Engineering Transactions, 53, 3, pp. 235–316, 2005
10.24423/engtrans.443.2005

The Thermodynamical Theory of Elasto-Yiscoplasticity (Review Paper)

P. Perzyna
Polish Academy of Sciences
Poland

The main objective of this paper is to survey some recent developments in the constitutive modelling of inelastic polycrystalline solids, which may be used for the description of important problems in modern manufacturing processes, and particularly for mesomechanical issues. This description is needed for the investigation by using the numerical methods how to avoid unexpected plastic strain localization and fracture phenomena in manufacturing technology. Since modern manufacturing processes lead to very complex states of stress and deformation for a solid body under consideration, then in the description we have to take into account the influence of stress triaxiality and plastic spin effects. In this paper emphasis is laid on experimental and physical foundations as well as on mathematical constitutive modelling for the description of localization of plastic deformation and various modes of fracture phenomena in polycrystalline solids. The description of kinematics of finite deformations and the stress tensors is given. The development of a thermo-elasto-viscoplastic model within the thermo-dynamic framework of the rate-type covariance constitutive structure with finite set of the internal state variables is presented. Particular attention is focused on the determination of the evolution laws for the internal state variables. Fracture criterion based on the evolution of microdamage is formulated. By assuming that the mechanical relaxation time is equal to zero, the thermo-elasto-plastic (rate-independent) response of the damaged material can be accomplished. The thermodynamical theory of elasto-viscoplasticity of polycrystalline solids presented has important features as follows: (i) invariance with respect to diffeomorphism; (ii) finite plastic deformation and plastic spin effects; (iii) plastic non-normality; (iv) softening effects generated by microdamage mechanism; (v) plastic deformation-induced anisotropic effects; (vi) thermomechanical couplings (thermal plastic softening and thermal expansion); (vii) influence of stress triaxiality on the evolution of microdamage; (viii) ratę sensitivity; (ix) length scale sensitivity; (x) regularization of the evolution problem; (xi) dissipation and dispersion effects; (xii) synergetic effects generated by cooperative phenomena. Ali these fundamental features have been carefully discussed. It should be noted that the very important part of constitutive modelling is the identification procedure for the material functions and constants involved in the constitutive equations proposed.
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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

References

R. ABRAHAM, J.E. MARSDEN, Foundations of mechanics, Second Edition, Addison-Wesley, Reading Mass., 1978.

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

A. AGAH-TEHRANI, E.H. LEE, R.L. MALETT, E.T. ONAT, The theory of elastic-plastic deformation at finite strain with induced anisotropy modelled isotropic-kinematic hardening, J. Mech. Phys. Solids, 35, 43–60, 1987.

A. AGAH-TEHRANI, E.H. LEE, R.L. MALLETT, E.H. ONAT, The theory of elastic-plastic deformation at finite strain with induced anisotropy modelled as combined isotropic-kinematic hardening, J. Mech. Phys. Solids, 35, 519–539, 1987.

P.J. ARMSTRONG, C.O. FREDERIGK, A mathematical representation of the multiaxial Baushinger effect, CEGB Report RD/B/N731, Central Electricity Generating Board, 1966.

R. J. ASARO, Crystal plasticity, J. Appl. Mech., 50, 921–934, 1983.

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

R.J. ASARO, A. NEEDLEMAN, Texture development and strain hardening in rate dependent polycrystals, Acta Metall., 33, 923–953, 1985.

T.W. BARBEE, L. SEAMAN, R. CREWDSON, D. CURRAN, Dynamic fracture criteria for ductile and brittle metals, J. Mater., 7, 393–401, 1972.

J.D. CAMPBELL, W.G. FERGUSON, The temperature and strain-rate dependence of the shear strength of mild steel, Phil. Mag., 81, 63–82, 1970.

M.M. CARROLL, A.C. HOLT, Static and dynamic pore-collapse relations for ductile solids, J. Appl. Phys., 43, 1626–1636, 1972.

A.K. CHAKRABARTI, J. W. SPRETNAK, Instability of plastic flow in the direction of pure shear, Metallurgical Transactions, 6A, 733–747, 1975.

S. CHENGWEI, Z. SHIMING, W. YANPING, L. CANGLI, Dynamic fracture in metals at high strain rate, in: High–Pressure Shock Compression of Solids, II. Dynamic Fracture and Fragmentation, L. DAYISON, D.E. GRADY and M. SHAHINPOOR [Eds.], 71–89, Springer Verlag, New York 1996.

K. CHO, Y.C. Cm, J. DUFFY, Microscopic observations of adiabatic shear bands in three different steels, Brown University Report, 1989.

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.

D.R. CURRAN, L. SEAMAN, D.A. SHOCKEY, dynamic failure in solids, Physics Today, January, 46–55, 1977.

D.R. CURRAN, L. SEAMAN, D. A. SHOCKEY, Linking dynamic fracture to microstructural processes, in: Shock Waves and High-Strain Rate Phenomena in Metals: Concepts and Applications, M. A. MEYERS and L.E. MURR [Eds.], 129–167, Plenum Press, New York 1981.

D.R. CURRAN, L. SEAMAN, D. A. SHOCKEY, Dynamic failure of solids, Physics Reports, 147, 253–388, 1987.

Y.F. DAFALIAS, Corotational rates for kinematic hardening at large plastic deformations, J. Appl. Mech., 50, 561–565, 1983.

Y.F. DAFALIAS, The plastic spin concept and simple illustration of Us rule in final plastic transformations, Mech. Mater., 3, 223, 1984.

Y.F. DAFALIAS, The plastic spin, J. Appl. Mech., 52, 865–871, 1985.

Y.F. DAFALIAS, Issues on the constitutive formulation at large elastoplastic deformations, Part 1: Kinematics, Acta Mechanica, 69, 119, 1987.

Y.F. DAFALIAS, Issues on the constitutive formulation at large elastoplastic deformations, Part 2: Kinetics, Acta Mechanica, 73, 121, 1988.

R. DAUTRAY, J.-L. LIONS, Mathematical analysis and numerical methods for science and technology, Vol. 6. Evolution Problems II, Springer, Berlin 1993.

W. DORNOWSKI, P. PERZYNA, Constitutive modelling of inelastic solids for plastic flow processes under cyclic dynamic loadings, Transaction of the ASME, J. Eng. Materials and Technology, 121, 210–220, 1999.

W. DORNOWSKI, P. PERZYNA, Numerical solutions of thermos-viscoplastic flow processes under cyclic dynamic loadings, in: Proc. Euromech Colloquium 383, Inelastic Analysis of Structures under Variable Loads: Theory and Engineering Applications, D. WEICHERT and G. MAIER [Eds.], 69–94, Kluwer Academic Publishers, 2000.

W. DORNOWSKI, P. PERZYNA, Localization phenomena in thermo–viscoplastic flow processes under cyclic dynamic loadings, Computer Assisted Mechanics and Engineering Sciences, 7, 117–160, 2000.

W. DORNOWSKI, P. PERZYNA, Analysis of the influence of various effects on cycle fatigue damage in dynamic processes, Arch. Applied Mechanics, 72, 418–438, 2002.

W. DORNOWSKI, P. PERZYNA, Localized fracture phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings, Acta Mechanica, 155, 233–255, 2002.

W. DORNOWSKI, P. PERZYNA, Numerical analysis of macrocrack propagation along a bimaterial interface under dynamic loading processes, Int. J. Solids and Structures, 39, 4949–4977, 2002.

A.R. DOWLING, J. HARDING, J.D. CAMPBELL, The dynamic punching of metals, J. Inst. of Metals, 98, 215–224, 1970.

M.K. DUSZEK, P. PERZYNA, Plasticity of damaged solids and shear band localization, Ing. Arch., 58, 330–392, 1988.

M.K. DUSZEK, P. PERZYNA, Influence of the kinematic hardening on the plastic flow localization in damaged solids, Arch. Mech., 40, 595–609, 1988.

M.K. DUSZEK, P. PERZYNA, On combined isotropic and kinematic hardening effects in plastic flow processes, Int. J. Plasticity, 7, 351–363, 1991.

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, K. KORBEL, P. PERZYNA, Adiabatic shear band localization in single crystals under dynamic loading processes, Arch. Mechanics, 49, 1069–1090, 1997.

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

M.K. DUSZEK-PERZYNA, P. PERZYNA, Analysis of the influence of different effects on criteria for adiabatic shear band localization in inelastic solids, in: Material Instabilities: Theory and Applications, ASME Congress, Chicago, 9–11 November 1994, R.C. BATRA and H.M. ZBIB [Eds.], 59–85, AMD–Vol. 183/MD–Vol.50, ASME, New York, 1994.

M.K. DUSZEK-PERZYNA, P. PERZYNA, Acceleration waves in analysis of adiabatic shear band localization, in: Nonlinear Waves in Solids, Proc. IUTAM Symposium, August 15–20, 1993, Victoria, Canada; J.L. WEGNER and F.R. NORWOOD [Eds.], 128–135, ASME Book No AMR 137, 1995.

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, Analysis of anisotropy and plastic spin effects on localization phenomena, Arch. Appl. Mechanics, 68, 352–374, 1998.

J. EFTIS, Constitutive modelling of spall fracture, in: High-Pressure Shock Compression of Solids, II. Dynamic Fracture and Fragmentation, L. DAYISON, D.E. GRADY and M. SHAHINPOOR [Eds.], 399–451, Springer Verlag, New York 1996.

J. EFTIS, J.A. NEMES, Constitutive modelling of spall fracture, Arch. Mech., 43, 399–435, 1991.

J. EFTIS, J.A. NEMES, Evolution equation for the void volume growth rate in a viscoplastic–damage constitutive model, Int. J. Plasticity, 7, 275–293, 1991.

J. EFTIS, J.A. NEMES, Modelling of impact-induced spall fracture and post spall behaviour of a circular plate, Int. J. Fracture, 53, 301–324, 1992.

A.G. EVANS, R.D. RAWLINGS, 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 and M.A. MEYERES [Eds.], 451–480, Marcel Dekker, New York 1986.

K.O. FRIEDRICHS, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7, 345–392, 1954.

K.O. FRIEDRICHS, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11, 333–418, 1958.

I. GILATH, Laser–induced spallation and dynamic fracture at ultrahigh strain rate, in: High-Pressure Shock Compression of Solids, II. dynamic Fracture and Fragmentation, L. DAYISON, D.E. GRADY and M. SHAHINPOOR [Eds.], 90–120, Springer Verlag, New York 1996.

J.H. GIOYANOLA, Adiabatic shear banding under pure shear loading, Mechanics of Materials, 7, 59–87, 1988.

A. GLEMA, T. ŁODYGOWSKI, P. PERZYNA, Interaction of deformation waves and localization phenomena in inelastic solids, Computer Methods in Applied Mechanics and Engineering, 183, 123–140, 2000.

A. GLEMA, T. ŁODYGOWSKI, P. PERZYNA, The role of dispersion for the description of strain localization in materials under impact loading, European Conference on Computational Mechanics, June 26–29, Cracow, Poland, 2001.

A. GLEMA, T. ŁODYGOWSKI, P. PERZYNA, Localization of plastic deformations as a result of wave interaction, CAM&ES, 3, 81–91, 2003.

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

H.A. GREBE, H.R. PAK, M.A. MEYER, Adiabatic shear band localization in titanium and Ti-6PctAl-4PctV alloy, Met. Trans., 16A, 761–775, 1985.

A.L. GURSON, Plastic flow and fracture behaviour of ductile materials incorporating void nucleation, growth, and interaction, PhD Thesis, Brown University, 1975.

A.L. GURSON, Continuum theory of ductile rupture by void nucleation and growth – Part I – Yield criteria and flow rules for porous ductile media, J. Eng. Mater. Technology, 99, 2–15, 1977.

B. GUSTAFSSON, H.O. KREISS, J. OLIGER, Time dependent problems and difference methods, John Wiley, New York 1995.

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.

K.J. HALE, dynamic systems and stability, J. Math. Anal. Appl., 26, 39–59, 1969.

B. HALPHEN, Sur le champ des vitesses en thermoplasticité finie, Int. J. Solids Structures, 11, 947, 1975.

K.A. HARTLEY, J.DUFFY, R.H. HAWLEY, Measurement of the temperature profile during shear band formulation in steels deforming at high strain rates, J. Mech. Phys. Solids, 35, 283–301, 1987.

F.E. HAUSER, J. A. SIMMONS, J.E. DORN, Strain rate effects in plastic wave propagation,

in: Response of Metals to High Velocity Deformation, 93–114, Wiley (Interscience), New York 1961.

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

R. HILL, J.R. RICE, Elastic potentials and the structure of inelastic constitutive laws, SIAM J. Appl. Math., 25, 448–461, 1973.

K. HOHENEMSER, W. PRAGER, Über die Ansätze der Mechanik isotroper Kontinua, ZAMM, 12, 216–226, 1932.

T.J.R. HUGHES, T. KATO, J.E. MARSDEN, Well-posed quasilinear second order hyperbolic system with application to nonlinear elastodynamics and general relativity, Arch. Rat. Mech. Anal., 63, 273–294, 1977.

T. J.R. HUGHES, J.E. MARSDEN, Classical elastodynamics as a linear symmetric hyperbolic systems, J. Elasticity, 8, 97–110, 1978.

K. IKEGAMI, Experimental plasticity on the anisotropy of metals, in: Proc. Euromech Colloquium 115, Mechanical Behaviour of Anisotropic Solids, J.P. BIEHLER [Ed.], 201–242, 1982.

I.R. IONESCU, M. SOFONEA, Functional and numerical methods in viscoplasticity, Oxford 1993.

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

J.N. JOHNSON, dynamic fracture and spallation in ductile solids, J. Appl. Phys., 52, 2812–2825, 1981.

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

J. KRATOCHYIL, Finite-strain theory of crystalline elastic-inelastic materials, J. Appl. Phys., 42, 1104, 1971.

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

U.S. LINDHOLM, Some experiments with the split Hopkinson pressure bar, J. Mech. Phys. Solids, 12, 317–335, 1964.

U.S. LINDHOLM, in: Mechanical behaviour of materials under dynamic loads, U.S. LIND HOLM [Ed.], 77–95, Springer Verlag 1968.

U.S. LINDHOLM, L.M. YEAKLEY, dynamic deformation of single and polycrystalline aluminium, J. Mech. Phys. Solids, 13, 41–53, 1965.

B. LORET, On the effects of plastic rotation in the finite deformation of anisotropic elasto-plastic materials, Mech. Mater., 2, 287–304, 1983.

T. ŁODYGOWSKI, P. PERZYNA, Numerical modelling of localized fracture of inelastic solids in dynamic loading processes, Int. J. Num. Meth. Engng., 40, 4137–4158, 1997.

T. ŁODYGOWSKI, P. PERZYNA, Localized fracture of inelastic polycrystalline solids under dynamic loading processes, Int. J. Damage Mechanics, 6, 364–407, 1997.

L.E. MALYERN, The propagation of longitudinal waves of plastic deformation in a bar of material exhibiting a strain–rate effects, J. Appl. Mech., 18, 203–208, 1951.

J. MANDEL, Plasticite classiqué et viscoplasticite, CISM Lecture Notes No. 97, Udine, Springer Verlag, Vien 1971.

J. MANDEL, Equations constitutives et directeurs dans les milieux plastigues et viscoplastiques, Int. J. Solids Structures, 9, 725–740, 1973.

J. MANDEL, Définition d’un repère privilégé pour l’etude des transformations anélastiques du polycrystal, 3. Méc. Théo. Appl., l, 7, 1982.

A. MARCHAND, K. CHO, J. DUFFY, The formation of adiabatic shear bands in an AISI 1018 cold–rolled steel, Brown University Report 1988.

A. MARCHAND, J. DUFFY, An experimental study of the formation process of adiabatic shear bands in a structural steel, J. Mech. Phys. Solids, 36, 251–283, 1988.

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, J. A. ROSAKIS, R. RAYICHANDRAN, 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.

M.E. MEAR, J.E. HUTCHINSON, Influence of yield surface curvature on flow localization in dilatant plasticity, Mech. Mater., 4, 395–407, 1985.

H.C. MEYERS, Dynamic behaviour of materials, John Wiley, New York 1994.

M. A. MEYERS, C.T. AIMONE, Dynamic fracture (spalling) of metals, Prog. Mater. Sci., 28, 1–96, 1983.

S.K. MITRA, J.E. DORN, On the nature of strain hardening in face-centered cubic metals, Trans. AIME, 224, 1062–1071, 1962.

F.R.N. 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.

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

P. PERZYNA, The constitutive equations for rate sensitive plastic materials, Quart. Appl. Math., 20, 321–332, 1963.

P. PERZYNA, Fundamental problems in viscoplasticity, Advances in Applied Mechanics, 9, 243–377, 1966.

P. PERZYNA, Thermodynamic theory of viscoplasticity, Advances in Applied Mechanics, 11, 313–354, 1971.

P. PERZYNA, Thermodynamics of a unique material structure, Arch. Mechanics, 27, 791–806, 1975.

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

P. PERZYNA, Modified theory of viscoplasticity. Application to advanced flow and instability phenomena, Arch. Mechanics, 32, 403–420, 1980.

P. PERZYNA, Thermodynamics of dissipative materials, in: Recent Developments in Thermodynamics of Solids, G. LEBON and P. PERZYNA [Eds.], 95–220, Springer, Wien 1980.

P. PERZYNA, Stability phenomena of dissipative solids with internal defects and imperfections, in: Proc. XV-th IUTAM Congress, Toronto, August 1980, Theoretical and Applied Mechanics, F.P.J. RIMROTT and B. TABARROK [Eds.], 369–376, North-Holland, Amsterdam 1981.

P. PERZYNA, Stability problems for inelastic solids with defects and imperfections, Arch. Mechanics, 33, 587–602, 1981.

P. PERZYNA, Application of dynamical system methods to flow processes of dissipative solids, Arch. Mechanics, 34, 523–539, 1982.

P. PERZYNA, Stability of flow processes for dissipative solids with internal imperfections, ZAMP, 35, 848–867, 1984.

P. PERZYNA, Constitutive modelling of dissipative solids for post-critical behaviour and fracture, ASME J. Eng. Materials and Technology, 106, 410–419, 1984.

P. PERZYNA, Dependence of fracture phenomena upon the evolution of constitutive structure of solids, Arch. Mechanics, 37, 485–501, 1985.

P. PERZYNA, Internal state variable description of dynamic fracture of ductile solids, Int. J. Solids Structures, 22, 797–818, 1986.

P. PERZYNA, Constitutive modelling for brittle dynamic fracture in dissipative solids, Ach. Mechanics, 38, 725–738, 1986.

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

P. PERZYNA, Influence of anisotropic effects on micro-damage process in dissipative solids, in: Proc. IUTAM/ICM Symposium on Yielding, Damage and Failure of Anisotropic Solids, Villerd-de-Lance, August 1987, 483–507, Mech. Eng. Publl. Limited, London 1990.

P. PERZYNA, Constitutive equations for thermoplasticity and instability phenomena in thermodynamic flow processes, in: Progress in Computational Analysis of Inelastic Structures, E. STEIN [Ed.], 1–78, Springer Verlag, Wien 1993.

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

P. PERZYNA, Interactions of elastic-viscoplastic waves and localization phenomena in solids, IUTAM Symposium on Nonlinear Waves in Solids, August 15–20, 1993, Victoria, Canada; J.L. IVEGNER and F.R. NORWOOD [Eds.], 114–121, ASME 1995.

P. PERZYNA, Constitutive modelling of dissipative solids for localization and fracture, in: Localization and Fracture Phenomena in Inelastic Solids, P. PERZYNA [Ed.], 99–242, Springer, Wien, New York 1998.

P. PERZYNA, Thermo-elasto-viscoplasticity and damage, in: Handbook of Materials Behaviour Models, J. LEMAITRE [Ed.], 821–834, Academic Press, New York 2001.

P. PERZYNA, Thermodynamical theory of inelastic single crystals, Engineering Transactions, 50, 107–164, 2002.

P. PERZYNA, A. DRABIK, Influence of thermal effects on micro-damaged solids, Arch. Mechanics, 40, 795–805, 1988.

P. PERZYNA, A. DRABIK, Description of micro-damage process by porosity parameter for nonlinear viscoplasticity, Arch. Mechanics, 41, 895–908, 1989.

P. PERZYNA, A. DRABIK, Micro-damage mechanism in adiabatic processes, Engineering Transactions, (in print), 2005.

P. PERZYNA, M.K. DUSZEK-PERZYNA, Constitutive modelling of inelastic single crystals for localization phenomena, in: Constitutiye Laws: Experiments and Numerical Implementation, A.M. RAJENDRAN and 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 variuos effects on criteria for adiabatic shear band localization in single crystals, Acta Mechanica, 129, 31–62, 1998.

A. PHILIPS, W. Y. Lu, Ań experimental investigation of yield surface and loading surface of pure aluminium with stress-controlled and strain-controlled paths of loading, ASME J. Eng. Mater. Technol., 106, 349–354, 1984.

W. PRAGER, The theory of plasticity: a survey of recent achievements, (J. Clayton Lecture). Proc. Inst. Mech. Eng., 169, 41–57, 1955.

W. PRAGER, Introduction to mechanics of continua, Gin and Co., New York 1961.

R.D. RICHTMYER, Principles of advanced mathematical physics, Vol. I, Springer, New York 1978.

R.D. RICHTMYER, K.W. MORTON, Difference methods for initial-value problems, John Wiley, New York 1967.

H.C. ROGERS, C.V. SHASTRY, Material factors in adiabatic shearing in steels, in: Shock Waves and High-Strain-Rate Phenomena in Metals, M.A. MEYERS and L.E. MURR [Eds.], 285–298, Plenum, New York 1981.

A.R. ROSENFIELD, G.T. HAHN, Numerical description of the ambient low-temperature, and high-strain rate flow and fracture behaviour of plain carbon steel, Trans. Am. Soc. Metals, 59, 962–980, 1966.

L. SEAMAN, T.W. BARBEE, D.R. CURRAN, Stanford Res. Inst. Tech. Rep., No. AFWL-TR-71-156, Dec. 1971.

L. SEAMAN, D.R. CURRAN, D. A. SHOCKEY, Stanford Res. Inst. Tech. Rep., No. AFWL-TR-71-156, Dec. 1971.

L. SEAMAN, D.R. CURRAN, D.A. SHOCKEY, J. Appl. Phys., 47, 4814–4820, 1976.

L. SEAMAN, D.R. CURRAN, W. J. MURRI, A continuum model for dynamic tensile microfracture and fragmentation, J. Appl. Mech., 52, 593–600, 1985.

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

A. SEEGER, Kristalplastizitat, in: Handbuch der Physik YII/2, S. FLUGGE [Ed.], 1–208, Springer Verlag 1958.

R.T. SHIELD, H. ZIEGLER, On Prager's hardening rule, ZAMP, 9a, 260–276, 1958.

D.A. SHOCKEY, L. SEAMAN, D.R. CURRAN, in: Metallurgical effects at high strain rates, R.W. ROHDE, B.M. BUTCHER, J.R. HOLLAND and C.H. KARBES [Eds.], 473, Plenum Press, New York 1973.

D. A. SHOCKEY, L. SEAMAN, D.R. CURRAN, The microstatistical fracture mechanics approach to dynamie fracture problem, Int. J. Fracture, 27, 145–157, 1985.

D.A. SHOCKEY, L. SEAMAN, K.C. DAO, D.R. CURRAN, Kinetics of void development in fracturing A533B tensile bars, Trans. ASME J. Pressure Vessel. Tech., 102, 14–21, 1980.

D. SIDEY, L.F. COFFIN, Low-cycle fatigue damage mechanisms at high temperature, in: Fatigue Mechanisms, Proc. ASTM STP 675 Symposium, Kansas City, Mo., May 1978, J.T. FONG [Ed.], 528–568, Baltimore 1979.

J.C. SIMO, T.J.R. HUGHES, Computational inelasticity, springer, New York 1998.

L.J. SLUYS, Wave propagation, localization and dispersion in softening solids, Doctoral Thesis, Delft University Press, Delft 1992.

G. STRANG, G. J. Fix, An analysis of the finite element method, Prentice-Hall, Englewood Cliffs 1973.

G.I. TAYLOR, H. GJUINNEY, The latent energy remaining in a metal after cold working, Proc. R. Soc. Lond., A143, 307–326, 1934 M.E. TAYLOR, Partial differential equations, vol. III Nonlinear Equations, Springer, New York 1996.

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

T. Y. THOMAS, Plastic fluid and fracture of solids, Academic Press, New York 1961.

S.P. TIMOTHY, The structure of adiabatic shear bands in metals: a critical review, Acta Metali., 35, 301–306, 1987.

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

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

V. TYERGAARD, Effects of yield surface curvature and void nucleation on plastic flow localization, J. Mech. Phys. Solids, 35, 43–60, 1987.

E. VAN DER GIESSEN, Continuum models of large deformation plasticity, Part I: Large deformation plasticity and the concept of a natural reference state, Eur. J. Mech., A/Solids, 8, 15, 1989.

E. VAN DER GIESSEN, Continuum models of large deformation plasticity, Part II: A kinematic hardening model and the concept of a plastically induced orientational structure, Eur. J. Mech., A/Solids, 8, 89, 1989.

E. VAN DER GIESSEN, Micromechanical and thermodynamic aspects of the plastic spin, Int. J. Plasticity, 7, 365–386, 1991.

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

C.L. WITTMAN, M.A. MEYERS, H.-R. PAK, Met. Trans., 21A, 707, 1990.

P. J. WRAY, Strain-rate dependence of the tensile failure of polycrystalline material at elevated temperatures, J. Appl. Phys., 46, 4018–4029, 1969.

A.T. YOKOBORI, T.JR. YOKOBORI, K. SATO, K. SYO, Fatigue crack growth under mixed modes I and II, Fatigue Fract. Engng. Mater. Struct., 8, 315–325, 1985.

S. ZAREMBA, Sur une formę 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 mécanigue physigue, Bull. Int. Acad. Sci. Cracovie, 614–621, 1903.

H. ZIEGLER, A modification of Prager's hardening rule, Quart. Appl. Math., 17, 55–65, 1959.




DOI: 10.24423/engtrans.443.2005