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Duality in Nonconvex Finite Deformation Theory: A Survey and Unified Approach

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From Convexity to Nonconvexity

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 55))

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Abstract

This paper presents a brief survey and a unified approach to the complementary-dual variational extremum principles in nonconvex finite deformation theory. By using an abstract deformation operator and its tangent decomposition, the well-known classical complementary energy principles can be written in a general form, and their extremum properties are clarified. Based on a new pure complementary energy principle (involving the stress only), it is shown that the nonlinear equilibrium equation in finite deformation problems can be transformed into a tensor equation. Hence a general analytic solution for 3-dimensional large deformation problems can be constructed. The properties of this general solution are characterized by a triality extremum principle.

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References

  1. Arthurs, AM (1980), Complementary Variational Principles, Second edi-tion, Clarendon Press.

    MATH  Google Scholar 

  2. Atluri, SN (1984), Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analysis of finitely deformed solids, with application to plates and shells -I, Theory, Computers & Structures, 18, no. 1, pp. 93–116.

    Article  MathSciNet  MATH  Google Scholar 

  3. Auchmuty, G (1983), Duality for non-convex variational principles, J. Diff. Equations, vol. 50, pp 80–145

    Article  MathSciNet  MATH  Google Scholar 

  4. Auchmuty, G (1989), Duality algorithms for nonconvex variational principles, Numer. Funct Anal. and Optim, 10, pp. 211–264.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bufler, H (1983), On the work theorems for finite and incremental elastic deformations with discontinuous fields: a unified treatment of different versions, Comput. Meth. Appl. Mech. Eng, 36, 95–124.

    Article  MathSciNet  MATH  Google Scholar 

  6. Curnier, A and Rakotomanana, L (1991), Generalized strain and stress measures: Critical survey and new results, Engineering Transactions, 39, no. 3–4, pp. 461–538.

    MathSciNet  Google Scholar 

  7. Dem’yanoV, VF, Stavroulakis, GE, Polyakova, LN and Panagiotopoulos, PD (1996), Quasidifferentiability and Nonsnгooth Modelling in Mechan-ics, Engineering and Economics, Kluwer Academic Publishers, Dordrecht/Boston/London, 348pp.

    Google Scholar 

  8. Dür, M and Horst, R (1997), Lagrange duality and partitioning techniques in nonconvex global optimization, J. Optim. Theory Appl, 95, no. 2, pp. 347–369.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ekeland, I and Temam, R (1976), Convex Analysis and Variational Problems, North-Holland, 1976

    MATH  Google Scholar 

  10. Felippa, C (1987), Will the force method come back? Trans. ASME, J. Appl. Mech 54, pp. 726–728.

    Article  MATH  Google Scholar 

  11. Fichera, G (1964), Problemi elastostatici con vincoli unilaterali; il problema di Signorini con ambigue condizioni al controno, Mem. Accad. Naz. Lincei,8, no. 7, pp. 91–140.

    MathSciNet  Google Scholar 

  12. Gallagher, RH (1993), Finite element structural analysis and complementary energy, Finite Element in Analysis and Design,13, pp. 115–126.

    Article  MATH  Google Scholar 

  13. Gao, DY (1988), Panpenalty finite element programming for limit analysis, Computers & Structures,28, pp. 749–755.

    Article  MATH  Google Scholar 

  14. Gao, DY (1992), Global extremum criteria for finite elasticity, ZAMP,43, 924–937.

    Article  MATH  Google Scholar 

  15. Gao, DY (1994), Stability and extremum principles for post yield analysis of finite plasticity, Acta Mech Sinica, 10, 311–325.

    Article  MATH  Google Scholar 

  16. Gao, DY, (1995), Duality theory in nonlinear buckling analysis for von Karman equations, Studies in Applied Mathematics,94, pp. 423–444.

    MathSciNet  MATH  Google Scholar 

  17. Gao, DY (1996a), Nonlinear elastic beam theory with application in contact problems and variational approaches, Mech. Research Commun, 23, no. 1, pp. 11–17.

    Article  MATH  Google Scholar 

  18. Gao, DY (1996b), Complementary finite-element method for finite deformation nonsmooth mechanics, J. Eng. Math, 30, pp. 339–353.

    Article  MATH  Google Scholar 

  19. Gao, DY (1997), Dual extremum principles in finite deformation theory with applications in post-buckling analysis of extended nonlinear beam model, Appl. Mech. Reviews, ASME, 50, no. 11, part 2, November, 564–571.

    Google Scholar 

  20. Gao, DY (1998), Complementary extremum principles in nonconvex parametric variational problems with applications, IMA J. of Applied Math, 61, 199–235

    Article  MATH  Google Scholar 

  21. Gao, DY (1999), Pure complementary energy principle and triality theory in finite elasticity, Mech. Research Commun, 26, 31–37

    Article  MATH  Google Scholar 

  22. Gao, DY (1999a), General analytic solutions and complementary variational principle for large deformation nonsmooth mechanics, Meccanica, 34,169–198

    MathSciNet  MATH  Google Scholar 

  23. Gao, DY (1999b), Duality Principles in Nonconvex systems, Theory, Methods and Applications, Kluwer Academic Publishers, 454 pp.

    Google Scholar 

  24. Gao, DY and Strang, G (1989а), Geometric nonlinearity: Potential energy, complementary energy and the gap function, Quartl Appl Math, 47(3), 487–504.

    MathSciNet  MATH  Google Scholar 

  25. Gao, DY and Strang, G (1989b), Dual extremum principles in finite deformation elastoplasitc analysis, Acta Applicandae Mathematicae, 17, 357–267

    Article  MathSciNet  Google Scholar 

  26. Ilan, W and Reddy, BD (1995), Computational plasticity: The variational basis and numerical analysis, Comput. Mechanics Advances, 2 pp. 283–400.

    Google Scholar 

  27. Haslinger, J and Panagiotopoulos, PD (1984), The reciprocal variational approach to the Signorini problem with friction, approximation results, Proc. Roy Soc. Edinburgh Sect A, 98 pp. 365–383.

    MathSciNet  MATH  Google Scholar 

  28. Hoger, A (1987), The stress conjugate to logarithmic strain, Int. J. Solids and Struct,23 no. 12, 1645–1656.

    Google Scholar 

  29. Koiter, WT (1976), On the complementary energy theorem in nonlinear elasticity theory, Trends in Appl of Pure Math to Mech, Ed by G Fichera, Pitman, London, pp 207–32.

    Google Scholar 

  30. Lee, SJ and Shield, RT (1980), Variational principles in finite elastics, J Appl Math Physics (ZAMP),31 pp 437–453.

    Article  MathSciNet  MATH  Google Scholar 

  31. Mosco, U (1972), Dual variational inequalities, J. Math. Analy. Appl, 40 202–206.

    Article  MathSciNet  MATH  Google Scholar 

  32. Nemat-Nasser, S (1972), General variational principles in nonlinear and linear elasticity with applications, Mechanics Today, 1, 214–61.

    Google Scholar 

  33. Oden, JT and Reddy, JN (1983), Variational Methods in Theoretical Me-chanics, Springer-Verlag.

    Google Scholar 

  34. Ogden, RW (1984), Non-linear elastic deformations, Ellis Horwood Ltd, Chichester, 417pp.

    Google Scholar 

  35. Panagiotopoulos, PD (1985), Inequality Problems in Mechanics and Applications, Birkhäuser, Boston.

    MATH  Google Scholar 

  36. Panagiotopoulos, PD (1993), Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer-Verlag, 451pp.

    MATH  Google Scholar 

  37. Pian, THH and Tong, P (1980), Reissner’s principle in finite element formulations. In: S. Nemat-Nasser (ed.) Mechanics Today, 5 Pergamon Press, pp. 377–395.

    Google Scholar 

  38. Reddy, BD (1992), Mixed variational inequalities arising in elastoplasticity, Nonlinear Analysis, Theory, Methods & Applications, 19 no. 11, pp. 1071–1089.

    Article  MathSciNet  MATH  Google Scholar 

  39. Reissner, E. (1996), Selected works in applied mechanics and mathematics, Jones and Bartlett Publishers, Boston, MA, 601pp.

    MATH  Google Scholar 

  40. Repin, SI and Xanthis, LS (1996), Aposteriori error estimation for elasto-plastic problems based on duality theory, Comput. Methods Appl. Mech. Engrg 138 pp. 317–339.

    Article  MathSciNet  MATH  Google Scholar 

  41. Rockafellar, RT (1974), Conjugate Duality and Optimization, SIAM

    Google Scholar 

  42. Sewell, MJ (1987), Maximum and Minimum Principles,Cambridge Univ. Press, 468pp.

    Book  MATH  Google Scholar 

  43. Strang, G (1986), Introduction to Applied Mathematics, Wellesley-Cambridge Press, 758pp

    MATH  Google Scholar 

  44. Stumpf, H (1978), Dual extremum principles and error bounds in non-linear elasticity theory, J. Elasticity,8 no. 4, 425–438.

    Article  MathSciNet  MATH  Google Scholar 

  45. Toland, JF (1979), A duality principle for non-convex optimization and the calculus of variations, Arch Rational Mech Anal 71 41–61.

    Article  MathSciNet  MATH  Google Scholar 

  46. Tonti, E (1972), A mathematical model for physical theories, Rend. Accad. Lincei, Vol LII I, pp. 133–139; II, pp. 350–356.

    Google Scholar 

  47. Wataru, S and Atluri, SN (1995), On newly developed assumed stress finite element formulations for geometrically and materially nonlinear problems, Finite Elem. Anal. Des, 21, no. 1–2, pp. 75–110.

    Article  MathSciNet  MATH  Google Scholar 

  48. Wright, SJ (1996), Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA,289pp.

    Google Scholar 

  49. Yau, ST and Gao, DY (1992), Obstacle problem of von Karman equations, Adv. Appl. Math, 13, pp. 123–141.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute & State University, Blacksburg, VA, 24061, USA

    David Yang Gao

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  1. David Yang Gao
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Editor information

Editors and Affiliations

  1. University of Delaware, USA

    R. P. Gilbert

  2. University of Florida, USA

    P. M. Pardalos

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© 2001 Kluwer Academic Publishers

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Gao, D.Y. (2001). Duality in Nonconvex Finite Deformation Theory: A Survey and Unified Approach. In: Gilbert, R.P., Panagiotopoulos, P.D., Pardalos, P.M. (eds) From Convexity to Nonconvexity. Nonconvex Optimization and Its Applications, vol 55. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0287-2_6

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  • DOI: https://doi.org/10.1007/978-1-4613-0287-2_6

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7979-9

  • Online ISBN: 978-1-4613-0287-2

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