Creep and long-term strength of metals under unsteady complex stress states (Review)

A. M. Lokoshchenko ; L. V. Fomin ; W. V. Teraud ; Yu. G. Basalov ; V. S. Agababyan

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 2, pp. 275-318.

Résumé

This article is an analytical review of experimental and theoretical studies of creep and creep rupture strength of metals under unsteady complex stress states published over the past 60 years. The first systematic studies of the creep of metals under complex stress conditions were published in the late 50s and early 60s of the 20th century in the Soviet Union (L. M. Kachanov and Yu. N. Rabotnov) and Great Britain (A. E. Johnson). Pioneering work on creep rupture strength first appeared in the USSR (L. M. Kachanov and Yu. N. Rabotnov). Subsequently, Yu. N. Rabotnov developed the kinetic theory of creep and creep rupture strength, with the help of which it is possible to efficiently describe various features of the creep process of metals up to fracture under various loading programs. Different versions of the kinetic theory use either a scalar damage parameter, or a vector parameter, or a tensor parameter, or a combination of them. Following the work of M. Kachanov and Yu. N. Rabotnov mechanics of continuum destruction began to develop in Europe, in Asia, and then in the USA. The hypothesis of proportionality of stress deviators and deviators of creep strain rates is accepted as the main connection between the components of stress tensors and creep strains. When modeling experimental data, the proportionality coefficient in this dependence takes different forms. The main problem in the development of this direction is the difficulty in obtaining experimental data with arbitrary loading programs. This review provides the main results of studies conducted by scientists from different countries. Except Yu. N. Rabotnov and L. M. Kachanov, also a significant contribution to the development of the direction of science made by Russian scientists N. N. Malinin, A. A. Ilyushin, V. S. Namestnikov, S. A. Shesterikov, A. M. Lokoshchenko, Yu. P. Samarin, O. V. Sosnin, A. F. Nikitenko, et al.

Détail

Keywords: analytical review, creep, creep rupture strength, complex stress state, unsteady loading, stress relaxation, scalar damage parameter, vector damage parameter, tensor damage parameter.

Bibliographie

[1] Kachanov L. M., “On the time of fracture in creep”, Izv. Akad. Nauk SSSR. OTN, 1958, no. 8, 26–31 (In Russian) | Zbl

[2] Rabotnov Yu. N., “On the mechanism of long-term fracture”, Voprosy prochnosti materialov i konstruktsii [Problems of Strength of Materials and Structures], Izdat. AN SSSR, Moscow, 1959, 5–7 pp. (In Russian)

[3] Rabotnov Yu. N., Creep of Structural Members, North‐Holland Series in Applied Mathematics and Mechanics, North-Holland, Amsterdam, 1969, ix+822 pp.

[4] Kachanov L. M., Osnovy mekhaniki razrusheniia [Fundamentals of Fracture Mechanics], Nauka, Moscow, 1974, 312 pp. (In Russian)

[5] Trąmpczyński W., Badanie wpływu historii obciążenia na pełzanie metali w złożonym stanie naprężenia, Inst. Podstawowych Probl. Techn. Polsk. Akad. Nauk, Warszawa, 1985, 249 pp. (in Polish)

[6] Sosnin O. V, Gorev B. V., Nikitenko A. F., Energeticheskii variant teorii polzuchesti [Energy Variant of Creep Theory], Inst. of Hydrodynamics, USSR Acad. of Sci., Novosibirsk, 1986, 9 pp. (In Russian)

[7] Nikitenko A. F., Polzuchest' i dlitel'naia prochnost' metallicheskikh materialov [Creep and Long-Term Strength of Metal Materials], Novosibirsk State Univ. of Archit. and Civil Engin., Novosibirsk, 1997, 278 pp. (In Russian)

[8] Mikhalevich V. M., Tenzorni modeli nakopichannia poshkodzhen' [Tensor Models of Damage Accumulation], Universum-Vinnitsia, Vinnitsia, 1998, 195 pp. (In Ukrainian) http://ir.lib.vntu.edu.ua/handle/123456789/13977

[9] Betten J., Creep Mechanics, Springer-Verlag, Berlin, 2008, xvi+367 pp | DOI

[10] Lokoshchenko A. M., Creep and long-term strength of metals, CRC Press, Boca, Raton, 2018, xviii+545 pp | DOI

[11] Johnson A. E., “Complex stress creep of metals”, Intern. Met. Rev., 30:1 (1985), 447–506 | DOI

[12] Shesterikov S. A., Lokoschenko A. M., “Creep and long-term strength of metals”, Science and Technology Results, Mechanics of Deformable Solids, v. 13, VINITI, Moscow, 1980, 3–104 (in Russian)

[13] Lemaître J., “Local approach of fracture”, Eng. Fract. Mech., 25:5–6 (1986), 523–537 | DOI

[14] Krajcinovic D., “The continuous damage theory: why, how and where?”, Spominski zbornik Antona Kuhlja, S.n., Ljubljana, 1982, 95–109

[15] Krajcinovic D., “On the basic structure of continuum damage models”, Fragmentation, Form and Flow in Fractured Media, Progr. F3-conf., Neve Ilan, Jan. 6–9, 1986, Hilger, Bristol, Jerusalem, 1986, 190–204

[16] Krajcinovic D., “Damage mechanics accomplishments, trends and needs”, Int. J. Solids Struct., 37:1–2 (2000), 267–277 | DOI | Zbl

[17] Chaboche J. L., “Continuum Damage Mechanics: Part I—General Concepts”, J. Appl. Mech., 55:1 (1988), 59–64 | DOI

[18] Betten J., “Mathematical modelling of materials behavior under creep conditions”, Appl. Mech. Rev., 54:2 (2001), 107–132 | DOI

[19] Yao Hua-Tang, Xuan Fu-Zhen, Wang Zhengdong, Tu Shan-Tung. , “A review of creep analysis and design under multi-axial stress states”, Nucl. Eng. Des., 237:18 (2007), 1969–1986 | DOI

[20] Rabotnov Yu. N., “Experimental data on creep of engineering alloys and phenomenological theories of creep. A review”, J. Appl. Mech. Tech. Phys., 6:1 (1965), 137–154 | DOI

[21] Namestnikov B. S., “On a hypothesis in the theory of triaxial creep”, no. 2, 1960, 3–14 (In Russian)

[22] Namestnikov V. S., Rabotnov Yu. N., “On the hypothesis of the equation of state for creep”, Prikl. Mekh. Tekh. Fiz. [J. Appl. Mech. Tech. Phys.], 1961, no. 3, 101–102 (In Russian)

[23] Namestnikov V. S., “On creep under variable loads at complex stress state”, Izv. Akad. Nauk SSSR. OTN, 1957, no. 10, 83–85 (In Russian)

[24] Namestnikov V. S., “A hypothesis on proportionality of deviators under creep conditions at varied loads and constant principal axes”, Prikl. Mekh. Tekh. Fiz. [J. Appl. Mech. Tech. Phys.], 1960, no. 3, 212–214 (In Russian)

[25] Johnson A. E., Henderson J., Mathur V. D., “Creep under changing complex stress systems. II”, Engineer, Lond., 206:5351 (1958), 251–257

[26] Johnson A. E., Khan B., “Creep under changing complex-stress systems in copper at 250$^\circ\mathrm{C}$”, Int. J. Mech. Sci., 7:12 (1965), 791–810 | DOI

[27] Johnson A. E., Henderson J., Mathur V. D., “Creep under changing complex stress systems. III”, Engineer, Lond., 206:5352 (1958), 287–291

[28] Johnson A. E., Henderson J., Mathur V. D., “Creep under changing complex stress systems. I”, Engineer, Lond., 206:5350 (1958), 209–216

[29] Ohashi Y., Ohno N., Kawai M., “Evaluation of creep constitutive equations for type 304 stainless steel under repeated multiaxial loading”, J. Eng. Mater. Technol., 104:3 (1982), 155–164 | DOI

[30] Kawai M., “Alternative form of the auxiliary hardening rule for multiaxial repeated creep”, Trans. Jap. Soc. Mech. Eng. A, 59:566 (1993), 2440–2447 | DOI

[31] Kawai M., “Creep hardening rule under multiaxial repeated stress changes”, JSME Int. J., Ser. A, 38:2 (1995), 201–212 | DOI

[32] Dyson B. F., McLean D., “Creep of Nimonic 80A in torsion and tension”, Met. Sci., 11:2 (1977), 37–45 | DOI

[33] Sosnin O. V., Gorev B. V., Nikitenko A. F., “Enegry variant of theory of creep. 1. Basic hypotheses and their experimental verification”, Strength Mater., 8:11 (1976), 1255–1260 | DOI

[34] Chu S. C., Sidebottom O. M., “Creep of metal torsion-tension members subjected to nonproportionate load changes”, Exp. Mech., 10:6 (1970), 225–232 | DOI

[35] Sidebottom O. M., “Elevated-temperature creep and relaxation of torsion-tension members”, Exp. Mech., 18:4 (1978), 121–126 | DOI

[36] Mark R., Findley W. N., “Concerning a creep surface derived from a multiple integral representation for 304 stainless steel under combined tension and torsion”, J. Appl. Mech., 45:4 (1978), 773–779 | DOI

[37] Findley W. N., Lai J. S., “Creep of 2618 aluminum under side-steps of tension and torsion and stress reversal predicted by a viscous-viscoelastic model”, J. Appl. Mech., 48:1 (1981), 47–54 | DOI

[38] Ding J. L., Lee S. R., “Development of viscoplastic constitutive equation through biaxial material testing”, Exp. Mech., 28:3 (1988), 304–309 | DOI

[39] Volkov I. A., Igumnov L. A., Kazakov D. A., Shishulin D. N., Smetanin I. V., “Defining relations of transient creep under complex stress state”, Problems of Strength and Plasticity, 78:4 (2016), 436–451 (In Russian) | DOI

[40] Oytana C., Delobelle P., Mermet A., “Constitutive equations study in biaxial stress experiments”, J. Eng. Mater. Technol., 104:1 (1982), 1–11 | DOI

[41] Lokoshchenko A. M., “Vibrocreep of metals in uniaxial and complex stress states”, Mech. Solids, 49:4 (2014), 453–460 | DOI

[42] Lokoshchenko A. M., Fomin L. V., Basalov Yu. G., Agababyan V. S., “Simulation of metal creep in nonstationary complex stress state”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:1 (2019), 69–89 (In Russian) | DOI | Zbl

[43] Aktaa J., Schinke B., “The influence of the hardening state on time dependent damage and its consideration in a unified damage model”, Fatigue Fract. Engng. Mater. Struct., 19:9 (1996), 1143–1151 | DOI

[44] Lucas G. E., Pelloux R. M. N., “Texture and stress state dependent creep in Zircaloy-2”, Met. Trans. A, 12:7 (1981), 1321–1331 | DOI

[45] Murakami S., Kawai M., Ohmi Y., “Effects of strain amplitude history and temperature history on multiaxial cyclic hardening of type 316 stainless steel”, Trans. Jap. Soc. Mech. Eng. A, 54:501 (1988), 1131–1139 | DOI

[46] Murakami S., Kawai M., Aoki K., Ohmi Y., “Temperature-dependence of multiaxial non-proportional cyclic behavior of type 316 stainless steel”, J. Eng. Mater. Technol., 111:1 (1989), 32–39 | DOI

[47] Ohashi Y., Tokuda M., “Precise measurement of plastic behaviour of mild steel tubular specimens subjectsd to combined torsion and axial force”, J. Mech. Phys. Solids, 21:4 (1973), 241–261 | DOI

[48] Ohashi Y., Kawai M., Momose T., “Effects of prior plasticity on subsequent creep of type 316 stainless steel at elevated temperature”, J. Eng. Mater. Technol., 108:1 (1986), 68–74 | DOI

[49] Murakami S., Kawai M., Yamada Y., “Creep after cyclic-plasticity under multiaxial conditions for type 316 stainless steel at elevated temperature”, J. Eng. Mater. Technol., 112:3 (1990), 346–352 | DOI

[50] Dyson B. F., Verma A. K. Szkopiak Z. C., “The influence of stress state on creep resistance: Experiments and modelling”, Acta Metallurgica, 29:9 (1981), 1573–1580 | DOI

[51] Cho U. W., Findley W. N., “Creep and plastic strains under stress reversal in torsion with and without simultaneous tension for 304 stainless steel at 593$^\circ\mathrm{C}$”, J. Appl. Mech., 50:3 (1983), 587–592 | DOI

[52] Johnson A. E., Henderson J., Khan B., “Complex stress creep fracture of copper at 250$^\circ\mathrm{C}$ under vibratory stress”, Engineer, Lond., 212:5509 (1961), 304–308

[53] Sosnin O. V., “Energy version of the theory of creep and long-term (creep) strength. Creep and rupture of nonstrengthening materials. I.”, Strength Mater., 5:5 (1973), 564–568 | DOI

[54] Sosnin O. V., “Version of creep theory with energy parameters of hardening”, Mekhanika deformiruemykh tel i konstruktsii [Mechanics of Deformable Solids and Structures], Mashinostroenie, Moscow, 1975, 460–463 (In Russian)

[55] Sosnin O. V., Lyubashevskaya I. V., Novoselya I. V., “Comparative estimation of high-temperature creep and rupture of structural materials”, J. Appl. Mech. Tech. Phys., 49:2 (2008), 261–266 | DOI

[56] Zolochevskii A. A., “Energy version of theory of creep and creep rupture of materials with different resistances to tension and contraction”, Izv. Vyssh. Uchebn. Zaved., Mashinostroenie, 1986, no. 12, 7–10 (In Russian)

[57] Kulagin D. A., Lokoshchenko A. M., “Modeling of influence of aggressive environment on creep and creep rupture of metals under complex stress-strain state”, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela [Mech. Solids], 2004, no. 1, 188–199 (In Russian)

[58] Leckie F. A., Hayhurst D. R, “Creep rupture of structures”, Proc. R. Soc. Lond. A, 340:1622 (1974), 323–347 | DOI | Zbl

[59] Leckie F. A., Hayhurst D. R., “Constitutive equations for creep rupture”, Acta Metallurgica, 25:9 (1977), 1059–1070 | DOI

[60] Kowalewski Z. L., Lin J., Hayhurst D. R., “Investigation of a high accuracy uni-axial creep testpiece with slit extensometer ridges”, Arch. Mech., 47:2 (1995), 261–279

[61] Othman A. M., Dyson B. F., Hayhurst D. R., Lin J., “Continuum damage mechanics modelling of circumferentially notched tension bars undergoing tertiary creep with physically-based constitutive equations”, Acta Metall. Mater., 42:3 (1994), 597–611 | DOI

[62] Vakili–Tahami F., Hayhurst D. R., Wong M T., “High-temperature creep rupture of low alloy ferritic steel butt-welded pipes subjected to combined internal pressure and end loadings”, Phil. Trans. R. Soc. Lond. A, 363:1836 (2005), 2629–2661 | DOI

[63] Xu Q., Hayhurst D. R., “The evaluation of high-stress creep ductility for 316 stainless steel at 550$^\circ\mathrm{C}$ by extrapolation of constitutive equations derived for lower stress levels”, Int. J. Pres. Ves. Pip., 80:10 (2003), 689–694 | DOI

[64] Rzhanitsyn A. R., “Theory of creep rupture under arbitrary uniaxial and biaxial loading”, Stroit. Mekh. Raschet Sooruzh., 1975, no. 4, 25–29 (In Russian)

[65] Lemaître J., Chaboche J.-L., “Aspect phénoménologique de la rupture par endommagement”, J. Méc. Appl., 2:3 (1978), 317–365

[66] Lemaître J., Sermage J. P., “One damage law for different mechanisms”, Comput. Mech., 20:1–2 (1997), 84–88 | DOI | Zbl

[67] Shesterikov S. A., Lebedev S. Yu., Yumashev M. A., “New functional relationships to describe the processes of creep and long-term strength”, Proc. of the 9th Conf. on Strength and Ductility (Moscow, January 22–26, 1996), v. 3, Inst. of Problems of Mechanics, Moscow, 1996, 130–134 (In Russian)

[68] Chrzanowski M., Madej J., “Isochronous creep rupture curves in plane stress”, Mech. Res. Commun., 7:1 (1980), 39–40 | DOI | Zbl

[69] Murakami S., Mizuno M., “A constitutive equation of creep, swelling and damage under neutron irradiation applicable to multiaxial and variable states of stress”, Int. J. Solids Struct., 29:19 (1992), 2319–2328 | DOI | Zbl

[70] Cane B. J., “Creep damage accumulation and fracture under multiaxial stresses”, D. Francois et al (eds.), Advances in Fracture Research, v. 3, Pergamon Press, New York, 1981, 1285–1293

[71] Dyson B. F., Loveday M. S., “Creep fracture in Nimonic 80A under triaxial tensile stressing”, A.R.S. Ponter, D.R. Hayhurst (eds.), Creep in Structures. International Union of Theoretical and Applied Mechanics, Springer, Berlin, Heidelberg, 1981, 406–421 | DOI

[72] Hayhurst D. R., Brown P. R., Morrison C. J., “The role of continuum damage in creep crack growth”, Phil. Trans. R. Soc. Lond. A, 311:1516 (1984), 131–158 | DOI

[73] Hayhurst D. R., Felce I. D., “Creep rupture under tri-axial tension”, Eng. Fract. Mech., 25:5–6 (1986), 645–664 | DOI

[74] Trivaudey F., Delobelle P., “High temperature creep damage under biaxial loading—Part I: Experiments”, J. Eng. Mater. Technol., 112:4 (1990), 442–449 | DOI

[75] Tvergaard V., “On the stress state dependence of creep rupture”, Acta Metallurgica, 34:2 (1986), 243–256 | DOI

[76] Trivaudey F., Delobelle P., “High temperature creep damage under biaxial loading—Part II: Model and simulations”, J. Eng. Mater. Technol., 112:4 (1990), 450–455 | DOI

[77] Trąmpczyński W., Hayhurst D. R., “Creep deformation and rupture under non-proportional loading”, A.R.S. Ponter, D.R. Hayhurst (eds.), Creep in Structures. International Union of Theoretical and Applied Mechanics, Springer, Berlin, Heidelberg, 388–405 | DOI

[78] Leckie F. A., Onat E. T., “Tensorial nature of damage measuring internal variables”, J. Hult, J. Lemaitre (eds.), Physical Non-Linearities in Structural Analysis. International Union of Theoretical and Applied Mechanics, Springer, Berlin, Heidelberg, 1981, 140–155 | DOI

[79] Getsov L. B., “Kinetic equations of failure in complex programs of cyclic loading”, Strength Mater., 10:7 (1978), 767–775 | DOI

[80] Pavlov P. A., Kurilovich N. N., “Delayed fracture of heat-resistant steels with nonsteady loading”, Strength Mater., 14:2 (1982), 188–191 | DOI

[81] Xu Q., “The development of validation methodology of multi-axial creep damage constitutive equations and its application to 0.5Cr0.5Mo0.25V ferritic steel at 590$^\circ\mathrm{C}$”, Nuc. Eng. Des., 228:1–3 (2004), 97–106 | DOI

[82] Lin J., Kowalewski Z. L., Cao J., “Creep rupture of copper and aluminium alloy under combined loadings—experiments and their various descriptions”, Int. J. Mech. Sci., 47:7 (2005), 1038–1058 | DOI

[83] Trąmpczyński W., Hayhurst D. R., Leckie F. A., “Creep rupture of copper and aluminium under non-proportional loading”, J. Mech. Phys. Solids, 29:5–6 (1981), 353–374 | DOI

[84] Kachanov L. M., “On brittle fracture in creep under complex loading”, Vestn. Leningrad. Univ., 1972, no. 1, 92–96 (In Russian)

[85] Namestnikova I. V., Shesterikov S. A., “Vector representation of the damage parameter”, Deformation and Fracture of Solids, Moscow State Univ., Moscow, 1985, 43–52 (In Russian)

[86] Peleshko V. A., “Using surface damage to describe creep and creep rupture under complex loading”, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela [Mech. Solids], 2003, no. 2, 124–138 (In Russian)

[87] Chow C. L., Yang X. J., Chu E., “Viscoplastic constitutive modeling of anisotropic damage under nonproportional loading”, J. Eng. Mater. Technol., 123:4 (2001), 403–408 | DOI

[88] Lokoshchenko A. M., “Study of creep rupture under complex stress using a kinetic approach”, Tr. Tsentr. Kotloturbin. Inst., 1986, no. 230, 107–109 (In Russian)

[89] Lokoshchenko A. M., “Methods for modeling the creep rupture of metals under stationary and nonstationary complex stress state”, Elasticity and Inelasticity, Proc. of the Int. Symp. Dedicated to the 100th Anniversary of the Birth of A. A. Il'yushin (Moscow, January 20–21, 2011), Moscow State Univ., Moscow, 2011, 389–393 (In Russian)

[90] Lokoshchenko A. M., Nazarov V. V., “Kinetic approach to studying the creep rupture of metals under biaxial tension”, Aviats.-Kosm. Tekh. Tekhnol., 2005, no. 10, 73–78 (In Russian)

[91] Lokoshchenko A. M., Nazarov V. V., “Analysis of the long-term strength of metals under complex stress using criterion and kinetic approaches”, Abstracts 9th All-Russia Congress on Theoretical and Applied Mechanics (Nizhny Novgorod, August 22–28, 2006), v. 3, Nizhegorod. Univ., Nizhny Novgorod, 2006, 135–136 (In Russian)

[92] Lokoshchenko A. M., Nazarov V. V., “Long-term strength of metals under an equiaxial plane stress state”, J. Appl. Mech. Tech. Phys., 50:4 (2009), 670–677 | DOI

[93] Dacheva M.D., Shesterikov S. A., Yumasheva M. A., “Damage under complex transient stress”, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela [Mech. Solids], 1998, no. 1, 44–47 (In Russian)

[94] Chizhik A. A., Petrenya Yu. K., “Fracture due to creep and mechanisms of microfracture”, Dokl. Akad. Nauk SSSR, 297:6 (1987), 1331–1333 (In Russian)

[95] Morachkovskii O. K., “On the creep fracture of anisotropic materials”, Probl. Mashinostroeniya, 1978, no. 6, 41–43 (In Russian)

[96] Chrzanovski M., Madej J., “Budowa granicznych krzywych zniszczenia w oparciu o koncepcję parametru uszkodzenia [The construction of failure limit curves be means of a damage]”, Mech. Teor. Stosow [J. Theor. Appl. Mech.], 18:4 (1980), 587–601 (In Polish)

[97] Khazhinskii G. M., Deformirovanie i dlitel'naia prochnost' metallov [Deformation and Long-Term Strength of Metals], Nauchnyi Mir, Moscow, 2008, 136 pp. (In Russian)

[98] Kachanov L. M., “Fracture in creep under complex loading”, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 1972, no. 5, 11–15 (In Russian)

[99] Hayhurst D. R., Trąmpczyński W., Leckie F. A., “Creep rupture under non-proportional loading”, Acta Metallurgica, 28:9 (1980), 1171–1183 | DOI

[100] Murakami S., Sanomura I., Saitoh K., “Formulation of cross-hardening in creep and its effects on the creep damage process of copper”, J. Engin. Mater. Technol., 108:2 (1986), 167–173 | DOI

[101] Lokoshchenko A. M., “Use of a vector damage parameter in modeling of long-term strength of metals”, Mech. Solids, 51:3 (2016), 315–320 | DOI

[102] Lokoshchenko A. M., “Modeling the long-term strength of metals in an unsteady complex stress state”, Mech. Solids, 53 (2018), 88–100 | DOI

[103] Leckie F. A., Wojewodzki W., “Estimates of rupture life-constant load”, Int. J. Solids Struct., 11:12 (1975), 1357–1365 | DOI | Zbl

[104] Rabotnov Yu. N., “Creep rupture”, M. Hetényi, W.G. Vincenti (eds.), Applied Mechanics. International Union of Theoretical and Applied Mechanics, Springer, Berlin, Heidelberg, 342–349 | DOI

[105] Tamuzh V. P., “A possible theory of prolonged failure”, Strength Mater., 3:2 (1971), 177–183 | DOI

[106] Altenbach H., Schiesse P., “Modelling of the constitutive behaviour of damaged materials”, Advances in Fracture Resistance and Structural Integrity, Selec. Pap. 8th Int. Conf. Fract. (Kyiv, June 8–14, 1993), Pergamon Press, Oxford, 1994, 51–57

[107] Il'yushin A. A., “One theory of creep rupture”, Inzh. Zh., Mekh. Tverd. Tela, 1967, no. 3, 21–35 (In Russian)

[108] Zavoichinskaya E. B., Kiiko I. A., Vvedenie v teoriiu protsessov razrusheniia tverdykh tel [Introduction to the Fracture of Solids], Moscow State Univ., Moscow, 2004, 168 pp. (In Russian)

[109] Pobedrya B. E., “On damage models for rheonomic media”, Mech. Solids, 33:4 (1998), 108–124

[110] Tamuzh V. P., Lagzdyn'sh A. Z., “A variant of the phenomenological theory of fracture”, Polymer Mech., 4:4–6 (1968), 493–500 | DOI

[111] Lagzdin' A. Z., Tamuzh V. P., “Construction of a phenomenological theory of fracture of anisotropic media”, Polymer Mech., 7:4 (1971), 563–571 | DOI

[112] Kopnov V. A., “Long-term strength of anisotropic materials with a complex stress state”, Strength Mater., 14:2 (1982), 183–187 | DOI

[113] Lebedev A. O., Mikhalevich V. M., “On theory of creep rupture”, Dop. NANU, 1998, no. 5, 57–62 (In Ukrainian)

[114] Lebedev A. A., Mikhalevich V. M., “Criterial relationships for residual life assessment of materials”, Strength Mater., 38:4 (2006), 348–353 | DOI

[115] Mikhalevich V. M., “Tensor models of rupture strength. Report no. 3. Criterional relations for loading with a change in stress state and the directions of the principal stresses”, Strength Mater., 28:3 (1996), 238–246 | DOI

[116] Betten J., “Net-stress analysis in creep mechanics”, Ing. Arch., 52:6 (1982), 405–419 | DOI | Zbl

[117] Betten J., “Damage tensors in continuum mechanics”, J. Mec. Theor. Appl., 2:1 (1983), 13–22

[118] Chow C. L., Wang J., “An anisotropic theory of continuum damage mechanics for ductile fracture”, Eng. Fract. Mech., 27:5 (1987), 547–558 | DOI

[119] Bodner S. R., “A procedure for including damage in constitutive equations for elastic-viscoplastic work-hardening materials”, J. Hult, J. Lemaitre (eds.), Physical Non-Linearities in Structural Analysis. International Union of Theoretical and Applied Mechanics, Springer, Berlin, Heidelberg, 1981, 21–28 | DOI

[120] Liu Y., Kageyama Y., Murakami S., “Creep fracture modeling by use of continuum damage variable based on Voronoi simulation of grain boundary cavity”, Int. J. Mech. Sci., 40:2–3 (1998), 147–158 | DOI | Zbl

[121] Murakami S., “Mechanical modeling of material damage”, J. Appl. Mech., 55:2 (1988), 280–286 | DOI

[122] Murakami S., Imaizumi T., “Mechanical description of creep damage state and its experimental verification”, J. Mec. Theor. Appl., 1:5 (1982), 743–761 | Zbl

[123] Murakami S., Ohno N., “A continuum theory of creep and creep damage”, A.R.S. Ponter, D.R. Hayhurst (eds.), Creep in Structures. International Union of Theoretical and Applied Mechanics, Springer, Berlin, Heidelberg, 1981, 422–444 | DOI

[124] Astaf'ev V. I., “Description of fracture processes under creep conditions”, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 1986, no. 4, 164–169

[125] Krajcinovic D., “Continuous damage mechanics revisited: Basic concepts and definitions”, J. Appl. Mech., 52:4 (1985), 829–834 | DOI

[126] Krajcinovic D., Rinaldi A., “Statistical damage mechanics—Part I: Theory”, J. Appl. Mech., 72:1 (2005), 76–85 | DOI | Zbl

[127] Krajcinovic D., Selvaraj S., “Creep rupture of metals—An analytical model”, J. Eng. Mater. Technol., 106:4 (1984), 405–409 | DOI

[128] Man'kovskii V. A., “Criteria for damage and long-term strength of structural materials”, Mashinoved., 1985, no. 1, 87–94 (In Russian)

[129] Delobelle P., Trivaudey F., Oytana C., “High temperature creep damage under biaxial loading: INCO 718 and 316 (17–12 SPH) steels”, Nucl. Eng. Des., 114:3 (1989), 365–377 | DOI

[130] Lemaître J., “A three-dimensional ductile damage model applied to deep-drawing forming limits”, Mech. Behav. Mater., Proc. of the 4th Int. Conf. (Stockholm, Sweden, 15–19 August 1983), Pergamon Press, Oxford, 1984, 1047–1053 | DOI

[131] Lokoshchenko A. M., “Application of kinetic theory to the analysis of high-temperature creep rupture of metals under complex stress (review)”, J. Appl. Mech. Tech. Phys., 53:4 (2012), 599–610 | DOI | Zbl

[132] Agakhi K. A., Georgievskii D. V., “Tensor nonlinear constitutive relations of isotropic creep theorywith tensor measure of damage”, Izv. Tulsk. Gos. Univ. Estestv. Nauki, 2013, no. 2, 2–9 (In Russian)

[133] Murakami S., Sanomura Y., “Creep and creep damage of copper under multiaxial states of stress”, A. Sawczuk and B. Bianchi (eds.), Elasticity Today-Modeling, Methods and Applications, Elsevier, London–New York, 1985, 535–551 pp.

[134] Lokoshchenko A. M., Polzuchest' i dlitel'naia prochnost' metallov v agressivnykh sredakh [Creep and LongTerm Strength of Metals in Corrosive Media], Moscow State Univ., Moscow, 2000, 178 pp. (In Russian)

[135] Lokoshchenko A. M., “Creep and long-term strength of metals in corrosive media (Review)”, Mater. Sci., 37:4 (2001), 559–572 | DOI

[136] Lokoshchenko A. M., “Methods of modeling the influence of the ambient medium on creep and long-term strength of metals”, Usp. Mekh., 1:4 (2002), 90–121 (In Russian)

[137] Lokoshchenko A. M., Il'in A. A., Mamonov A. M., Nazarov V. V., “Experimental and theoretical study of the effect of hydrogen on the creep and long-term strength of VT6 titanium alloy”, Russ. Metall., 2008, no. 2, 142–147 | DOI

[138] Lokoshchenko A. M., Il'in A. A., Mamonov A. M., Nazarov V. V., “Analysis of the creep and long-term strength of VT6 titanium alloy with preliminarily injected hydrogen”, Mater. Sci., 44:5 (2008), 700–707 | DOI