Circular and annular two-phase plates of minimal compliance

• Autorzy: Kolanek, K. , Lewiński, T.
• Konferencja: Optimal design of materials and structures OPTY-2001 (August 27-29, 2001 ; Poznań ; Polska)
• Języki publikacji: EN

Abstrakty

EN

The paper deals with optimal design of thin plates. The plate thickness assumes two possible values: h1 and h2 and the plate volume is given. The problem of minimizing the plate compliance needs relaxation. The relaxed formulation was found by Gibiansky and Cherkaev in 1984. In the present paper a finite element approximation of this problem is presented in the framework of rotationally symmetric bending of circular and annular plates. The problem is composed of a nonlinear equilibrium problem coupled with a minimum compliance problem. The aim of the present paper is to analyze the forms of the optimal solutions, in particular, to look into the underlying microstructures. It is proved that in some solutions a ribbed microstructure occurs with ribs non-coinciding with both the radial and circumferential directions. Due to non-uniqueness of the sign of an angle of inclination of ribs the appearance of this microstructure does not contrasts with the radial symmetry of the problem. In the degenerated problem when the smallest thickness h1 vanishes the above interpretation of the inclined ribbed microstructure becomes incorrect; in these regions one can assume that the plate is solid but with a varying thickness. The degenerated case of h1=0 was considered in the papers by Rozvany et al. and Ong et al. but there such a microstructure was not taken into account. One of the aims of the paper is to re-examine these classical and frequently cited results.

Słowa kluczowe

PL

płyta kołowa; podatność minimalna

EN

circular plate; compliance minimal

• Czasopismo: Computer Assisted Mechanics and Engineering Sciences
• Rocznik: 2003
• Tom: Vol. 10, No. 2
• Strony: 177-199
• Opis fizyczny: Bibliogr. 29 poz., rys.,wykr.

Twórcy

autor

Kolanek, K.

• Polish Academy of Sciences, Institute of Fundamental Technological Research, Świętokrzyska 21, 00-049 Warsaw, Poland

autor

Lewiński, T.

• Warsaw University if Technology, Faculty of Civil Engineering, Institute of Structural Mechanics, Al. Armii Ludowej 16, 00-637 Warsaw, Poland

Bibliografia

• [1] G. Allaire, R.V. Kohn. Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions. Q. Appl. Math., 51: 675-699, 1993.
• [2] G. Allaire, R.V. Kohn. Optimal design for minimum weight and compliance in plane stress using extremal microstructures, Eur. J. Mech., A / Solids, 12: 839-878, 1993.
• [3] F. Belblidia, S.M. Afonso, E. Hinton, G.C.R. Antonino. Integrated design optimization of stiffened plate structures. Eng. Computations, 16: 934-951, 1999.
• [4] M.P. Bendsøe. Optimization of Structural Topology, Shape and Material. Springer, Berlin, 1995.
• [5] M.P. Bendsøe, O. Sigmund. Material interpolation schemes in topology optimization. Arch. Appl. Mech., 69: 635-654, 1999.
• [6] G. Cheng. On non-smoothness in optimal design of solid elastic plates. Int. J. Solids. Struct., 17: 795-810, 1981.
• [7] G. Cheng, N. Olhoff. An investigation concerning optimal design of solid elastic plates. Int . J. Solids. Struct., 16: 305-323, 1981.
• [8] G. Cheng, N. Olhoff. Regularized formulation for optimal design of axisymmetric plates. Int. J. Solids. Struct., 18: 153-169, 1982.
• [9] A. Cherkaev, Variational Methods for Structural Optimization. Springer, New York, 2000.
• [10] S. Czarnecki, T. Lewiński. Optimal layouts of a two-phase isotropic material in thin elastic plates. In: Z. Waszczyszyn, J. Pamin, eds., Proc. 2nd European Conference on Computational Mechanics, ECCM-2001, Kraków, 26-29 June 2001. CD-ROM, 2001.
• [11] A. Diaz, R. Lipton, C.A. Soto. A new formulation of the problem of optimum reinforcement of Reissner-Mindlin plates. Comp. Meth. Appl. Mech. Engrg., 123: 121-139. 1995.
• [12] H.A. Eschenauer, V.V. Kobelev, A. Schumacher. Bubble method for topology and shape optimization of structures, Struct. Optimiz., 8: 42-51, 1994.
• [13] L.V. Gibiansky, A.V. Cherkaev. Designing composite plates of extremal rigidity (in Russian). Fiziko¬Tekhnicheskiy Inst. im. A.F. Ioffe, AN SSSR, preprint No. 914. Leningrad 1984. English translation in: A.V. Cherkaev, R.V. Kohn, eds., Topics in the Mathematical Modelling of Composite Materials. Birkhauser, Boston 1997.
• [14] M. Kleiber, C. Woźniak. Nonlinear Mechanics of Structures. Polish Scientific Publishers, Warsaw, 1991.
• [15] M. Kleiber. Lectures on computer methods in the non-linear thermo-mechanics of deformable bodies (in Polish). Available from http://www.ippt.gov.pl, 2001.
• [16] R.V. Kohn, G. Strang. Optimal design and relaxation of variational problems. Comm. Pure Appl. Math., 39: 113-137, 139-183, 353-379, 1986.
• [17] K. Kolanek, T. Lewiński, Thin circular plates of minimal compliance. In: W. Szcześniak, ed., Theoretical Foundations of Civil Engineering, VII, 316-325. Oficyna Wydawnicza PW, Warszawa, 1999.
• [18] W. Kozłowski, Z. Mróz. Optimal design of solid plates. Int. J. Solids. Struct., 5: 781-794, 1969.
• [19] T. Lewiński, J.J. Telega. Plates, Laminates and Shells. Asymptotic Analysis and Homogenization. World Scientific. Series on Advances in Mathematics for Applied Sciences, vol. 52, Singapore, New Jersey, London, Hong Kong, 2000.
• [20] T. Lewiński, J.J. Telega. Michell-like grillages and structures with locking, Arch. Mech., 53: 303-331, 2001.
• [21] Q.Q. Liang, Y.M. Xie, G.P. Steven. A performance index for topology and shape optimization of plate bending problems with displacement constraints. Struct. Multidisc. Optimiz., 21: 393-399, 2001.
• [22] R. Lipton. On a saddle-point theorem with application to structural optimization. J. Optim. Theory. Appl., 81: 549-568, 1994.
• [23] K.A. Lurie, A.V. Cherkaev. Effective characteristics of composite materials and optimum design of structural members (in Russian). Adv. Mech. (Uspekhi Mekhaniki), 9: 3-81, 1986.
• [24] N. Olhoff, K.A. Lurie, A.V. Cherkaev, A.V. Fedorov. Sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates. Int. J. Solids. Struct., 17: 931-948, 1981.
• [25] T.G. Ong, G.I.N. Rozvany, W.T. Szeto. Least-weight design of perforated plates for given compliance: non-zero Poisson's ratio. Comp. Meth. Appl. Mech. Eng., 66: 301-322, 1988.
• [26] G.I.N. Rozvany, N. Olhoff, M.P. Bendsøe, T.G. Ong, R. Sandler, W.T. Szeto. Least-weight design of perforated elastic plates. I,II. Int. J. Solids. Struct., 23: 521-536, 537-550, 1987.
• [27] G.I.N. Rozvany, N. Olhoff, K.-T. Cheng, J.E. Taylor. On the solid plate paradox in structural optimization. J. Struct. Mech., 10: 1-32, 1982.
• [28] L. Tartar. An introduction to the homogenization method in optimal design. In: B. Kawohl, O. Pironneau, L. Tartar, J.-P. Zolesio, eds., Optimal Shape Design, 47-156, Springer, Berlin, 2000.
• [29] J.J. Telega, T. Lewiński. On a saddle-point theorem in minimum compliance design. J. Optimiz. Th. Appl., 106: 441-450, 2000.