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A Kinematic Method for Shakedown and Limit Analysis of Periodic Heterogeneous Media

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Inelastic Behaviour of Structures under Variable Repeated Loads

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 432))

Abstract

In this Chapter the kinematic (second, Koiter’s) shakedown theorem is applied to the representative volume of periodic heterogeneous media with Huber-Mises local plastic behavior. The adopted formulation of shakedown analysis is based on periodicity boundary conditions, conventional finite element modeling and penalization enforcement of plastic incompressibility. A cost-effective iterative solution procedure is discussed and computationally tested. Numerical tests and engineering applications are presented with reference to perforated plates and metal-matrix unidirectional fiber-reinforced composites.

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References

  • Bonnet, M. Maier, G. and Polizzotto, C. (1998). Symmetric Galerkin boundary element method. Appl. Mech. Reviews ASME 51: 669–704.

    Article  Google Scholar 

  • Carvelli, V., Cen, Z., Liu, Y. and Maier, G. (1999a). Shakedown analysis of defective pressure vessels by a kinematic approach. Arch. Appl. Mech. 69: 751–764.

    Article  MATH  Google Scholar 

  • Carvelli, V., Maier, G. and Taliercio, A. (1999b). Shakedown analysis of periodic heterogeneous materials by a kinematic approach. J. Mech. Eng. (Strojnicky ÄŒasopis) 50: 229–240.

    Google Scholar 

  • Carvelli, V., Maier, G. and Taliercio, A. (2000). Kinematic limit analysis of periodic heterogeneous media. Comp. Mod. Eng. Sci. 1: 15–26.

    Google Scholar 

  • Cohn, M. Z., Maier, G. and Grierson, D. eds. (1979). Engineering plasticity by mathematical programming. Pergamon Press, New York.

    MATH  Google Scholar 

  • Comi, C., Maier, G., Perego, U. (1992). Generalized variable finite element modelling and extremum theorems in stepwise holonomic elastoplasticity with internal variables. Comput. Meth. Appl. Mech. Eng. 96: 133–171.

    Article  MathSciNet  Google Scholar 

  • Corigliano, A., Maier, G. and Pycko, S. (1995). Dynamic shakedown analysis and bounds for elastoplastic structures with nonassociative, internal variable constitutive laws. Int. J. Solids and Structures 32: 3145–3166.

    Article  MathSciNet  MATH  Google Scholar 

  • Corradi, L. and Maier, G. (1981). Finite element elastoplastic and limit analysis: some consistency criteria and their implications. In Wunderlich W., Stein E. and Bathe K.J., eds., Nonlinear Finite Element Analysis in Structural Mechanics. Springer Verlag, 290–306.

    Chapter  Google Scholar 

  • de Buhan, P. and Taliercio, A. (1988). Critère de résistance macroscopique pour les matériaux composites à fibres. C. R. Acad. Sci. Paris Série II 307: 227–232.

    Google Scholar 

  • Druyanov, B. and Roman, I. (1999). Conditions for shakedown of damaged elastic plastic bodies. European J. Mech. A/Solids 18: 641–651.

    Article  MATH  Google Scholar 

  • Dvorak, G. J., Lagoudas, D. C. and Huang, C. M. (1994). Fatigue damage and shakedown in metal matrix composite laminates. Mech. Composite Mat. Struct. 1: 171–202.

    Article  Google Scholar 

  • Fiacco, A, V. and McCormick, G. P. (1968). Nonlinear programming sequential unconstrained minimization techniques. Wiley Publ. Co., New York. Reprinted by SIAM Publications in 1990.

    MATH  Google Scholar 

  • Francescato, P. and Pastor, J. (1997). Lower and upper numerical bounds to the off-axis strength of unidirectional fiber-reinforced composites by limit analysis methods. European J. Mech. A/Solids 16: 213–234.

    Google Scholar 

  • Gao, D. Y., Ogden, R. W. and Stavroulakis, G.E. (2001). Nonsmooth-nonconvex mechanics. Kluwer Akademic Publ., Dordrecht.

    Book  MATH  Google Scholar 

  • Hachemi, A. and Weichert, D. (1997). Application of shakedown theory to damaging inelastic material under mechanical and thermal loads. Int. J. Mech. Sci. 39: 1067–1076.

    Article  MATH  Google Scholar 

  • Kaliszky, S. (1989). Plasticity: theory and engineering applications. Elsevier, Amsterdam, North Holland.

    MATH  Google Scholar 

  • Koiter, W. T. (1960). General theorems for elastic-plastic solids. In Sneddon, J. N. and Hill, R., eds., Progress in solid mechanics. North Holland, Amsterdam. 165–221.

    Google Scholar 

  • König, J. A. (1987). Shakedown of elastic plastic structures. Elsevier, Amsterdam.

    Google Scholar 

  • König, J. A. and Kleiber, M. (1978). On a new method of shakedown analysis. Bulletin de l’Academie Polonaise des Sciences–Sér. des Scien. Techn. 26: 165–171.

    MATH  Google Scholar 

  • Liu, Y. H., Cen, Z. Z. and Xu, B. Y. (1995). A numerical method for plastic limit analysis of 3-D structures. Int. J. Sol. Struct. 32: 1645–1658.

    Article  MATH  Google Scholar 

  • Lubliner, J. (1990). Plasticity theory. Mc Milian Publ., New York.

    MATH  Google Scholar 

  • Maier, G., Carvelli, V., Cocchetti, G. (2000). On direct methods for shakedown and limit analysis, Eur. J. Mech. A/Solids. 19: S79–S100.

    Google Scholar 

  • Maier, G., Carvelli, V., Taliercio, A. (2001). Limit and shakedown analysis of periodic heterogeneous media. In Lemaitre, J, ed., Handbook of Material Behaviour Models, Academic Press, San Diego, 10.8: 1017–1028.

    Google Scholar 

  • Maier, G. and Nappi, A. (1984). On bounding post-shakedown quantities by the boundary element method. Engineering Analysis 1: 223–229.

    Article  Google Scholar 

  • Nagtegaal, J. C., Parks, D. M. and Rice, J. R. (1974). On numerically accurate finite element solutions in the fully-plastic range. Comp. Meth. Appl. Mech. Eng. 4: 153–177.

    Article  MathSciNet  MATH  Google Scholar 

  • Polizzotto, C., Borino, G. and Fuschi, P. (1996) An extended shakedown theory for elastic-plasticdamage material models. European J. Mech.–A/Solids 15: 825–858.

    MathSciNet  MATH  Google Scholar 

  • Ponter, A. R. S. and Leckie, F. A. (1998a). Bounding properties of metal-matrix composites subjected to cyclic loading. J. Mech. Phys. Solids 46: 697–717.

    Article  MathSciNet  MATH  Google Scholar 

  • Ponter, A. R. S. and Leckie, F. A. (1998b). On the behaviour of metal matrix composites subjected to cyclic thermal loading. J. Mech. Phys. Solids 46: 2183–2199.

    Article  MathSciNet  MATH  Google Scholar 

  • Suquet, P. (1982). Analyse limite et homogénéisation. C. R. Acad. Sci. Paris Série II 296: 1355–1358.

    MathSciNet  Google Scholar 

  • Taliercio, A. (1992). Lower and upper bounds to the macroscopic strength domain of a fiber-reinforced composite material. Int. J. Plasticity 8: 741–762.

    Article  MATH  Google Scholar 

  • Tarn, J. Q., Dvorak, G. J. and Rao, M. S. M. (1975). Shakedown of unidirectional composites. Int. J. Sol. Struct. 11: 751–764.

    Article  MATH  Google Scholar 

  • Weichert, D., Hachemi, A. and Schwabe, F. (1999a). Application of shakedown analysis to the plastic design of composites. Arch. Appl. Mech. 69: 623–633.

    Article  MATH  Google Scholar 

  • Weichert, D., Hachemi, A. and Schwabe, F. (1999b). Shakedown analysis of composites. Mech. Res. Comm. 26: 309–318.

    Article  MATH  Google Scholar 

  • Zhang, Y.G., Zhang, P. and Lu, M. W. (1993). Computational limit analysis of rigid-plastic bodies in plane strain. Acta Mech. Sol. Sinica 6: 341–348.

    Google Scholar 

  • Zhang, Y.G. (1995). An iterative algorithm for kinematic shakedown analysis. Comp. Meth. Appl. Mech. Eng. 127: 217–226.

    Article  MATH  Google Scholar 

  • Zhang, Y.G. and Lu, M. W. (1995). An algorithm for plastic limit analysis. Comp. Meth. Appl. Mech. Eng. 126: 333–341.

    Article  Google Scholar 

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Authors and Affiliations

  1. Technical University (Politecnico) of Milan, Milan, Italy

    Giulio Maier & Valter Carvelli

Authors
  1. Giulio Maier
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  2. Valter Carvelli
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Editors and Affiliations

  1. Institut für Allgemeine Mechanik, Rheinisch-Westfälische Technische Hochschule Aachen, Templergraben, 64, 52056, Aachen, Germany

    Dieter Weichert

  2. Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Giulio Maier

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Maier, G., Carvelli, V. (2002). A Kinematic Method for Shakedown and Limit Analysis of Periodic Heterogeneous Media. In: Weichert, D., Maier, G. (eds) Inelastic Behaviour of Structures under Variable Repeated Loads. International Centre for Mechanical Sciences, vol 432. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2558-8_7

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  • DOI: https://doi.org/10.1007/978-3-7091-2558-8_7

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-83687-3

  • Online ISBN: 978-3-7091-2558-8

  • eBook Packages: Springer Book Archive

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