Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant–Kirchhoff Material

  • David Yang Gao
  • Eldar Hajilarov
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 37)


This paper presents a detailed study on analytical solutions to a general nonlinear boundary-value problem in finite deformation theory. Based on canonical duality theory and the associated pure complementary energy principle in nonlinear elasticity proposed by Gao in (Mech Res Commun 26:31–37, 1999, [6], Wiley Encyclopedia of Electrical and Electronics Engineering, 1999, [7], Meccanica 34:169–198, 1999, [8]), we show that the general nonlinear partial differential equation for deformation is actually equivalent to an algebraic (tensor) equation in stress space. For St Venant–Kirchhoff materials, this coupled cubic algebraic equation can be solved principally to obtain all possible solutions. Our results show that for any given external source field such that the statically admissible first Piola–Kirchhoff stress field has nonzero eigenvalues, the problem has a unique global minimal solution, which is corresponding to a positive-definite second Piola–Kirchhoff stress \(\mathbf{T}\), and at most eight local solutions corresponding to negative-definite \(\mathbf{T}\). Additionally, the problem could have 15 unstable solutions corresponding to indefinite \(\mathbf{T}\). This paper demonstrates that the canonical duality theory and the pure complementary energy principle play fundamental roles in nonconvex analysis and finite deformation theory.



This research was supported by the US Air Force Office of Scientific Research under the grant AFOSR FA9550-17-1-0151. Results presented in Sect. 3 were discussed with Professor Ray Ogden from University of Glasgow.


  1. 1.
    Cai, K., Gao, D.Y., Qin, Q.H.: Post-buckling solutions of hyper-elastic beam by canonical dual finite element method. Math. Mech. Solids 19(6), 659–671 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ciarlet, P.G.: Mathematical Elasticity. Volume I: Three-Dimensional Elasticity. Elsevier Science Publishers B.V., North-Holland (1988)Google Scholar
  3. 3.
    Gao, D.Y.: Global extremum criteria for nonlinear elasticity. J. Appl. Math. Phys. (ZAMP) 43, 924–937 (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Gao, D.Y.: Dual extremum principles in finite deformation theory with applications to post-buckling analysis of extended nonlinear beam theory. Appl. Mech. Rev. 50, S64–S71 (1997)CrossRefGoogle Scholar
  5. 5.
    Gao, D.Y.: Duality, triality and complementary extremum principles in nonconvex parametric variational problems with applications. IMA J. Appl. Math. 61, 199–235 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gao, D.Y.: Pure complementary energy principle and triality theory in finite elasticity. Mech. Res. Commun. 26, 31–37 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gao, D.Y.: Duality-Mathematics. Wiley Encyclopedia of Electrical and Electronics Engineering, vol. 6, pp. 68–77. Wiley, New York (1999)Google Scholar
  8. 8.
    Gao, D.Y.: General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics. Meccanica 34, 169–198 (1999)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications, xviii + 454pp. Kluwer Academic Publishers, Boston (2000)Google Scholar
  10. 10.
    Gao, D.Y.: Analytic solution and triality theory for nonconvex and nonsmooth variational problems with applications. Nonlinear Anal. 42, 1161–1193 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gao, D.Y.: Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Global Optim. 17, 127–160 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gao, D.Y.: Nonconvex semi-linear problems and canonical dual solutions. Gao, D.Y., Ogden, R.W. (eds.) Advances in Mechanics and Mathematics, vol. II, pp. 261–312. Kluwer Academic Publishers, Boston (2003)Google Scholar
  13. 13.
    Gao, D.Y.: Perfect duality theory and complete set of solutions to a class of global optimization. Optimization 52, 467–493 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gao, D.Y.: Complementary variational principle, algorithm, and complete solutions to phase transitions in solids governed by Landau-Ginzburg equation. Math. Mech. Solids 9, 285–305 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gao, D.Y.: Canonical duality theory: unified understanding and generalized solutions for global optimization. Comput. Chem. Eng. 33, 1964–1972 (2009)CrossRefGoogle Scholar
  16. 16.
    Gao, D.Y., Ogden, R.W.: Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation. Q. J. Mech. Appl. Math. 61, 497–522 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gao, D.Y., Ogden, R.W.: Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem. ZAMP 59, 498–517 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gao, D.Y., Ruan, N.: Solutions to quadratic minimization problems with box and integer constraints. J. Global Optim. 47, 463–484 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gao, D.Y., Strang, G.: Geometric nonlinearity: potential energy, complementary energy, and the gap function. Q. Appl. Math. 47, 487–504 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gao, D.Y., Yu, H.F.: Multi-scale modelling and canonical dual finite element method in phase transitions of solids. Int. J. Solids Struct. 45, 3660–3673 (2008)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hellinger, E.: Die allgemeine Ansätze der Mechanik der Kontinua. Encyklopädie der Mathematischen Wissenschaften IV 4, 602–694 (1914)Google Scholar
  22. 22.
    Koiter, W.T.: On the complementary energy theorem in nonlinear elasticity theory. In: Fichera, G. (ed.) Trends in Applications of Pure Mathematics to Mechanics. Pitman, London (1976)Google Scholar
  23. 23.
    Lee, S.J., Shield, R.T.: Variational principles in finite elastics. J. Appl. Math. Phys. (ZAMP) 31, 437–453 (1980)CrossRefzbMATHGoogle Scholar
  24. 24.
    Lee, S.J., Shield, R.T.: Applications of variational principles in finite elasticity. J. Appl. Math. Phys. (ZAMP) 31, 454–472 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Levinson, M.: The complementary energy theorem in finite elasticity. J. Appl. Mech. 87, 826–828 (1965)CrossRefGoogle Scholar
  26. 26.
    Li, S.F., Gupta, A.: On dual configuration forces. J. Elast. 84, 13–31 (2006)CrossRefzbMATHGoogle Scholar
  27. 27.
    Li, C., Zhou, X., Gao, D.Y.: Stable trajectory of logistic map. Nonlinear Dyn. (2014). doi: 10.1007/s11071-014-1433-y MathSciNetzbMATHGoogle Scholar
  28. 28.
    Oden, J.T., Reddy, J.N.: Variational Methods in Theoretical Mechanics. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  29. 29.
    Ogden, R.W.: A note on variational theorems in non-linear elastostatics. Math. Proc. Camb. Philos. Soc. 77, 609–615 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Reissner, E.: On a variational theorem for finite elastic deformations. J. Math. Phys. 32, 129–135 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ruan, N., Gao, D.Y.: Global optimal solutions to a general sensor network localization problem. Perform. Eval. 75–76, 1–16 (2014)CrossRefGoogle Scholar
  32. 32.
    Ruan, N., Gao, D.Y.: Canonical duality approach for nonlinear dynamical systems. IMA J. Appl. Math. 79, 313–325 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ruan, N., Gao, D.Y., Jiao, Y.: Canonical dual least square method for solving general nonlinear systems of equations. Comput. Optim. Appl. 47, 335–347 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Santos, H.A.F.A., Gao, D.Y.: Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam. Int. J. Nonlinear Mech. 47, 240–247 (2011). doi: 10.1016/j.ijnonlinmec.2011.05.012 CrossRefGoogle Scholar
  35. 35.
    Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  36. 36.
    Veubeke, B.F.: A new variational principle for finite elastic displacements. Int. J. Eng. Sci. 10, 745–763 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhang, J., Gao, D.Y., Yearwood, J.: A novel canonical dual computational approach for prion AGAAAAGA amyloid fibril molecular modeling. J. Theor. Biol. 284, 149–157 (2011). doi: 10.1016/j.jtbi.2011.06.024 MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Science and TechnologyFederation University AustraliaBallaratAustralia

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