Cytoskeleton to Connective Tissue,” Soft Matter, 10(12), pp. 1864–1884. [CrossRef] [PubMed]
Gibson, L. J. , and Ashby, M. F. , 1999, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK.
Islam, M. R. , Tudryn, G. , Bucinell, R. , Schadler, L. , and Picu, R. C. , 2017, “ Morphology and Mechanics of Fungal Mycelium,” Sci. Rep., 7(1), p. 13070.
Heussinger, C. , and Frey, E. , 2007, “ Role of Architecture in the Elastic Response of Semiflexible Polymer and Fiber Networks,” Phys. Rev. E, 75(1), p. 011917. [CrossRef]
Zhu, H. , Knott, J. , and Mills, N. , 1997, “ Analysis of the Elastic Properties of Open-Cell Foams With Tetrakaidecahedral Cells,” J. Mech. Phys. Solids, 45(3), p. 319327. [CrossRef]
Warren, W. , and Kraynik, A. , 1991, “ The Nonlinear Elastic Behavior of Open-Cell Foams,” ASME J. Appl. Mech., 58(2), pp. 376–381. [CrossRef]
Jang, W.-Y. , Kraynik, A. M. , and Kyriakides, S. , 2008, “ On the Microstructure of Open-Cell Foams and Its Effect on Elastic Properties,” Int. J. Solids Struct., 45(7–8), pp. 1845–1875. [CrossRef]
Zhu, H. , and Windle, A. , 2002, “ Effects of Cell Irregularity on the High Strain Compression of Open-Cell Foams,” Acta Mater., 50(5), pp. 1041–1052. [CrossRef]
Li, K. , Gao, X. L. , and Subhash, G. , 2006, “ Effects of Cell Shape and Strut Cross-Sectional Area Variations on the Elastic Properties of Three-Dimensional Open-Cell Foams,” J. Mech. Phys. Solids, 54(4), pp. 783–806. [CrossRef]
Markaki, A. , and Clyne, T. , 2001, “ The Effect of Cell Wall Microstructure on the Deformation and Fracture of Aluminium-Based Foams,” Acta Mater., 49(9), pp. 1677–1686. [CrossRef]
Storm, C. , Pastore, J. J. , MacKintosh, F. C. , Lubensky, T. C. , and Janmey, P. A. , 2004, “ Nonlinear Elasticity in Biological Gels,” preprint arXiv: cond-mat/0406016. https://arxiv.org/ftp/cond-mat/papers/0406/0406016.pdf
Roeder, B. A. , Kokini, K. , Sturgis, J. E. , Robinson, J. P. , and Voytik-Harbin, S. L. , 2002, “ Tensile Mechanical Properties of Three-Dimensional Type I Collagen Extracellular Matrices With Varied Microstructure,” ASME J. Biomech. Eng., 124(2), pp. 214–222. [CrossRef]
Onck, P. R. , Koeman, T. , van Dillen, T. , and van der Giessen, E. , 2005, “ Alternative Explanation of Stiffening in Cross-Linked Semiflexible Networks,” Phys. Rev. Lett., 95(17), p. 178102. [CrossRef] [PubMed]
Stein, A. M. , Vader, D. A. , Weitz, D. A. , and Sander, L. M. , 2011, “ The Micromechanics of Three-Dimensional Collagen-I Gels,” Complexity, 16(4), pp. 22–28. [CrossRef]
Vader, D. , Kabla, A. , Weitz, D. , and Mahadevan, L. , 2009, “ Strain-Induced Alignment in Collagen Gels,” PLoS One, 4(6), p. e5902. [CrossRef] [PubMed]
Kim, O. V. , Liang, X. , Litvinov, R. I. , Weisel, J. W. , Alber, M. S. , and Purohit, P. K. , 2016, “ Foam-like Compression Behavior of Fibrin Networks,” Biomech. Model. Mechanobiol., 15(1), pp. 213–228. [CrossRef] [PubMed]
Kim, O. V. , Litvinov, R. I. , Weisel, J. W. , and Alber, M. S. , 2014, “ Structural Basis for the Nonlinear Mechanics of Fibrin Networks Under Compression,” Biomaterials, 35(25), pp. 6739–6749. [CrossRef] [PubMed]
Ban, E. , Barocas, V. H. , Shephard, M. S. , and Picu, R. C. , 2016, “ Softening in Random Networks of Non-Identical Beams,” J. Mech. Phys. Solids, 87, pp. 38–50. [CrossRef] [PubMed]
Licup, A. J. , Münster, S. , Sharma, A. , Sheinman, M. , Jawerth, L. M. , Fabry, B. , Weitz, D. A. , and MacKintosh, F. C. , 2015, “ Stress Controls the Mechanics of Collagen Networks,” Proc. Natl. Acad. Sci., 112(31), pp. 9573–9578. [CrossRef]
Huisman, E. M. , van Dillen, T. , Onck, P. R. , and Van der Giessen, E. , 2007, “ Three-Dimensional Cross-Linked F-Actin Networks: Relation Between Network Architecture and Mechanical Behavior,” Phys. Rev. Lett., 99(20), p. 208103. [CrossRef] [PubMed]
Sharma, A. , Licup, A. , Rens, R. , Vahabi, M. , Jansen, K. , Koenderink, G. , and MacKintosh, F. , 2016, “ Strain-Driven Criticality Underlies Nonlinear Mechanics of Fibrous Networks,” Phys. Rev. E, 94(4), p. 042407. [CrossRef] [PubMed]
Licup, A. J. , Sharma, A. , and MacKintosh, F. C. , 2016, “ Elastic Regimes of Subisostatic Athermal Fiber Networks,” Phys. Rev. E, 93(1), p. 012407. [CrossRef] [PubMed]
Head, D. A. , Levine, A. J. , and MacKintosh, F. , 2003, “ Deformation of Cross-Linked Semiflexible Polymer Networks,” Phys. Rev. Lett., 91(10), p. 108102. [CrossRef] [PubMed]
Wilhelm, J. , and Frey, E. , 2003, “ Elasticity of Stiff Polymer Networks,” Phys. Rev. Lett., 91(10), p. 108103. [CrossRef] [PubMed]
Picu, R. , 2011, “ Mechanics of Random Fiber Networks—A Review,” Soft Matter, 7(15), pp. 6768–6785. [CrossRef]
Islam, M. , Tudryn, G. J. , and Picu, C. R. , 2016, “ Microstructure Modeling of Random Composites With Cylindrical Inclusions Having High Volume Fraction and Broad Aspect Ratio Distribution,” Comput. Mater. Sci., 125, pp. 309–318. [CrossRef]
Version, A. , 2013, “6.13, Analysis User's Manual,” Dassault Systèmes Simulia Corp, Providence, RI.
Shahsavari, A. , and Picu, R. , 2012, “ Model Selection for Athermal Cross-Linked Fiber Networks,” Phys. Rev. E, 86(1 Pt. 1), p. 011923. [CrossRef]
Žagar, G. , Onck, P. R. , and van der Giessen, E. , 2015, “ Two Fundamental Mechanisms Govern the Stiffening of Cross-Linked Networks,” Biophys. J., 108(6), pp. 1470–1479. [CrossRef] [PubMed]
Picu, R. C. , Deogekar, S. , and Islam, M. R. , 2018, “ Poisson Contraction and Fiber Kinematics in Tissue: Insight From Collagen Network Simulations,” ASME J. Biomech. Eng., 140(2), p. 021002.
Gaitanaros, S. , Kyriakides, S. , and Kraynik, A. M. , 2012, “ On the Crushing Response of Random Open-Cell Foams,” Int. J. Solids Struct., 49(19–20), pp. 2733–2743. [CrossRef]
Head, D. A. , Levine, A. J. , and MacKintosh, F. C. , 2003, “ Distinct Regimes of Elastic Response and Deformation Modes of Cross-Linked Cytoskeletal and Semiflexible Polymer Networks,” Phys. Rev. E, 68(6), p. 061907. [CrossRef]
Verhille, G. , Moulinet, S. , Vandenberghe, N. , Adda-Bedia, M. , and Le Gal, P. , 2017, “ Structure and Mechanics of Aegagropilae Fiber Network,” Proc. Natl. Acad. Sci., 114(18), pp. 4607–4612. [CrossRef]
Wu, X. F. , and Dzenis, Y. A. , 2005, “ Elasticity of Planar Fiber Networks,” J. Appl. Phys., 98(9), p. 093501. [CrossRef]
Bancelin, S. , Lynch, B. , Bonod-Bidaud, C. , Ducourthial, G. , Psilodimitrakopoulos, S. , Dokládal, P. , Allain, J.-M. , Schanne-Klein, M.-C. , and Ruggiero, F. , 2015, “ Ex Vivo Multiscale Quantitation of Skin Biomechanics in Wild-Type and Genetically-Modified Mice Using Multiphoton Microscopy,” Sci. Rep., 5(1), p. 17635.
Mauri, A. , Ehret, A. E. , Perrini, M. , Maake, C. , Ochsenbein-Kölble, N. , Ehrbar, M. , Oyen, M. L. , and Mazza, E. , 2015, “ Deformation Mechanisms of Human Amnion: Quantitative Studies Based on Second Harmonic Generation Microscopy,” J. Biomech., 48(9), pp. 1606–1613. [CrossRef] [PubMed]
Mills, N. , and Gilchrist, A. , 2000, “ High Strain Extension of Open-Cell Foams,” ASME J. Eng. Mater. Technol., 122(1), pp. 67–73. [CrossRef]
Toll, S. , 1998, “ Packing Mechanics of Fiber Reinforcements,” Polym. Eng. Sci., 38(8), pp. 1337–1350. [CrossRef]


Grahic Jump Location
Fig. 1

Illustration of (a) cellular and (b) fibrous network (green lines represent fibers and red lines are cross-links) in 3D. Both networks have volume fraction ϕ=3% and other parameters as shown in Table 1.

Grahic Jump Location
Fig. 2

Nominal stress–stretch behavior of cellular (dashed lines) and cross-linked networks (solid lines) for three volume fractions: (a) uniaxial tension and (b) uniaxial compression. Each stress–strain curve is obtained by averaging the response of three replicas of the stochastic microstructure with the same set of network parameters (ϕ,lb,Nc). The bars shown for volume fraction (ϕ=3%, blue squares in the two figures) represent the range of the three realizations. Similar level of variability is observed for the other volume fractions, but bars are not shown for clarity. The inset in (b) shows a detail of the compression curves of cellular networks. The vertical axis is normalized by fiber modulus (Ef) in both (a) and (b).

Grahic Jump Location
Fig. 6

(a) Effect of interfiber contacts on the compressive stress–strain response for cellular and fibrous networks with volume fraction ϕ=3%, p = 1 and lb/lc=0.036 and (b) the evolution of the density of interfiber contacts (pc) with stretch for cellular (black circles) and cross-linked networks (blue triangles), compared with Toll's model of Eq. (4) (continuous line).

Grahic Jump Location
Fig. 7

Variation of the mean contact force (CF) at interfiber contacts during compression for cellular and fibrous networks with ϕ=3%. The contact force is normalized by D2 and by the modulus of the fiber material.

Grahic Jump Location
Fig. 3

Scaling of the small strain modulus (E0) with the network density (ρ) and bending length (lb) for (a) cellular and (b) fibrous networks. Data in (b) are shown for two cross-link densities: fully cross-linked case with p = 1 (circles) and partially cross-linked with p = 0.69 (triangles). The vertical dashed lines and change of symbol color indicate the transition from bending to stretching-dominated regimes and corresponding scalings.

Grahic Jump Location
Fig. 4

Variation of tangent modulus (Et) as a function of stress under tension for (a) cellular and (b) fibrous networks with various volume fractions. The vertical axis is normalized by the small strain network modulus, E0 and the horizontal axis is normalized by the stress at transition from regime I to II. Results for the fibrous network with ϕ=3% and with various values of p are shown in (c). The axes of the plots in the insets are normalized by the fiber modulus, Ef, in order to avoid distorting the actual shape of the curves.

Grahic Jump Location
Fig. 5

Variation of the (a) incremental Poisson's ratio, νi, and (b) orientation index, P2, under uniaxial tension for both network types (here, p = 1 for fibrous networks). The arrows indicated the transitions between the three regimes.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In