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Ill-posedness of the initial and boundary value problems in non-associative plasticity. (English) Zbl 0868.73029

A clear criterion is established for the well-posedness of the initial, boundary, and initial-boundary value problems when the plasticity theory is associative as well as non-associative. The authors show cases where non-associativity leads to ill-posedness of these problems, even when the material is not at failure. Specifically, they demonstrate that the initial-boundary and boundary value problems either have no solution or, if they do, the solution is not unique.

MSC:

74C99 Plastic materials, materials of stress-rate and internal-variable type
35Q72 Other PDE from mechanics (MSC2000)
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[1] Richmond, O., Spitzig, W. A.: Pressure dependence and dilatancy of plastic flow. XV International Congress of Theoretical and Applied Mechanics, Toronto, Canada 1980 (Rimrott, F. P. J., Tabbarok, B., eds.), pp. 377-386. Amsterdam: North-Holland 1980.
[2] De Borst, R.: Non-linear analysis of frictional materials. Doctoral dissertation. Institut TNO voor Bouwmaterialen en Bouwconstructies, Delft, Holland 1986.
[3] Maier, G., Hueckel, T.: Non-associated and coupled flow rules in rock-like materials. Int. J. Rock Mech. Min. Sci. Geom. Abstr.16, 77-92 (1979). · doi:10.1016/0148-9062(79)91445-1
[4] Vermeer, P. A.: Formulation and analysis of sand deformation problems. Doctoral dissertation. Delft University of Technology. Delft, Holland 1980.
[5] Willam, K., Etse, G.: Failure diagnostics of non-associated elastoplastic material models. In: Developments in theoretical and applied mechanics Vol. XV (Hanagud, S. V., et al., eds.), pp. 887-897. XV Southeastern Conference of Applied Mechanics, Atlanta: Georgia, 1990 Atlanta: Georgia Inst. of Technology 1990.
[6] Sandler, I. S.: The consequences of non-associated plasticity in dynamic problems. In: Constitutive laws for engineering materials: theory and applications Vol. I (Desai, C. S., et al., eds.), pp. 345-352. New York: Elsevier 1987.
[7] Sandler, I. S.: Issues related to computational methods in continuum mechanics. In: Developments in theoretical and applied mechanics Vol. XV (Hnagud S. V., et al., eds.), pp. 898-907. XV Southeastern Conference of Applied Mechanics, Atlanta, Georgia 1990. Atlanta: Georgia Inst. of Technology 1990.
[8] Hill, R.: A general theory of uniqueness and stability of elastic-plastic solids. J. Mech. Phys. Solids6, 236-249 (1958). · Zbl 0091.40301 · doi:10.1016/0022-5096(58)90029-2
[9] Drucker, D. C.: A more fundamental approach to plastic stress-strain relations Proceedings of the First National Congress of Applied Mechanics, ASME, 487-491 (1951).
[10] Drucker, D. C.: A definition of a stable inelastic material. J. Appl. Mech.26, 101-106 (1959). · Zbl 0088.17103
[11] Mandel, J.: Condition de stabilité et postulat de Drucker. In: Rheology and soil mechanics (Kravtchenko, J., Sirieys, P. M., eds.), pp. 56-68. IUTAM Symposium, Grenoble, France 1964. Berlin Göttingen Heidelberg: Springer 1964.
[12] Schaeffer, D. G.: Instability and ill-posedness in the deformation of granular materials. Int. J. Num. Anal. Methods Geomech.14, 253-278 (1990). · Zbl 0727.73026 · doi:10.1002/nag.1610140403
[13] Ralston, A., Rabinowitz, P.: A First course in numerical analysis, 2nd ed. Tokyo: McGraw-Hill, 1978. · Zbl 0408.65001
[14] Courant, R., Hilbert, D.: Methods of mathematical physics, Vol. 1, New York: Interscience 1953. · Zbl 0051.28802
[15] Mroz, Z.: Non-associated flow laws in plasticty. J. Mech.2, 21-42 (1963).
[16] Valanis, K. C.: On the uniqueness of solution of the initial value problem in softening materials. J. Appl. Mech.52, 649-653 (1985). · Zbl 0577.73008 · doi:10.1115/1.3169115
[17] Hvorslev, M. J.: Über die Festigkeitseigenschaften gestörter bindiger Böden. In: Critical state soil mechanics (Schofield, A., Wroth, P., eds.), pp. 209-215. New York: McGraw-Hill 1968.
[18] Casagrande, A.: Characteristics of cohesionless soils affecting the stability of slopes and earth fills. J. Boston Soc. Civil Eng.26, 257-276 (1936).
[19] Poorooshab, H. B., Hobulec, I., Sherbourne, A. N.: Yielding and flow of sand in trixial compression: Part I. Can. Geotech. J.3, 179-190 (1966). · doi:10.1139/t66-023
[20] Lade, P. V.: The stress-strain and strength characteristics of cohesionless soils. Ph. D. Thesis. Berkley: University of California 1972.
[21] Lade, P. V., Nelson, R. B., Yto, Y. M.: Non-associated flow and stability of granular materials. J. Eng. Mech. ASCE11, 1302-1318 (1987). · doi:10.1061/(ASCE)0733-9399(1987)113:9(1302)
[22] Peters, J. F.: Discussion of instability of granular materials with non-associated flow. J. Eng. Mech. ASCE117, 179-190 (1991). · doi:10.1061/(ASCE)0733-9399(1991)117:4(934)
[23] Lade, P. V., Bopp, P. A., Peters, J. F.: Instability of dilating sand. Mech. Mat.16, 249-264 (1993). · doi:10.1016/0167-6636(93)90056-W
[24] Thomas, T. Y.: Plastic flow and fracture in solids. New York: Academic Press 1961. · Zbl 0095.38902
[25] Hill, R.: Acceleration waves in solids. J. Mech. Phys. Solids10, 1-16 (1962). · Zbl 0111.37701 · doi:10.1016/0022-5096(62)90024-8
[26] Rudnicki, J. M., Rice, J. R.: Conditions for the localization of deformation in pressure sensitive dilatant materials. J. Mech. Solids23, 371-394 (1975). · doi:10.1016/0022-5096(75)90001-0
[27] Vermeer, P. A.: A simple shear band analysis using compliances. In: Deformation and failure of granular materials (Vermeer, P. A., Luger, H. J., eds.), pp. 493-499. IUTAM Symposium, Delft, Holland 1982. Rotterdam: Balkema 1982.
[28] Molenkamp, F.: Comparisons of frictional material models with respect to shear band initiation. Geotechnique35, 127-142 (1985). · doi:10.1680/geot.1985.35.2.127
[29] Valanis, K. C.: Banding and stability in plasticmaterials. Acta Mech.79, 113-141 (1989). · Zbl 0688.73068 · doi:10.1007/BF01181483
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