On the homogenization of metal matrix composites using strain gradient plasticity. (English) Zbl 1346.74157
Summary: The homogenized response of metal matrix composites (MMC) is studied using strain gradient plasticity. The material model employed is a rate independent formulation of energetic strain gradient plasticity at the micro scale and conventional rate independent plasticity at the macro scale. Free energy inside the micro structure is included due to the elastic strains and plastic strain gradients. A unit cell containing a circular elastic fiber is analyzed under macroscopic simple shear in addition to transverse and longitudinal loading. The analyses are carried out under generalized plane strain condition. Micro-macro homogenization is performed observing the Hill-Mandel energy condition, and overall loading is considered such that the homogenized higher order terms vanish. The results highlight the intrinsic size-effects as well as the effect of fiber volume fraction onthe overall response curves, plastic strain distributions and homogenized yield surfaces under different loading conditions. It
is concluded that composites with smaller reinforcement size have larger initial yield surfaces and furthermore, they exhibit more kinematic hardening.
MSC:
| 74Q05 | Homogenization in equilibrium problems of solid mechanics |
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
| 74E30 | Composite and mixture properties |
References:
| [1] | McDanels, D.: Analysis of stress-strain, fracture, ductility behavior of aluminum matrix composites containing discontinuous silicon carbide reinforcement. Metallurgical Transactions A (Physical Metallurgy and Materials Science) 16A, 1105–1115 (1985) · doi:10.1007/BF02811679 |
| [2] | Bao, G., Hutchinson, J.W., McMeeking, R.M.: Particle reinforcement of ductile matrices against plastic flow and creep. Acta Metal. et Mater. 39, 1871–1882 (1991) · doi:10.1016/0956-7151(91)90156-U |
| [3] | Legarth, B.N.: Effects of geometrical anisotropy on failure in a plastically anisotropic metal. Engineering Fracture Mechanics 72, 2792–2807 (2005) · doi:10.1016/j.engfracmech.2005.06.004 |
| [4] | Legarth, B.N., Kuroda, M.: Particle debonding using different yield criteria. European Journal of Mechanics–A/Solids 23, 737–751 (2004) · Zbl 1058.74626 · doi:10.1016/j.euromechsol.2004.05.002 |
| [5] | Tvergaard, V.: Analysis of tensile properties for a whisker-reinforced metal-matrix. omposite. Acta Metallurgica 38, 185–194 (1990) · doi:10.1016/0956-7151(90)90048-L |
| [6] | Tvergaard, V.: Fibre debonding and breakage in a whisker-reinforced metal. Materials Science and Engineering: A 190, 215–222 (1995) · doi:10.1016/0921-5093(95)80005-0 |
| [7] | Lloyd, D.J.: Particle reinforced aluminium and magnesium matrix composites. International Materials Reviews 39, 1–23 (1994) · doi:10.1179/imr.1994.39.1.1 |
| [8] | Fleck, N.A., Ashby, M.F., Hutchinson, J.W.: The role of geometrically necessary dislocations in giving material strengthening. Scripta Materialia 48, 179–183 (2003) · doi:10.1016/S1359-6462(02)00338-X |
| [9] | Gao, H., Huang, Y.: Geometrically necessary dislocation and size-dependent plasticity. Scripta Materialia 48, 113–118 (2003) · doi:10.1016/S1359-6462(02)00329-9 |
| [10] | Mughrabi, H.: The effect of geometrically necessary dislocations on the flow stress of deformed crystals containing a heterogeneous dislocation distribution. Materials Science and Engineering A319–321, 139–143 (2001) · doi:10.1016/S0921-5093(01)01003-6 |
| [11] | Hutchinson, J.W.: Plasticity at the micron scale. International Journal of Solids and Structures 37, 225–238 (2000) · Zbl 1075.74022 · doi:10.1016/S0020-7683(99)00090-6 |
| [12] | Acharya, A., Bassani, J.L.: Lattice incompatibility and a gradient theory of crystal plasticity. Journal of the Mechanics and Physics of Solids 48, 1565–1595 (2000) · Zbl 0963.74010 · doi:10.1016/S0022-5096(99)00075-7 |
| [13] | Bassani, J.L.: Incompatibility and a simple gradient theory of plasticity. Journal of the Mechanics and Physics of Solids 49, 1983–1996 (2001) · Zbl 1030.74015 · doi:10.1016/S0022-5096(01)00037-0 |
| [14] | Fleck, N.A., Hutchinson, J.W.: Strain Gradient Plasticity. In: Hutchinson, J.W., Wu, T.Y., eds. Advances in Applied Mechanics 33, 295–361. Academic Press, USA (1997) · Zbl 0894.73031 · doi:10.1016/S0065-2156(08)70388-0 |
| [15] | Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 49, 2245–2271 (2001) · Zbl 1033.74006 · doi:10.1016/S0022-5096(01)00049-7 |
| [16] | Fleck, N.A., Willis, J.R.: A mathematical basis for straingradient plasticity theory–Part I: Scalar plastic multiplier. Journal of the Mechanics and Physics of Solids 57, 161–177 (2009) · Zbl 1195.74020 · doi:10.1016/j.jmps.2008.09.010 |
| [17] | Fleck, N.A., Willis, J.R.: A mathematical basis for strain-gradient plasticity theory–Part II: Tensorial plastic multiplier. Journal of the Mechanics and Physics of Solids 57, 1045–1057 (2009) · Zbl 1173.74316 · doi:10.1016/j.jmps.2009.03.007 |
| [18] | Gao, H., Huang, Y., Nix, W.D., et al.: Mechanism-based strain gradient plasticity–I. Theory. Journal of the Mechanics and Physics of Solids 47, 1239–1263 (1999) · Zbl 0982.74013 · doi:10.1016/S0022-5096(98)00103-3 |
| [19] | Gudmundson, P.: A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379–1406 (2004) · Zbl 1114.74366 · doi:10.1016/j.jmps.2003.11.002 |
| [20] | Gurtin, M.E.: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids 50, 5–32 (2002) · Zbl 1043.74007 · doi:10.1016/S0022-5096(01)00104-1 |
| [21] | Gurtin, M.E., Anand, L.: A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. Journal of the Mechanics and Physics of Solids 53, 1624–1649 (2005) · Zbl 1120.74353 · doi:10.1016/j.jmps.2004.12.008 |
| [22] | Lele, S.P., Anand, L.: A small-deformation strain-gradient theory for isotropic viscoplastic materials. Philosophical Magazine 88, 1478–6435 (2008) · doi:10.1080/14786430802087031 |
| [23] | Eshelby, J.D.: The determination of the field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A 241, 376–396 (1957) · Zbl 0079.39606 · doi:10.1098/rspa.1957.0133 |
| [24] | Budiansky, B.: On elastic moduli of some heterogeneous materials. Journal of The Mechanics and Physics of Solids 13, 223–227 (1965) · doi:10.1016/0022-5096(65)90011-6 |
| [25] | Hill, R.: A self-consistent mechanics of composite materials. Journal of The Mechanics and Physics of Solids 13, 213–222 (1965) · doi:10.1016/0022-5096(65)90010-4 |
| [26] | Hashin, Z.: Analysis of composite materials–A survey. J. Appl. Mech. 50, 481–505 (1983) · Zbl 0542.73092 · doi:10.1115/1.3167081 |
| [27] | Ghosh, S., Lee, K., Moorthy, S.: Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method. International Journal of Solids and Structures 32, 27–62 (1995) · Zbl 0865.73060 · doi:10.1016/0020-7683(94)00097-G |
| [28] | Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics and Physics of Solids 11, 127–140 (1963) · Zbl 0108.36902 · doi:10.1016/0022-5096(63)90060-7 |
| [29] | Keller, R.R., Phelps, J.M., Read, D.T.: Preprocessing and post-processing for materials based on the homogenization method with adaptive finite element methods. Computer Methods in Applied Mechanics and Engineering 83, 143–198 (1990) · Zbl 0737.73008 · doi:10.1016/0045-7825(90)90148-F |
| [30] | Kouznetsova, V., Brekelmans, W.A.M., Baaijens, F.P.T.: An approach to micro-macro modeling of heterogeneous materials. Comput. Mech. 27, 37–48 (2001) · Zbl 1005.74018 · doi:10.1007/s004660000212 |
| [31] | Suquet, P.M.: Local and global aspects in the mathematical theory of plasticity. In: Sawczuk, A., Biachi G., eds. Plasticity Todays: Modelling, Methods and Applications, Elsvier Applied Science Publishers, London. 279–310 (1985) |
| [32] | Terada, K., Hori, M., Kyoya, T., et al.: Simulation of the multi-scale convergence in computational homogenization approaches. Journal of Solids and Structures 37, 2285–2311 (2000) · Zbl 0991.74056 · doi:10.1016/S0020-7683(98)00341-2 |
| [33] | Wieckowski, Z.: Dual finite element methods in homogenization for elastic-plastic fibrous composite material. International Journal of Plasticity 16, 199–221 (2000) · Zbl 0969.74070 · doi:10.1016/S0749-6419(99)00070-4 |
| [34] | Niordson, C.F.: Strain gradient plasticity effects in whisker-reinforced metals. Journal of the Mechanics and Physics of Solids 51, 1863–1883 (2003) · Zbl 1077.74511 · doi:10.1016/S0022-5096(03)00003-6 |
| [35] | Niordson, C.F., Tvergaard, V.: Nonlocal plasticity effects on the tensile properties of a metal matrix composite. European Journal of Mechanics–A/Solids 20, 601–613 (2001) · Zbl 1014.74020 · doi:10.1016/S0997-7538(01)01149-4 |
| [36] | Niordson, C.F., Tvergaard, V.: Nonlocal plasticity effects on fibre debonding in a whisker-reinforced metal. European Journal of Mechanics–A/Solids 21, 239–248 (2002) · Zbl 1023.74041 · doi:10.1016/S0997-7538(01)01190-1 |
| [37] | Kouznetsova, V.G., Geers, M.G.D., Brekelmans, W.A.M.: Multi-scale second-order computational homogenization of multi-phase materials: A nested finite element solution strategy. Comput. Methods Appl. Mech. Engrg. 193, 5525–5550 (2004) · Zbl 1112.74469 · doi:10.1016/j.cma.2003.12.073 |
| [38] | Hill, R.: Elastic properties of reinforced solids: Some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372 (1963) · Zbl 0114.15804 · doi:10.1016/0022-5096(63)90036-X |
| [39] | Azizi, R., Niordson, C.F., Legarth, B.N.: Size-effects on yield surfaces for micro reinforced composites. International Journal of Plasticity 27, 1817–1832 (2011) · Zbl 1426.74067 · doi:10.1016/j.ijplas.2011.05.006 |
| [40] | Fredriksson, P., Gudmundson, P., Mikkelsen, L. P.: Finite element implementation and numerical issues of strain gradient plasticity with application to metal matrix composites. International Journal of Solids and Structures 46, 3977–3987 (2009) · Zbl 1183.74273 · doi:10.1016/j.ijsolstr.2009.07.028 |
| [41] | Borg, U., Niordson, C.F., Kysar, J.W.: Size effects on void growth in single crystals with distributed voids. International Journal of Plasticity 24, 688–701 (2008) · Zbl 1131.74009 · doi:10.1016/j.ijplas.2007.07.015 |
| [42] | Hussein, M.I., Borg, U., Niordson, C.F., et al.: Plasticity size effects in voided crystals. Journal of the Mechanics and Physics of Solids 56, 114–131 (2008) · Zbl 1162.74338 · doi:10.1016/j.jmps.2007.05.004 |
| [43] | Taya, M., Lualy, K.E., Wakashima, K., et al.: Bauschinger effect in particulate SiC-6061 aluminum composites. Material Science and Engineering 124, 103–111 (1990) · doi:10.1016/0921-5093(90)90140-X |
| [44] | Anand, L., Gurtin, M.E., Lele, S.P., et al.: A one-dimensional theory of strain-gradient plasticity: Formulation, analysis, numerical results. Journal of the Mechanics and Physics of Solids 53, 1789–1826 (2005) · Zbl 1120.74350 · doi:10.1016/j.jmps.2005.03.003 |
| [45] | Niordson, C.F., Legarth, B.N.: Strain gradient effects on cyclic plasticity. Journal of the Mechanics and Physics of Solids 58, 542–557 (2010) · Zbl 1244.74032 · doi:10.1016/j.jmps.2010.01.007 |
| [46] | Choi, H.S, Jang, Y.H.: Micromechanical behavior of a unidirectional composite subjected to transverse shear loading. Appl Compos. Mater. 18, 127–148 (2011) · doi:10.1007/s10443-010-9136-0 |
| [47] | Barai, P., Weng, G.J.: A theory of plasticity for carbon nanotube reinforced composites. International Journal of Plasticity 27, 539–559 (2011) · Zbl 1426.74086 · doi:10.1016/j.ijplas.2010.08.006 |
| [48] | Kim, K.T., Cha, S.I., Hong, S.H., et al.: Microstructure and tensile behavior of carbon nanotubes reinforced Cu matrix nanocomposites. Mater. Sci. Eng. / A–Struct. Mater. Prop. Microstruct. and Proces. 430, 27–33 (2006) · doi:10.1016/j.msea.2006.04.085 |
| [49] | Shu, J.Y., Barlow, C.Y.: Strain gradient effects on microscopic strain field in a metal matrix composite. International Journal of Plasticity 16, 563–591 (2000) · Zbl 1010.74005 · doi:10.1016/S0749-6419(99)00088-1 |
| [50] | Lissenden, C.J., Arnold, S.M.: Effect of microstructural architecture on flow/damage surfaces for metal matrix composites. Damage Mechanics in Engineering Materials 46, 385–400 (1998) · doi:10.1016/S0922-5382(98)80054-8 |
| [51] | Yan, Y.W., Geng, L., Li, A.B.: Experimental and numerical studies of the effect of particle size on the deformation behavior of the metal matrix composites. Materials Science and Engineering 448, 315–325 (2007) · doi:10.1016/j.msea.2006.10.158 |
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