Mathematical model and analysis of the strength of particle reinforced ideally plastic composites
Author:
Guillermo H. Goldsztein
Journal:
Quart. Appl. Math. 75 (2017), 769-782
MSC (2010):
Primary 74C05, 74Q05, 74Q15, 74Q20
DOI:
https://doi.org/10.1090/qam/1469
Published electronically:
March 10, 2017
MathSciNet review:
3686520
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Additional Information
Abstract: We consider fiber reinforced ideally plastic composites. We analyze a mathematical model valid for microstructures and applied stresses that lead to both microscopic and macroscopic anti-plane shear deformations. We obtain a bound on the yield set of the reinforced material in terms of the shapes of the cross section of the fibers, their volume fraction, and the yield stresses of the matrix. We construct examples showing that our bound is sharp.
References
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- Dominique Jeulin, Wei Li, and Martin Ostoja-Starzewski, On the geodesic property of strain field patterns in elastoplastic composites, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464 (2008), no. 2093, 1217–1227. MR 2386643, DOI https://doi.org/10.1098/rspa.2007.0192
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- P.-M. Suquet, Overall potentials and extremal surfaces of power law or ideally plastic composites, J. Mech. Phys. Solids 41 (1993), no. 6, 981–1002. MR 1220788, DOI https://doi.org/10.1016/0022-5096%2893%2990051-G
- D. R. S. Talbot and J. R. Willis, Variational principles for inhomogeneous nonlinear media, IMA J. Appl. Math. 35 (1985), no. 1, 39–54. MR 820896, DOI https://doi.org/10.1093/imamat/35.1.39
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
- L. Tartar, The compensated compactness method applied to systems of conservation laws, Systems of Nonlinear Partial Differential Equations (J.M. Ball, ed.), 1983, pp. 263–285.
- G. Taylor, Plastic strains in metals, J. Inst. Metals 62 (1938), 307–324.
- S. Kozlov V. Jikov and O. Oleinik, Homogenization of differential operations and integral functionals, Springer-Verlag, New York, 1994.
- V. Nesi, V. P. Smyshlyaev, and J. R. Willis, Improved bounds for the yield stress of a model polycrystalline material, J. Mech. Phys. Solids 48 (2000), no. 9, 1799–1825. MR 1765196, DOI https://doi.org/10.1016/S0022-5096%2899%2900100-3
- J.R. Willis, The overall elastic response of composite materials, J. Appl. Mech. 50 (1983), 1202–1209.
References
- A. Garroni, V. Nesi, and M. Ponsiglione, Dielectric breakdown: optimal bounds, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2014, 2317–2335. MR 1862657, DOI https://doi.org/10.1098/rspa.2001.0803
- J. F. W. Bishop and R. Hill, A theory of the plastic distortion of a polycrystalline aggregate under combined stresses, Philos. Mag. (7) 42 (1951), 414–427. MR 0041691
- G. Bouchitté and P. Suquet, Homogenization, plasticity and yield design, Composite media and homogenization theory (Trieste, 1990) Progr. Nonlinear Differential Equations Appl., vol. 5, Birkhäuser Boston, Boston, MA, 1991, pp. 107–133. MR 1145747, DOI https://doi.org/10.1007/978-1-4684-6787-1_7
- P. Ponte Castañeda, The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids 39 (1991), no. 1, 45–71. MR 1085622, DOI https://doi.org/10.1016/0022-5096%2891%2990030-R
- P. Ponte Castañeda, New variational principles in plasticity and their application to composite materials, J. Mech. Phys. Solids 40 (1992), no. 8, 1757–1788. MR 1190849, DOI https://doi.org/10.1016/0022-5096%2892%2990050-C
- P. Ponte Castaneda and G. deBotton, On the homogenized yield strength of two-phase composites, Proc. R. Soc. Lond. A 438 (1992), 419–431.
- P. Ponte Castaneda and P. Suquet, Nonlinear composites, Advances in Appl. Mech. 34 (1997), 171–302.
- Dominique Jeulin, Wei Li, and Martin Ostoja-Starzewski, On the geodesic property of strain field patterns in elastoplastic composites, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464 (2008), no. 2093, 1217–1227. MR 2386643, DOI https://doi.org/10.1098/rspa.2007.0192
- P. de Buhan, Lower bound approach to the macroscopic strength properties of a soil reinforced by columns, C. R. Acad. Sci. Paris, Serie II 317 (1993), 287–293.
- P. de Buhan and A. Taliercio, A homogenization approach to the yield strength of composite materials, European J. Mech. A. Solids 10 (1991), 129–154.
- G. de Botton, The effective yield strength of fiber-reinforced composites, Internat. J. Solids Structures 32 (1995), no. 12, 1743–1757. MR 1331003, DOI https://doi.org/10.1016/0020-7683%2894%2900203-9
- Adriana Garroni and Robert V. Kohn, Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2038, 2613–2625. MR 2011358, DOI https://doi.org/10.1098/rspa.2003.1152
- Z. Hashin, Failure criteria for unidirectional fiber composites, J. Appl. Mech. 47 (1980), 329–334.
- W. Huang, Plastic behavior of some composite materials, J. Comp. Mat. 5 (1971), 320–338.
- Robert V. Kohn and Thomas D. Little, Some model problems of polycrystal plasticity with deficient basic crystals, SIAM J. Appl. Math. 59 (1999), no. 1, 172–197. MR 1647809, DOI https://doi.org/10.1137/S0036139997320019
- Guoan Li and P. Ponte Castañeda, The effect of particle shape and stiffness on the constitutive behavior of metal-matrix composites, Internat. J. Solids Structures 30 (1993), no. 23, 3189–3209. MR 1252753, DOI https://doi.org/10.1016/0020-7683%2893%2990109-K
- Graeme W. Milton and Sergey K. Serkov, Bounding the current in nonlinear conducting composites, J. Mech. Phys. Solids 48 (2000), no. 6-7, 1295–1324. MR 1766404, DOI https://doi.org/10.1016/S0022-5096%2899%2900083-6
- François Murat, Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 69–102 (French). MR 616901
- T. Olson, Improvements on taylor’s upper bound for rigid-plastic composites, Mater. Sci. Eng. A 175 (1994), 15–20.
- W. Prager, Plastic failure of fiber reinforced materials, J. Appl. Mech. 36 (1969), 542–544.
- G. Sachs, Zur ableitung einer fleissbedingun, Z. Ver. Dtsch. Ing. 72 (1928), 734–736.
- L.S. Shu and B.W. Rosen, Strength of fiber-reinforced composites by limit analysis methods, J. Composite Mater. 1 (1967), 366–381.
- Pierre Suquet, Analyse limite et homogénéisation, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 296 (1983), no. 18, 1355–1358 (French, with English summary). MR 720280
- P.-M. Suquet, Overall potentials and extremal surfaces of power law or ideally plastic composites, J. Mech. Phys. Solids 41 (1993), no. 6, 981–1002. MR 1220788, DOI https://doi.org/10.1016/0022-5096%2893%2990051-G
- D. R. S. Talbot and J. R. Willis, Variational principles for inhomogeneous nonlinear media, IMA J. Appl. Math. 35 (1985), no. 1, 39–54. MR 820896, DOI https://doi.org/10.1093/imamat/35.1.39
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
- L. Tartar, The compensated compactness method applied to systems of conservation laws, Systems of Nonlinear Partial Differential Equations (J.M. Ball, ed.), 1983, pp. 263–285.
- G. Taylor, Plastic strains in metals, J. Inst. Metals 62 (1938), 307–324.
- S. Kozlov V. Jikov and O. Oleinik, Homogenization of differential operations and integral functionals, Springer-Verlag, New York, 1994.
- V. Nesi, V. P. Smyshlyaev, and J. R. Willis, Improved bounds for the yield stress of a model polycrystalline material, J. Mech. Phys. Solids 48 (2000), no. 9, 1799–1825. MR 1765196, DOI https://doi.org/10.1016/S0022-5096%2899%2900100-3
- J.R. Willis, The overall elastic response of composite materials, J. Appl. Mech. 50 (1983), 1202–1209.
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Additional Information
Guillermo H. Goldsztein
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry Steert, Atlanta, Georgia 30332-0160
Email:
ggold@math.gatech.edu
Keywords:
Microstructures,
yield condition,
fiber-reinforced composite material,
ideally plastic material,
variational calculus
Received by editor(s):
August 16, 2016
Received by editor(s) in revised form:
February 14, 2017
Published electronically:
March 10, 2017
Article copyright:
© Copyright 2017
Brown University
