The Local Conditions of Uniqueness and Plastic Strain Localization Part II: Comparison of the Local Uniqueness Condition with the Rice-Rudnicki Local Plastic Strain Localization
Restricted accessResearch articleFirst published online January, 2011
The Local Conditions of Uniqueness and Plastic Strain Localization Part II: Comparison of the Local Uniqueness Condition with the Rice-Rudnicki Local Plastic Strain Localization
Based on previous work of the first author, the local sufficient condition of uniqueness of the solution of the incremental boundary-value problem in generalized elastoplasticity which excludes the possibility of bifurcation state is compared with the local necessary condition for the localization of plastic deformations as a Rice—Rudnicki localization plane. The analytical results of limitations imposed on the isothermal hardening functions (moduli) by the Rice—Rudnicki condition of plastic strain localization are compared with those resulting from the local uniqueness condition. It is shown that the local sufficient condition of uniqueness of the solution of the incremental boundary-value problem with an equals sign becomes a local necessary condition of non-uniqueness or local necessary condition of the possible appearance of a bifurcation state.
Śloderbach, Z.Generalized coupled thermoplasticity. Part I. Fundamental equations and identities. Archives of Mechanics , 35, 337-349 (1983).
2.
Śloderbach, Z.Generalized coupled thermoplasticity. Part II. On the uniqueness and bifurcations criteria. Archives of Mechanics, 35, 351-367 (1983).
3.
Śloderbach, Z. and Pajak, J.Generalized coupled thermoplasticity taking into account large strains: part I. Conditions of uniqueness of the solution of boundary-value problem and bifurcation criteria. Mathematics and Mechanics of Solids2010, in press.
4.
Hill, R.A general theory of uniqueness and stability in elasto-plastic solids. Journal of the Mechanics and Physics of Solids, 6, 236-249 ( 1958).
5.
Hill, R.Bifurcation and uniqueness in non-linear mechanics of continua, Problem of Continuum Mechanics (N.I. Muskhelishwili Anniversary Volume), SIAM, Philadelphia, PA, 1961, pp. 155-164.
6.
Hill, R.Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1986.
7.
Hueckel, T.Coupling of elastic and plastic deformations of bulk solids. Acta Mechanica, 11, 227-235 (1976).
8.
Hueckel, T. and Konig, J.A.Some problems in elastoplasticity. Academia Pollaca Della Scienze, Conference 74, Ossolineum, Warszawa, 1979.
9.
Hueckel, T. and Maier, G.Incremental boundary value problems in the presence of coupling of elestic and plastic deformations. A rock mechanics oriented theory. International Journal of Solids and Structures, 13, 1-15 ( 1977).
10.
Hueckel, T. and Maier, G.Non-associated and coupled flow rules of elastoplasticity for geotechnical media. Proceeding 9th International Conference of Soil Mechanics Foundation of Engineering (JCSFE), Speciality session 7, Constitutive relations for soils, Tokyo, 1977, pp. 129-142.
11.
Maier, G.A minimum principle for incremental elastoplasticity, with non-associated plastic flow laws. Journal of the Mechanics and Physics of Solids, 18, 319-330 (1970).
12.
Marciniak, Z.Limit deformations in sheet metal stamping. WNT, Warszawa, 1971.
13.
Mróz, Z.On forms of constitutive laws for elastic-plastic solids. Archives of Applied Mechanics, 18, 3-35 ( 1966).
14.
Perzyna, P.Instability phenomena and adiabatic shear band localisation in thermoplastic flow processes. Acta Mechanica, 106, 173-205 ( 1986).
15.
Perzyna, P. and Duszek, M.K.The localisation of plastic deformation in thermoplastic solids, International Journal of Solids and Structures, 27, 1419-1443 (1991).
16.
Petryk, H.On constitutive inequalities and bifurcation in elastic-plastic solids with a yield-surface vertex. Journal of the Mechanics and Physics of Solids, 37, 265-291 (1989).
17.
Pęcherski, R.B.Finite deformation plasticity with strain induced anisotropy and shear banding. Journal of Materials Processing Technology, 60, 35-44 (1996).
18.
Raniecki, B.Uniqueness criteria in solids with non-associated plastic flow laws at finite deformations. Bulletin De l’Academie Polonaise des Sciences, Serie des Sciences Techniques, 27(8-9), 391-399 (1979).
19.
Raniecki, B. and Bruhns, O.T.Bounds to bifurcation stress in solids with non-associated plastic flow at finite strain. Journal of the Mechanics and Physics of Solids, 29, 153-172 (1981).
20.
Rudnicki, J.W. and Rice J.R., Condition for the localisation of deformation in pressure-sensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23, 371-394 (1975).
21.
Rice, J.R.The localisation of plastic deformation, in ed. W.T. Koiter, Theoretical and Applied Mechanics, North-Holland, Amsterdam, 1976, pp. 207-220.
22.
Śloderbach, Z. and Sawicki, T.Determination of the critical adiabatical twisting moment in the case of thick and thin-walled metal tubes. Engineering Transactions, 31, 447-457 (1983).
23.
Izbicki, J.R. and Mróz, Z.Carrying capacity limit in soil and rock mechanics [in Polish], IFTR - PAS (Institute of Fundamental Technological Research - Polish Academy of Sciences), PWN, Warszawa-Poznań, 1976.
24.
Lubliner, J.Plasticity Theory, Macmillan, New York, 1990.
25.
Macha, E.Simulation investigations of the position of fatigue fracture plane in materials with biaxial loads. Materialwissenschaft und Werkstofftechnik20(I,4/89), 132-136, (II, 5/89), 153-163 ( 1989).
26.
Macha, E.A review of energy-based multiaxial fatigue failure criteria. The Archive of Mechanical Engineering, 48, 71-101 (2001).
27.
Lubliner, J., On the thermodynamic foundations of non-linear solid mechanics. International Journal of Non-Linear Mechanics , 7, 237-254 (1972).
28.
Życzkowski, M. and Szuwalski, K.On the termination of the process of finite plastic deformations. Journal de Mécanique Theorique et Appliqee, 1, 175-186 (1982).
