On the uniqueness conditions and bifurcation criteria in coupled thermo-elasto-plasticity

• Autorzy: Śloderbach Z.

Identyfikatory

DOI

10.1515/ijame-2017-0012

• Języki publikacji: EN

Abstrakty

EN

The global and local conditions of uniqueness and the criteria excluding a possibility of bifurcation of the equilibrium state for small strains are derived. The conditions and criteria are derived analyzing the problem of uniqueness of solution of the basic incremental boundary problem of coupled generalized thermo-elasto-plasticity. This paper is a continuation of some previous works by the author, but contains new derivation of the global and local criteria excluding a possibility of bifurcation of the equilibrium state for a comparison body dependent on statically admissible fields of stress velocity. All the thermal elastoplastic coupling effects, non-associated laws of plastic flow and influence of plastic strains on thermoplastic properties of a body were taken into account in this work. Thus, the mathematical problem considered here is not a self-conjugated problem. The paper contains four Appendices A, B, C and D where the local necessery and sufficient conditions of uniqueness have been derived.

Słowa kluczowe

EN

bifurcation of the equilibrium state; conditions and criteria of uniqueness; boundary-value problem; generalized coupled thermo-elasto-plasticity; comparison bodies; family of local conditions of uniqueness; parameter of the optimization

PL

termoplastyczność; elastoplastyczność; odkształcenie plastyczne

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mechanics and Engineering
• Rocznik: 2017
• Tom: Vol. 22, no. 1
• Strony: 199--227
• Opis fizyczny: Bibliogr. 41 poz.

Twórcy

autor

Śloderbach Z.

• Opole University of Technology Faculty of Applications of Chemistry and Mechanics 45-036 Opole, Luboszycka 7, POLAND

Bibliografia

• [1] Śloderbach Z. (1980): Bifurcations criteria for equilibrium states in generalized thermoplasticity [in Polish], Ph.D. thesis. IFTR Reports, Institute of Fundamental Technological Research, Polish Academy of Sciences, nr 37/1980, Warsaw, pp.1-100.
• [2] Śloderbach Z. (1983): Generalized coupled thermoplasticity. Part I. Fundamental equations and identities. Archives of Mechanics, No.35, vol.3, pp.337-349.
• [3] Śloderbach Z. (1983): Generalized coupled thermoplasticity. Part II. On the uniqueness and bifurcations criteria. Archives of Mechanics, No.35, vol.3, pp.351-367.
• [4] Mróz Z. and Raniecki B. (1976): A derivation of the uniqueness conditions in coupled thermoplasticity. Int. J. Eng. Sci., No.14, pp.395-401.
• [5] Mróz Z. and Raniecki B. (1976): On the uniqueness problem in coupled thermoplasticity. Int. J. Eng. Sci., No.14, pp.211-221.
• [6] Raniecki B. (1977): Problems of Applied Thermoplasticity [in Polish], DSc Thesis. IFTR Reports, Institute of Fundamental Technological Research, Polish Academy of Sciences, nr 29/1977, Warsaw, pp.1-120.
• [7] Śloderbach Z. (2016): Closed set of the uniqueness conditions and bifurcations criteria in generalized coupled thermoplasticity for small deformations. Continuum Mechanics and Thermodynamics, vol.28, pp.633-654.
• [8] Śloderbach Z. (2016): Closed system of coupling effects in generalized thermo-elastoplasticity. International Journal of Applied Mechanics and Engineering, vol.21, No.2, pp.461-483.
• [9] Gyarmati J. (1970): Non-equilibrium thermodynamics. Field theory and variational principles. Berlin, New York: Springer-Verlag.
• [10] Armero F. and Simo J.C. (1993): A priori estimates and unconditionelly stable product formula alghorithms for nonlinear coupled thermoplasticity. Int. J. Plasticity, vol.9, pp.749-782.
• [11] Benall A. and Bigoni D. (2004): Effects of temperature and thermo-mechanical couplings on material instabilities and strain localization of inelastic materials. J. Mech. Phys. Solids, vol.52, No.3, pp.725-753.
• [12] Bertram A. (2003): A finite thermoplasticity based on isomorphisms. Int. J. Plasticity, vol.19, pp.2027-2050.
• [13] Candija M. and Brnic J. (2004): Associative coupled thermoplasticity at finite strain with temperature-dependent material parameters. Int. J. Plasticity, vol.20, No.10, pp.1851-1874.
• [14] Casey J. (1998): On elastic-thermo-plastic materials at finite deformations. Int. J. Plasticity, vol.14, No.1-3, pp.173-191.
• [15] Epstein M. and Maugin G.A. (2000): Thermomechanics of volumetyric growth in uniform bodies. Int. J. Plasticity, vol.16, No.7-8, pp.951-978.
• [16] Hakansson P., Wallin M. and Ristinmaa M. (2002): Comparison of isotropic hardening and kinematic hardening in thermoplasticity. Int. J. Plasticity, vol.18, pp.379-397.
• [17] Huttel C. and Matzenmiller A. (1999): Extension of generalized plasticity to finite deformations and structures.Int. J. Plasticity, vol.36, pp.5255-5276.
• [18] Itskov M. (2004): On the application of the additive decomposition of generalized strain measures in large strain plasticity. Mechanics Res. Commun., vol.31, No.5, pp.507-517.
• [19] Lehmann Th. (1991): Thermodynamical foundations of large inelastic deformations of solid bodies including damage. Int. J. Plasticity, vol.7, pp.79-98.
• [20] Mahnken R. and Stein E. (1997): Parametr identification for finite deformation elasto-plasticity in principal directions. Computer Method Applied Mech. Engineering, vol.147, No.1-2, July 30, pp.17-39.
• [21] Marakin A.A. and Sokolova M.Y. (2002): Thermomechanical models of irreversible finite deformation of anisotropic solids, strength of materials. Vol.34, No.6, pp.529-535.
• [22] Maugin G.A. and Epstein M. (1998): Geometrical material structure of elastoplasticity. Int. J. Plasticity, vol.14, pp.109-115.
• [23] Meyers A., Bruhns O.T. and Xiao H. (2000): Large strain response of kinematic hardening elastoplasticity with the logarythmic rate: Swift effect in torsion. Meccanica, vol.35, No.3, pp.229-247.
• [24] Miehe C. (1995): A theory of large strain isotropic thermoplasticity based on metric transformation tensors. Archive of Applied Mechanics, vol.66, No.1/2, pp.45-64.
• [25] Srinivasa A.R. (2001): Large deformation plasticity and the poynting effect Int. J. Plasticity, vol.17, No.9, pp.1189-1214.
• [26] Xiao H., Bruhns O.T. and Meyers A. (2007): Thermodynamic laws and consistent Eulerian formulation of finie elastoplasticity with thermal effects. J. Mech. Phys. Solids, vol.55, No.2, pp.338-365.
• [27] Nguyen H.V. (1999): Thermomechanical Couplings in Elasto-Plastic Metals in the Case of Finie Deformations, DSc Thesis [in Polish]. IFTR-Reports, No 10/1999, pp.1-116, Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw.
• [28] Śloderbach Z. and Pajak J. (2010): 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 Solids, vol.15, No.3, pp.308-327.
• [29] Śloderbach Z., Pajak J. (2010): Generalized coupled thermoplasticity taking into account large strains: Part II. A model of non-compressible elastic-plastic solid with non-associated plastic flow law. Mathematics and Mechanics of Solids, vol.15, No.3, pp.28-341.
• [30] Śloderbach Z. (2014): Thermomechanical problems of deformable bodies for small deformations. Opole University of Technology, ISBN 978-8364056-54-3, pp.1-272.
• [31] Maugin G.A. (1990): Internal variables and dissipative structure. J. of Non-Equilibrium Thermodynamics, vol.15, pp.173-192.
• [32] Simo J. and Miehe C. (1992): Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Computer Method Applied Mech. Engineering, vol.98, pp.41-104.
• [33] Raniecki B. (1979): Uniqueness criteria in solids with non-associated plastic flow laws at finite deformation.Bulletin De l’Acadademie Polonaise des Sciences, Serie des Sciences Techniques, vol.27, pp.391-399.
• [34] Raniecki B. and Bruhns O.T. (1981): Bounds to bifurcation stress in solids with non-associated plastic flow At finite strain. J. Mech. Phys. Solids, vol.29, No.2, pp.153-172.
• [35] Raniecki B. and Sawczuk A. (1975) Thermal effects in plasticity. Part I: Coupled theory. Zeitschrift für Angewandte Mathematik und Mechanik - ZAMM, vol.55, No.6, pp.232-241.
• [36] Raniecki B. and Sawczuk A. (1975): Thermal effects in plasticity. Part II: Uniqueness and applications. Zeitschrift für Angewandte Mathematik und Mech.-ZAMM, vol.55, No.7/8, pp.363-373.
• [37] Mróz Z. (1963): Non-associated flow laws in plasticity. J. de Mecanique, vol.2, No.1, pp.21-42.
• [38] Mróz Z. (1966): On forms of constutive laws for elastic-plastic solids. Arch. Applied Mechanics, vol.18, No.1, Warszawa, pp.3-35.
• [39] Hueckel T. and Maier G. (1977): Incremental boundary value problems in the presence of coupling of elestic and plastik deformations. A rock mechanics oriented theory. International Journal of Solids and Structures, vol.13, pp.1-15.
• [40] Hueckel T. and Maier G. (1977): Non-associated and coupled flow rules of elastoplasticity for geotechnical media. Proc. 9-th International Conference of Soil Mechanics Foundation of Engineering (JCSFE), Tokyo, Speciality Session 7, Constutive relations for soils, pp.129-142.
• [41] Maier G. (1970): A minimum principle for incremental elastoplasticity, with non-associated flow laws. Journal Mechanics of Physics Solids, vol.18, pp.319-330.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017)