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Research Papers

Effect of Fiber Crimp on the Elasticity of Random Fiber Networks With and Without Embedding Matrices

[+] Author and Article Information
Ehsan Ban

Department of Mechanical, Aerospace,
and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Jonsson Engineering Center, Room 2049,
110 8th Street,
Troy, NY 12180;
Scientific Computation Research Center,
Rensselaer Polytechnic Institute,
Low Center for Industrial Innovation,
CII-4011 110 8th Street,
Troy, NY 12180
e-mail: bane@rpi.edu

Victor H. Barocas

Department of Biomedical Engineering,
University of Minnesota,
7-105 Nils Hasselmo Hall,
312 Church Street SE,
Minneapolis, MN 55455
e-mail: baroc001@umn.edu

Mark S. Shephard

Scientific Computation Research Center,
Rensselaer Polytechnic Institute,
Low Center for Industrial Innovation,
CII-4011 110 8th Street,
Troy, NY 12180
e-mail: shephard@rpi.edu

Catalin R. Picu

Department of Mechanical,
Aerospace, and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Jonsson Engineering Center, Room 2048,
110 8th Street,
Troy, NY 12180;
Scientific Computation Research Center,
Rensselaer Polytechnic Institute,
Low Center for Industrial Innovation,
CII-4011 110 8th Street,
Troy, NY 12180
e-mail: picuc@rpi.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 9, 2015; final manuscript received January 11, 2016; published online January 27, 2016. Assoc. Editor: M. Taher A. Saif.

J. Appl. Mech 83(4), 041008 (Jan 27, 2016) (7 pages) Paper No: JAM-15-1605; doi: 10.1115/1.4032465 History: Received November 09, 2015; Revised January 11, 2016

Fiber networks are assemblies of one-dimensional elements representative of materials with fibrous microstructures such as collagen networks and synthetic nonwovens. The mechanics of random fiber networks has been the focus of numerous studies. However, fiber crimp has been explicitly represented only in few cases. In the present work, the mechanics of cross-linked networks with crimped athermal fibers, with and without an embedding elastic matrix, is studied. The dependence of the effective network stiffness on the fraction of nonstraight fibers and the relative crimp amplitude (or tortuosity) is studied using finite element simulations of networks with sinusoidally curved fibers. A semi-analytic model is developed to predict the dependence of network modulus on the crimp amplitude and the bounds of the stiffness reduction associated with the presence of crimp. The transition from the linear to the nonlinear elastic response of the network is rendered more gradual by the presence of crimp, and the effect of crimp on the network tangent stiffness decreases as strain increases. If the network is embedded in an elastic matrix, the effect of crimp becomes negligible even for very small, biologically relevant matrix stiffness values. However, the distribution of the maximum principal stress in the matrix becomes broader in the presence of crimp relative to the similar system with straight fibers, which indicates an increased probability of matrix failure.

FIGURES IN THIS ARTICLE
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Copyright © 2016 by ASME
Topics: Fibers , Stiffness
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References

Figures

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Fig. 2

(a) Contour map of the normalized overall network stiffness, EN, function of the crimp amplitude, c, and fraction of crimped fibers, f, for networks which are not embedded in matrix. The network stiffness values are normalized by the stiffness of the same network with straight fibers, EN0. (b) Data selected from (a) for four values of parameter c: 0 (filled circles), 0.25 (filled squares), 0.5 (triangles), 0.66 (open squares), and 1 (open circles), corresponding to tortuosity values of τ=1, 1.17, 1.32, 1.40, and 1.56, respectively. The solid curve represents a lower bound for EN/EN0 predicted for networks in the axially dominated regimes (Appendix).

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Fig. 1

(a) Individual crimped fiber and (b) snapshot of an undeformed 3D Voronoi network of 1331 naturally curved fibers. A normalized fiber crimp amplitude c=0.2 (τ=1.14) is used in both panels.

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Fig. 3

Estimations for the normalized overall stiffness of a network as a function of the normalized crimp amplitude or tortuosity for the case f=1. The symbols represent data from Fig. 2 for a 3D Voronoi network in the axially dominated regime. The solid line represents the prediction of Eq. (3) truncated to the first-order, while the dashed line represents the fit of Eq. (3) truncated to the third-order. The inset shows the normalized network stiffness of a 2D Voronoi, bending-dominated network (symbols), with the curve being the prediction of Eq. (3) truncated to the first-order.

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Fig. 4

Normalized tangent stiffness, Et, against normalized true stress, T, for networks with crimped fibers. Stress and stiffness values are normalized by the stiffness of the network with straight fibers. In all cases, f=1, except as indicated in the legend. The star marks the approximate strain of transition to the hardening regime.

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Fig. 5

(a) Contour map of the normalized sample stiffness, EN, for different normalized crimp amplitudes, c, and fractions of crimped fibers, f, for network of fibers embedded in an elastic matrix. The network stiffness values are normalized by the stiffness of systems with straight fibers. The matrix stiffness is Em=10−4 kPa. The network parameters are indicated in the text. (b) Section of the probability density function of the normalized maximum principal stress in the matrix, σ1. The full range of this distribution is shown in the inset. This set of data corresponds to a network with c=0.25 (τ=1.17) and f=1. These values are normalized by the matrix stiffness of 10 −2 kPa.

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Fig. 6

Scaling of normalized network stiffness against curving threshold length, lp, divided by the average fiber length, l¯ using the fiber removing model. The network stiffness values are normalized by the stiffness of networks with straight fibers. The inset shows the exponential normalized fiber length distribution in a network. The fiber length is normalized by the dimension of the simulated networks, L.

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