Abstract
This investigation is concerned with the interaction—as far as load-absorption is concerned—of a pair of identical parallel elastic filaments in a fiber-reinforced composite material. The filaments are assumed to have uniform circular cross-sections, are taken to be semi-infinite, and are supposed to be continuously bonded to an all-around infinite matrix of distinct elastic properties. At infinity the matrix is subjected to uniaxial tension parallel to the filaments. Two separate but related problems are treated. In the first both filaments extend to infinity in the same direction and their terminal cross-sections are coplanar. In the second problem the filaments extend to infinity in opposite directions and their terminal cross-sections need no longer be coplanar, the two filaments being permitted to overlap partly. An approximate scheme based in part on three-dimensional linear elasticity and developed originally by Muki and Sternberg is employed in the analysis. The problems are ultimately reduced to Fredholm integral equations which characterize the distribution of the axial filament force. The integral equations are analyzed asymptotically and numerically. Results are presented which show the variation of filament force with position and the effect on this variation of various relevant geometrical and material parameters. One result is of particular interest. In the second problem, involving the overlapping filaments, for certain cases the filament force exceeds its far-field asymptotic value for a portion of the filament length. Stated another way, this means that the filament is loaded in excess of that which one would calculate by equating axial strains.
Zusammenfassung
Die Wechselwirkung eines identischen parallelen elastischen Fadenpaares in einem faserverstärkten Verbundstoff wird in Bezug auf die Lastaufnahme untersucht. Man nimmt an, dass die Fäden einen gleichmässigen runden Querschnitt haben, halbunendlich sind, und dass sie ohne Unterbrechung an einem überall unendlichen Grundstoff eindeutiger elastischer Eigenschaften anhaften. Im Unendlichen wird der Grundstoff einem einachsigen Zug parallel zu den Fäden ausgezetzt. Es werden zwei verschiedene, aber in Verbindung stehende, Fragenstellungen behandelt. In der Ersten erstrecken sich beide Fäden unendlich lang in derselben Richtung und ihre Endquerschnitte sind koplanar. In der Zweiten erstrecken sie sich unendlich lang in entgegengesetzten Richtungen und ihre Endquerschnitte müssen nicht koplanar sein, obwohl die Fäden sich teilweise überlagern dürfen. Die Analyse wird mit einem von Muki und Sternberg entwickelten Näherungsverfahren durchgeführt, welches zum Teil auf einer dreidimensionalen linearen Elastizitätstheorie beruht. Letzten Endes nehmen diese Probleme die Form Fredholmscher Integralgleichungen an, welche die Verteilung der Fadenlängskraft kenzeichnen. Die Integralgleichungen werden asymptotisch und zahlenmässig analysiert.
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References
Dow, N. F.,Study of stresses near a discontinuity in a filament-reinforced composite metal, Technical Information Series, Report R63SD61, General Electric Company (1963).
Rosen, B. W.,Mechanics of composite strengthening, Chapter 3 in Fiber Composite Materials, American Society for Metals (1965).
Sternberg, E., and R. Muki, Load-absorption by a filament in a fiber-reinforced material,Zeitschrift für angewandte Mathematik und Physik, 21/4 (1970) 552.
Muki, R., and E. Sternberg, On the diffusion of an axial load from an infinite cylindrical bar embedded in an elastic medium,International Journal of Solids and Structures, 5/6 (1969) 587.
Cohen, L. J., and J. P. Romualdi, Stress, strain and displacement fields in a composite material reinforced with discontinuous fibers,Journal of the Franklin Institute, 284/6 (1967) 388.
Chen, P. E., Strength properties of discontinuous fiber composites,Polymer Engineering and Science, 11/4 (1971) 51.
Love, A. E. H.,A treatise on the mathematical theory of elasticity, Fourth Edition, Dover, New York 1944.
Muki, R., and E. Sternberg, Elastostatic load-transfer to a half-space from a partially embedded axially loaded rod,International Journal of Solids and Structures, 6/1 (1970) 69.
Watson, G. N.,A treatise on the theory of Bessel functions, Second Edition, Cambridge University Press, 1962.
Weber, H., Über die Besselschen Funktionen und ihre Anwendungen auf die Theorie der elektrischen Ströme,J. reine angew. Math., LXXX (1873) 75.
Eason, G., Noble, B., and I. N. Sneddon,On certain integrals of Lipschitz-Hankel type involving products of Bessel functions, Philosophical Transactions of the Royal Society of London, Series A, 247, (1955) 529.
Bateman Manuscript Project,Table of integral transforms, vol. 1, McGraw-Hill, New York 1954.
Muki, R., and E. Sternberg, Note on an asymptotic property of solutions to a class of Fredholm integral equations,Quarterly of Applied Mathematics, 28/2 (1970) 277.
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The results communicated in this paper were obtained in the course of an investigation supported in part by the Office of Naval Research under Contract N00014-67-A-0094-0020. The work was carried out during the author's tenure of a National Science Foundation Traineeship.
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Carne, T.G. Load absorption and interaction of two adjacent filaments in a fiber-reinforced material. J Elasticity 6, 1–38 (1976). https://doi.org/10.1007/BF00135174
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DOI: https://doi.org/10.1007/BF00135174