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Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. (English) Zbl 1058.74070
Summary: A gradient-enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general nonlinear framework. In this approach, the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non-uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum, and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher-order stress tensor are derived based on an extension of Hill-Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher-order macrocontinuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects.

MSC:
74Q15 Effective constitutive equations in solid mechanics
74Q05 Homogenization in equilibrium problems of solid mechanics
74M25 Micromechanics of solids
74E05 Inhomogeneity in solid mechanics
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[1] Eshelby, Proceedings of the Royal Society of London Series A 241 pp 376– (1957) · Zbl 0079.39606
[2] Willis, Advances in Applied Mechanics 21 pp 1– (1981) · Zbl 0476.73053
[3] Ponte Casta├▒eda, Advances in Applied Mechanics 34 pp 171– (1998)
[4] Asymptotic Analysis for Periodic Structures. North-Holland: Amsterdam, 1978.
[5] Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer: Berlin, 1980. · Zbl 0432.70002
[6] Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier: Amsterdam, 1993. · Zbl 0924.73006
[7] (eds). Fundamentals of Metal-Matrix Composites. Butterworth-Heinemann: Boston, 1993.
[8] Huet, Journal of the Mechanics and Physics of Solids 38 pp 813– (1990)
[9] Local and global aspects in the mathematical theory of plasticity. In Plasticity Today: Modelling, Methods and Applications, (eds). Elsevier Applied Science Publishers: London, 1985; 279-310.
[10] Guedes, Computer Methods in Applied Mechanics and Engineering 83 pp 143– (1990) · Zbl 0737.73008
[11] Nonlinear homogenization method for practical applications. In Computational Methods in Micromechanics. AMD-vol. 212/MD-vol. 62 (eds). ASME: New York, 1995; 1-16.
[12] Ghosh, International Journal of Solids and Structures 32 pp 27– (1995) · Zbl 0865.73060
[13] Ghosh, Computer Methods in Applied Mechanics and Engineering 132 pp 63– (1996) · Zbl 0892.73061
[14] Smit, Computer Methods in Applied Mechanics and Engineering 155 pp 181– (1998) · Zbl 0967.74069
[15] Miehe, Computer Methods in Applied Mechanics and Engineering 171 pp 387– (1999) · Zbl 0982.74068
[16] Michel, Computer Methods in Applied Mechanics and Engineering 172 pp 109– (1999) · Zbl 0964.74054
[17] Feyel, Computer Methods in Applied Mechanics and Engineering 183 pp 309– (2000) · Zbl 0993.74062
[18] Terada, Computer Methods in Applied Mechanics and Engineering 190 pp 5427– (2001) · Zbl 1001.74095
[19] Kouznetsova, Computational Mechanics 27 pp 37– (2001) · Zbl 1005.74018
[20] Moulinec, Computer Methods in Applied Mechanics and Engineering 157 pp 69– (1998) · Zbl 0954.74079
[21] Ghosh, International Journal of Solids and Structures 38 pp 2335– (2001) · Zbl 1015.74058
[22] Smyshlyaev, Journal of the Mechanics and Physics of Solids 42 pp 1851– (1994) · Zbl 0819.73046
[23] Smyshlyaev, Proceedings of the Royal Society of London Series A 451 pp 795– (1995) · Zbl 0872.73035
[24] Smyshlyaev, Journal of the Mechanics and Physics of Solids 44 pp 465– (1996) · Zbl 1054.74553
[25] Drugan, Journal of the Mechanics and Physics of Solids 44 pp 497– (1996) · Zbl 1054.74704
[26] Boutin, International Journal of Solids and Structures 33 pp 1023– (1996) · Zbl 0920.73282
[27] Triantafyllidis, Journal of the Mechanics and Physics of Solids 44 pp 1891– (1996) · Zbl 1054.74585
[28] Smyshlyaev, Journal of the Mechanics and Physics of Solids 48 pp 1325– (2000) · Zbl 0984.74065
[29] Peerlings, Journal de Physique IV 11 pp 153– (2001)
[30] Zhu, Acta Mechanica 121 pp 165– (1997) · Zbl 0885.73044
[31] Forest, Mechanics Research Communications 25 pp 449– (1998) · Zbl 0949.74054
[32] van der Sluis, International Journal of Solids and Structures 36 pp 3193– (1999) · Zbl 0976.74055
[33] Ostoja-Starzewski, Mechanics Research Communications 26 pp 387– (1999) · Zbl 0954.74049
[34] Bouyge, International Journal of Solids and Structures 38 pp 1721– (2001) · Zbl 1003.74006
[35] Homogenization for materials with microstructure. In Recent Advances in Solids/Structures and Application of Metallic Materials. vol. PVP-vol. 369 ASME: New York, 1997; 33-38.
[36] Yuan, Mechanics Research Communications 28 pp 265– (2001) · Zbl 0978.74532
[37] Forest, Modelling and Simulation in Materials Science and Engineering 7 pp 829– (1999)
[38] Forest, International Journal of Solids and Structures 38 pp 4585– (2001) · Zbl 1033.74038
[39] Forest, International Journal of Solids and Structures 37 pp 7105– (2000) · Zbl 0998.74019
[40] Toupin, Archive for Rational Mechanics and Analysis 11 pp 385– (1962) · Zbl 0112.16805
[41] Koiter, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen Series B 67 pp 17– (1964)
[42] Mindlin, International Journal of Solids and Structures 4 pp 109– (1968) · Zbl 0166.20601
[43] Fleck, Advances in Applied Mechanics 33 pp 295– (1997)
[44] Shu, International Journal for Numerical Methods in Engineering 44 pp 373– (1999) · Zbl 0943.74072
[45] Terada, International Journal of Solids and Structures 37 pp 2285– (2000) · Zbl 0991.74056
[46] van der Sluis, Mechanics of Materials 32 pp 449– (2000)
[47] Hill, Journal of the Mechanics and Physics of Solids 11 pp 357– (1963) · Zbl 0114.15804
[48] Bodner, Journal of Applied Mechanics 42 pp 385– (1975)
[49] Leonov, Rheologica Acta 15 pp 85– (1976) · Zbl 0351.73001
[50] Baaijens, Rheologica Acta 30 pp 284– (1991)
[51] van der Aa, Mechanics of Materials 32 pp 423– (2000)
[52] Mindlin, Archive for Rational Mechanics and Analysis 11 pp 415– (1962) · Zbl 0112.38906
[53] Fleck, Acta Metallurgica et Materialia 42 pp 475– (1994)
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