Hencky Strain and Hencky Model: Extending History and Ongoing Tradition

Multidiscipline Modeling in Materials and Structures

Volumn 1, Issue 1, 2005, Pages 1-52

  Xiao, H ^a  

^a RUHR UNIVERSITY BOCHUM   (Germany)

Author keywords

Consistency criteria; Elasticity; Eulerain rate formulations; Finite deformations; Hencky's logarithmic strain

Indexed keywords

EID: 84993104389     PISSN: 15736105     EISSN: 15736113     Source Type: Journal    
DOI: 10.1163/1573611054455148     Document Type: Review

References (209)

1. Anand, L., On H. Hencky's approximate strain-energy function for moderate deformations. J. Appl. Mech. 46, 78-82 (1979).
2. Anand, L., Moderate deformations in extension-torsion of incompressible isotropic elastic materials. J. Mech. Phys. Solids 34, 293-304 (1986).
3. Atluri, S.N., On constitutive relations at finite strain: hypoelasticity and elastoplasticity with isotropic or kinematic hardening. Compt. Meth. Appl. Mech. Eng. 43, 137-171 (1984).
4. Backmann, M.E., Form for the relation between stress and finite elastic and plastic strains under impulsive loading. J. Appl. Phys. 35, 2524-2533 (1964).
5. Bathe, K.J., Finite Element Procedures. Prentice-Hall, Englewood Cliffs, NJ (1996).
6. Bazant, Z.P., Easy-to-compute tensors with symmetric inverse approximating Hencky finite strain and its rate. J. Engng. Mater. Techn. 120, 131-136 (1998).
7. Bernstein, B., Hypoelasticity and elasticity. Arch. Rat. Mech. Anal. 6, 89-104 (1960).
8. Bernstein, B., Ericksen, J.L., Work functions in hypoelasticity. Arch. Rat. Mech. Anal. 1, 396-409 (1958).
9. Bertram, A., Axiomatische Einfiihrungin die Kontinuumsmechanik. B.I. Wissenschafts- verlag, Mannheim/Wien/Ziirich (1989).
10. Bertram, A., Svendsen, B., On material objectivity and reduced constitutive equations. Arch. Mech. 53, 613-636 (2001).
11. Bilby, B.A., Lardner, L.R.T., Stroh, A.N., Continuum theory of dislocations and the theory of plasticity. In: Proc. 10th ICTAM, Vol. 8, pp. 35-44. Universite de Bruxelles, Bruxelle (1957).
12. Billington, E.W., Non-linear mechanical response of various metals: II Permanent length changes in twisted tubes. J. Phys. D: Appl. Phys. 9, 533-552 (1976).
13. Bonet, J., Wood, R.D., Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge (1997).
14. Bongmba, C.N., Ein Enites anisotropes Materialmodell auf der Basis der Hencky-Dehnung und der logarithmischen Rate zur Beschreibung duktiler Schadigung. Ph D thesis, Ruhr-Universitat, Bochum (2001).
15. Bongmba, C.N., Schiitte, H., Ein nichtlineares anisotropes Materialmodell auf der Basis der Hencky-Dehnung und der logarithmischen Rate zur Beschreibung duktiler Schadigung. Technische Mechanik 22, 205-222 (2002).
16. Bruhns, O.T., Die Beriicksichtigung einer isotropen WerkstofFverfestigung bei der elastisch-plastischen Blechbiegung mit endlichen Formanderungen. Ing.-Arch. 39, 63-72 (1970).
17. Bruhns, O.T., Bifurcation problems in plasticity. In: The Constitutive Law in Ther- moplasticity., CISM Courses and Lectures No. 281, Th. Lehmann (ed.), pp. 465-540, Springer-Verlag, Wien-New York (1984).
18. Bruhns, O.T., Bongmba, C.N., On the numerical implementation of a finite strain anisotropic damage model based upon the logarithmic rate. Eur. J. Finite Elements 10, 385-400 (2001).
19. Bruhns, O.T., Diehl, H., An internal variable theory of inelastic behaviour at high rates of strain. Arch. Mech. 41, 427-460 (1989).
20. Bruhns, O.T., Gupta, N.K., Meyers, A., Xiao, H., Bending of an elastoplastic strip with isotropic and kinematic hardening. Arch. Appl. Mech. 72, 759-778 (2003).
21. Bruhns, O.T., Lehmann, Th., Optimum deformation rate in large inelastic deformations. In: MetaJ Forming Plasticity, H. Lippmann (ed.), pp. 120-138, Springer-Verlag, Berlin (1978).
22. Bruhns, O.T., Meyers, A., Xiao, H., On non-corotational rates of Oldroyd's type and relevant issues in rate constitutive formulations. Proc. Roy. Soc. London ser. A 460, 909-928 (2003).
23. Bruhns, O.T., Thermann, K., Elastisch-plastische Biegung eines Plattenstreifens bei endlichen Formanderungen. Ing.-Arch. 38, 141-152 (1969).
24. Bruhns, O.T., Xiao, H., Meyers, A., Self-consistent Eulerian rate type elastoplasticity models based upon the logarithmic stress rate. Int. J. Plasticity 15, 479-520 (1999).
25. Bruhns, O.T., Xiao, H., Meyers, A., Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky's logarithmic strain tensor. Proc. Roy. Soc. London ser. A 457, 2207-2226 (2001).
26. Bruhns, O.T., Xiao, H., Meyers, A., Large simple shear and torsion problems in kinematic hardening elastoplasticity with logarithmic rate. Int. J. Solids Structures 38, 8701-8722 (2001).
27. Bruhns, O.T., Xiao, H., Meyers, A., A self-consistent Eulerian rate type model for finite deformation elastoplasticity with isotropic damage. Int. J. Solids Structures 38, 657-683 (2001).
28. Bruhns, O.T., Xiao, H., Meyers, A., Finite bending of a rectangular block of an elastic Hencky material. J. Elasticity 66, 237-256 (2002).
29. Casey, J., Naghdi, P.M., On the use of invariance requirements for intermediate config- urations associated with the polar decomposition of a deformation gradient. Quart. Appl. Math. 41, 339-342 (1983).
30. Chen, W.F., Saleeb, A.P., Constitutive Equations for Engineering Materials. Vol.1: Elasticity and Modeling-Revised Edition. Elsevier, New York (1994).
31. Cho, H.W., Dafalias, Y.P., Distortional and orientational hardening at large viscoplas- tic deformations. Int. J. Plasticity 12, 903-925 (1996).
32. Cleja-Tigoiu, S., Soos, E., Elastoviscoplastic models with relaxed configurations and internal state variables. Appl. Mech. Rev. 43, 131-151 (1990).
33. Colak, O.U., Modeling of large simple shear using a viscoplastic overstress model and classical plasticity model with different objective stress rates. Acta Mechanica 167, 171-187(2004).
34. Colak, O.U., Krempl, E., Modelling of the monotonic and cyclic Swift effects using an isotropic, finite viscoplasticity theory based on overstress (PVBO). Int. J. Plasticity 21, 573-588 (2005).
35. Cotter, B.A., Rivlin, R.S., Tensors associated with time-dependent stress. Quart. AppL Math. 13, 177-182 (1955).
36. Criscione, J.C., Rivlin's representation formula is ill-conceived for the determination of response functions via biaxial testing. J. Elasticity 70, 129-147 (2003).
37. Criscione, J.C., Humphrey, J.D., Douglas, A.S., William, C.H., An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity. J. Mech. Phys. Solids 48, 2445-2465 (2000).
38. Dafalias, Y.P., Corotational rates for kinematic hardening at large plastic deformations. J. Appl Mech. 50, 561-565 (1983).
39. Dafalias, Y.P., The plastic spin concept and a simple illustration of its role in finite plastic transformations. Mechanics of Materials 3, 223-233 (1984).
40. Dafalias, Y.P., A missing link in the macroscopic constitutive formulation of large plastic deformations. In: Plasticity Today: Medelling, Methods and Applications, A. Sawczuk and G. Bianchi (eds.), pp. 135-151, Elsevier, London (1985).
41. Dafalias, Y.P., The plastic spin. J. AppL Mech. 52, 865-871 (1985).
42. Davis, E.A,, Plastic behaviour of metals in the strain-hardening range, part II. J. AppL Phys. 8, 213-217 (1937).
43. De Boer, R., Die elastisch-plastische Biegung eines Plattenstreifens aus inkompress-iblem WerkstofF bei endlichen Pormanderungen. Ing.-Arch. 36, 145-154 (1967).
44. De Boer, R., Bruhns, O.T., Zur Berechnung der Eigenspannungen bei einem durch endliche Biegung verformten inkompressiblen Plattenstreifen. Acta Mech. 8, 146-159 (1969).
45. Dienes, J.K., On the analysis of rotation and stress rate in deforming bodies. Acta Mechanica 32, 217-232 (1979).
46. Doyle, T.C., Ericksen, J.L., Nonlinear elasticity. Adv. Appl. Mech. 4, 53-115 (1956).
47. Drucker, D.C., Conventional and unconventional plastic response and representation. Appl. Mech. Rev. 41, 151-167 (1988).
48. Dubey, R.N., Choice of tensor-rates-a methodology. SM Archives 12, 233-244 (1987).
49. Eckart, C., The thermodynamics of irreversible processes. IV. The theory of elasticity and anelasticity. Phys. Rev. 73, 373-382 (1948).
50. Eglit, M.E., Tensorial characteristics of finite deformations. Prikl. Mat. Mekh. 24, 1432-1438 (1960).
51. Ericksen, J.L., Hypoelastic potentials. Q. Mech. Appl. Math. 11, 67-72 (1958).
52. Eterovic, A.L., Bathe, K.J., A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures. Int. J. Num. Meth. Engng. 30, 1099-1115 (1990).
53. Fitzjerald, S., A tensorial Hencky measure of strain and strain rate for finite deformation. J. Appl. Phys. 51, 5111-5115 (1980).
54. Freed, E., Natural strain. J. Engng. Mater. Techn. 117, 379-385 (1995).
55. Pressengeas, C., Molinari, A., Representations du comportement plastique anisotrope aux grandes deformations. Arch. Mech. 36, 483 (1984).
56. Freund, L.B., Constitutive equations for elastic-plastic materials at finite strain. Int. J. Solids Structures 6, 1193-1209 (1970).
57. Green, A.E., Simple extension of hypoelastic body of grade zero. J. Rat. Mech. Anal. 5, 637-642 (1956).
58. Green, A.E., 1956b. Hypo-elasticity and plasticity. Proc. Roy. Soc. London A234, 46-59 (1956).
59. Green, A.E., Hypoelasticity and plasticity. Arch. Rat. Mech. Anal. 5,725-734 (1956).
60. Green, A.E., Mclnnis, B.C., Generalized hypoelasticity. Proc. Roy. Soc. Edinburgh A57, 220-230 (1967).
61. Green, A.E., Naghdi, P.M., A general theory of an elastic-plastic continuum. Arch. Rational Mech. Anal 18, 251-281 (1965).
62. Green, A.E., Naghdi, P.M., A thermodynamic development of elastic-plastic continua. In: Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids, H. Parkus and L.I. Sedov (eds.), pp. 117-131, Springer-Verlag, Wien, New York (1966).
63. Green, A.E., Naghdi, P.M., Some remarks on elastic-plastic deformation at finite strain. Int. J. Engng. Sci. 9, 1219-1229 (1971).
64. Grioli, S., Mathematical Theory of Elastic Equilibrium [Recent Results). Srpinger-Verlag, Berlin, etc. (1962).
65. Guo, Z.-H., Time derivatives of tensor fields in nonlinear continuum mechanics. Arch. Mech. 15, 131-163 (1963).
66. Guo, Z.-H., Some notes on hypoelasticity. Arch. Mech. 15, 683-690 (1963).
67. Guo, Z.-H., On the constitutive equation of hypoelasticity. Bull. Acad. Polon. Sci. Ser. Sci. Tech. 11, 301-304.
68. Curtin, M.E., On the hypoelastic formulation of plasticity using the past maximum of stress. ASMEJ. Appl Mech. 50, 894-896 (1983).
69. Curtin, M.E., Spear, K., On the relationship between the logarithmic strain rate and the stretching tensor. Int. J. Plasticity 19, 437-444 (1983).
70. Haupt, P., Continuum Mechanics and Theories of Materials. Springer-Verlag (2002).
71. Haupt, P., Tsakmakis, C., On kinematic hardening and large plastic deformation. Int. J. Plasticity 2, 279-293 (1986).
72. Haupt, P., Tsakmakis, C., On the application of dual variables in continuum mechanics. Continuum Mech. Thermodyn. 1, 165-196 (1989).
73. Heiduschke, K., The logarithmic strain space description. Int. J. Solids Structures 32, 1047-1062 (1995).
74. Hencky, H., Uber die Form des Elastizitasgesetzes bei ideal elastischen StofFen. Z. Techn. Phys. 9, 215-220; 457 (1928).
75. Hencky, H., Welche Umstande bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Korpern? Z. Phys. 55, 145-155 (1929).
76. Hencky, H., Das Superpositionsgesetz eines endlich deformierten relaxationslaigen elastischen Kontinuums und seine Bedeutung fiir eine exakte Ableitung der Gleichun-gen fur die zahe Fliissigkeit in der Eulerschen Form. Ann. Physik 5, 617-630 (1929).
77. Hencky, H., The law of elasticity for isotropic and quasi-isotropic substances by finite deformations. J. Rheology 2, 169-176 (1931).
78. Hencky, H., A simple model explaining the hardening effect in polycrystalline metals. J. Rheology 3, 30-36 (1932).
79. Hencky, H., On propagation of elastic waves in materials under high hydrostatic pressure. Philos. Mag. 14, 254-258 (1932).
80. Hencky, H., The elastic behaviour of vulcanised rubbers. Rubber Chem. Techn. 6, 217-224 (1933).
81. Hencky, H., Stresses in rubber tires. Mech. Eng. 27, 149-153 (1935).
82. Hill, R., A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids 6, 236-249 (1958).
83. Hill, R., Some basic principles in the mechanics of solids without a natural time. J. Mech. Phys. Solids 7, 209-225 (1959).
84. Hill, R., The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Mech. 15, 79-95 (1967).
85. Hill, R., On the classical constitutive relations for elastic-plastic solids. In: Recent Progress in Applied Mechanics, Folke Odqvist Volume, B. Bromberg, J. Hult and F. Niordson (eds.), pp. 241-249, Almqvist and Wiksell, Stockholm (1967).
86. Hill, R., On constitutive inequalities for simple materials. J. Mech. Phys. Mech. 16, 229-242; 315-322 (1968).
87. Hill, R., Constitutive inequalities for isotropic elastic solids at finite strain. Proc. Roy. Soc. London ser. A 326, 131-147 (1970).
88. Hill, R., Aspects of invariance in solid mechanics. Adv. Appl. Mech. 18, 1-75 (1978).
89. Hoger, A., The material time derivative of logarithmic strain tensor. Int. J. Solids Structures 22, 1019-1032 (1986).
90. Hoger, A., The stress conjugate to logarithmic strain. Int. J. Solids Structures 23, 1645-1656 (1987).
91. Hutchinson, J.W., Finite strain analysis of elastic-plastic solids and structures. In: Numerical Solutions of Nonlinear Structural Problems, R.F. Hartung (ed.), ASME, pp. 17-29 (1973).
92. Hutchinson, J.W., Neale, K., Finite strain J2 deformation theory. In: Finite Elasticity, D.E. Carlson and R.T. Shield (eds.), pp. 237-247. Martinus Nijhoff, The Hague (1980).
93. Jaumann, G., Geschlossenes System physikalischer und chemischer Differenzialgesetze. Sitz. Acad. Wiss. Wien (Ila) 120, 385-530 (1911).
94. Johnson, G.C., Bammann, D.J., A discussion of stress rates in finite deformation problems. Int. J. Solids Structures 20, 725-737 (1984).
95. Khan, A.S., Huang, S.J., Continuum Theory of Plasticity. John Wiley & Sons, New York (1995).
96. Kleiber, M., On errors inherent in commonly accepted rate forms of the elastic consti- tutive law. Arch. Mech. 38, 271-279 (1986).
97. Kojic, M., Bathe, K.J., Studies of finite element procedures-stress solution of a closed elastic strain path with stretching and shearing using the updated Lagrangian-Jaumann formulation. Coiiipt. Struct. 26, 175-179 (1987).
98. Kollmann, F.G., Sansour, C., Viscoplastic shells: theory and numerical analysis. Arch. Mech. 49, 477-511 (1997).
99. Kroner, E., Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Rational Mech. Anal. 4, 273-334 (1960).
100. Lee, E.H., Elastic-plastic deformations at finite strains. J. Appl. Mech. 36, 1-6 (1969).
101. Lee, E.H., Finite deformation effects in plasticity analysis. In: Plasticity Today: Medelling, Methods and Applications, A. Sawczuk and G. Bianchi (eds.), pp. 61-74, Elsevier, London (1985).
102. Lee, E.H., Liu, D.T., Finite strain elastic-plastic theory with application to plane-wave analysis. J. Appl. Phys. 38, 19-27 (1967).
103. Lee, E.H., Liu, D.T., Finite strain elastic-plastic theory. In: Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids. H. Parkus and L.I. Sedov (eds.), pp. 213-221, Springer-Verlag, Wien, New York (1968).
104. Lee, E.H., Mallett, R.L., Wertheimer, T.B., Stress analysis for anisotropic hardening in finite deformation plasticity. J. Appl. Mech. 50, 554-560 (1983).
105. Lehmann, Th., Einige Betrachtung zu den Grundlagen der Umformtechnik. Ing.-Arch. 29, 1-21 (1960).
106. Lehmann, Th., Anisotrope plastische Formanderungen. Rheologica Acta 4, 281-285 (1962).
107. Lehmann, Th., Zur Beschreibung grofier plastischen Formanderungen unter Beriicksich-tigung der WerkstofFverfestigung. Rheologica Acta 2, 247-254 (1964).
108. Lehmann, Th., Anisotrope plastische Formanderungen. Romanian J. Tech. Sci. Appl Mech. 17, 1077-1086 (1972).
109. Lehmann, Th., Einige Bemerkungen zu einer allgemeinen Klasse von StofFgesetzen fiir grofie elastoplastische Formanderungen. Ing.-Arch. 41, 297-310 (1972).
110. Lehmann, Th., On large deformation elastoplasticity. In: Foundations of Plasticity, A. Sawczuk (ed.), pp. 571-585, NoordhofF, Leyden (1973).
111. Lehmann, Th. (ed.). The Constitutive Laws in Thermoplasticity, CISM Courses and Lectures No. 281. Springer-Verlage, Wien, New York (1984).
112. Lehmann, Th., On the balance of energy and entropy at inelastic deformations of solid bodies. Eur. J. Mech. A/Solids 8, 235-251 (1989).
113. Lehmann, Th., Guo, Z.-H., Liang, H.Y., The conjugacy between Cauchy stress and logarithm of the left stretch tensor. Eur. J. Mech. A/Solids 10, 395-404 (1991).
114. Lin, R.C., Viscoelastic and Elastic-viscoelastic-elastoplastic Constitutive Characterizations of Polymers at Finite Strains: Theoretical and Numerical Aspects. Ph D thesis, Universitat der Bundeswehr Hamburg, Hamburg (2002).
115. Lin, R.C., Numerical study of consistency of rate constitutive equations with elasticity at finite deformation. Int. J. Numer. Methods Engrg. 55, 1053-1077 (2002).
116. Lin, R.C., Hypoelasticity-based analytical stress solutions in the simple shear process. Z. Angew. Math. Mech. 83, 163-171 (2003).
117. Lin, R.C., Schomburg, U., Kletschkowski, T., Analytical stress solutions of a closed deformation path with stretching and shearing using the hypoelastic formulations. Eur. J. Mech. A/Solids 22, 443-461 (2003).
118. Liu, C.S., Lie symmetries of finite strain elastic-perfectly plastic models and exactly consistent schemes for numerical integrations. Int. J. Solids Structures 41, 1823-1853 (2004).
119. Liu, C.S., Hong, H.K., Non-oscillation criteria for hypoelastic models under simple shear deformation. J. Elasticity 57, 201-241 (1999).
120. Liu, C.S., Hong, H.K., Using comparison theorem to compare corotational stress rates in the model of perfect plasticity. Int. J. Solids Structures 38, 2969-2987 (2001).
121. Loret, B., On the effect of plastic rotation in the finite deformation of anisotropic elastoplastic materials. Mech. Mater. 2, 287-304 (1983).
122. Lubarda, V.A., Elastoplasticity Theory. CRC Press, New York (2001).
123. Lubliner, J., Plasticity Theory. Macmillan, New York(1990).
124. Ludwik, P., Elemente der technologischen Mechanik. Springer-Verlag, Berlin (1909).
125. Mazur, E.F., On the definition of stress rate. Q. Appl. Math. 19, 160-163 (1961).
126. Meyers, A., On the consistency of some Eulerian strain rates. Z. Angew. Math. Mech. 79, 171-177 (1999).
127. Meyers, A., Schiefie, P., Bruhns, O.T., Some comments on objective rates of symmetric Eulerian tensors with application to Eulerian strain rate. Acta Mechanica 139, 91-103 (2000).
128. Meyers, A., Xiao, H., Bruhns, O.T., Elastic stress ratchetting and corotational stress rates. Technische Mechanik 23, 92-102 (2003).
129. Meyers, A., Xiao, H., Bruhns, O.T., Comparison of objective stress rates in single parameter strain cycles (accepted) (2004).
130. Miehe, C., Apel, N., Lambrecht, M., Anisotropic additive plasticity in the logarithmic strain space. Modular kinematic formulation and implementation based on incremental minimization principles for standard materials. Compt. Meth. Appl. Mech. Engrg. 191, 5383-5425 (2004).
131. Miehe, C., Stein, E., Wagner, W., Associative multiplicative elasto-plasticity: Formulation and aspects of the numerical implementation including stability analysis. Computer k Structures 52, 969-978 (1994).
132. Montheillet, F., Cohen, M., Jonas, J.J., Axial stress and texture development during the torsion testing of Al, Cu and a-Fe. Acta Metall. 32, 2077-2089 (1984).
133. Moss, C., On instabilities in large deformation simple shear loading. Compt. Meth. Appl. Mech. Engng. 46, 329-338 (1984).
134. Miiller, C., Modellierung des Materialverhaltens polykristalliner Forrngedachtnis-legierungen bei Eniten Deformationen. PhD Thesis, Ruhr-Universitat Bochum (2003).
135. Murnaghan, F.D., The compressibility of solids under extreme pressures. In: Theodore von Karman Anniversary Volume, pp.121-136. California Institute of Technology, Passadena (1941).
136. Nadai, A., Plastic behavior of metals in the strain-hardening range. Part I. J. Appl. Phys. 8, 205-213 (1937).
137. Naghdi, P.M., A critical review of the state of finite plasticity. Z. angew. Math. Phys. 41, 315-394 (1990).
138. Naghdi, P.M., Casey, J., A prescription for the identification of finite plastic strain. Int. J. Engng. 30, 1257-1278 (1992).
139. Naghdi, P.M., Wainwright, W.L., On the time derivative of tensors in mechanics of continua. Quart. Appl. Math. 19, 95-109 (1961).
140. Neale, K.W., Phenomenological constitutive laws in finite plasticity. SM Archives 6, 70-128 (1981).
141. Nagtegaal, J.C., de Jong, J.E., Some aspects of non-isotropic work-hardening in finite strain plasticity. In: Proceedings of the workshop on plasticity of metals at finite strain: theory, experiment and computation, E.H. Lee and R.L. Mallett (eds.), pp. 65-102, Stanford University, Stanford (1982).
142. Needleman, A., Tvergaard, V., Necking of biaxially stretched elastoplastic circular plates. J. Mech. Phys. Solids 25, 159-183 (1977).
143. Nemat-Nasser, S., On finite deformation elastoplasticity. Int. J. Solids Structures 18, 857-872 (1982).
144. Nemat-Nasser, S., On finite plastic fiow of crystalline solids and geomaterials. J. Appl Mech. 50, 1114-1126 (1983).
145. Nemat-Nasser, S., Phenomenological theories of elastoplasticity and strain localization at high strain rates. Appl. Mech. Rev. 45, 519-545 (1992).
146. Noll, W., On the continuity of the solid and fiuid state. J. Rational Mech. Anal. 4, 3-81 (1955).
147. Ogden, R.W., Non-linear Elastic Deformations. Ellis Horwood, Chichester (1984).
148. Oldroyd, J.G., On the formulation of rheological equations of state. Proc. Roy. Soc. London ser. A 200, 523-541 (1950).
149. Olsen, E.T., Bernstein, B., A class of hypo-elastic non-elastic materials and their thermodynamics. Arch. Rat. Mech. Anal 86, 291-303 (1984).
150. Paulun, J.E., Pecherski, Study of corotational rates for kinematic hardening in finite deformation plasticity. Arch. Mech. 37, 661-678 (1985).
151. Paulun, J.E., Pecherski, On the application of the plastic spin concept for the description of anisotropic hardening in finite deformation plasticity. Int. J. Plasticity 3, 303-314 (1987).
152. Peric, D., Owen, D.R.J., Honnor, M.E., A model for finite strain elastoplasticity based on logarithmic strains: Computational issues. Compt. Meth. AppL Mech. Engrg. 94, 35-61 (1992).
153. Prager, W., A new method of analyzing stresses and strains in work-hardening plastic solids. J. Appl. Mech. 23, 493-496 (1956).
154. Prager, W.,. An elementary discussion of definition of stress rate. Quart. Appl. Math. 18, 403-407 (1960).
155. Prager, W., Introduction to Mechanics of Continua. Ginn, Boston (1961).
156. Prager, W., On higher rates of stress and deformation. J. Mech. Phys. Solids 10, 133-138 (1962).
157. Quent, T., Ein mikromechanisch begriindetes Modell zur Beschreibung des duktilen Verhaltens metallischer Werkstoffe bei endlichen Deformationen unter Beriicksich-tigung von Porenschadigung. Ph D Thesis, Ruhr-Universitat, Bochum (2001).
158. Raniecki, B., Bruhns, O.T., Bounds to bifurcation stresses in solids with nonassociated plastic fiow law at finite strain. J. Mech. Phys. Solids 29, 153-172 (1981).
159. Raniecki, B., Nguyen, H.V., Isotropic elasto-plastic solids at finite strain and arbitrary pressure. Arch. Mech. 36, 687-704 (1984).
160. Reed, K.W., Atluri, S.N., Constitutive modelling and computational implementation for finite strain plasticity. Int. J. Plasticity 1, 63-87 (1985).
161. Reinhardt, W.D., Dubey, R.N., Eulerian strain-rate as a rate of loarithmic strain. Mech. Res. Commun. 22, 165-170 (1995).
162. Reinhardt, W.D., Dubey, R.N., Coordinate-independent representations of spins in continuum mechanics. J. Elasticity 42, 133-144 (1996).
163. Rice, J.R., Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Constitutive Equations in Plasticity, Ali S. Argon (ed.), pp. 21-79(1975).
164. Richter, H., Das isotrope Elastizitatsgesetz. Z. Angew. Math. Mech. 28, 202-209 (1948).
165. Richter, H., Verzerrungstensor, Verzerrungsdeviator und Spannungstensor bei endlichen Formanderungen. Z. Angew. Math. Mech. 29, 65-75 (1949).
166. Rudnicki, J.W., Rice, J.R., Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23, 371-394 (1975).
167. Sansour, C., On the dual variable of the logarithmic strain tensor, the dual variable of the Cauchy stress tensor, and related issues. Int. J. Solids Structures 38, 9221-9232 (2001).
168. Sansour, C., Bednarczyk, H., A study on rate-type constitutive equations and the existence of a free energy function. Acta Mechanica 100, 205-221 (1993).
169. Schieck, B., Stumpf, H., The appropriate corotational rate, exact formula for the plastic spin and constitutive model for finite elastoplasticity. Int. J. Solids Structures 32, 3643-3667 (1995).
170. Sedov, L.I., Different definitions of the rates of change of tensors, PrikL Mat. Mekh. 14, 579-586 (1960).
171. Seth, B.R., Generalized strain measures with applications to physical problems. In: Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, M. Reiner and D. Abir (eds.), pp. 162-172. Pergamon Press, Oxford (1964).
172. Simo, J.C., Hughes, T.J.R., Computational Inelasticity. Springer-Verlag, Berlin (1998).
173. Simo, J.C., Pister, K.S. Remarks on rate constitutive equations for finite deformation problem: computational implications. Compt. Meth. Appl. Mech. Engng. 46, 201-215 (1984).
174. Sowerby, R., Chu, E., Rotations, stress rates and strain measures in homogeneous deformation processes. Int. J. Solids Structures 20, 1037-1048 (1984).
175. Storen, S., Rice, J.R., Localized necking in thin sheets. J. Mech. Phys. Solids 23, 421-441 (1975).
176. Stumpf, H., Schieck, B., Theory and analysis of shells undergoing finite elastic-plastic strains and rotations. Acta Mechanica 106, 1-21 (1994).
177. Svendsen, B., Bertram, A., On frame-indifFerence and form-invariance in constitutive theory. Acta Mechanica 132, 195-207 (1999).
178. Swift, W., Length changes in metals under torsional overstrain. Engineering 163, 253-257 (1947).
179. Szabo, L., Balia, M., Comparison of stress rates. Int. J. Solids Structures 25, 279-297 (1989).
180. Tanner, R.L, Tanner, E.T., Heinrich Hencky: a rheological pioneer. Rheol. Acta 42, 93-101 (2003).
181. Thomas, T.Y., On the structure of stress-strain relation. Proc. Nat. Acad. Sci. U.S.A. 41, 716-720 (1955).
182. Thomas, T.Y., Combined elastic and Prantl-Reuss stress-strain relations. Proc. Natl. Acad. Sci. U.S.A. 41, 720-726 (1955).
183. Thomas, T.Y., Combined elastic and von Mises stress-strain relations. Proc. Natl. Acad. Sci. U.S.A. 41, 908-910 (1955).
184. Tokuoka, T., Yield conditions and fiow rules derived from hypo-elasticity. Arch. Rat. Mech. Anal 42, 239-252 (1971).
185. Truesdell, C., The mechanical foundations of elasticity and fiuid dynamics. J. Rational Mech. Anal 1, 125-300 (1952).
186. Truesdell, C., The simplest rate theory of pure elasticity. Comm. Pure Appl. Math. 8, 123-132 (1955).
187. Truesdell, C., Hypo-elasticity. J. Rat. Mech. Anal. 4, 83-133 (1955).
188. Truesdell, C., Hypo-elastic shear. J. Appl. Phys. 27, 441-447 (1956).
189. Truesdell, C., Noll, W., The nonlinear field theories of mechanics. In: Handbuch der Physik, S. Fliigge, Bd. III/3, Springer-Verlag, Berlin (1965).
190. Truesdell, C., Toupin, R.A., The classical field theories. In: Handbuch der Physik, S. Fliigge, Bd. III/l, Springer-Verlag, Berlin (1960).
191. Tvergaard, V., Effect of Kinematic hardening on localized necking in biaxially stretched sheets. Int. J. Mech. Sci. 20, 651-658 (1978).
192. Weber, G., Anand, L., Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Compt. Meths. AppL Mech. Engrg. 70, 173-202 (1990).
193. Willis, J.R., Some constitutive equations applicable to problems of large dynamic plastic deformation. J. Mech. Phys. Solids 17, 359-369 (1969).
194. Xia, Z., Ellyin, P., A stress rate measure for finite elastic plastic deformation. Acta Mechanica 98, 1-14 (1993).
195. Xiao, H., Unified explicit basis-free expressions for time rate and conjugate stress of an arbitrary Hill's strain. Int. J. Solids Structures 32, 3327-3340 (1995).
196. Xiao, H., Bruhns, O.T., Meyers, A., A new aspect in kinematics of large deformations. In: Plasticity and impact mechanics, N.K. Gupta (ed.), pp. 100-109. New Age Publ. Ltd., New Delhi (1996).
197. Xiao, H., Bruhns, O.T., Meyers, A., Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124, 89-105 (1997).
198. Xiao, H., Bruhns O.T., Meyers, A., Hypoelasticity model based upon the logarithmic stress rate. J. Elasticity 47, 51-68 (1997).
199. Xiao, H., Bruhns, O.T., Meyers, A., Strain rates and material spins. J. Elasticity 52, 1-41 (1998).
200. Xiao, H., Bruhns, O.T., Meyers, A., On objective corotational rates and their defining spin tensors. Int. J. Solids Structures 35, 4001-4014 (1998).
201. Xiao, H., Bruhns, O.T., Meyers, A., Existence and uniqueness of the integrable-exactly hypoelastic equation t * = X{tiD)I + 2fj.D and its significance to finite inelasticity. Acta Mechanica 138, 31-50 (1999).
202. Xiao, H., Bruhns, O.T., Meyers, A., A natural generation of hypoelasticity and Eulerian rate type formulation of hyperelasticity. J. Elasticity 56, 59-63 (1999).
203. Xiao, H., Bruhns, O.T., Meyers, A., The choice of objective rates in finite elasto-plasticity: general results on the uniqueness of the logarithmic rate. Proc. Roy. Soc. London ser. A 456, 1865-1882 (2000).
204. Xiao, H., Bruhns, O.T., Meyers, A., A consistent finite elastoplasticity theory combining additive and multiplicative decomposition of the stretching and the deformation gradient. Int. J. Plasticity 16, 143-177 (2000).
205. Xiao, H., Bruhns, O.T., Meyers, A., Large strain response of perfect elastoplasticity and kinematic hardening plasticity with the logarithmic rate: Swift effect in torsion. Int. J. Plasticity 17, 211-235 (2001).
206. Xiao, H., Bruhns, O.T., Meyers, A., A closed-form solution to finite bending of a compressible elastic-perfectly plastic rectangular block. Defence Science Journal 53, 25-39 (2003).
207. Yang, W., Cheng, L., Hwang, K.C., Objective corotational rates and shear oscillation. Int. J. Plasticity 8, 643-656 (1992).
208. Zbib, H.M., Aifantis, E.G., On the concept of relative and plastic spins and its implication to large deformation theories. Part II: anisotropic hardening plasticity. Acta Mechanica 75, 35-56 (1988).
209. Zaremba, S., Sur une forme perfectionee de la theorie de la relaxation. Anz. Akad. Krakau, math.-naturw. Klasse, 594-614 (1903).