Mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading

International Journal for Numerical Methods in Engineering

Volumn 76, Issue 7, 2008, Pages 1044-1064

  Fish, Jacob ^a   Fan, Rong ^a  

^a RENSSELAER POLYTECHNIC INSTITUTE   (United States)

Author keywords

Large deformation; Macro; Mathematical homogenization; Micro; Multiscale; Nonperiodic; Unit cell

Indexed keywords

ASYMPTOTIC ANALYSIS; DEFORMATION;

LARGE DEFORMATION; MATHEMATICAL HOMOGENIZATION; MICRO; MULTISCALE; NONPERIODIC; UNIT CELL;

DIRECT NUMERICAL SIMULATION;

EID: 60949094606     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2355     Document Type: Article

References (40)

1. Babuska I. Homogenization and application. Mathematical and computational problems. In Numerical Solution of Partial Differential Equations - III, SYNSPADE, Hubbard B (ed.). Academic Press: New York, 1975.
2. Benssousan A, Lions JL, Papanicoulau G. Asymptotic Analysis for Periodic Structures. North-Holland: Amsterdam, 1978.
3. Sanchez-Palencia E. Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer: Berlin, 1980.
4. Bakhvalov NS, Panassenko GP. Homogenization: Averaging Processes in Periodic Media. Kluwer Academic Publishers: Dordrecht, 1989.
5. Hill R. A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society of London, Series A 1948; 193:281-297.
6. Private communications with M. G. D. Geers, 2007.
7. Hill R. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 1963; 11:357-372.
8. Terada K, Kikuchi N. Nonlinear homogenization method for practical applications. In Computational Methods in Micromechanics, vol. AMD-212/MD-62, Ghosh S, Ostoja-Starzewski M (eds). ASME: New York, 1995; 1-16.
9. Terada K, Kikuchi N. A class of general algorithms for multi-scale analysis of heterogeneous media. Computer Methods in Applied Mechanics and Engineering 2001; 190:5427-5464.
10. Matsui K, Terada K, Yuge K. Two-scale finite element analysis of heterogeneous solids with periodic microstructures. Computers and Structures 2004; 82(7-8):593-606.
11. Smit RJM, Brekelmans WAM, Meijer HEH. Prediction of the mechanical behavior of nonlinear heterogeneous systems by multilevel finite element modeling. Computer Methods in Applied Mechanics and Engineering 1998; 155:181-192.
12. Miehe C, Koch A. Computational micro-to-macro transition of discretized microstructures undergoing small strain. Archives of Applied Mechanics 2002; 72:300-317.
13. Kouznetsova V, Brekelmans W-A, Baaijens FP-T. An approach to micro-macro modeling of heterogeneous materials. Computational Mechanics 2001; 27:37-48.
14. Feyel F, Chaboche J-L. FE2 multiscale approach for modeling the elastoviscoplastic behavior of long fiber SiC/Ti composite materials. Computer Methods in Applied Mechanics and Engineering 2000; 183:309-330.
15. 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 1995; 32:27-62.
16. Ghosh S, Lee K, Moorthy S. Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and Voronoi cell finite element model. Computer Methods in Applied Mechanics and Engineering 1996; 132:63-116.
17. Michel J-C, Moulinec H, Suquet P. Effective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechanics and Engineering 1999; 172:109-143.
18. Geers MGD, Kouznetsova V, Brekelmans WAM. Gradient-enhanced computational homogenization for the micro-macro scale transition. Journal of Physics IV 2001; 11:145-152.
19. McVeigh C, Vernerey F, Liu WK, Brinson LC. Multiresolution analysis for material design. Computer Methods in Applied Mechanics and Engineering 2006; 195:5053-5076.
20. Miehe C. Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. International Journal for Numerical Methods in Engineering 2002; 55:1285-1322.
21. Zohdi TI, Oden JT, Rodin GJ. Hierarchical modeling of heterogeneous bodies. Computer Methods in Applied Mechanics and Engineering 1996; 138:273-298.
22. Feyel F, Chaboche JL. Fe2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Computer Methods in Applied Mechanics and Engineering 2000; 183:309-330.
23. Weinan E, Engquist B et al. Heterogeneous multiscale method: a general methodology for multiscale modeling. Physical Review B 2003; 67(9):1-4.
24. Hou TY, Wu XH. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of Computational Physics 1997; 134(1):169-189.
25. Loehnert S, Belytschko T. A multiscale projection method for macro/ microcrack simulations. International Journal for Numerical Methods in Engineering 2007; 71(12):1466-1482.
26. Belytschko T, Loehnert S, Song J. Multiscale aggregating discontinuities: a method for circumventing loss of material stability. International Journal for Numerical Methods in Engineering 2008; 73(6):869-894.
27. Fish J, Shek K, Pandheeradi M, Shephard MS. Computational plasticity for composite structures based on mathematical homogenization: theory and practice. Computer Methods in Applied Mechanics and Engineering 1997; 148:53-73.
28. Fish J, Shek KL. Finite deformation plasticity of composite structures: computational models and adaptive strategies. Computer Methods in Applied Mechanics and Engineering 1999; 172:145-174.
29. Fish J, Yu Q. Multiscale damage modeling for composite materials: theory and computational framework. International Journal for Numerical Methods in Engineering 2001; 52(1-2):161-192.
30. Clayton JD, Chung PW. An atomistic-to-continuum framework for nonlinear crystal mechanics based on asymptotic homogenization. Journal of Mechanics and Physics of Solids 2006; 54:1604-1639.
31. Fish J, Shek KL. Finite deformation plasticity of composite structures: computational models and adaptive strategies. Computer Methods in Applied Mechanics and Engineering 1999; 172:145-174.
32. Yuan Z, Fish J. Towards realization of computational homogenization in practice. International Journal for Numerical Methods in Engineering 2008; 73:361-380.
33. Mesarovic SD, Padbidri J. Minimal kinematic boundary conditions for simulations of disordered microstructures. Philosophical Magazine 2005; 85(1):65-78.
34. Belytschko T, Liu WK, Moran B. Nonlinear Finite Elements for Continua and Structures. Wiley: New York, 2000.
35. Miehe C. Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy. Computer Methods in Applied Mechanics and Engineering 2003; 192:559-591.
36. Belytschko T, Bindeman LP. Assumed strain stabilization of the eight node hexahedral element. Computer Methods in Applied Mechanics and Engineering 1993; 105(2):225-260.
37. Belytschko T, Lin JI, Chen-Shyh T. Explicit algorithms for the nonlinear dynamics of shells. Computer Methods in Applied Mechanics and Engineering 1984; 42(2):225-251.
38. Nemat-Nasser S, Guo WG. Thermomechanical response of DH-36 structural steel over a wide range of strain rates and temperatures. Mechanics of Materials 2003; 35:1023-1047.
39. Nemat-Nasser S. Experimental characterization of polyurea with constitutive modeling and simulations. Presentation at ERC ACTD Workshop, Massachusetts Institute of Technology, 2004.
40. Liechti KM, Wu J-D. Mixed-mode, time-dependent rubber/metal debonding. Journal of the Mechanics and Physics of Solids 2001; 49:1039-1072.