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TECHNICAL PAPERS

Structure, Mechanics and Failure of Stochastic Fibrous Networks: Part II—Network Simulations and Application

[+] Author and Article Information
C. W. Wang, A. M. Sastry

Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, MI 48109-2125

J. Eng. Mater. Technol 122(4), 460-468 (May 30, 2000) (9 pages) doi:10.1115/1.1288768 History: Received May 26, 2000; Revised May 30, 2000
Copyright © 2000 by ASME
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References

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Drugan,  W. J., and Willis,  J. R., 1996, “A Micromechanics-Based Nonlocal Constitutive Equation and Estimates of Representative Volume Element Size for Elastic Composites,” J. Mech. Phys. Solids, 44, pp. 497–524.
Cheng,  X., and Sastry,  A. M., 1999, “On Transport in Stochastic, Heterogeneous Fibrous Domains,” Mech. Mater., 31, pp. 765–786.
Wang,  C. W., Cheng,  X., Sastry,  A. M., and Choi,  S. B., 1999, “Investigation of Failure Processes in Porous Battery Substrates: Part I—Experimental Findings,” ASME J. Eng. Mater. Technol., 121, pp. 503–513.
Kirkpatrick,  S., 1973, “Percolation and conduction,” Rev. Mod. Phys., 45, No. 4, pp. 574–588.
Niskanen, K., 1993, “Strength and Fracture of Paper,” Products of Papermaking: Transactions of the Tenth Fundamental Research Symposium, Baker, C. F., ed., PIRA International, United Kingdom, Vol. 2, pp. 641–725.
Backer,  S., and Petterson,  D. R., 1960, “Some Principles of Nonwoven Fabrics,” Text. Res. J., 30, pp. 704–711.
Hearle,  J. W. S., and Newton,  A., 1968, “Part XV: The Application of the Fiber Network Theory,” Text. Res. J., 1, pp. 343–351.
Hearle, J. W. S., 1980, “The mechanics of Dense Fibre Assemblies,” The Mechanics of Flexible Fibre Assemblies, Hearle, J. W. S., Thwaites, J. J., and Amirbayat, J., eds., Sijthoff and Noordhoff, New York, pp. 51–86.
Lu,  W., Carlsson,  L. A., and Andersson,  Y., 1995, “Micro-Model of paper, Part 1: bounds on elastic properties,” Tappi J., 78, pp. 155–164.
Lu,  W., and Carlsson,  L. A., 1996, “Micro-Model of paper, Part 2: statistical analysis of the paper structure,” Tappi J., 79, pp. 203–210.
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Sastry,  A. M., Choi,  S. B., and Cheng,  X., 1998, “Damage in Composite NiMH Positive Electrodes,” ASME J. Eng. Mater. Technol., 120, pp. 280–283.
Sastry,  A. M., Cheng,  X., and Wang,  C. W., 1998, “Mechanics of Stochastic Fibrous Networks,” J. Thermopl. Compos. Mater., 11, pp. 288–296.
Cheng,  X., Wang,  C. W., Sastry,  A. M., and Choi,  S. B., 1999, “Investigation of Failure Processes in Porous Battery Substrates: Part II—Simulation Results and Comparisons,” ASME J. Eng. Mater. Technol., 121, pp. 514–523.
Cheng, X., Sastry, A. M., and Layton, B. E., 2000, “Transport in Stochastic Fibrous Networks,” ASME J. Eng. Mater. Technol., accepted.
Cheng,  X., Wang,  C. W., Sastry,  A. M., and Choi,  S. B., 1999, “Investigation of Failure Processes in Porous Battery Substrates: Part II—Simulation Results and Comparisons,” ASME J. Eng. Mater. Technol., 121, pp. 514–523.
Wang,  C. W., Berhan,  L., and Sastry,  A. M., 2000, “Structure, Mechanics and Failure of Stochastic Fibrous Networks: Part I—Microscale Considerations,” ASME J. Eng. Mater. Technol., 122, pp. 450–459.

Figures

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Original (a) and reduced (b) networks, based on elimination of nonload-bearing structures for unidirectional (y-direction) loading only. Calculated parameters on the network as generated are as follows: original volume fraction –20 percent, reduced volume fraction –16 percent, number of intersection points –216; number of segments –361; average segment length –0.053; standard deviation, segment length –0.054. The fibers are randomly placed, and have aspect ratio l/d=100, with uniform staple length l=1, with length of cell lc=1.
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Connectivity in beam networks. For connectivity calculated via a 2D assumption for beams, only three cases arise, practically speaking, wherein (i) two, (ii) three, or (iii) four beams, respectively, intersect at a single point.
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Frequency distribution plots for segment size, for three aspect ratios of fibers, (a) l/d=100, (b) l/d=50 and (c) l/d=10
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Comparison of the Euler-Bernoulli beams and Timoshenko beams in simulations of effective network modulus, for a range of volume fractions and three aspect ratios (100,50,10). For all cases, l/lc=1.
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Comparison of segment strain versus normalized segment length in a single network with l/d=100,l/lu=1, and original volume fraction of 10 percent
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Schematic of network failure criteria. The microstructure in (a), comprised of 7 fiber segments and 5 nodes, is deformed until a local failure initiates at node b. The least conservative failure progression assumption (b), the beam assumption, removes only fiber segment 2, since stress is maximized theoretically at its end at point b. The most conservative assumption, the node assumption, is shown in (c), wherein the entire node b fails, resulting in loss of beam segments 1, 2, 6, and 7.
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Comparison of failure assumptions in predicting peak stress, with peak stress shown in the typical network stress-strain curve shown in (a). Peak stresses are shown for both Euler-Bernoulli and Timoshenko beam assumptions in (b), for a range of volume fractions and three aspect ratios (100,50,10). For all cases, l/lc=1.
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Schematic of torsion springs at a fiber-fiber “joint” wherein staple fibers are preserved as an elemental unit rather than segments (as shown in Fig. 2). Only connections between segments from different fibers are modeled with torsion spring connections.
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Connectivity versus effective modulus in simple bilinear networks. Schematics of the illustrative cases are shown wherein segments (whose lengths sum to 1) are joined by torsion springs of variable stiffness.
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Effective moduli (a) and maximum stresses (b) are given for α=30 and α=150 deg, for a range of torsion spring constants and a variable number of segments for arrays as shown in Fig. 9
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Normalized effective moduli (a) and peak stresses (b), for two normalized torsion spring constants. Plots were for networks comprised of fibers with uniform aspect ratio of 10, with representative cell edge length of lc=1.2.
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Normalized effective moduli (a) and peak stresses (b), for three normalized torsion spring constants, and for three representative cell sizes (lc=0.5, 1.0, 10). Plots were for networks comprised of fibers with uniform aspect ratio of 100.

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