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TECHNICAL PAPERS

Structure, Mechanics and Failure of Stochastic Fibrous Networks: Part I—Microscale Considerations

[+] Author and Article Information
C. W. Wang, L. Berhan, 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), 450-459 (May 30, 2000) (10 pages) doi:10.1115/1.1288769 History: Received May 26, 2000; Revised May 30, 2000
Copyright © 2000 by ASME
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References

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.
Wang, C. W., and Sastry, A. M., 2000 “Structure, Mechanics and Failure of Stochastic Fibrous Networks: Part II-Network Simulations and Application,” ASME J. Eng. Mater. Technol., 122 .
Sastry,  A. M., Cheng,  X., and Wang,  C. W., 1998, “Mechanics of Stochastic Fibrous Networks,” J. Thermoplast. Compos. Mater., 11, pp. 288–296.
Hashin,  Z., 1962, “The elastic moduli of heterogeneous material,” ASME J. Appl. Mech., 29, pp. 143–150.
Christensen,  R. M., and Lo,  K. H., 1979, “Solutions for effective shear properties in three phase sphere and cylinder models,” J. Mech. Phys. Solids, 27, pp. 315–330.
Meridith, R. E., and Tobias, C. W., 1962, “II Conduction in Heterogeneous Systems,” Advances in Electrochemistry and Electrochemical Engineering, Interscience, New York, pp. 15–47.
Cox,  H. L., 1952, “The Elasticity and Strength of Paper and Other Fibrous materials,” Br. J. Appl. Phys., 3, pp. 72–79.
Backer,  S., and Petterson,  D. R., 1960, “Some Principles of Nonwoven Fabrics,” Text. Res. J., 30, pp. 704–711.
Niskanen, K., 1993, “Strength and Fracture of Paper,” Products of Papermaking: Transactions of the Tenth Fundamental Research Symposium, PIRA International, United Kingdom, Vol. 2, pp. 641–725.
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.
Ostoja-Starzewski,  M., Sheng,  P. Y., and Jasiuk,  I., 1994. “Influence of Random Geometry on Effective Properties and Damage Formation in Composite Materials,” ASME J. Eng. Mater. Technol., 116, pp. 384–391.
Alzebdah,  K., and Ostoja-Starzewski,  M., 1996, “Micromechanically based stochastic finite elements: length scales and anisotropy,” Probab. Eng. Mech., 11, pp. 205–214.
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.
Lu,  W., Carlsson,  L. A., and de Ruvo,  A., 1996, “Micro-Model of paper, Part 3: Mosaic model,” Tappi J., 79, pp. 197–205.
Curtin,  W. A., 1990, “Lattice Trapping of Cracks,” J. Mater. Res., 5, pp. 1549–1560.
Cheng,  X., and Sastry,  A. M., 1999, “On Transport in Stochastic, Heterogeneous Fibrous Domains,” Mech. Mater., 31, pp. 765–786.
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Figures

Grahic Jump Location
Results for the virtual displacement method. Plots of moduli in compliant zones 1 and 2 versus normalized torsion spring constant, for the specific case of b2=b1/10. The solution for E2 is singular as K→0 (load applied to segment 2).
Grahic Jump Location
Two-beam assembly of beams joined by a torsion spring at B
Grahic Jump Location
A reduction of a physically realistic bond between fibers in a fiber/particle network to a 2D beam assembly (a), and notation for the two-beam assembly of rigidly joined beams, with each beam having a “compliant zone” of lengths b2 (b)
Grahic Jump Location
Maximum loads in two-beam assemblies, with notation of Fig. 3, for α=30, α=90 and α=150 deg, plotted for varying beam lengths as log(l1/l2). Segments are joined by torsion springs, (a) for normalized spring constant 1.0 and 0.1, and (b) for normalized spring constant 0.01, 0.001, and 0.0001. Euler-Bernoulli and Timoshenko beam results are compared in each case; two-beam assemblies are comprised of segments of diameter d=0.2.
Grahic Jump Location
Two-beam structural moduli, with notation of Fig. 3, for α=30, α=90 and α=150 deg, plotted for varying beam lengths as log(l1/l2). Segments are joined by torsion springs, (a) for normalized spring constants 1.0 and 0.1, and (b) for normalized spring constants 0.01, 0.001, and 0.0001. Euler-Bernoulli and Timoshenko beam results are compared in each case; two-beam assemblies are comprised of segments of diameter d=0.2.
Grahic Jump Location
Maximum loads in two-beam assemblies, with notation of Fig. 3, for α=30, 90 and 150 deg, plotted for varying beam lengths as log(l1/l2). Euler-Bernoulli and Timoshenko beam results are compared in each case; two-beam assemblies are comprised of segments of diameter d=0.2.
Grahic Jump Location
Two-beam structural moduli, with notation of Fig. 3, for α=30, 90 and 150 deg, plotted for varying beam lengths as log(l1/l2). Nodes between segments are rigid. Euler-Bernoulli and Timoshenko beam results are compared in each case; two-beam assemblies are comprised of segments of diameter d=0.2.
Grahic Jump Location
Two-beam network analyses notation
Grahic Jump Location
Network generation approach, with a single fiber shown for simplicity. Fiber is placed in the unit cell (a) whereupon periodic boundary conditions are applied, effectively “wrapping” overlapping ends back into unit cell (b), and nondomain-spanning segments are removed, as they do not bear network loads (c). Notation for a two-element case, with fixed end. Location and numbering of nodes used in calculating maximum stress are shown below the two-beam schematic.
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SEM image (50×) of an NiMH positive plate substrate, produced by National Standard (Fibrex), containing 50/50 fiber/powder by weight ratio, 97 percent pure nickel by mass; calculated porosity: 82 percent; fiber diameter: 30 mm; staple lengths: 0.64–1.27 cm; content

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