Variational Foundations of Large Strain Multiscale Solid Constitutive Models: Kinematical Formulation

Advanced Computational Materials Modeling: From Classical to Multiâ-Scale Techniques

Volumn , Issue , 2010, Pages 341-378

  De Souza Neto, Eduardo A ^a   Feijóo, Raúl A ^b  

^a SWANSEA UNIVERSITY   (United Kingdom)

^b LABORATÓRIO NACIONAL DE COMPUTAÇÃO CIENTÍFICA   (Brazil)

Author keywords

Dissipative solids; Hill Mandel principle; Large strains; Multiscale modeling; Representative volume element; Variational formulation

Indexed keywords

EID: 84887203435     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9783527632312.ch9     Document Type: Chapter

References (51)

1. Hori, M. and Nemat-Nasser, S. (1999)On two micromechanics theories for determining micro-macro relations in heterogeneous solids. Mechanics of Materials,31, 667-682.
2. Michel, J.C., Moulinec, H., and Suquet, P. (1999)Effective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechnics and Engineering,172, 109-143.
3. Nemat-Nasser, S. (1999)Averaging theorems in finite deformation plasticity. Mechanics of Materials,31, 493-523.
4. Ibrahimbegović, A. and Marković, D. (2003)Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures. Computer Methods in Applied Mechnics and Engineering,192, 3089-3107.
5. Miehe, C., Shotte, J., and Lambrecht, M. (2002) Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals. Journal of the Mechanics and Physics of Solids, 50(10), 2123-2167.
6. Miehe, C. (2003)Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy. Computer Methids in Applied Mechanics and Engineering,192, 559-591.
7. Miehe, C. (2002)Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. International Journal of Numerical Methods in Engineering,55, 1285-1322.
8. Miehe, C. and Koch, A. (2002)Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Archive of Applied Mechanics,72, 300-317.
9. Clayton, J.D. and McDowell, D.L. (2003)A multiscale multiplicative decomposition for elastoplasticity of polycrystals. International Journal of Plasticity,19, 1401-1444.
10. Reese, S. (2003)Meso-macro modelling of fibre-reinforced rubber-like composites exhibiting large elastoplastic deformations. International Journal of Solids and Structures,40, 951-980.
11. Terada, K. and Kikuchi, N. (2001)A class of general algorithms for multi-scale analysis of heterogeneous media. Computer Methods in Applied Mechanics and Engineering,190, 5427-5464.
12. Terada, K., Saiki, I., Matsui, K., and Yamakawa, Y. (2003)Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strains. Computer Methods in Applied Mechanics and Engineering,192, 3531-3563.
13. Kouznetsova, V.G., Geers, M.G.D., and Brekelmans, W.A.M. (2004)Multi-scale second order computational homogenization of multi-phase materials: a nested finite element solution strategy. Computer Methods in Applied Mechnics and Engineering,193, 5525-5550.
14. Matsui, K., Terada, K., and Yuge, K. (2004)Two-scale finite element analysis of heterogeneous solids with periodic microstructure. Computers and Structures,82, 593-606.
15. Michel, J.C. and Suquet, P. (2004)Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis. Computer Methods in Applied Mechnics and Engineering,193, 5477-5502.
16. Yang, Q.-S. and Becker, W. (2004)Effective stiffness and microscopic deformation of an orthotropic plate containing arbitrary holes. Computers and Structures,82, 2301-2307.
17. Bilger, N., Auslender, F., Bornert, M., Michel, J.-C., Moulinec, H., Suquet, P., and Zaoui, A. (2005)Effect of a nonuniform distribution of voids on the plastic response of voided materials: a computational and statistical analysis. International Journal of Solids and Structures,42, 517-538.
18. Goktepe, S. and Miehe, C. (2005)A micro-macro approach to rubber-like materials. Part III: the micro-sphere model of anisotropic mullins-type damage. Journal of the Mechanics and Physics of Solids,53, 2259-2283.
19. Ibrahimbegović, A., Grešovnik, I., Markovič, D., Melnyk, S., and Rodič, T. (2005) Shape optimization of two-phase inelastic material with microstructure. Engineering Computations, 22(5-6), 605-645.
20. Li, X. and Wienan, E. (2005)Multiscale modeling of the dynamics of solids at finite temperature. Journal of the Mechanics and Physics of Solids,53, 1650-1685.
21. Okumura, D., Higashi, Y., Sumida, K., and Ohno, N. (2007) A homogenization theory of strain gradient single crystal plasticity and its finite element discretization. International Journal of Plasticity, 23(7), 1148-1166.
22. Nakamachi, E., Tam, N.N., and Morimoto, H. (2007) Multi-scale finite element analyses of sheet metals by using SEM-EBSD measured crystallographic RVE models. International Journal of Plasticity, 23(3), 450-489.
23. Speirs, D.C.D., de Souza Neto, E.A., and Perić, D. (2008)An approach to the mechanical constitutive modelling of arterial tissue based on homogenization and optimization. Journal of Biomechanics,41, 2673-2680.
24. Giusti, S.M., Novotny, A.A., de Souza Neto, E.A., and Feijóo, R.A. (2008)Sensitivity of the macroscopic elasticity tensor to topological micrsotructural changes. Journal of the Mechanics and Physics of Solids,57, 555-570.
25. Giusti, S.M., Novotny, A.A., and de Souza Neto, E.A. (2010) Sensitivity of the macroscopic response of elastic microstructures to the insertion of holes. Proceedings of the Royal Society A, 466(2118), 1703-1723.
26. Hill, R. (1965) A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13(4), 213-222.
27. Mandel, J. (1971) Plasticit ́e Classique et Viscoplasticit ́e, CISM Lecture Notes, Udine, Italy, Springer-Verlag.
28. Gurson, A.L. (1977)Continuum theory of ductile rupture by void nucleation and growth - Part I: yield criteria and flow rule for porous media. Journal of Engineering Materials and Technology,99, 2-15.
29. Bensoussan, A., Lions, J.L., and Papanicolau, G. (1978) Asymptotic Analysis of Periodic Structures, North-Holland.
30. Germain, P., Nguyen, Q.S., and Suquet, P. (1983)Continuum thermodynamics. ASME Journal of Applied Mechanics,50, 1010-1020.
31. Sanchez-Palencia, E. and Zaoui, A. (eds) (1987) Homogenization Techniques for Composite Media, Lecture Notes in Physics, vol. 272, Springer-Verlag, Berlin.
32. Holzapfel, G., Gasser, T.C., and Ogden, R.W. (2000)A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal Of Elasticity,61, 1-48.
33. Suquet, P.M. (1987) Elements of homogenization for inelastic solid mechanics, in Homogenization Techniques for Composite Media (eds E. Sanchez-Palencia and A. Zaoui), Lecture Notes in Physics, vol. 272, Springer-Verlag, Berlin, pp. 194-278.
34. Simo, J.C. and Hughes, T.J.R. (1998) Computational Inelasticity, Springer-Verlag, New York.
35. de Souza Neto, E.A., Perić, D., and Owen, D.R.J. (2008) Computational Methods for Plasticity: Theory and Application, Wiley, Chichester.
36. Miehe, C., Schotte, J., and Schr̈oder, J. (1999)Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Computational Materials Science,16, 372-382.
37. Truesdell, C. (1969) Rational Thermodynamics, McGraw-Hill, New York.
38. Lubliner, J. (1990) Plasticity Theory, Macmillan, New York.
39. Lemaitre, J. and Chaboche, J.L. (1990) Mechanics of Solid Materials, Cambridge University Press.
40. Sanchez-Palencia, E. (1980) Non-homogeneous Media and Vibration, Theory Lecture Notes in Physics, Springer, Berlin.
41. Gurtin, M.E. (1981) An Introduction to Continuum Mechanics, Academic Press.
42. de Souza Neto, E.A. and Feijóo, R.A. (2006) Variational foundations of multi-scale constitutive models of solid: small and large strain kinematical formulation, LNCC Research & Development Report, No. 16/2006, National Laboratory for Scientific Computing, Petr ́opolis, Brazil.
43. Oden, J.T. (1979) Applied Functional Analysis, Prentice-Hall, Englewood Cliffs, New Jersey.
44. Larsson, F. and Runesson, K. (2007)RVE computations with error control and adaptivity: the power of duality. Computational Mechanics,39, 647-661.
45. Taylor, G.I. (1938)Plastic strains in metals. Journal Institute of Metals,62, 307-324.
46. de Souza Neto, E.A. and Feijóo, R.A. (2008) On the equivalence between spatial and material volume averaging of stress in large strain multi-scale solid constitutive models. Mechanics of Materials, 40(10), 803-811.
47. Hughes, T.J.R. and Pister, K. (1978)Consistent linearization in mechanics of solids and structures. Computers and Structures,8, 391-397.
48. Simo, J.C. and Taylor, R.L. (1985)Consistent tangent operators for rate-independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering,48, 101-118.
49. Crisfield, M.A. (1997) Non-linear Finite Element Analysis of Solids and Structures, vol. 2, Advanced Topics, Wiley, Chichester.
50. Blanco, P.J. (2008) Kinematical incompatibility, domain immersion and constitutive modelling: Connection to the modelling of the cardiovascular system, PhD Thesis, LNCC, Petr ́opolis, Brazil (in portuguese).
51. Haug, E.J. Céa, J. (1981) Proceedings: Optimization of Distributed Parameters Structures, EUA, Iowa.