Flutter Instability and Ill-Posedness in Solids and Fluid-Saturated Porous Media

  • B. Loret
  • F. M. F. Simões
  • J. A. C. Martins
Conference paper
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 414)


For elastoplasticity with locally smooth yield surface and plastic potential, the nature, real or complex, of the squares of the acceleration wave-speeds is analyzed. Emphasis is laid on the effect that some features of the constitutive equations have on that nature. Specifically, our reference problem contemplates an infinite elastic-plastic body endowed with elastic isotropy and deviatoric associativity. In a second step, individual or simultaneous deviations with respect to these reference properties are analyzed. The performed linearized analyses are essentially intended to detect, in the course of a loading process, the onset of wave-speeds whose squares are complex, a phenomenon called flutter. Finite element simulations show how perturbations evolve in such a situation, and, as expected, the plastic loading condition plays a crucial role.

The problem is reconsidered in the context of fluid-saturated porous media whose solid skeleton is elastic-plastic. Unlike for single phase solids, flutter is not excluded in the reference situation where elasticity is isotropic and deviatoric plasticity holds. The propagation of waves of different lengths leads to make a distinction between the two phenomena of flutter instability and flutter ill-posedness. The introdution of a characteristic length in the elastic-plastic behaviour of the solid skeleton may imply the existence of an infimum wavelength below which instability modes do not exist, and well-posedness of the dynamic problem is recovered.


Flow Rule Solid Skeleton Decay Coefficient Acceleration Wave Associative Flow Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Achenbach, J.D. (1975). Wave propagation in elastic solids, North-Holland, Amsterdam, The Netherlands.Google Scholar
  2. An, L. and Schaeffer, D.G. (1992). The flutter instability in granular flow. J. Mech. Phys. Solids, 40, 683–698.MathSciNetCrossRefMATHGoogle Scholar
  3. Asaro, R. and Rice, J.R. (1977). Strain localization in ductile single crystals. J. Mech. Phys. Solids, 25, 309–317.CrossRefMATHGoogle Scholar
  4. Atkin, R. (1968). Completeness theorems for linearized theories of interacting continua. Quart. J. Mech. Appl. Math., 21, 171–193.MathSciNetCrossRefMATHGoogle Scholar
  5. Baker, R. and Desai, C.S. (1982). Consequences of deviatoric normality in plasticity with isotropic strain-hardening. Int. J. Num. Anal. Meth. Geomechanics, 6, 383–390.CrossRefMATHGoogle Scholar
  6. Bazant, Z.P. and Krizek, R.J. (1975). Saturated sand as an inelastic two-phase medium. J. of Engng. Mechanics Div., Transactions of the ASCE, 101-EM4, 317–332.Google Scholar
  7. Bazant, Z.P. and Lin, F.B. (1988). Non-local yield limit degradation. Int. J. Num. Meth. Engineering, 26, 1805–1823.CrossRefMATHGoogle Scholar
  8. Bazant, Z.P. and Cedolin, L. (1991). Stability of Structures,Oxford University Press.Google Scholar
  9. Benallal, A., Billardon, R. and Geymonat, G. (1989). Conditions de bifurcation à l’intérieur et aux frontières pour une classe de matériaux non standards. Comptes-Rendus Académie des Sciences, Paris, t. 308, série II, 893–898.MathSciNetGoogle Scholar
  10. Benallal, A. (1992). Ill-posedness and localization in solid structures. Computational Plasticity, D.R.J. Owen et al. eds., Pineridge Press, Swansea, U.K., 581–592.Google Scholar
  11. Benallal, A., Billardon, R. and Geymonat, G. (1993). Bifurcation and localization in rate-independent materials. Some general considerations. Bifurcation and Stability of Dissipative Systems, CISM course 327, Q.S. Nguyen ed., Springer-Verlag, Wien-New York, 1–44.Google Scholar
  12. Bigoni, D. and Hueckel, T. (1991). Uniqueness and localization -I. Associative and non-associative elastoplasticity. Int. J. Solids Structures, 28, 197–213.MathSciNetCrossRefMATHGoogle Scholar
  13. Bigoni, D. and Willis, J.R. (1994). A dynamical interpretation of flutter instability. Localisation and Bifurcation Theory for Soils and Rocks, R. Chambon, J. Desrues and I. Vardoulakis eds., Balkema, Rotterdam, The Netherlands, 51–58.Google Scholar
  14. Bigoni, D. and Loret, B. (1999). Effects of elastic anisotropy on strain localization and flutter instability in plastic solids. J. Mech. Phys. Solids, 47 (7), 1409–1436.MathSciNetCrossRefMATHGoogle Scholar
  15. Biot, M.A. (1956). Theory of propagation of elastic waves in a fluid saturated porous solid-I. Low-frequency range. J. of Acoustical Society of America, 28 (1), 168–179;MathSciNetCrossRefGoogle Scholar
  16. Biot, M.A. (1956). II. Higher frequency range. J. of Acoustical Society of America, 28 (1), 179–191.MathSciNetCrossRefGoogle Scholar
  17. Biot, M.A. and Willis, D.G. (1957). The elastic coefficients of the theory of consolidation. J. of Applied Mechanics, Transactions of the ASME, 24, 594–601.MathSciNetGoogle Scholar
  18. Biot, M.A. (1973). Nonlinear and semilinear rheology of porous solids. J. of Geophysical Research, 78 (23), 4924–4937.CrossRefGoogle Scholar
  19. Beskos, D.E. (1989). Dynamics of saturated rocks -I. Equations of motion. J. of Engng. Mechanics Div., Transactions of the ASCE, 115, 982–995.Google Scholar
  20. Bowen, R.M (1976). Theory of mixtures. Continuum Physics, vol. 3, 1–127, A.C. Eringen ed., Academic Press, New York.Google Scholar
  21. Bowen, R.M. and Reinicke, K.M. (1978). Plane progressive waves in a binary mixture of linear elastic materials. J. of Applied Mechanics, Transactions of the ASME, 45, 493–499.CrossRefMATHGoogle Scholar
  22. Bowen, R.M. (1982). Compressible porous media models by the use of the theory of mixtures. Int. J. Eng. Science, 20, 697–735.CrossRefMATHGoogle Scholar
  23. Bowen, R.M. and Lockett, R.R. (1983). Inertial effects in poroelasticity. J. of Applied Mechanics, Transactions of the ASME, 50, 334–342.CrossRefMATHGoogle Scholar
  24. Carroll, M.M. (1979). An effective stress law for anisotropic elastic deformation. J. of Geophysical Research, 84 (B13), 7510–7512.CrossRefGoogle Scholar
  25. Chen, P.J. (1973). Growth and decay of waves in solids. Mechanics of Solids, vol. 3, 303–401, C. Truesdell ed., Springer-Verlag, Berlin-Heidelberg, Germany.Google Scholar
  26. Cho, H. and Barber, J.R. (1999). Stability of the three-dimensional Coulomb friction law. Proc. R. Soc. Lond., A-455(1983), 839.Google Scholar
  27. Dowaikh, M.A. and Ogden, R.W. (1990). On surface waves and deformations in a pre-stressed incompressible elastic solid. IMA J. Appl. Math., 44, 261–284.MathSciNetCrossRefMATHGoogle Scholar
  28. Dowaikh, M.A. and Ogden, R.W. (1991). Interfacial waves and deformations in pre-stressed elastic media. Proc. R. Soc. Lond., A-433, 313–328.Google Scholar
  29. Eringen, A.C. and Ingram, J.D. (1965). A continuum theory for chemically reacting media -I. Int. J. Eng. Science, 3, 197–212.CrossRefGoogle Scholar
  30. Green, A.E. and Naghdi, P.M. (1969). On basic equations for mixtures. Quart. J. Mech. Appl. Math., 22 (4), 427–438.MathSciNetCrossRefMATHGoogle Scholar
  31. Hilber, H.M., Hughes, T.J.R. and Taylor, R.M. (1977). Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engineering and Structural Dynamics, 5, 283–292.CrossRefGoogle Scholar
  32. Hill, R. (1962). Acceleration waves in solids. J. Mech. Phys. Solids, 10, 1–16.MathSciNetCrossRefMATHGoogle Scholar
  33. Holmes, P.J. (1977). Bifurcation to divergence and flutter in flow-induced oscillations: a finite dimensional analysis. J. of Sound and Vibrations, Transactions of the ASME, 55 (4), 471–503.Google Scholar
  34. Huseyin, K. (1978). Vibrations and Stability of Multiple Parameter Systems, Pitman, London, U.K.Google Scholar
  35. Lade, P.V. and Musante, H.M. (1977). Failure conditions in sand and remolded clay. Proceedings of the 9th International Conference Soil Mechanics and Foundation Engineering, Tokyo, Japan, vol. 1, 181–186.Google Scholar
  36. Loret, B. (1990). Acceleration waves in elastic-plastic porous media: interlacing and separation properties. Int. J. Eng. Science, 28 (12), 1315–1320.MathSciNetCrossRefMATHGoogle Scholar
  37. Loret, B., Prévost J.H. and Harireche, O. (1990). Loss of hyperbolicity in elastic-plastic solids with deviatoric associativity. Eur. J. Mechanics-A/Solids, 9 (3), 225–231.MATHGoogle Scholar
  38. Loret, B. and Harireche, O. (1991). Acceleration waves, flutter instabilities and stationary discontinuities in inelastic porous media. J. Mech. Phys. Solids, 39 (5), 569–606.MathSciNetCrossRefMATHGoogle Scholar
  39. Loret, B. and Prévost, J.H. (1991). Dynamic strain localization in fluid-saturated porous media. J. of Engng. Mechanics Div., Transactions of the ASCE, 117 (4), 907–922.Google Scholar
  40. Loret, B. (1992). Does deviation from deviatoric associativity lead to the onset of flutter instability ?. J. Mech. Phys. Solids, 40 (6), 1363–1375.MathSciNetCrossRefMATHGoogle Scholar
  41. Loret, B., Martins, J.A.C. and Simóes, F.M.F. (1995). Surface boundary conditions trigger flutter instability in non-associative elastic-plastic solids, Int. J. Solids Structures, 32 (15), 2155–2190.CrossRefMATHGoogle Scholar
  42. Loret, B., Simóes, F.M.F. and Martins, J.A.C. (1997). Growth and decay of acceleration waves in non-associative elastic-plastic fluid-saturated porous media. Mt. J. Solids Structures, 34 (13), 1583–1608.MathSciNetCrossRefMATHGoogle Scholar
  43. Loret, B. and Rizzi, E. (1997). Qualitative analysis of strain localization. Part II: Transversely isotropic elasticity and plasticity. Int. J. Plasticity, 13 (5), 501–519.CrossRefGoogle Scholar
  44. Mandel, J. (1962). Ondes plastiques dans un milieu indéfini à trois dimensions. J. de Mécanique, 1, 3–30.MathSciNetGoogle Scholar
  45. Mandel, J. (1964a). Propagation des surfaces de discontinuité dans un milieu élasto-plastique, IUTAM Symposium Stress-waves in anelastic solids, Providence, H. Kolsky and W. Prager eds., Springer-Verlag, Berlin, Germany, 331–340.Google Scholar
  46. Mandel, J. (1964b). Conditions de stabilité et postulat de Drucker, IUTAM Symposium Rheology and Soil Mechanics, Grenoble, G. Kravchenko and P. Sirieys eds., Springer-Verlag, Berlin, Germany, 58–68.Google Scholar
  47. Martins, J.A.C. and Faria, L.O. (1991). Friction-induced surface instabilities and oscillations in nonlinear elasticity. Rencontres Scientifiques du Cinquantenaire, Rapport interne 124 du Laboratoire de Mécanique et Acoustique, Marseille, France, 31–50.Google Scholar
  48. Martins, J.A.C., Faria, L.O. and Guimarâes, J. (1992). Dynamic surface solutions in linear elasticity with frictional boundary conditions. Friction-Induced Vibration, Chatter, Squeal and Chaos, DE-vol. 49, R.A. Ibrahim and A. Soom eds., The American Society of Mechanical Engineers, New York, USA, 33–39.Google Scholar
  49. Martins, J.A.C., Faria, L.O. and Guimarâes, J. (1995). Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions. J. of Vibration and Acoustics, Transactions of the ASME, 117, 445–451.Google Scholar
  50. Martins, J.A.C., Simóes, F.M.F, Gastaldi, F. and Monteiro Marques, M. (1995). Dissipative graph solutions for a 2 degree-of-freedom quasistatic frictional contact problem. Int. J. Eng. Science, 33, 1959–1986.CrossRefMATHGoogle Scholar
  51. Martins, J.A.C., Barbarin, S., Raous, M. and Pinto da Costa, A. (1999). Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction. Computer Methods in Applied Mechanics and Engineering, 177 (34), 289–328.MathSciNetCrossRefMATHGoogle Scholar
  52. Moirot, F. (1998). Étude de la stabilité d’un équilibre en présence de frottement de Coulomb, application au crissement des freins à disque, Thèse de Doctorat, École Polytechnique, Palaiseau, France.Google Scholar
  53. Molenkamp, F. (1991). Material instability for drained and undrained behaviour -I. Shear-band generation. Int. J. Num. Anal. Meth. Geornechanics, 15, 147–168;CrossRefMATHGoogle Scholar
  54. Molenkamp, F. (1991). II. Combined uniform deformation and shear-band generation. Int. J. Num. Anal. Meth. Geomechanics, 15, 169–180.CrossRefMATHGoogle Scholar
  55. Mühlhaus, H.B. and Aifantis, E. (1991). A variational principle for gradient plasticity. Int. J. Solids Structures, 28, 845–857.CrossRefMATHGoogle Scholar
  56. Needleman, A. and Ortiz, M. (1991). Effect of boundaries and interfaces on shear-band localization. Int. J. Solids Structures, 28 (7), 859–877.CrossRefMATHGoogle Scholar
  57. Neilsen, M.K. and Schreyer, H.L. (1993). Bifurcations in elastic-plastic materials. Int. J. Solids Structures, 30 (4), 521–544.CrossRefMATHGoogle Scholar
  58. Nguyen, Q.-S. (1994). Bifurcation and stability in dissipative media (plasticity, friction, fracture), Part 1. Applied Mechanics Review, Transactions of the ASME, 47 (1), 1–31.Google Scholar
  59. Nur, A. and Byerlee, J.D. (1971). An exact effective stress law for elastic deformation of rock with fluids. J. of Geophysical Research, 76 (26), 6414–6419.CrossRefGoogle Scholar
  60. O’Malley, R.E., Jr. (1991). Singular Perturbation Methods for Ordinary Differential Equations. Applied Mathematical Sciences, 89, Springer-Verlag, New York.Google Scholar
  61. Ottosen, N.S. and Runesson, K. (1991a). Acceleration waves in elastoplasticity. Int. J. Solids Structures, 28, 135–159.MathSciNetCrossRefMATHGoogle Scholar
  62. Ottosen, N.S. and Runesson, K. (1991b). Properties of discontinuous bifurcation solutions in elastoplasticity. Int. J. Solids Structures, 27 (4), 401–421.MathSciNetCrossRefMATHGoogle Scholar
  63. Petryk, H. (1993). General theory of bifurcation and stability in time-independent plasticity. Bifurcation and Stability of Dissipative Systems, Q.S. Nguyen, ed., International Centre for Mechanical Sciences, Courses and Lectures, No. 327, Springer-Verlag, Wien, New York.Google Scholar
  64. Rice, J.R. (1975). On the stability of dilatant hardening for saturated rock masses. J. of Geophysical Research, 80, 1531–1536.CrossRefGoogle Scholar
  65. Rice, J.R. (1976). The localization of plastic deformation, Theoretical and Applied Mechanics, 14th IUTAM Congress, Delft, W.T. Koiter ed., North-Holland, Amsterdam, The Netherlands, 207–220.Google Scholar
  66. Rice, J.R. and Rudnicki, J.W. (1980). A note on some features of the theory of localization of deformation. Int. J. Solids Structures, 16, 597–605.MathSciNetCrossRefMATHGoogle Scholar
  67. Rizzi, E. and Loret, B. (1997). Qualitative analysis of strain localization. Part I: Transversely isotropic elasticity and isotropic plasticity. Int. J. Plasticity, 13 (5), 461–499.CrossRefMATHGoogle Scholar
  68. Rizzi, E. and Loret, B. (1999). Strain localization in fluid-saturated anisotropic elastic-plastic porous media. Int. J. Eng. Science, 37, 235–251.MathSciNetCrossRefMATHGoogle Scholar
  69. Rudnicki, J.W. and Rice, J.R. (1975) Conditions for the localization of deformations in pressure-sensitive dilatant materials. J. Mech. Phys. Solids, 23, 371–394.CrossRefGoogle Scholar
  70. Schaeffer, D. G. (1990). Instability and ill-posedness in the deformation of granular materials. Int. J. Num. Anal. Meth. Geornechanics, 14, 253–278.MathSciNetCrossRefMATHGoogle Scholar
  71. Simo, J.C., and Taylor, R.L. (1985). Consistent tangent operators for rate-independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering, 48, 101–118.CrossRefMATHGoogle Scholar
  72. Simo, J.C., and Hughes, T.J.R. (1987). General return mapping algorithms for rate-independent plasticity. Constitutive Laws for Engineering Materials: Theory and Applications, C.S. Desai et al. eds., Elsevier Science, Amsterdam, The Netherlands, 221–231.Google Scholar
  73. Simóes, F.M.F. (1997). Instabilidades em problemas nâo associados da Mecânica dos Sólidos. Ph. D. dissertation, 12/12/1997, Universidade Técnica de Lisboa, Portugal.Google Scholar
  74. Simóes, F.M.F. and Martins, J.A.C. (1998). Instability and ill-posedness in sonic friction problems. Int. J. Eng. Science, 36, 1265–1293.CrossRefMATHGoogle Scholar
  75. Simóes, F.M.F., Martins, J.A.C. and Loret, B. (1999). Instabilities in elastic-plastic fluid-saturated porous media: harmonic wave versus acceleration wave analyses. Int. J. Solids Structures, 36, 1277–1295.CrossRefMATHGoogle Scholar
  76. Stoer, J. and Bulirsch, R. (1980). Introduction to numerical analysis, Springer-Verlag, New York, USA.Google Scholar
  77. Suo, Z., Ortiz, M. and Needleman, A. (1992). Stability of solids with interfaces. J. Mech. Phys. Solids, 40, 613–640.MathSciNetCrossRefMATHGoogle Scholar
  78. Truesdell, C. and Toupin, R. (1960). The Classical Field Theories, Encyclopedia of Physics, vol. IIl/l, S. Flügge ed., Springer-Verlag, Berlin, Germany.Google Scholar
  79. Truesdell, C. and Noll, W. (1965). The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics, vol. III/3, S. Flügge ed., Springer-Verlag, Berlin, Germany.Google Scholar
  80. Vermeer, P.A. and Brinkgreve, R.B.J. (1994). A new effective non-local strain-measure for softening plasticity. Localisation and Bifurcation Theory for Soils and Rocks, R. Chambon, J. Desrues and I. Vardoulakis eds., Balkema, Amsterdam, The Netherlands, 89–100.Google Scholar
  81. Vidyasagar, M. (1993). Nonlinear Systems Analysis, Prentice-Hall International Editions, Englewood Cliffs, New Jersey.Google Scholar
  82. Wilkinson, J.H. (1965). The algebraic eigenvalue problem, Oxford University Press, Oxford. Ziegler, H. (1968). Principles of Structural Stability, Blaisdell, Waltham, Massachussets, USA.Google Scholar

Copyright information

© Springer-Verlag Wien 2000

Authors and Affiliations

  • B. Loret
    • 1
  • F. M. F. Simões
    • 2
  • J. A. C. Martins
    • 2
  1. 1.Laboratoire Sols, Solides, StructuresGrenobleFrance
  2. 2.Instituto Superior TécnicoLisbonPortogallo

Personalised recommendations