Detecting stress fields in an optimal structure Part I: Two-dimensional case and analyzer

Structural and Multidisciplinary Optimization

Volumn 26, Issue 1-2, 2004, Pages 1-15

  Cherkaev, A ^a   Kucuk, I ^b  

^a UNIVERSITY OF UTAH   (United States)

^b AMERICAN UNIVERSITY OF SHARJAH   (United Arab Emirates)

Author keywords

Analyzer; Composite; Invariant norm; Necessary and sufficient conditions; Nonconvex; Stress fields; Suboptimal projects

Indexed keywords

APPROXIMATION THEORY; COMPOSITE MATERIALS; DIFFERENTIATION (CALCULUS); EIGENVALUES AND EIGENFUNCTIONS; ELASTIC MODULI; MICROSTRUCTURE; OSCILLATORS (MECHANICAL); PROBLEM SOLVING; STATISTICAL METHODS; STRESS ANALYSIS; TENSORS; TOPOLOGY; VECTORS;

INVARIANT NORM; NECESSARY AND SUFFICIENT CONDITIONS; NONCONVEX; STRESS FIELDS; SUBOPTIMAL PROJECTS;

STRUCTURAL OPTIMIZATION;

EID: 1142288317     PISSN: 1615147X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00158-003-0327-2     Document Type: Article

References (26)

1. Allaire, G.; Kohn, R.V. 1994: Optimal lower bounds on the elastic energy of a composite made from two non well-ordered isotropic materials. Q. Appl. Math. LII(2), 311-333
2. Atkin, R.J.; Fox, N. 1980: An Introduction to the Theory of Elasticity. New York: Longman Mathematical Texts
3. Bendsøe, M.P.; Díaz, A.R.; Kikuchi, N. 1993: Topology and generalized layout optimization of elastic structures. In: Topology design of structures (Sesimbra, 1992), pp. 159-205. Dordrecht: Kluwer Academic Publishers Group
4. Bendsoøe, M.P. 1995: Optimization of Structural Topology, Shape, and Material. New York, Berlin, Heidelberg: Springer-Verlag
5. Cherkaev, A. V. 2000: Variational Methods for Structural Optimization, Vol. 140. New York, Berlin, Heidelberg: Springer-Verlag
6. Cherkaev, A.V.; Grabovsky, Y.; Movchan, A.B.; Serkov, S.K. 1998a: The cavity of the optimal shape under shear stresses. Int. J. Solids Struct. 35(33), 4391-4410
7. Cherkaev, A.V.; Kohn, R.V. (eds.) 1997: Topics in the mathematical modelling of composite materials. Cambridge, MA: Birkhäuser Boston
8. Cherkaev, A.V.; Krog, L.A.; Kucuk, I. 1998b: Staple optimal design of two-dimensional elastic structures. Control Cybern. 27(2), 265-282
9. Cherkaev, A.V.; Lurie, K.A.; Milton, G.W. 1992: Invariant properties of the stress in plane elasticity and equivalence classes of composites. Proc. R. Soc. London, Ser. A 4381904, 438: 519-529
10. Eschenauer, H.A.; Olhoff, N. 2001: Topology optimization of continuum structures: A review. Appl. Mech. Rev. 54(4), 331-389
11. Francfort, G.; Murat, F. 1986: Homogenization and optimal bounds in linear elasticity. Arch. Rat. Mech. Anal. 94, 307-334
12. Fung, Y.C. 1965: Foundation of Solid Mechanics. Upper Saddle River: Prentice-Hall
13. Gibiansky, L.V.; Cherkaev, A.V. 1984: Design of composite plates of extremal rigidity. Report 914, Ioffe Physico-technical Institute, Acad. of Sci., USSR, Leningrad, USSR. English translation in Gibiansky and Cherkaev (1997)
14. Gibiansky, L.V.; Cherkaev, A.V. 1997: Design of composite plates of extremal rigidity. In: Cherkaev, A.V.; Kohn, R.V. (eds.) Topics in the Mathematical Modelling of Composite Materials, Vol. 31 of Progress in Nonlinear Differential Equations and Their Applications, pp. 95-137. Boston, MA: Birkhäuser Boston Translation. The original publication in Gibiansky and Cherkaev (1984)
15. Grabovsky, Y. 1996a: Bounds and extremal microstructures for two-component composites: A unified treatment based on the translation method. Proc. R. Soc. London, Ser. A 4521947, 452: 945-952
16. Grabovsky, Y. 1996b: Explicit solution of an optimal design problem with non-affine displacement boundary conditions. Proc. R. Soc. London, Ser. A 452, 909-918
17. Grabovsky, Y.; Kohn, R.V. 1995: Anisotropy of the Vigdergauz microstructure. J. Appl. Mech. 62(4), 1063-1065
18. Hashin, Z.; Shtrikman, S. 1962: A variational approach to the theory of the elastic behavior of polycrystals. J. Mech. Phys. Solids, 10, 343-352
19. Küçük, I. 2001: Variational Approach to Optimization of Elastic Structures. PhD thesis, University of Utah
20. Lurie, K.A. 1975: Optimal Control in Problems of Mathematical Physics. Moscow: Nauka (in Russian)
21. Lurie, K.A.; Cherkaev, A.V. 1978: Nonhomogeneous bar of extremal torsional rigidity. In: Nonlinear Problems in Structural Mechanics. Structural Optimization, pp. 64-68. Kiev: Naukova Dumka
22. Milton, G.W. 1981: Bounds on the complex permittivity of a two-component composite material. J. Appl. Phys. 52, 5286-5293
23. Milton, G.W. 1986: Modelling the properties of composites by laminates. In: Ericksen, J. et al. (eds.) Homogenization and Effective Moduli of Materials and Media (Minneapolis, Minn., 1984/1985), pp. 150-175, New York, Berlin, Heidelberg: Springer-Verlag
24. Milton, G.W. 2001: The Theory of Composites. 6. Cambridge Monographs on Applied and Computational Mathematics. Cambridge: Cambridge University Press
25. Timoshenko, S.P. 1970: Theory of Elasticity. New York: McGraw-Hill
26. Vigdergauz, S.B. 1989: Regular structures with extremal elastic properties. MTT, 24(3), 57-63