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Bifurcation and Instability of Non-Associative Elastoplastic Solids

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Material Instabilities in Elastic and Plastic Solids

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 414))

Abstract

Global and local uniqueness and stability criteria for elastoplastic solids with non-associative flow rules are presented. Hill’s general theory is developed in the form generalized by Raniecki to non-associativity. Local stability criteria are presented and systematically discussed in a critical way. These are: positive definiteness and non-singularity of the constitutive operator, and positive definiteness (strong ellipticity) and non-singularity (ellipticity) of the acoustic tensor. The former criteria are particularly relevant for homogeneous deformation of solids subject to all-round controlled nominal surface tractions. Dually, the latter criteria are particularly relevant for homogeneous deformation of solids subject to displacements prescribed on the entire boundary. Flutter instability as related to complex conjugate eigenvalues of the acoustic tensor is also briefly discussed.

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References

  • An, L. and Schaeffer, D. (1990). The flutter instability in granular flow. J. Mech. Phys. Solids 40: 683–698.

    Article  MathSciNet  Google Scholar 

  • Beatty, M.F. (1987). Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues-with examples. Appl. Mech. Rev. 40: 1699–1734.

    Article  Google Scholar 

  • Benallal, A. Billardon, R. and Geymonat, G. (1990). Phènoménes de localisation a la frontière d’un solide. C. R. Acad. Sci., Paris 310: 670–684.

    MathSciNet  Google Scholar 

  • Bigoni, D. (1995). On flutter instability in elastoplastic constitutive models. Int. J. Solids Structures 32: 3167–3189.

    Article  MathSciNet  MATH  Google Scholar 

  • Bigoni, D. (1996). On smooth bifurcations in non-associative elastoplasticity. J. Mech. Phys. Solids 44: 1337–1351.

    Article  Google Scholar 

  • Bigoni, D. and Hueckel, T. (1990). A note on strain localization for a class of non-associative plasticity rules. Ingenieur-Archiv 60: 491–499.

    Google Scholar 

  • Bigoni, D. and Hueckel, T. (1991a). Uniqueness and localization-I. Associative and non-associative elastoplasticity. Int. J. Solids Structures 28: 197–213.

    Article  MathSciNet  MATH  Google Scholar 

  • Bigoni, D. and Hueckel, T. (1991b). Uniqueness and localization-II. Coupled elastoplasticity. Int. J. Solids Structures 28: 215–224.

    Article  MathSciNet  MATH  Google Scholar 

  • Bigoni, D. and Laudiero, F. (1989). The quasi-static finite cavity expansion in a non-standard elastoplastic medium. Int. J. Mech. Sci. 31: 825–837.

    Article  Google Scholar 

  • Bigoni, D. and Loret, B. (1999). Effects of elastic anisotropy on strain localization and flutter instability in plastic solids. J. Mech. Phys. Solids 47: 1409–1436.

    Article  MathSciNet  MATH  Google Scholar 

  • Bigoni, D., Loret, B. and Radi, E. (2000). Localization of deformation in plane elastic-plastic solids with anisotropic elasticity. J. Mech. Phys. Solids, Special issue dedicated to Prof. J.R. Willis,in press.

    Google Scholar 

  • Bigoni, D. and Willis, J.R. (1994). A dynamical interpretation of flutter instability. In: Chambon, R., Desrues, J. and Vardoulakis, I., eds., Localisation and Bifurcation of Rocks and Soils Rotterdam: A.A. Balkema Scientific Publishers. 51–58.

    Google Scholar 

  • Bigoni, D. and Zaccaria, D. (1992a). Strong ellipticity of comparison solids in elastoplasticity with volumetric non-associativity. Int. J. Solids Structures 29: 2123–2136.

    Article  MathSciNet  MATH  Google Scholar 

  • Bigoni, D. and Zaccaria, D. (1992b). Loss of strong ellipticity in non-associative elastoplasticity. J. Mech. Phys. Solids 40: 1313–1331.

    Article  MathSciNet  MATH  Google Scholar 

  • Bigoni, D. and Zaccaria, D. (1994a). On eigenvalues of the acoustic tensor in elastoplasticity. Eur. J. Mechanics-A/Solids 13: 621–638.

    MathSciNet  MATH  Google Scholar 

  • Bigoni, D. and Zaccaria, D. (1994b). Eigenvalues of the elastoplastic constitutive operator. ZAMM 74: 355–357.

    Article  MathSciNet  MATH  Google Scholar 

  • Biot, M.A. (1965) Mechanics of incremental deformations. New York: Wiley.

    Google Scholar 

  • Boehler, J.P. and Willis J.R. (1991). An analysis of localization in highly pre-deformed sheet steel. Unpublished.

    Google Scholar 

  • Brannon, R.M. and Drugan, W.J. (1993). Influence of non-classical elastic-plastic constitutive features on shock wave existence and spectral solutions. J. Mech. Phys. Solids 41: 297–330.

    Article  MathSciNet  MATH  Google Scholar 

  • Bruhns, O. and Raniecki, B. (1982). Ein Schrankenverfahren bei Verzweigungsproblemen in-elastischer Formänderungen. ZA MM 62: T111–T113.

    MATH  Google Scholar 

  • Cattaneo, C. (1946). Su un teorema fondamentale nella teoria delle onde di discontinuità. Atti Acad. Naz. Lincei (parts I and II) I:67–72 and 728–734.

    Google Scholar 

  • Chadwick, P. and Powdrill B. (1965). Singular surfaces in linear thermoelasticity. Int. J. Eng. Science 3: 561–595.

    Article  MathSciNet  Google Scholar 

  • Chau, K.T. (1992). Non-normality and bifurcation in a compressible pressure-sensitive circular cylinder under axisymmetric tension and compression. Int. J. Solids Structures 29: 801–824.

    Article  MATH  Google Scholar 

  • Chau, K.T. (1995). Buckling, barrelling, and surface instabilities of a finite, transversely isotropic circular cylinder. Quart. Appl. Math. 53: 225–244.

    MathSciNet  MATH  Google Scholar 

  • Chau, K.T. and Rudnicki, J.W. (1990). Bifurcations of compressible pressure-sensitive materials in plane strain tension and compression. J. Mech. Phys. Solids 38: 875–898.

    Article  MATH  Google Scholar 

  • Cheng, Y.S. and Lu, W.D. (1993). Uniqueness and bifurcation in elastic-plastic solids. Int. J. Solids Structures 30: 3073–3084.

    Article  MathSciNet  MATH  Google Scholar 

  • Christoffersen, J. (1991). Hyperelastic relations with isotropic rate forms appropriate for elastoplasticity. Eur. J. Mechanics-A/Solids 10: 91–99.

    MATH  Google Scholar 

  • Piero, G. (1979). Some properties of the set of fourth-order tensors, with applications to elasticty. J. Elasticity 9: 245–261.

    Article  MathSciNet  MATH  Google Scholar 

  • Drucker, D.C. (1954). Coulomb friction plasticity and limit ‘dads. J. Appl. Mech. 76: 71–74.

    Google Scholar 

  • Curtin, M.E. (1972). The linear theory of Elasticity. In Fliigge, S., ed., Encyclopedia of Physics VIa/2. Berlin: Springer. 1–295.

    Google Scholar 

  • Curtin, M.E. (1981). An introduction to continuum mechanics. New York: Academic Press.

    Google Scholar 

  • Hayes, M. (1966). On the displacement boundary-value problem in linear elastostatics. Quart. J. Mech. Appl. Math. XIX: 151–155.

    Article  MathSciNet  Google Scholar 

  • Hadamard, J. (1903). Leçons sur la Propagation des Ondes et les Équations de l’ Hydrodynamique. Paris: Hermann.

    MATH  Google Scholar 

  • Hill, R. (1950). The mathematical theory of plasticity. Oxford: Clarendon Press.

    MATH  Google Scholar 

  • Hill, R. (1952). On discontinuous plastic states, with special reference to localized necking in thin sheets. J. Mech. Phys. Solids 1: 19–30.

    Article  MathSciNet  Google Scholar 

  • Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. J. Mech,. Phys. Solids 6: 236–249.

    Article  MATH  Google Scholar 

  • Hill, R. (1959). Some basic principles in the mechanics of solids without a natural time. 1. Mech. Phys. Solids 7: 209–225.

    Article  MATH  Google Scholar 

  • Hill, R. (1961). Discontinuity relations in mechanics of solids. In Sneddon, I.N. and 1Ii11, R., eds., Progress in Solid Mechanics II. Amsterdam: North-Holland. 247–276.

    Google Scholar 

  • Hill, R. (1962). Acceleration waves in solids. J. Mech. Phys. Solids 10: 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, R. (1967a). Eigenmodal deformations in elastic/plastic continua.1. Mech. Phys. Solids 15: 371–386.

    Article  Google Scholar 

  • Hill, R. (1967b). On the classical constitutive laws for elastic/plastic solids. In Broberg, B., ed., Recent Progress in Applied Mechanics, The Folke Odkvist Volume Stocklrolrn:Alingvist and Wiksell. 241–249.

    Google Scholar 

  • Hill, R. (1968). On constitutive inequalities for simple materials. J. Mech. Phys. Solids 16: 229–242.

    Article  MATH  Google Scholar 

  • Hill, R. (1978) Aspects of invariance in solid mechanics. In Yih, C.-S., ed., Advances in. Applied Mechanics 18. New York: Academic Press. 1–75.

    Google Scholar 

  • Hill, R. and Hutchinson, J. W. (1975). Bifurcation phenomena in the plane tension test. J. Mech. Phys. Solids 23: 239–264.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, R. and Rice, J. R. (1973). Elastic potentials and the structure of inelastic constitutive laws. SIAM J. Appl. Math. 25: 448–461.

    Article  MathSciNet  MATH  Google Scholar 

  • Horgan, C.O. and Polignone, D.A. (1995). Cavitation in nonlinearly elastic solids: A review. Appi. Mech. Rev. 48: 471–485.

    Article  Google Scholar 

  • Huang, K., Hutchinson, J.W. and Tvergaard, V. (1991). Cavitation instabilities in elastic-plastic solids. J. Mech. Phys. Solids 39: 223–241.

    Article  Google Scholar 

  • Hueckel, T. (1976). Coupling of elastic and plastic deformation of bulk solids. Meccanica 11: 227–235.

    Article  MATH  Google Scholar 

  • Hutchinson, J. W. (1973). Post-bifurcation behavior in the plastic range. J. Mech. Phys. Solids 21: 163–190.

    Article  MATH  Google Scholar 

  • Hutchinson, J. W. and Miles, J.P. (1974). Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension. J. Mech. Phys. Solids 22: 61–71.

    Article  MATH  Google Scholar 

  • Kleiber, M. (1984) Numerical study on necking-type bifurcations in void-containing elasticplastic material. Int. J. Solids Structures 20: 191–210.

    Article  MATH  Google Scholar 

  • Kleiber, M. (1986) On plastic localization and failure in plane strain and round void containing tensile bars. Int. J. Plasticity 2: 205–221.

    Article  MATH  Google Scholar 

  • Loret, B. (1992). Does deviation from deviatoric associativity lead to the onset of flutter instability?. 1. Mech. Phys. Solids 40: 1363–1375.

    Article  MathSciNet  MATH  Google Scholar 

  • Loret, B., Martins, J.A.C. and Simes, F.M.F. (1995). Surface boundary conditions trigger flutter instability in non-associative elastic-plastic solids. Int. J. Solids Structures 32: 2155–2190.

    Article  MATH  Google Scholar 

  • Loret, B., Prevost, J.H. and Harireche, O. (1990). Loss of hyperbolicity in elastic-plastic solids with deviatoric associativity. Ear. J. Mechanics-A/Sol ds 9: 225–231.

    MathSciNet  MATH  Google Scholar 

  • Maier, G. and Hueckel, T. (1979). Non associated and coupled flow-rules of elastopla,sticity for rock-like materials. Int., I. Rock 1llech. Min. Sci. 16: 77–92.

    Article  Google Scholar 

  • Mandel, J. (1966). Conditions de stabilité et postulat de Drucker. In Kravtchenko, J. and Sirieys. P.M., eds., Rheology and Soil Mechanics. Berlin: Springer. 58–68.

    Google Scholar 

  • Mclan, E. (1938). Zur Plastizitiit des räumliche Kontiunuuns. Ingcnicus-.4rchie 9: 116–126.

    Google Scholar 

  • Miles, J.P. (1973). Fluid-pressure cigeastates and bifurcation in tension specimens under lateral pressure. J. Mech. Phys. Solids 21: 145–162.

    Article  MATH  Google Scholar 

  • Miles, J.P. and Nnwayhid, U.A. (1985). Bifurcation in compressible elastic/plastic cylinders under nniaxial tension. Appl. Sci. lies. 42: 33–514.

    MATH  Google Scholar 

  • Mrdz, Z. (1963). Non-associated flow laws in plasticity.1. de Mechaniguc 2: 21–42.

    Google Scholar 

  • Mrdz, Z. (1966). On fornns of constitutive laws for elastic-plastic solids. Arch. Alcch. Stesowane) 18: 1–34.

    Google Scholar 

  • Nadai, A. (1931) Pla.st.icit.y. New York:MrGi:nv-Ilill.

    Google Scholar 

  • Nadai, A. (1950) Theory of flow and frn.elon of solids. New York: McGraw-Hill.

    Google Scholar 

  • Neale, K.W. (1981). Phenomenological constitutive laws in finite plasticity SM Archives 6: 79–128.

    MATH  Google Scholar 

  • Needleman, A. (1979). Non-normality and bifurcation in plane strain tension or compression. 1. Mech. Phys. Solids 27: 231–2514.

    Article  MathSciNet  MATH  Google Scholar 

  • Needleman, A. and Ortiz, M. (1991). Elfects of ularies ant interfaces on shear-band localization. Int. J. Solids Structures 28: 859–877.

    Article  MATH  Google Scholar 

  • Nguyen, S.Q. and Ttiantafyllidis, N. (1989). Plastic bifurcation and postbifurca,tion analysis for generalized standard continua. 1. Alcch. Plrys. Solids 37: 515–566.

    Google Scholar 

  • Nikolaevskii, V.N. and Rice, H. (1979). Current. topics in non-elastic deformation of geological materials. lu ‘l’inunerhans, N.D. and Barber, ALS., eds., Proocedirrgs of the Smith. AIR:I PT C.’onfercnce: lligh. Iiïssarc Science and Technology. New York: Plenuum. 2: 455–464.

    Google Scholar 

  • Ogden, R.W. (1984). Non-linear elastic deformations. Chichester:Ellis Norwood

    Google Scholar 

  • Ogden, W. (1985). Local:unl global bifurcation phenomena in plane-strain finite elasticity. ha. J. Solids Structures 21: 121–132.

    Article  MATH  Google Scholar 

  • Ottosen, N.S. and Bunesson, H. (1991). Acceleration waves in elastoplasticit.y. Irrt. J. Solids Stractnres 28: 135–159.

    Article  MATH  Google Scholar 

  • Petryk, H. (1985a). On energy criteria of plastic instability. In Plastic Instability, Proc. Considère Memorial. Paris: Ecole Nat. Ponts Chauss. Press. 215–226.

    Google Scholar 

  • Petryk, H. (1985b). On stability and symmetry conditions in time-independent plasticity. Arch. Mech. 37: 503–520.

    MATH  Google Scholar 

  • Petryk, H. (1991). The energy criteria of instability in time-independent inelastic solids. Arch. Mech. 43: 519–545.

    MathSciNet  MATH  Google Scholar 

  • Petryk, H. (1992). Material instability and strain-rate discontinuities in incrementally nonlinear continua. J. Mech. Phys. Solids 40: 1227–1250.

    Article  MathSciNet  MATH  Google Scholar 

  • Petryk, H. (1993a). Theory of bifurcation and instability in time-independent plasticity. In Nguyen, Q.S., ed., CISM Lecture Notes No. 327, Udine 1991. Wien: Springer. 95–152.

    Google Scholar 

  • Petryk, H. (1993b). Stability and constitutive inequalities in plasticity. In Muschik, W., ed., CISM Lecture Notes No. 336, Udine 1992. Wien: Springer. 255–329.

    Google Scholar 

  • Petryk, H. (1999). General conditions for uniqueness in materials with multiple mechanisms of inelastic deformation. J. Mech. Phys. Solids in press.

    Google Scholar 

  • Petryk, H. and Thermann, K. (1985). Second-order bifurcation in elastic-plastic solids. J. Mech. Phys. Solids 33: 577–593.

    Article  MathSciNet  MATH  Google Scholar 

  • Prager, W. (1954). Discontinuous fields of plastic stress and flow. In 2nd Nat. Congr. Appl. Mech., Ann Arbor, Michigan, 21–32.

    Google Scholar 

  • Radi, E., Bigoni, D. and Tralli, A. (1999). On uniqueness for frictional contact rate problems. J. Mech. Phys. Solids 47: 275–296.

    Article  MathSciNet  MATH  Google Scholar 

  • Raniecki, B. (1979). Uniqueness criteria in solids with non-associated plastic flow laws at finite deformations, Bull. Acad. Polon. Sci. ser. sci. tech. XXVII: 391–399.

    MathSciNet  Google Scholar 

  • Raniecki, B. and Bruhns, O.T. (1981). Bounds to bifurcation stresses in solids with nonassociated plastic flow law at finite strain. J. Mech. Phys. Solids 29: 153–171.

    Article  MathSciNet  MATH  Google Scholar 

  • Rice, J. R. (1977). The localization of plastic deformation. In Koiter, W.T., ed., Theoretical and Applied Mechanics. Amsterdam: North-Holland. 207–220.

    Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Runesson, K. and Mróz, Z. (1989). A note on non-associated plastic flow rules. Int. J. Plasticity 5: 639–658.

    Article  MATH  Google Scholar 

  • Ryzhak, E. I. (1987). Necessity of Hadamard conditions for stability of elastic-plastic solids. Izv. AN SSSR MTT (Mechanics of Solids) 99–102.

    Google Scholar 

  • Ryzhak, E. I. (1993). On stable deformation of “unstable” materials in a rigid triaxial testing machine. J. Mech. Phys. Solids 41: 1345–1356.

    Article  MATH  Google Scholar 

  • Ryzhak, E. I. (1994). On stability of homogeneous elastic bodies under boundary conditions weaker than displacement conditions. Q. Jl. Mech. appl. Math. 47: 663–672.

    Article  MathSciNet  MATH  Google Scholar 

  • Simoes, F.M.F. (1997). Instabilities in non-associated problems of solid mechanics. Ph.D. Thesis, Technical University of Lisbon, in Portuguese.

    Google Scholar 

  • Szabo, L. (1994). Shear band formulation in finite elastoplasticity. Int. J. Solids Structures 31: 1291–1308.

    Article  MATH  Google Scholar 

  • Thomas, T.Y. (1953). The effect of compressibility on the inclination of plastic slip bands in flat bars. Proc. Nat. Acad. Sci. 39: 266–273.

    Article  MathSciNet  MATH  Google Scholar 

  • Thomas, T.Y. (1961) Plastic flows and fracture of solids. New York: Academic Press.

    Google Scholar 

  • Tomita, Y., Shindo, A. and Fatnassi, A. (1988). Bounding approach to bifurcation point of annular plates with nonassociated flow law subjected to uniform tension at their outer edges. Int. J. Plasticity 4: 251–263.

    Article  MATH  Google Scholar 

  • Truesdell, C. and Noll, W. (1965). The non-linear field theories of mechanics. In Flügge, S., ed., Encyclopedia of Physics:III/3. Berlin: Springer-Verlag.

    Google Scholar 

  • Tvergaard, V. (1982). Influence of void nucleation on ductile shear fracture at a free surface. J. Mech. Phys. Solids 30: 399–425

    Article  MATH  Google Scholar 

  • Hove, L. (1947). Sur l’extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues. Proc. Sect. Sci. K. Akad. van Wetenschappen, Amsterdam, 50: 18–23.

    MATH  Google Scholar 

  • Vardoulakis, I. (1981). Bifurcation analysis of the plane rectilinear deformation on dry sand samples. Int. J. Solids Structures 11: 1085–1101.

    Article  Google Scholar 

  • Vardoulakis, I. (1983). Rigid granular plasticity model and bifurcation in the triaxial test. Acta Mechanica 49: 57–79.

    Article  MATH  Google Scholar 

  • Young, N.J.B. (1976). Bifurcation phenomena in the plane compression test. J. Mech. Phys. Solids 24: 77–91.

    Article  MATH  Google Scholar 

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  1. University of Trento, Trento, Italy

    D. Bigoni

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  1. Polish Academy of Sciences, Poland

    Henryk Petryk

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Bigoni, D. (2000). Bifurcation and Instability of Non-Associative Elastoplastic Solids. In: Petryk, H. (eds) Material Instabilities in Elastic and Plastic Solids. CISM International Centre for Mechanical Sciences, vol 414. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2562-5_1

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