| ||Mechanics of Solids|
A Journal of Russian Academy of Sciences
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936
Archive of Issues
|Total articles in the database:|| ||10864|
|In Russian (Èçâ. ÐÀÍ. ÌÒÒ):|| ||8009|
|In English (Mech. Solids):|| ||2855|
|D. V. Georgievskii, D. M. Klimov, and B. E. Pobedrya, "Specific features of the behavior of viscoelastic models," Mech. Solids. 39 (1), 88-120 (2004)|
||Specific features of the behavior of viscoelastic models|
||D. V. Georgievskii (Moscow)|
D. M. Klimov (Moscow)
B. E. Pobedrya (Moscow)
||The major problem of solid mechanics is the modeling of
the deformation processes. The models can be divided into
two classes-scleronomic models and rheonomic models.
The constitutive relations of the scleronomic models are
time-invariant. The rheonomic models are described by operator
relations that explicitly depend on time. The simplest rheonomic
physically linear model in the solid mechanics is the model
of a viscoelastic body that describes relaxation (the reduction
in the stress at constant strain) and creep (the increase
in the strain at constant stress). The model of a linear viscoelastic
body takes into account the energy dissipation due to heat release during
the deformation of the body, as well as other effects that are not accounted
for by other models.|
To a great extent, the foundation of the modern theory of viscoelasticity
has been laid by an outstanding scientist Yu. N. Rabotnov . This theory
has the extensive literature; see, e.g., the monographs [1-17] many of which
have become classical textbooks. The enhancement in engineering requires the
possibilities for the modeling to be extended. This is the case, in particular,
for the model of linear viscoelastic body. To give a review of such a modeling
is just the purpose of the present paper.
|1. ||Yu. N. Rabotnov, The Creep in Structural Members [in Russian],
Nauka, Moscow, 1966.|
|2. ||L. M. Kachanov, The Theory of Creep [in Russian],
Fizmatgiz, Moscow, 1960.|
|3. ||D. R. Bland, The Theory of Linear Viscoelasticity,
Pergamon Press, London, 1960.|
|4. ||J. D. Ferry, Viscoelastic Properties of Polymers,
Wiley, New York, London, 1961.|
|5. ||M. Reiner, Rheology, Springer, Berlin, 1958.|
|6. ||A. R. Rzhanitsyn, The Theory of Creep [in Russian],
Stroiizdat, Moscow, 1968.|
|7. ||A. A. Il'yushin and B. E. Pobedrya, Fundamentals
of the Mathematical Theory of Thermoviscoelasticity
[in Russian], Nauka, Moscow, 1970.|
|8. ||Yu. N. Rabotnov and S. T. Mileiko, Short-term Creep
[in Russian], Nauka, Moscow, 1970.|
|9. ||W. A. Day, The Thermodynamics of Simple Materials with
Fading Memory, Springer, Berlin, Heidelberg, New York, 1972.|
|10. ||V. V. Moskvitin, The Strength of Viscoelastic Materials
as Applied to Solid Fuel Rocket Engine Charges
[in Russian], Nauka, Moscow, 1972.|
|11. ||I. I. Bugakov, The Creep of Polymer Materials
[in Russian], Nauka, Moscow, 1973.|
|12. ||F. K. G. Odquist, Mathematical Theory of Creep and
Creep Rupture, Clarendon Press, Oxford, 1974.|
|13. ||N. N. Malinin, Applied Theory of Plasticity and Creep
[in Russian], Mashinostroenie, Moscow, 1975.|
|14. ||P. M. Ogibalov, V. A. Lomakin, B. P. Kishkin,
Mechanics of Polymers
[in Russian], Izd-vo MGU, Moscow, 1975.|
|15. ||V. A. Postnov, Theory of Plasticity and Creep
[in Russian], Izd-vo Leningradsk.
Korablestroit. In-ta, Leningrad, 1975.|
|16. ||W. Fluegge, Viscoelasticity, Springer, New York, 1975.|
|17. ||M. A. Koltunov, Creep and Relaxation
[in Russian], Vysshaya Shkola, Moscow, 1976.|
|18. ||Yu. N. Rabotnov, Fundamentals of Hereditary Mechanics of Solids
[in Russian], Nauka, Moscow, 1977.|
|19. ||G. N. Savin, Mechanics of Solids
[in Russian], Naukova Dumka, Kiev, 1970.|
|20. ||H. Kraus, Creep Analysis,
Wiley, New York, 1980.|
|21. ||A. K. Malmeister, V. P. Tamuzh, and G. A. Teters,
Strength of Polymer and Composite Materials
[in Russian], Zinatne, Riga, 1980.|
|22. ||M. E. Gurtin, An Introduction to Continuum Mechanics,
Acad. Press, New York, 1981.|
|23. ||K. N. Rusinko, Theory of Plasticity and Unsteady Creep
[in Russian], Vishcha Shkola, Lvov, 1981.|
|24. ||R. M. N. Christensen, Theory of Viscoelasticity. An Introduction,
Acad. Press, New York, London, 1971.|
|25. ||Yu. S. Urzhumtsev, Prediction of Long-term Strength in Polymer
Materials [in Russian], Nauka, Moscow, 1982.|
|26. ||H. Ruesch, D. Junfwirth and H. Hilsdorf,
Creep and Shrinkage: Their Effect on the Behavior,
Springer, New York, 1983.|
|27. ||N. Kh. Arutyunyan and V. B. Kolmanovskii,
Theory of Creep of Inhomogeneous Bodies
[in Russian], Nauka, Moscow, 1983.|
|28. ||J. Čadek, Creep kovových materiálu, Academia, Praha, 1984.|
|29. ||A. S. Kravchuk, V. P. Maiboroda, and Yu. S. Urzhumtsev,
Mechanics of Polymer and Composite Materials
[in Russian], Nauka, Moscow, 1985.|
|30. ||A. C. Pipkin, Lectures on Viscoelasticity Theory,
Springer, New York, Berlin, 1986.|
|31. ||G. Z. Sharafutdinov, Photoviscoelasticity
[in Russian], Izd-vo MGU, Moscow, 1987.|
|32. ||A. A. Il'yushin, Continuum Mechanics
[in Russian], Izd-vo MGU, Moscow, 1990.|
|33. ||I. I. Bugakov and I. I. Demidova,
The Method of Photothermoviscoelasticity
[in Russian], Izd-vo SPbGU, St. Petersburg, 1970.|
|34. ||B. E. Pobedrya, Numerical Methods in Elasticity and Plasticity
[in Russian], Izd-vo MGU, Moscow, 1993.|
|35. ||A. D. Drozdov, Viscoelastic Structures,
Acad. Press, San Diego, 1998.|
|36. ||S. G. Samko, A. A. Kilbas, and O. I. Marichev,
Fractional Derivatives and Integrals and some their Applications
[in Russian], Nauka i Tekhnika, Minsk, 1987.|
|37. ||I. Podlubny, Fractional Differential Equations. An Introduction
to Fractional Derivatives, Fractional Differential Equations, to Methods
of their Solution and some of their Applications, Acad. Press,
San Diego, 1999.|
|38. ||L. B. Eldred, W. P. Baker, and A. N. Palazotto,
"Kelvin-Voigt vs fractional derivative models as constitutive
relations for viscoelastic materials,"
AIAA Journal, Vol. 33, No. 3, pp. 547-550, 1995.|
|39. ||N. Shimizu and M. Iijima, "Fractional differential model
of viscoelastic material," Trans. Japan Soc. Mech. Eng., Ser. C,
Vol. 62, No. 604, pp. 4447-4451, 1996.|
|40. ||Y. A. Rossikhin and M. V. Shitikova,
"Application of fractional derivatives to the analysis
of damped vibrations of viscoelastic single mass systems,"
Acta Mech., Vol. 120, No. 1-4, pp. 109-125, 1997.|
|41. ||A. D. Drozdov, "Fractional differential models in
Acta Mech., Vol. 124, No. 1-4, pp. 155-180, 1997.|
|42. ||M. Enelund, L. Maehler, K. Runesson, and B. L Josefson,
"Formulation and integration of the standard linear viscoelastic
solids with fractional order rate laws,"
Intern J. Solids and Structures, Vol. 36, No. 16, pp. 2417-2442, 1999.|
|43. ||S. W. J. Welch, R. A. L. Rorrer, and R. G. Duren,
"Application of time-based fractional calculus methods to
viscoelastic creep and stress relaxation of materials,"
Mech. Time-depend. Mater., Vol. 3, No. 3, pp. 279-303, 1999.|
|44. ||T. A. Surguladze, "One application of fractional Green's
Izv. AN. MTT [Mechanics of Solids], No. 1, pp. 53-60, 2001.|
|45. ||T. A. Surguladze, "On certain applications of fractional calculus
to viscoelasticity," J. Math. Sci., Vol. 112, No. 5, pp. 4517-4557, 2002.|
|46. ||M. Enelund, A. Fenander, and P. Olsson,
"Fractional integral formulation of constitutive equations
AIAA Journal, Vol. 35, No. 8, pp. 1356-1362, 1997.|
|47. ||B. E. Pobedrya, "Energy dissipation in viscoelasticity,"
Vestnik MGU [Bulletin of the Moscow State University],
Ser. 1. Matematika, Mekhanika, No. 4, pp. 35-46, 2003.|
|48. ||M. Fabrizio, "On the inversion of a linear viscoelastic
Math. Appl., Vol. 3, No. 2, pp. 141-148, 1992.|
|49. ||G. Matarazzo, "Time-irreversibility and existence
and uniqueness of solutions of problems in linear viscoelasticity,"
Ukr. Mat. Zh., Vol. 52, No. 7, pp. 923-930, 2000.|
|50. ||S. W. Park and R. A. Schapery, "Methods of interconversion
between linear viscoelastic material functions. Part 1. A numerical
method based on Prony series,"
Intern J. Solids and Structures, Vol. 36, No. 11, pp. 1653-1675, 1999.|
|51. ||R. A. Schapery and S. W. Park, "Methods of interconversion
between linear viscoelastic material functions. Part 2. An approximate
Intern J. Solids and Structures, Vol. 36, No. 11, pp. 1677-1699, 1999.|
|52. ||V. A. Pal'mov, "Rheological models in nonlinear solid mechanics,"
Uspekhi Mekhaniki [Advances in Mechanics], Vol. 3, No. 3, pp. 75-115,
|53. ||J. Egan, "A new look at linear visco-elasticity,"
Mater. Letters, Vol. 31, No. 3-6, pp. 351-357, 1997.|
|54. ||A. A. Il'yushin and P. M. Ogibalov,
"Some generalizations of the Voigt and Maxwell models,"
Mekhanika Polimerov, No. 2, pp. 190-196, 1966.|
|55. ||G. Del Piero and L. Deseri, "Monotonic, completely monotonic,
and exponential relaxation functions in linear viscoelasticity,"
Quart. Appl. Math., Vol. 53, No. 2, pp. 273-300, 1995.|
|56. ||A. Morro and M. Fabrizio, "Further inequalities for viscoelastic
relaxation functions," Mech. Res. Comm., Vol. 22, No. 4, pp. 349-353, 1995.|
|57. ||S. Hazanov, "New class of creep-relaxation functions,"
Intern J. Solids and Structures, Vol. 32, No. 2, pp. 165-172, 1995.|
|58. ||F. Akyildiz, R. S. Jones, and K. Walters, "On the spring-dashpot
representation of linear viscoelastic behavior,"
Rheol. Acta, Vol. 29, No. 5, pp. 482-484, 1990.|
|59. ||G. V. Vorontsov and A. I. Reznichenko, "On the relationship
between the matrices of linear differential and integral models of
a viscoelastic body subject to the spatial stress-strain state,"
Izv. Vuzov Severo-Kavkaz. Regiona. Tekhn. Nauki, No. 3, pp. 99-100, 1998.|
|60. ||G. N. Savin and Ya. Ya. Rushchitskii, Fundamentals of Mechanics
of Hereditary Media [in Russian], Vishcha Shkola, Kiev, 1976.|
|61. ||P. Mazilu, "On the constitutive law of Boltzmann-Volterra,"
Rev. Roum. Math. Pure Appl., Vol. 18, No. 7, pp. 1067-1069, 1973.|
|62. ||N. I. Malinin, "Creep of structural members from polymer materials,"
Zh. Prikl. Mekhaniki i Tekhn. Fiziki, No. 2, pp. 109-125, 1970.|
|63. ||P. S. Theocaris, "Creep and relaxation contraction ratio
of linear viscoelastic materials,"
J. Mech. and Phys. Solids, Vol. 12, No. 3, pp. 125-138, 1964.|
|64. ||G. N. Maslov, "Thermal stress state of concrete blocks
with the creep of the concrete being taken into account,"
Izv. NIIG, Vol. 28, pp. 175-188, 1941.|
|65. ||I. I. Ulitskii, Creep of Concrete
[in Russian], Ukrsgostekhizdat, Kiev, Lvov, 1948.|
|66. ||V. A. Florin, "One-dimensional problem of the compaction of a soil,
with the ageing, nonlinear creep, and fracture of the structure being taken
Izv. AN SSSR. ONT, No. 9, pp. 1229-1234, 1953.|
|67. ||A. A. Il'yushin, "An experimental method fro solving one integral
equation of viscoelasticity,"
Mekhanika Polimerov, No. 4, pp. 584-587, 1969.|
|68. ||N. N. Malinin, Creep Design of Engineering Structural Elements
[in Russian], Mashinostroenie, Moscow, 1981.|
|69. ||Yu. E. Yakubovskii, "On the determination of the creep kernels
for ageing materials,"
Vol. 27, No. 6, pp. 37-44, 1991.|
|70. ||A. D. Drozdov, "On constitutive laws for ageing materials at finite strains,"
Europ. J. Mech., Ser. A, Vol. 12, No. 3, pp. 305-324, 1993.|
|71. ||V. M. Pestrikov, "On the constitutive relations for ageing materials
with the physical and chemical processes being taken into account,"
Izv. AN. MTT [Mechanics of Solids], No. 4, pp. 134-140, 1999.|
|72. ||V. A. Palmov, "General principles of nonlinear rheology
and typical mistakes in applications,"
in Transactions of SPbAN on Strength
Issues [in Russian], Vol. 1, pp. 126-139, 1997.|
|73. ||A. I. Leonov, On the Description of Rheological Behavior of Elastoviscous
Media at Large Elastic Strains. Preprint 34 [in Russian], Institute for Problems
in Mechanics of the USSR Academy of Sciences, Moscow, 1973.|
|74. ||V. G. Gromov, Instability, Bifurcations, and Catastrophes
of Steady-state Motions of Hereditary Solids [in Russian],
D.Sc. Dissertation, Tul'sk. Politekhn. In-t, Tula, 1984.|
|75. ||V. G. Gromov, "A method for constructing constitutive relations
for viscoelastic bodies subject to finite strains,"
Doklady AN SSSR, Vol. 285, No. 1, pp. 69-73, 1985.|
|76. ||D. Durban, D. G. Zeitoun, and H. E. Banaim,
"Finite linear viscoelasticity,"
J. Eng. Mech., Vol. 116, No. 11, pp. 2449-2462, 1990.|
|77. ||V. A. Palmov, "Large strains in viscoelastoplasticity,"
Acta Mech., Vol. 125, No. 1-4, pp. 129-139, 1997.|
|78. ||S. Reese and S. Govindjee, "A theory of finite viscoelasticity
and numerical aspects,"
Int. J. Solids and Structures, Vol. 35, No. 26-27, pp. 3455-3482, 1998.|
|79. ||R. A. Schapery, "Nonlinear viscoelastic solids,"
Int. J. Solids and Structures, Vol. 37, No. 1-2, pp. 359-366, 1999.|
|80. ||V. A. Palmov, "Comparison of different approaches in
viscoelastoplasticity for large strains,"
ZAMM, Vol. 80, No. 11-12, pp. 801-806, 2000.|
|81. ||J. Bonet, "Large strain viscoelastic constitutive models,"
Int. J. Solids and Structures, Vol. 38, No. 1-2, pp. 2953-2968, 2001.|
|82. ||A. A. Il'yushin and P. M. Ogibalov, "Quasilinear theory
of viscoelasticity and the small parameter method,"
Mekhanika Polimerov, No. 2, pp. 170-189, 1966.|
|83. ||B. E. Pobedrya, "On the state equations in nonlinear viscoelasticity,"
Mekhanika Polimerov, No. 3, pp. 427-435, 1967.|
|84. ||B. E. Pobedrya, "Mathematical theory of nonlinear viscoelasticity,"
in Elasticity and Inelasticity [in Russian], Issue 3, pp. 95-173, Izd-vo MGU,
|85. ||B. E. Pobedrya, "Numerical methods in viscoelasticity,"
Mekhanika Polimerov, No. 3, pp. 417-428, 1973.|
|86. ||Yu. A. Basistov and Yu. G. Yanovskii, "On the identification
of mathematical models of viscoelastic media in rheology and electrorheology,"
Mekhanika Kompozitsionnykh Materialov i Konstruktsii,
Vol. 7, No. 1, pp. 114-130, 2001.|
|87. ||M. Grasselli, "Identifying relaxation kernels in linearly viscoelastic
bodies," J. Inverse Ill-posed Probl., Vol. 4, No. 5, pp. 391-407, 1996.|
|88. ||S. A. Shesterikov and A. M. Lokoshscenko, "Creep
and long-term strength of metals," in
Achievements in Science and Technology. Ser. Mechanics of Solids [in Russian],
Vol. 13, pp. 3-104, VINITI, Moscow, 1980.|
|89. ||O. E. Ol'khovik, "Different-modulus theory of creep,"
Izv. Vuzov. Mashinostroenie, No. 10-12, pp. 21-25, 1994.|
|90. ||L. A. Zlochevskaya and A. N. Sklepus, "Energy-based version
of the theory of creep of metals that possess different strength for tension
Problemy Prochnosti, No. 2, pp. 108-115, 2001.|
|91. ||Yu. G. Basalov, V. N. Kuznetsov, and S. A. Shesterikov,
"Constitutive relations for a rheonomous material,"
Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 69-81, 2000.|
|92. ||B. E. Pobedrya, "Thermodynamics of viscoelastic models,"
in Applied Mathematics and Programming [in Russian], Issue 1, pp. 75-86,
Izd-vo AN MSSR, Kishinev, 1969.|
|93. ||D. L. Bykov, A. A. Il'yushin, P. M. Ogibalov, and B. E. Pobedrya,
"Some basic problems of thermoviscoelasticity,"
Mekhanika Polimerov, No. 1, pp. 41-58, 1971.|
|94. ||I. I. Gol'denblat and V. L. Bazhanov, "Thermodynamic theory of creep,"
in Elasticity and Inelasticity [in Russian], Issue 2, pp. 202-218,
Izd-vo MGU, Moscow, 1971.|
|95. ||B. E. Pobedrya, "On coupled problems in solid mechanics,"
in Elasticity and Inelasticity [in Russian], Issue 2, pp. 224-253,
Izd-vo MGU, Moscow, 1971.|
|96. ||S. Hassani, A. Alauoi Soulimani, and A. Ehrlacher,
"A nonlinear viscoelastic model: The pseudo-linear model,"
Europ. J. Mech., Ser. A, Vol. 17, No. 4, pp. 567-598, 1998.|
|97. ||D. L. Bykov and D. N. Konovalov, "Determination of material functions
in the nonlinear theory of
thermoviscoelasticity using its hierarchical structure,"
Izv. AN. MTT [Mechanics of Solids], No. 5, pp. 189-205, 1999.|
|98. ||A. I. Murdoch, "Remarks on the foundations of linear viscoelasticity,"
J. Mech. Phys. Solids, Vol. 40, No. 7, pp. 1559-1568, 1992.|
|99. ||Z. Sobotka, "Differential equations and their integrals
following from viscoelasticity,"
Acta Techn. CSAV, Vol. 39, No. 6, pp. 675-710, 1994.|
|100. ||Z. Sobotka, "Związk róžnickowe i calcowe w
Zecz. Nauk. Mech., No. 115, pp. 367-372, 1994.|
|101. ||G. A. Sviridyuk and T. G. Sukacheva, "Remarks on linear models
of viscoelastic media,"
Vestnik Chelyabinsk. Un-ta, Ser. Matematika i Mekhanika, No. 1,
pp. 135-147, 1996.|
|102. ||J. Brilla, "Laplace transform and new mathematical theory
of viscoelasticity," Meccanica, Vol. 32, No. 3, pp. 187-195, 1997.|
|103. ||A. A. Kaminskii and I. Yu. Podil'chuk,
"A method for solving boundary-value problems of linear viscoelasticity,"
Prikladnaya Mekhanika, Vol. 34, No. 12, pp. 77-85, 1998.|
||30 September 2003|