DOI QR코드

DOI QR Code

A Variational Numerical Method of Linear Elasticity through the Extended Framework of Hamilton's Principle

확장 해밀턴 이론에 근거한 선형탄성시스템의 변분동적수치해석법

  • Kim, Jinkyu (School of Civil, Environmental and Architectural Engineering, Korea Univ.)
  • 김진규 (고려대학교 건축사회환경공학부)
  • Received : 2014.01.22
  • Accepted : 2014.01.29
  • Published : 2014.02.28

Abstract

The extended framework of Hamilton's principle provides a new rigorous weak variational formalism for a broad range of initial boundary value problems in mathematical physics and mechanics in terms of mixed formulation. Based upon such framework, a new variational numerical method of linear elasticity is provided for the classical single-degree-of-freedom dynamical systems. For the undamped system, the algorithm is symplectic with respect to the time step. For the damped system, it is shown to be accurate with good convergence characteristics.

References

  1. Agrawal, O.P. (2001) A New Lagrangian and a New Lagrange Equation of Motion for Fractionally Damped Systems, Journal of Applied Mechanics, 68, pp.339-341. https://doi.org/10.1115/1.1352017
  2. Agrawal, O.P. (2002) Formulation of Euler-Lagrange Equations for Fractional Variational Problems, Journal of Mathematical Analysis and Applications, 272, pp.368-379. https://doi.org/10.1016/S0022-247X(02)00180-4
  3. Bretherton, F.P. (1970) A Note on Hamiltons Principle for Perfect Fluids, Journal of Fluid Mechanics, 44, pp.19-31. https://doi.org/10.1017/S0022112070001660
  4. Agrawal, O.P. (2008) A General Finite Element Formulation for Fractional Variational Problems, Journal of Mathematical Analysis and Applications, 337, pp.1-12. https://doi.org/10.1016/j.jmaa.2007.03.105
  5. Atanackovic, T.M. Konjik, S., Pilipovic, S. (2008) Variational Problems with Fractional Derivatives: Euler-Lagrange Equations, Journal of Physics A-Mathematical and Theoretical, 41, pp. 095201. https://doi.org/10.1088/1751-8113/41/9/095201
  6. Baleanu, D., Muslih, S.I. (2005) Lagrangian Formulation of Classical Fields within Riemann-Liouville Fractional Derivatives, Physica Scripta, 72, pp.119-121. https://doi.org/10.1238/Physica.Regular.072a00119
  7. Cresson, J. (2007) Fractional Embedding of Differential Operators and Lagrangian Systems, Journal of Mathematical Physics, 48, pp.033504 https://doi.org/10.1063/1.2483292
  8. Gossick, B.R. (1967) Hamilton's Principle and Physical Systems, Academic Press, New York.
  9. Gurtin, M.E. (1964a) Variational Principles for Linear Elastodynamics, Archive for Rational Mechanics and Analysis, 16, pp.34-50.
  10. Gurtin, M.E. (1964b) Variational Principles for Linear Initial-value Problems, Quarterly of Applied Mathematics, 22, pp.252-256.
  11. Hamilton, W.R. (1834) On a General Method in Dynamics, Philosophical Transactions of the Royal Society of London, 124, pp.247-308. https://doi.org/10.1098/rstl.1834.0017
  12. Hamilton, W.R. (1835) Second Essay on a General Method in Dynamics, Philosophical Transactions of the Royal Society of London, 125, pp.95-144. https://doi.org/10.1098/rstl.1835.0009
  13. Kim, J., Dargush, G.F., Ju, Y.K. (2013) Extended Framework of Hamilton's Principle for Continuum Dynamics, International Journal of Solids and Structures, 50, pp.3418-3429. https://doi.org/10.1016/j.ijsolstr.2013.06.015
  14. Landau, L.E., Lifshits, E.M. (1975) The Classical Theory of Fields, Pergamon Press, Oxford.
  15. Rayleigh, J.W.S. (1877) The Theory of Sound, Dover, New York.
  16. Riewe, F. (1996) Nonconservative Lagrangian and Hamiltonian Mechanics, Physical Review E, 53, pp.1890-1899. https://doi.org/10.1103/PhysRevE.53.1890
  17. Riewe, F. (1997) Mechanics with Fractional Derivatives, Physical Review E, 55, pp.3581-3592. https://doi.org/10.1103/PhysRevE.55.3581
  18. Slawinski, M.A. (2003) Seismic Waves and Rays in Elastic Media, Pergamon, Amsterdam.
  19. Tiersten, H.F. (1967) Hamiltons principle for Linear Piezoelectric Media, Proceedings of the IEEE, 55, pp.1523-1524. https://doi.org/10.1109/PROC.1967.5887
  20. Tonti, E. (1973) On the Variational Formulation for Linear Initial Value Problems, Annali di Matematica Pura Applicata, 95, pp.331-359. https://doi.org/10.1007/BF02410725