Engineering Transactions, 50, 3, pp. 107–164, 2002

Thermodynamical Theory of Inelastic Single Crystals

Piotr Perzyna
Institute of Fundamental Technological Research, Polish Academy of Sciences

The paper aims at the development of the thermodynamic theory of elasto-viscoplasticity of single crystals which takes account of the evolution of the dislocation substructure. The next objective is the application of the theory developed for the investigation of the adiabatic shear-band formation in single crystals under dynamic loading processes. The description of the kinematics of finite elasto-viscoplastic deformations of single crystal is based on notions of the Riemannian space of manifolds and the tangent space. A multiplicative decomposition of the deformation gradient is adopted and the Lie derivative is used to define all objectives rates for the introduced vectors and tensors. A general constitutive model is developed within the thermodynamic framework of the rate-type covariance constitutive structure with finite set of the internal state variables, and takes account of the effects as follows: (i) thermomechanical coupling; (ii) influence of covariance terms, lattice deformations and rotations and plastic spin; (iii) evolution of the dislocation substructure; (iv) deviation from the Schmidt rule of a critical resolved shear stress for slip; (v) rate sensitivity (viscosity). A notion of covariance is understood in the sense of invariance under arbitrary spatial diffeomorphisms. The developed thermo-viscoplasticity theory of single crystals is based on the axioms as follows: (i) existence of the free energy function; (ii) invariance with respect to any diffeomorphism (any superposed motion); (iii) assumption of the entropy production inequality; (iv) assumption of the evolution equations for the internal state variables in the particular rate-dependent form. To describe the evolution of the dislocation substructure, a finite set of the internal state variables is interpreted as follows: the density of mobile dislocations, the density of obstacle dislocations and the concentration of the point defects. Physical foundations and experimental motivations are given. Two fundamental constitutive equations of the rate-type for the Kirchhoff stress tensor and temperature are formulated. To show that the thermodynamic theory of viscoplasticity of single crystals takes account of all the mentioned effects, an analysis of the thermomechanical couplings and internal dissipation is presented. Particular attention is focused on synergetic effects, generated by cooperative phenomena of thermomechanical couplings and the influence of the evolution of the dislocation substructure. The initial boundary value problem (the evolution problem) for rate-dependent elasto-plastic single crystal has been proved to be well posed. Criteria for adiabatic shear-band localization of plastic deformation are obtained by assuming that some eigenvalue of the instantaneous adiabatic acoustic tensor for rate-independent response is equal to zero. The formation of the adiabatic shear-band is investigated. It has been found that the synergetic effects generated by cooperative phenomena of thermomechanical couplings and the influence of the evolution of the dislocation substructure play a fundamental role in the inception of localization. The results obtained are compared with available experimental observations.
Full Text: PDF


R. ABRAHAM, J.E. MARSDEN and T. RATIU, Manifolds, tensor analysis and applications, Springer, Berlin 1988.

R.J. ASARO, Micromechanics of crystals and polycrystals, Adv. Appl. Mech., 23, 1–115, 1983.

R.J. ASARO, J.R. RICE, Strain localization in ductile single crystals, J. Mech. Phys. Solids, 25, 309–338, 1977.

H. BALKE, Y. ESTRIN, Micromechanical modelling of shear-banding in single crystals, Int. J. Plast., 10, 133–147, 1994.

S.J. BASIŃSKI, Z.S. BASIŃSKI, Plastic deformation and work hardening, Dislocations in Solids, Vol.4 Dislocations in Metallurgy, F.R.N. NABARRO [Ed.], 261–362, Nort-Holland, Amsterdam 1979.

J.L. BASSANI, Plastic flow of crystals, Adv. Appl. Mech., 30, 191–258, 1994.

Y.W. CHANG, R.J. ASARO, Lattice rotations and shearing in crystals, Arch. Mech., 32, 369–388, 1980.

Y.W. CHANG, R.J. ASARO, An experimental study of shear localization in aluminum-copper single crystals, Acta Metall., 29, 241–257, 1981.

B.D. COLEMAN, M.E. GURTIN, Thermodynamics with internal state variables, J. Chem. Phys, 47, 597–613, 1967.

B.D. COLEMAN, W. NOLL, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal, 13, 167–178, 1963.

H. CONRAD, Thermally activated deformation of metals, J. Metals, 16, 582–588, 1964.

W.A. DAY, M. SILHAVY, Efficiency and the existence of entropy in classical thermodynamics, Arch. Rational Mech. Anal, 64, 205–219, 1977.

M.K. DUSZEK, P. PERZYNA, The localization of plastic deformation in thermoplastic solids, Int. J. Solids Structures, 27, 1419–1443, 1991.

M.K. DUSZEK-PERZYNA, P. PERZYNA, Adiabatic shear-band localization in elastic-plastic single crystals, Int. J. Solids Structures, 30, 61–89, 1993.

M.K. DUSZEK-PERZYNA, P. PERZYNA, Adiabatic shear-band localization of inelastic single crystals in symmetric double slip process, Archive of Applied Mechanics, 66, 369–384, 1996.

M.K. DUSZEK-PERZYNA, P. PERZYNA and E. STEIN, Adiabatic shear-band localization in elastic-plastic damaged solids, Int. J. Plasticity, 8, 361–384, 1992.

Y. ESTRIN, L.P. KUBIN, Load strain hardening and nonuniformity of plastic deformation, Acta Metall., 34, 2455–2464, 1986.

A.G. EVANS , R.D. KUMBLE, The thermally activated deformation of crystalline materials, Phys. Stat. Sol, 34, 9–31, 1969.

P.S. FOLLANSBEE, Metallurgical applications of shock–wave and high-strain-rate phenomena, L.E. MURR, K.P. STAUDHAMMER, M.A. MEYERES [Eds.], 451–480, Marcel Dekker, New York 1986.

P.S. FOLLANSBEE, U.F. KOCKS, A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable, Acta Met., 36, 81–93, 1988.

P. GLANSDORFF, I. PRIGOGINE, Thermodynamic theory of structure, stability and fluctuations, Wiley-Interscience, London 1977.

J. A. GORMAN, D.S. WOOD and T. VREELAND, Mobility of dislocation in aluminium, J. Appl. Phys., 40, 833–841, 1969.

A.E. GREEN, P.M. NAGHDI, On thermodynamics and the nature of the second law, Proc. R. Soc. Lond., A357, 253–270, 1977.

A.E. GREEN, P.M. NAGHDI, On thermodynamics and the nature of the second law for mixtures of interacting continua, Quart. J. Mech. Appl. Maths., 31, 265–293, 1978.

M.E. GURTIN, Thermodynamics and stability, Arch. Rational Mech. Anal., 59, 63–96, 1975.

J. HADAMARD, Lecons sur la propagation des on des et les equations de l'hydrodynamique, Chap. 6, Paris 1903.

H. HAKEN, Cooperative phenomena in systems far from thermal equilibrium and in non-physical systems, Reviews of Modern Physics, 47, 67–121, 1975.

H. HAKEN, Advanced Synergetics, Springer, Berlin 1987.

H. HAKEN, Information and self-organization, Springer, Berlin 1988.

R. HILL, Acceleration wave in solids, J. Mech. Phys. Solids, 10, 1–16, 1962.

R. HILL, J.R. RICE, Constitutive analysis of elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids, 20, 401–413, 1972.

G. JAUMANN, Geschlossenes System physikalischer und chemischer Differentialgesetze, Sitzungsber. Akad. Wiss. Wien (Ha), 120, 385–530, 1911.

U.F. KOCKS, A.S. ARGON and M.F. ASHBY, Thermodynamics and kinetics of slip, Pergamon Press, 1975.

A. KUMAR, R.G. KUMBLE, Viscous drag on dislocations at high strain-rates in copper, J. Appl. Physics, 40, 3475–3480, 1969.

L.L. LISIECKI, D.R. NELSON and R.J. ASARO, Lattice rotations, necking and localized deformation in f.c.c. single crystals, Scripta Met., 16, 441–449, 1982.

J.E. MARSDEN, T. J.R. HUGHES, Mathematical foundations of elasticity, Prentice-Hall, Englewood Cliffs, New York 1983.

W.P. MASON, Phonon viscosity and its effect on acoustic wave attenuation and dislocation motion, J. Acoustical Soc. Amer., 32, 458–472, 1960.

J.J. MASON, A. J. ROSAKIS and R. RAVICHANDRAN, On the strain and strain-rate dependence of the fraction of plastic work converted to heat: an experimental study using high speed infrared detectors and the Kolsky bar, Mechanics of Materials, 17, 135–145, 1994.

H. MECKING, U.F. KOCKS, Kinetics of flow and strain-hardening, Acta Metall., 29, 1865–1875, 1981.

I. MÜLLER, On the entropy inequality, Arch. Rational Mech. Anal., 26, 118–141, 1967.

I. MÜLLER, Thermodynamics of mixtures of fluids, J. Mécanique, 14, 267–303, 1975.

F.N.R. NABARRO, Theory of crystal dislocations, Oxford 1967.

S. NEMAT-NASSER, Phenomenological theories of elastoplasticity and strain localization at high strain-rates, Appl. Mech. Rev., 45, S19–S45, 1992.

G. NICOLIS, I. PRIGOGINE, Self-organization in nonequilibrium systems, Wiley-Interscience, New York 1977.

J. OLDROYD, On the formulation of rheological equations of state, Proc. R. Soc. Lond., A200, 523–541, 1950.

D. PEIRCE, J.R. ASARO and A. NEEDLEMAN, An analysis of nonuniform and localized deformation in ductile single crystals, Acta Metall., 30, 1087–1119, 1982.

D. PEIRCE, J.R. ASARO and A. NEEDLEMAN, Material rate dependence and localized deformation in crystalline solids, Acta Metall., 31, 1951–1976, 1983.

P. PERZYNA, Coupling of dissipative mechanisms of viscoplastic flow, Arch. Mechanics, 29, 607–624, 1977.

P. PERZYNA, Temperature and rate-dependent theory of plasticity of crystalline solids, Revue Phys. Appl., 23, 445–459, 1988.

P. PERZYNA, Instability phenomena and adiabatic shear-band localization in thermoplastic flow processes, Acta Mechanica, 106, 173–205, 1994.

P. PERZYNA, Thermodynamics of crystal viscoplasticity and instability phenomena, Material Instability in Solids, (R. de Borst, E. van Giessen, [Eds.]), 65–89, John Wiley and Sons, New York 1998.

P. PERZYNA, M.K. DUSZEK-PERZYNA, Constitutive modelling of inelastic single crystals for localization phenomena, Constitutive Laws: Experiments and Numerical Implementation, A.M. RAJENDRAN, R.C. BATRA [Eds.], 70–83., CIMME, Barcelona 1995.

P. PERZYNA, K. KORBEL, Analysis of the influence of substructure of crystal on the localization phenomena of plastic deformation, Mechanics of Materials, 24, 141–158, 1996.

P. PERZYNA, K. KORBEL, Analysis of the influence of various effects on criteria for adiabatic shear-band localization in single crystals, Acta Mechanica, 129, 31–62, 1998.

Q. QIN, J.L. BASSANI, Non-Schmid yield behavior in single crystals, J. Mech. Phys. Solids, 40, 813–833, 1992.

Q. QIN, J.L. BASSANI, Non-associated plastic flow in single crystals, J. Mech. Phys. Solids, 40, 835–862, 1992.

M.M. RASHID, G.T. GRAY and S. NEMAT-NASSER, Heterogeneous deformations n copper single crystals at high and Iow strain-rates, Philosophical Magazine A, 65, 707–735, 1992.

J.R. RICE, The localization of plastic deformation, Theoretical and Applied Mechanics, W.T. KOITER [Ed.], 207–220. North-Holand, 1976.

J. W. RUDNICKI, J.R. RICE, Conditions for the localization of deformation in pressure-sensitive dilatant materials, J. Mech. Phys. Solids, 23, 371–394, 1975.

A. SEEGER, The generation of lattice defects by moving dislocations and its application to the temperature dependence of the flow-stress of f.c.c. crystals, Phil. Mag., 46, 1194–1217, 1955.

A. SEEGER, Kristalplastizitat, Handbuch der Physik VII/2, S. FLÜGGE [Ed.], 1–208, Springer-Verlag, 1958.

J. SERRIN, Conceptual analysis of the classical second laws of thermodynamics, Arch. Rational Mech. Anal., 70, 355–371, 1979.

I.S. SOKOLNIKOFF, The mathematical theory of elasticity, (2nd. ed.), Mc Graw-Hill, New York 1956.

W.A. SPITZIG, Deformation behaviour of nitrogenated Fe-Ti-Mn and Fe-Ti single crystals, Acta Metali., 29, 1359–1377, 1981.

G.I. TAYLOR, Analysis of plastic strain in a crystal, Stephen Timoshenko 60th Anniversary Volume, J.M. LESSELS [Ed.], MacMillan, New York 1938.

G.I. TAYLOR, H. QUINNEY, The latent energy remaining in a metal after cold working, Proc. R. Soc. Lond., A143, 307–326, 1934.

C. TEODOSIU, F. SIDOROFF, A theory of finite elastoplasticity of single crystals, Int. J. Engng. Sci., 14, 165–176, 1976.

C. TRUESDELL, Rational Thermodynamics, Mc Graw-Hill, New York 1969.

C. TRUESDELL, W. NOLL, The non-linear field theories of mechanics, Handbuch der Physik III/3, S. FLUGGE [Ed.], Springer-Verlag, Berlin 1965.

J.C. WILLEMS, Dissipative dynamical systems, Arch. Rat. Mech. Anal., 45, 321–393, 1972.

S. ZAREMBA, Sur une forme perfectionnée de la théorie de la relaxation, Bull. Int. Acad. Sci. Cracovie, 594–614, 1903.

S. ZAREMBA, Le principe des mouvements relatifs et les éguations de la mécanique physigue, Bull. Int. Acad. Sci. Cracovie, 614–621, 1903.

Copyright © 2014 by Institute of Fundamental Technological Research
Polish Academy of Sciences, Warsaw, Poland