March  2011, 15(2): 391-400. doi: 10.3934/dcdsb.2011.15.391

Bound on the yield set of fiber reinforced composites subjected to antiplane shear

1. 

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, United States

Received  September 2009 Revised  February 2010 Published  December 2010

We consider fiber reinforced composites where both the matrix and the fibers are made of ideally plastic materials with the fibers being much stronger than the matrix. We restrict our attention to microstructures and applied stresses that lead to both microscopic and macroscopic antiplane shear deformations. We obtain a bound on the yield set of the composite in terms of the shapes of the fibers, their volume fraction and the yield set of the matrix.
Citation: Guillermo H. Goldsztein. Bound on the yield set of fiber reinforced composites subjected to antiplane shear. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 391-400. doi: 10.3934/dcdsb.2011.15.391
References:
[1]

G. Bao, J. W. Hutchinson and R. M. McMeeking, Particle reinforcement of ductile matrices against plastic flow and creep,, Acta Metall. Mater., 39 (1991), 1871. doi: 10.1016/0956-7151(91)90156-U. Google Scholar

[2]

M. Berveiller and A. Zaoui, An extension of the self consistent scheme to plastically-flowing polycrystals,, J. Mech. Phys. Solids, 26 (1979), 325. doi: 10.1016/0022-5096(78)90003-0. Google Scholar

[3]

K. Bhattacharya and P. M. Suquet, A model problem concerning recoverable strains of shape-memory polycrystals,, Proc. R. Soc. Lond. A, 461 (2005), 2797. doi: 10.1098/rspa.2005.1493. Google Scholar

[4]

J. Bishop and R. Hill, A theory for the plastic distortion of a polycrystalline aggregate under combined stresses,, Phil. Mag. A, 42 (1951), 414. Google Scholar

[5]

G. Bouchitté and P. Suquet, Homogenization, plasticity and yield design,, in, (1991), 107. Google Scholar

[6]

T. W. Butler and E. J. Jr. Sullivan, On the transverse strength of fiber reinforced materials,, J. Appl. Mech., 40 (1973), 523. doi: 10.1115/1.3423017. Google Scholar

[7]

G. deBotton, The effective yield strength of fiber-reinforced composites,, Int. J. Solids Structures, 32 (1995), 1743. doi: 10.1016/0020-7683(94)00203-9. Google Scholar

[8]

G. deBotton and P. Ponte Castañeda, Elastoplastic constitutive relations for fiber reinforced solids,, Int. J. Solids Structures, 30 (1993), 1865. doi: 10.1016/0020-7683(93)90222-S. Google Scholar

[9]

P. de Buhan, Lower bound approach to the macroscopic strength properties of a soil reinforced by columns,, C. R. Acad. Sci. Paris, 317 (1993), 287. Google Scholar

[10]

P. de Buhan and A. Taliercio, A homogenization approach to the yield strength of composite materials,, European J. Mech. A. Solids, 10 (1991), 129. Google Scholar

[11]

F. Demengel and T. Qi, Convex function of a measure obtained by homogenization,, C. R. Acad. Sci. Paris, 303 (1986), 285. Google Scholar

[12]

I. Doghri and C. Friebel, Effective elasto-plastic properties of inclusion-reinforced composites. Study of shape, orientation and cyclic response,, Mech. Mat., 37 (2005), 45. doi: 10.1016/j.mechmat.2003.12.007. Google Scholar

[13]

I. Doghri and L. Tinel, Micromechanical modelling and computation of elasto-plastic materials reinforced with distributed-orientation fibers,, Int. J. Plasticity, 21 (2005), 1919. doi: 10.1016/j.ijplas.2004.09.003. Google Scholar

[14]

D. C. Drucker, The safety factor of an elastic-plastic body in plane strain,, J. Appl. Mech., 18 (1951), 371. Google Scholar

[15]

D. C. Drucker, Extended limit design theorems for continuous media,, Q. Appl. Math., 9 (1952), 381. Google Scholar

[16]

D. C. Drucker, On minimum weight design and strength of non-homogeneous plastic bodies,, In, (1959), 139. Google Scholar

[17]

D. C. Drucker, "Engineering and Continuum Aspects of High Strength Materials,", In Proc. 2nd Berkely Int. Mat. Conf. (ed. Zackay) (1965), (1965). Google Scholar

[18]

D. C. Drucker, W. Prager and H. J. Greenberg, Extended limit design theorems for continuous media,, Q. Appl. Math., 9 (1952), 381. Google Scholar

[19]

A. Garroni and R. V. Kohn, Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity,, Proc. R. Soc. Lond. A., 459 (2003), 2613. doi: 10.1098/rspa.2003.1152. Google Scholar

[20]

A. Garroni, V. Nesi and M. Ponsiglione, Dielectric breakdown: Optimal bounds,, Proc. R. Soc. Lond. A., 457 (2001), 2317. doi: 10.1098/rspa.2001.0803. Google Scholar

[21]

G. H. Goldsztein, Rigid perfectly plastic two-dimensional polycrystals,, Proc. R. Soc. Lond. A., 457 (2001), 2789. doi: 10.1098/rspa.2001.0839. Google Scholar

[22]

G. H. Goldsztein, Two-dimensional rigid polycrystals whose grains have one ductile direction,, Proc. R. Soc. Lond. A., 459 (2003), 1949. doi: 10.1098/rspa.2002.1099. Google Scholar

[23]

Z. Hashin, Failure criteria for unidirectional fiber composites,, J. Appl. Mech., 47 (1980), 329. doi: 10.1115/1.3153664. Google Scholar

[24]

R. Hill, A theory of the yielding and plastic flow of anisotropic metals,, Proc. R. Soc. Lond. A., 193 (1948), 281. doi: 10.1098/rspa.1948.0045. Google Scholar

[25]

R. Hill, Theory of mechanical properties of fibre-strengthened materials: II. Inelastic behavior,, J. Mech. Phys. Solids, 12 (1964), 213. doi: 10.1016/0022-5096(64)90020-1. Google Scholar

[26]

G. Hu, A method of plasticity for general aligned spheroidal void or fiber-reinforced composites,, Int. J. Plasticity, 12 (1996), 439. doi: 10.1016/S0749-6419(96)00015-0. Google Scholar

[27]

W. Huang, Plastic behavior of some composite materials,, J. Comp. Mat., 5 (1971), 320. doi: 10.1177/002199837100500303. Google Scholar

[28]

D. Jeulin, W. Li and M. Ostoja-Starzewski, On the geodesic property of strain field patterns in elastoplastic composites,, Proc. R. Soc. Lond. A, 464 (2008), 1217. doi: 10.1098/rspa.2007.0192. Google Scholar

[29]

B. Ji and T. Wang, Plastic constitutive behavior of short-fiber/particle reinforced composites,, Int. J. Plasticity, 19 (2003), 565. doi: 10.1016/S0749-6419(01)00041-9. Google Scholar

[30]

V. Jikov, S. Kozlov and O. Oleinik, "Homogenization of Differential Operations and Integral Functionals,", Translated from the Russian by G. A. Yosifian [G. A. Iosif'yan], (1994). Google Scholar

[31]

U. F. Kocks, The relation between polycrystal deformation and single-crystal deformation,, Metallurgical Transactions, 1 (1970), 1121. Google Scholar

[32]

R. V. Kohn and T. D. Little, Some model problems of polycrystal plasticity with deficient basic crystals,, SIAM J. Appl. Math., 59 (1998), 172. doi: 10.1137/S0036139997320019. Google Scholar

[33]

R. H. Lance and D. N. Robinson, A maximum shear stress theory of plastic failure of fibre-reinforced materials,, J. Mech. Phys. Solids, 19 (1971), 49. doi: 10.1016/0022-5096(71)90017-2. Google Scholar

[34]

B. J. Lee and M. E. Mear, On the yield strength of metals containing spheroidals inclusions or voids,, Mech. Mat., 12 (1991), 191. doi: 10.1016/0167-6636(91)90017-T. Google Scholar

[35]

G. Li and P. Ponte Castañeda, The effect of particle shape and stiffness on the constitutive behavior of metal-matrix composites,, Int. J. Solids Structures, 30 (1993), 3189. doi: 10.1016/0020-7683(93)90109-K. Google Scholar

[36]

S. Majumdar and Jr. McLaughlin, Upper bounds to in-plane shear strength of unidirectional fiber-reinforced composites,, J. Appl. Mech., 40 (1973), 824. doi: 10.1115/1.3423104. Google Scholar

[37]

S. Majumdar and Jr. McLaughlin, Effects of phase geometry and volume fraction on the plane stress limit analysis of a unidirectional fiber-reinforced composite,, Int. J. Solids Structures, 11 (1975), 777. doi: 10.1016/0020-7683(75)90001-3. Google Scholar

[38]

P. V. McLaughlin, Plastic limit behavior of filament-reinforced materials,, Int. J. Solids Structures, 8 (1972), 1299. doi: 10.1016/0020-7683(72)90081-9. Google Scholar

[39]

G. W. Milton, On characterizing the set of possible tensors of composites. The variational method and the translation method,, Communications on Pure and Applied Mathematics, 43 (1990), 63. doi: 10.1002/cpa.3160430104. Google Scholar

[40]

G. W. Milton and S. K. Serkov, Bounding the current in nonlinear conducting composites,, J. Mech. Phys. Solids, 48 (2000), 1295. doi: 10.1016/S0022-5096(99)00083-6. Google Scholar

[41]

F. Murat, Compacité par compensation: Condition necessaire et suffisante de continuité faible sous une hypothése de rang constant,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 68. Google Scholar

[42]

V. Nesi, V. P. Smyshlyaev and J. R. Willis, Improved bounds for the yield stress of a model polycrystalline material,, J. Mech. Phys. Solid, 48 (2000), 1799. doi: 10.1016/S0022-5096(99)00100-3. Google Scholar

[43]

T. Olson, Improvements on Taylor's upper bound for rigid-plastic composites,, Mater. Sci. Eng. A, 175 (1994), 15. doi: 10.1016/0921-5093(94)91039-1. Google Scholar

[44]

P. Ponte Castañeda, The effective mechanical properties of nonlinear isotropic composites,, J. Mech. Phys. Solids, 39 (1991), 45. doi: 10.1016/0022-5096(91)90030-R. Google Scholar

[45]

P. Ponte Castañeda, New variational principles in plasticity and their applications to composite materials,, J. Mech. Phys. Solids, 40 (1992), 1757. doi: 10.1016/0022-5096(92)90050-C. Google Scholar

[46]

P. Ponte Castañeda and G. deBotton, On the homogenized yield strength of two-phase composites,, Proc. R. Soc. Lond. A, 438 (1992), 419. doi: 10.1098/rspa.1992.0116. Google Scholar

[47]

P. Ponte Castañeda, Exact second-order estimates for the effective mechanical properties of nonlinear composite materials,, J. Mech. Phys. Solids, 44 (1996), 827. doi: 10.1016/0022-5096(96)00015-4. Google Scholar

[48]

P. Ponte Castañeda and M. Nebozhyn, Variational estimates of the self consistent type for some model nonlinear polycrystals,, Proc. R. Soc. Lond. A, 453 (1997), 2715. doi: 10.1098/rspa.1997.0144. Google Scholar

[49]

P. Ponte Castañeda and P. Suquet, Nonlinear composites,, Advances in Appl. Mech., 34 (1997), 171. doi: 10.1016/S0065-2156(08)70321-1. Google Scholar

[50]

W. Prager, Plastic failure of fiber reinforced materials,, J. Appl. Mech., 36 (): 542. Google Scholar

[51]

Y. P. Qiu and G. J. Weng, The influence of inclusion shape on the overall behavior of a two-phase isotropic composite,, Int. J. Solids Structures, 27 (1991), 1537. doi: 10.1016/0020-7683(91)90076-R. Google Scholar

[52]

K. Sab, Homogenization of non-linear random media by a duality method. Application to plasticity,, Asymptotic Anal., 9 (1994), 311. Google Scholar

[53]

G. Sachs, Zur Ableitung einer Fleissbedingun,, Z. Ver. Dtsch. Ing., (1928), 734. Google Scholar

[54]

L. S. Shu and B. W. Rosen, Strength of fiber-reinforced composites by limit analysis methods,, J. Composite Mater., 1 (1967), 366. doi: 10.1177/002199836700100405. Google Scholar

[55]

A. J. M. Spencer, Plasticity theory for fibre-reinforced composites,, J. Eng. Math., 26 (1992), 107. doi: 10.1007/BF00043230. Google Scholar

[56]

L. Z. Sun and J. W. Ju, Matrix composites containing randomly located and oriented spheroidal particles,, J. Appl. Mech. Trans. ASME, 71 (2004), 774. doi: 10.1115/1.1794699. Google Scholar

[57]

P. Suquet, Analyse limite et homogénéisation,, C. R. Acad. Sci. Ser. II, 296 (1983), 1355. Google Scholar

[58]

P. Suquet, Elements of homogenization for inelastic solid mechanics,, in, (1987), 193. doi: 10.1007/3-540-17616-0_15. Google Scholar

[59]

P. Suquet, Discontinuities and plasticity,, in, (1988), 278. Google Scholar

[60]

P. Suquet, On the overall mechanical behavior of nonlinear composites,, C. R. Acad. Sci. Ser. II, 315 (1992), 909. Google Scholar

[61]

P. Suquet, Overall potentials and extremal surfaces of power law or ideally plastic materials,, J. Mech. Phys. Solids, 41 (1993), 981. doi: 10.1016/0022-5096(93)90051-G. Google Scholar

[62]

D. R. S. Talbot and J. R. Willis, Variational principles for inhomogeneous nonlinear media,, IMA J. Appl. Math., 35 (1985), 39. doi: 10.1093/imamat/35.1.39. Google Scholar

[63]

L. Tartar, Compensated compactness and applications to partial differential equations,, in, IV (1979), 136. Google Scholar

[64]

L. Tartar, The compensated compactness method applied to systems of conservation laws,, in, (1983), 263. Google Scholar

[65]

G. Taylor, Plastic strains in metals,, J. Inst. Metals, 62 (1938), 307. Google Scholar

[66]

J. R. Willis, The overall elastic response of composite materials,, J. Appl. Mech., 50 (1983), 1202. doi: 10.1115/1.3167202. Google Scholar

show all references

References:
[1]

G. Bao, J. W. Hutchinson and R. M. McMeeking, Particle reinforcement of ductile matrices against plastic flow and creep,, Acta Metall. Mater., 39 (1991), 1871. doi: 10.1016/0956-7151(91)90156-U. Google Scholar

[2]

M. Berveiller and A. Zaoui, An extension of the self consistent scheme to plastically-flowing polycrystals,, J. Mech. Phys. Solids, 26 (1979), 325. doi: 10.1016/0022-5096(78)90003-0. Google Scholar

[3]

K. Bhattacharya and P. M. Suquet, A model problem concerning recoverable strains of shape-memory polycrystals,, Proc. R. Soc. Lond. A, 461 (2005), 2797. doi: 10.1098/rspa.2005.1493. Google Scholar

[4]

J. Bishop and R. Hill, A theory for the plastic distortion of a polycrystalline aggregate under combined stresses,, Phil. Mag. A, 42 (1951), 414. Google Scholar

[5]

G. Bouchitté and P. Suquet, Homogenization, plasticity and yield design,, in, (1991), 107. Google Scholar

[6]

T. W. Butler and E. J. Jr. Sullivan, On the transverse strength of fiber reinforced materials,, J. Appl. Mech., 40 (1973), 523. doi: 10.1115/1.3423017. Google Scholar

[7]

G. deBotton, The effective yield strength of fiber-reinforced composites,, Int. J. Solids Structures, 32 (1995), 1743. doi: 10.1016/0020-7683(94)00203-9. Google Scholar

[8]

G. deBotton and P. Ponte Castañeda, Elastoplastic constitutive relations for fiber reinforced solids,, Int. J. Solids Structures, 30 (1993), 1865. doi: 10.1016/0020-7683(93)90222-S. Google Scholar

[9]

P. de Buhan, Lower bound approach to the macroscopic strength properties of a soil reinforced by columns,, C. R. Acad. Sci. Paris, 317 (1993), 287. Google Scholar

[10]

P. de Buhan and A. Taliercio, A homogenization approach to the yield strength of composite materials,, European J. Mech. A. Solids, 10 (1991), 129. Google Scholar

[11]

F. Demengel and T. Qi, Convex function of a measure obtained by homogenization,, C. R. Acad. Sci. Paris, 303 (1986), 285. Google Scholar

[12]

I. Doghri and C. Friebel, Effective elasto-plastic properties of inclusion-reinforced composites. Study of shape, orientation and cyclic response,, Mech. Mat., 37 (2005), 45. doi: 10.1016/j.mechmat.2003.12.007. Google Scholar

[13]

I. Doghri and L. Tinel, Micromechanical modelling and computation of elasto-plastic materials reinforced with distributed-orientation fibers,, Int. J. Plasticity, 21 (2005), 1919. doi: 10.1016/j.ijplas.2004.09.003. Google Scholar

[14]

D. C. Drucker, The safety factor of an elastic-plastic body in plane strain,, J. Appl. Mech., 18 (1951), 371. Google Scholar

[15]

D. C. Drucker, Extended limit design theorems for continuous media,, Q. Appl. Math., 9 (1952), 381. Google Scholar

[16]

D. C. Drucker, On minimum weight design and strength of non-homogeneous plastic bodies,, In, (1959), 139. Google Scholar

[17]

D. C. Drucker, "Engineering and Continuum Aspects of High Strength Materials,", In Proc. 2nd Berkely Int. Mat. Conf. (ed. Zackay) (1965), (1965). Google Scholar

[18]

D. C. Drucker, W. Prager and H. J. Greenberg, Extended limit design theorems for continuous media,, Q. Appl. Math., 9 (1952), 381. Google Scholar

[19]

A. Garroni and R. V. Kohn, Some three-dimensional problems related to dielectric breakdown and polycrystal plasticity,, Proc. R. Soc. Lond. A., 459 (2003), 2613. doi: 10.1098/rspa.2003.1152. Google Scholar

[20]

A. Garroni, V. Nesi and M. Ponsiglione, Dielectric breakdown: Optimal bounds,, Proc. R. Soc. Lond. A., 457 (2001), 2317. doi: 10.1098/rspa.2001.0803. Google Scholar

[21]

G. H. Goldsztein, Rigid perfectly plastic two-dimensional polycrystals,, Proc. R. Soc. Lond. A., 457 (2001), 2789. doi: 10.1098/rspa.2001.0839. Google Scholar

[22]

G. H. Goldsztein, Two-dimensional rigid polycrystals whose grains have one ductile direction,, Proc. R. Soc. Lond. A., 459 (2003), 1949. doi: 10.1098/rspa.2002.1099. Google Scholar

[23]

Z. Hashin, Failure criteria for unidirectional fiber composites,, J. Appl. Mech., 47 (1980), 329. doi: 10.1115/1.3153664. Google Scholar

[24]

R. Hill, A theory of the yielding and plastic flow of anisotropic metals,, Proc. R. Soc. Lond. A., 193 (1948), 281. doi: 10.1098/rspa.1948.0045. Google Scholar

[25]

R. Hill, Theory of mechanical properties of fibre-strengthened materials: II. Inelastic behavior,, J. Mech. Phys. Solids, 12 (1964), 213. doi: 10.1016/0022-5096(64)90020-1. Google Scholar

[26]

G. Hu, A method of plasticity for general aligned spheroidal void or fiber-reinforced composites,, Int. J. Plasticity, 12 (1996), 439. doi: 10.1016/S0749-6419(96)00015-0. Google Scholar

[27]

W. Huang, Plastic behavior of some composite materials,, J. Comp. Mat., 5 (1971), 320. doi: 10.1177/002199837100500303. Google Scholar

[28]

D. Jeulin, W. Li and M. Ostoja-Starzewski, On the geodesic property of strain field patterns in elastoplastic composites,, Proc. R. Soc. Lond. A, 464 (2008), 1217. doi: 10.1098/rspa.2007.0192. Google Scholar

[29]

B. Ji and T. Wang, Plastic constitutive behavior of short-fiber/particle reinforced composites,, Int. J. Plasticity, 19 (2003), 565. doi: 10.1016/S0749-6419(01)00041-9. Google Scholar

[30]

V. Jikov, S. Kozlov and O. Oleinik, "Homogenization of Differential Operations and Integral Functionals,", Translated from the Russian by G. A. Yosifian [G. A. Iosif'yan], (1994). Google Scholar

[31]

U. F. Kocks, The relation between polycrystal deformation and single-crystal deformation,, Metallurgical Transactions, 1 (1970), 1121. Google Scholar

[32]

R. V. Kohn and T. D. Little, Some model problems of polycrystal plasticity with deficient basic crystals,, SIAM J. Appl. Math., 59 (1998), 172. doi: 10.1137/S0036139997320019. Google Scholar

[33]

R. H. Lance and D. N. Robinson, A maximum shear stress theory of plastic failure of fibre-reinforced materials,, J. Mech. Phys. Solids, 19 (1971), 49. doi: 10.1016/0022-5096(71)90017-2. Google Scholar

[34]

B. J. Lee and M. E. Mear, On the yield strength of metals containing spheroidals inclusions or voids,, Mech. Mat., 12 (1991), 191. doi: 10.1016/0167-6636(91)90017-T. Google Scholar

[35]

G. Li and P. Ponte Castañeda, The effect of particle shape and stiffness on the constitutive behavior of metal-matrix composites,, Int. J. Solids Structures, 30 (1993), 3189. doi: 10.1016/0020-7683(93)90109-K. Google Scholar

[36]

S. Majumdar and Jr. McLaughlin, Upper bounds to in-plane shear strength of unidirectional fiber-reinforced composites,, J. Appl. Mech., 40 (1973), 824. doi: 10.1115/1.3423104. Google Scholar

[37]

S. Majumdar and Jr. McLaughlin, Effects of phase geometry and volume fraction on the plane stress limit analysis of a unidirectional fiber-reinforced composite,, Int. J. Solids Structures, 11 (1975), 777. doi: 10.1016/0020-7683(75)90001-3. Google Scholar

[38]

P. V. McLaughlin, Plastic limit behavior of filament-reinforced materials,, Int. J. Solids Structures, 8 (1972), 1299. doi: 10.1016/0020-7683(72)90081-9. Google Scholar

[39]

G. W. Milton, On characterizing the set of possible tensors of composites. The variational method and the translation method,, Communications on Pure and Applied Mathematics, 43 (1990), 63. doi: 10.1002/cpa.3160430104. Google Scholar

[40]

G. W. Milton and S. K. Serkov, Bounding the current in nonlinear conducting composites,, J. Mech. Phys. Solids, 48 (2000), 1295. doi: 10.1016/S0022-5096(99)00083-6. Google Scholar

[41]

F. Murat, Compacité par compensation: Condition necessaire et suffisante de continuité faible sous une hypothése de rang constant,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 68. Google Scholar

[42]

V. Nesi, V. P. Smyshlyaev and J. R. Willis, Improved bounds for the yield stress of a model polycrystalline material,, J. Mech. Phys. Solid, 48 (2000), 1799. doi: 10.1016/S0022-5096(99)00100-3. Google Scholar

[43]

T. Olson, Improvements on Taylor's upper bound for rigid-plastic composites,, Mater. Sci. Eng. A, 175 (1994), 15. doi: 10.1016/0921-5093(94)91039-1. Google Scholar

[44]

P. Ponte Castañeda, The effective mechanical properties of nonlinear isotropic composites,, J. Mech. Phys. Solids, 39 (1991), 45. doi: 10.1016/0022-5096(91)90030-R. Google Scholar

[45]

P. Ponte Castañeda, New variational principles in plasticity and their applications to composite materials,, J. Mech. Phys. Solids, 40 (1992), 1757. doi: 10.1016/0022-5096(92)90050-C. Google Scholar

[46]

P. Ponte Castañeda and G. deBotton, On the homogenized yield strength of two-phase composites,, Proc. R. Soc. Lond. A, 438 (1992), 419. doi: 10.1098/rspa.1992.0116. Google Scholar

[47]

P. Ponte Castañeda, Exact second-order estimates for the effective mechanical properties of nonlinear composite materials,, J. Mech. Phys. Solids, 44 (1996), 827. doi: 10.1016/0022-5096(96)00015-4. Google Scholar

[48]

P. Ponte Castañeda and M. Nebozhyn, Variational estimates of the self consistent type for some model nonlinear polycrystals,, Proc. R. Soc. Lond. A, 453 (1997), 2715. doi: 10.1098/rspa.1997.0144. Google Scholar

[49]

P. Ponte Castañeda and P. Suquet, Nonlinear composites,, Advances in Appl. Mech., 34 (1997), 171. doi: 10.1016/S0065-2156(08)70321-1. Google Scholar

[50]

W. Prager, Plastic failure of fiber reinforced materials,, J. Appl. Mech., 36 (): 542. Google Scholar

[51]

Y. P. Qiu and G. J. Weng, The influence of inclusion shape on the overall behavior of a two-phase isotropic composite,, Int. J. Solids Structures, 27 (1991), 1537. doi: 10.1016/0020-7683(91)90076-R. Google Scholar

[52]

K. Sab, Homogenization of non-linear random media by a duality method. Application to plasticity,, Asymptotic Anal., 9 (1994), 311. Google Scholar

[53]

G. Sachs, Zur Ableitung einer Fleissbedingun,, Z. Ver. Dtsch. Ing., (1928), 734. Google Scholar

[54]

L. S. Shu and B. W. Rosen, Strength of fiber-reinforced composites by limit analysis methods,, J. Composite Mater., 1 (1967), 366. doi: 10.1177/002199836700100405. Google Scholar

[55]

A. J. M. Spencer, Plasticity theory for fibre-reinforced composites,, J. Eng. Math., 26 (1992), 107. doi: 10.1007/BF00043230. Google Scholar

[56]

L. Z. Sun and J. W. Ju, Matrix composites containing randomly located and oriented spheroidal particles,, J. Appl. Mech. Trans. ASME, 71 (2004), 774. doi: 10.1115/1.1794699. Google Scholar

[57]

P. Suquet, Analyse limite et homogénéisation,, C. R. Acad. Sci. Ser. II, 296 (1983), 1355. Google Scholar

[58]

P. Suquet, Elements of homogenization for inelastic solid mechanics,, in, (1987), 193. doi: 10.1007/3-540-17616-0_15. Google Scholar

[59]

P. Suquet, Discontinuities and plasticity,, in, (1988), 278. Google Scholar

[60]

P. Suquet, On the overall mechanical behavior of nonlinear composites,, C. R. Acad. Sci. Ser. II, 315 (1992), 909. Google Scholar

[61]

P. Suquet, Overall potentials and extremal surfaces of power law or ideally plastic materials,, J. Mech. Phys. Solids, 41 (1993), 981. doi: 10.1016/0022-5096(93)90051-G. Google Scholar

[62]

D. R. S. Talbot and J. R. Willis, Variational principles for inhomogeneous nonlinear media,, IMA J. Appl. Math., 35 (1985), 39. doi: 10.1093/imamat/35.1.39. Google Scholar

[63]

L. Tartar, Compensated compactness and applications to partial differential equations,, in, IV (1979), 136. Google Scholar

[64]

L. Tartar, The compensated compactness method applied to systems of conservation laws,, in, (1983), 263. Google Scholar

[65]

G. Taylor, Plastic strains in metals,, J. Inst. Metals, 62 (1938), 307. Google Scholar

[66]

J. R. Willis, The overall elastic response of composite materials,, J. Appl. Mech., 50 (1983), 1202. doi: 10.1115/1.3167202. Google Scholar

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