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Forced harmonic vibrations and dissipative heating-up of viscoelastic thin-walled elements (Review)

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Abstract

The paper presents a review of scientific studies on development, of the classical and refined models of the thermomechanical behavior of thin-walled single- and multilayer viscoelastic elements. Allowance is made for the temperature dependence of the properties of the material and physical and geometrical nonlinearities in the case of monoharmonic strain as one of the most typical types of deformation. Methods of solution of nonlinear connected problems of thermoviscoelasticity and results of solution of some specific problems on vibrations and heating-up of thin-walled rods, plates, and shells in quasistatic and dynamic formulations are discussed. A number of thermomechanical effects are noted. They are due to the coupling of mechanical and thermal fields and physical and geometrical nonlinearities, taken into account either separately or jointly.

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References

  1. I. Yu. Babich and N. P. Semenyuk, “Stability and initial supercritical behavior of composite shells,”Prikl. Mekh.,34, No. 6, 3–38 (1998) (Int. Appl. Mech.,34, No. 2, 507–537 (1998)).

    MATH  MathSciNet  Google Scholar 

  2. G. I. Barenblat, “Effects of small vibrations of polymers subject to deformation,”Prikl. Mat. Mekh.,30, No. 1, 73–81 (1966).

    Google Scholar 

  3. R. E. Bellman and R. E. Kalaba,Quasilinearization and Nonlinear Boundary-Value Problems [Russian translation], Mir, Moscow (1968).

    MATH  Google Scholar 

  4. V. L. Berdichevskii,Variation Principles of Continuum Mechanics [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  5. V. V. Bolotin, “The equations of nonstationary temperature fields in thin shells in the presence of heat sources,”Prikl. Mat. Mekh.,24, No. 2, 361–364 (1960).

    MathSciNet  Google Scholar 

  6. V. V. Bolotin and A. I. Litvinov, “Theory of vibrodamping polymeric coatings,”Mekh. Polim., No. 2, 269–277 (1978).

    Google Scholar 

  7. V. V. Bolotin and Yu. N. Novichkov,The Mechanics of Multilayer Structures [in Russian], Mashinostroyeniye, Moscow (1980).

    Google Scholar 

  8. A. S. Vol'mir,The Stability of Deformable Systems [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  9. A. S. Vol'mir,The Nonlinear Dynamics of Plates and Shells [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  10. I. I. Vorovich and V. G. Safronenko, “Application of variation principles to connected problems of thermoviscoelastic shells,”Izv. Akad. Nauk SSSR, Ser. Mekh. Tverd. Tela, No. 4, 166–173 (1980).

    MathSciNet  Google Scholar 

  11. L. A. Galin, “Action of vibrational load on polymeric materials,”Izv. Akad. Nauk SSSR, Ser. Mekh., No. 6, 11–15 (1965).

    Google Scholar 

  12. É. I. Grigolyuk and F. A. Kogan, “The present state of the theory of multilayer shells,”Prikl. Mekh.,8, No. 6, 3–18 (1972).

    Google Scholar 

  13. É. I. Grigolyuk and P. P. Chulkov,Stability and Vibrations of Three-Layer Shells [in Russian], Mashinostroyeniye, Moscow (1973).

    Google Scholar 

  14. Ya. M. Grigorenko and A. T. Vasilenko,Problems of the Statics of Anisotropic Inhomogeneous Shells [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  15. Ya. M. Grigorenko and A. T. Vasilenko, “Solution of problems and analysis of the stress-strain state of anisotropic shells (review),”Prikl. Mekh.,33, No. 11, 3–37 (1997) (Int. Appl. Mech.,33, No. 11, 851–880, (1997)).

    Google Scholar 

  16. Ya. M. Grigorenko and A. P. Mukoyed,Solution of Nonlinear Problems of the Theory of Shells on a Computer [in Russian], Vyshcha Shk., Kiev (1983).

    Google Scholar 

  17. Ya. M. Grigorenko, G. I. Bespalova, A. T. Vasilenko, et al.,Numerical Solution of Boundary-Value Problems of Orthotropic Shells of Revolution on an M-220 Computer [in Russian], Naukova Dumka, Kiev (1971).

    Google Scholar 

  18. B. P. Gumenyuk, “On the effect of flexural vibration on the temperature of a viscoelastic rod,”Visnyk Kyivskogo Univ., Ser. Mat. Mekh., No. 14, 100–105 (1972).

    Google Scholar 

  19. B. P. Gumenyuk, “Toward the effect of flexural vibration on the temperature of a viscoelastic rod,”Visnyk Kyivskogo Univ., Ser. Mat. Mekh., No. 15, 41–46 (1973).

    Google Scholar 

  20. B. P. Gumenyuk and V. G. Karnaukhov, “Thermal instability in connected dynamic problems of thermoviscoelasticity,”Dokl. Akad. Nauk Ukr. SSR., Ser. A, No. 7, 609–629 (1978).

    Google Scholar 

  21. B. P. Gumenyuk and V. G. Karnaukhov, “Thermal instability of viscoelastic bodies under dynamic cyclic loads,” in:Thermal Stresses of Structural Elements [in Russian], Issue 18 (1978), pp. 46–50.

  22. B. P. Gumenyuk and V. G. Karnaukhov, “Approximate calculation of critical thermal states in connected dynamic problems of thermoviscoelasticity,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 10, 905–908 (1977).

    Google Scholar 

  23. B. P. Gumenyuk and V. G. Karnaukhov, “Effect of thermomechanical conjugation on the dynamic behavior of a viscoelastic rod subject to forced longitudinal vibrations,”Dop. Akad. Nauk Ukr. RSR, Ser. A, No. 2, 128–132 (1978).

    Google Scholar 

  24. B. P. Gumenyuk and V. G. Karnaukhov, “Heat generation in a viscoelastic cylindrical shell under cyclic loading,” in:Boundary-Value Problems of the Theory of Elasticity and Viscoelasticity [in Russian], Sverdlovsk (1980), pp. 42–48.

  25. B. P. Gumenyuk, V. G. Karnaukhov, and I. F. Kirichok, “Connected dynamic problems of thermoviscoelasticity for plates and shells subject to cyclic loads,” in:Proc. 10th All-Union Conf. on The Theory of Shells and Plates [in Russian], Vol. 1, Kutaisi (1975), pp. 374–481.

  26. B. P. Gumenyuk, V. G. Karnaukhov, and I. F. Kirichok, “Using the quasilinearization and WKB methods to solve connected problems of thermoviscoelasticity in cyclic loads,” in:The Stress-Strain State of Elastic and Viscoelastic Structures [in Russian], Sverdlovsk (1977), pp. 14–24.

  27. B. P. Gumenyuk, V. G. Karnaukhov, and I. K. Senchenkov, “Influence of thermomechanical coupling on the dynamic behavior of viscoelastic bodies,”Zh. Prikl. Mekh. Tekh. Fiz., No. 3, 148–155 (1980).

    Google Scholar 

  28. A. Jagota and P. R. Dawson, “The influence of lateral wall vibrations on the ultrasonic welding of thin-walled parts,”Trans. ASME, Konstr. Tekh. Mash., Ser. B,11, 117–128 (1988).

    Google Scholar 

  29. A. A. Dudchenko, S. A. Lurie, and I. F. Obraztsov, “Anisotropic plates and shells,”Itogi Nauki Tekh. VINITI Ser. Mekh. Def. Tverd. Tela,15, 126–140 (1983).

    Google Scholar 

  30. A. A. Il'yushin and B. E. Pobedrya,Fundamentals of the Mathematical Theory of Thermoviscoelasticity [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  31. Kapania, “Methods of design of laminated shells (review),”Sovr. Mashinostr., Ser. B, No. 12, 1–11 (1995).

    Google Scholar 

  32. V. G. Karnaukhov,Connected Problems of thermoviscoelasticity [in Russian], Naukova Dumka, Kiev (1982).

    Google Scholar 

  33. V. G. Karnaukhov and B. P. Gumenyuk, “Vibrations of a viscoelastic rod with thermomechanical conjugation,”Inzh. Fiz. Zh.,35, No. 4, 692–697 (1978).

    Google Scholar 

  34. V. G. Karnaukhov and I. F. Kirichok,Connected Problems of the Theory of Viscoelastic Plates and Shells [in Russian], Naukova Dumka, Kiev (1986).

    Google Scholar 

  35. V. G. Karnaukhov and I. F. Kirichok, “The quasilinearization method in connected dynamic problems of thermoviscoelasticity,”Vopr. Dinam. Prochn., No. 42, 3–11 (1983).

    Google Scholar 

  36. V. G. Karnaukhov and I. F. Kirichok, “The thermomechanical theory of three-layer shells with a viscoelastic damping layer under cyclic loading,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 6, 46–49 (1981).

    Google Scholar 

  37. V. G. Karnaukhov and I. F. Kirichok, “The thermomechanical theory of three-layer orthotropic shells with a viscoelastic filler,”Prikl. Mekh.,19, No. 7, 31–37 (1983).

    Google Scholar 

  38. V. G. Karnaukhov and I. F. Kirichok, “One variant of constructing the thermomechanical theory of structural flexible viscoelastic shells subject to harmonic deformation,”Prikl. Mekh.,27, No. 5, 57–63 (1991).

    Google Scholar 

  39. V. G. Karnaukhov and I. F. Kirichok, “The thermomechanical theory of flexible shallow three-layer shells with a viscoelastic damping layer,” in:Energy Dissipation in Mechanical Systems Subject to Vibrations [in Russian], Naukova Dumka, Kiev (1982), pp. 31–40.

    Google Scholar 

  40. V. G. Karnaukhov and I. F. Kirichok, “Thermomechanical behavior of flexible viscoelastic plates and shells under cyclic loads,”Probl. Prochn., No. 3, 10–14 (1979).

    Google Scholar 

  41. V. G. Karnaukhov, I. F. Kirichok, and B. P. Gumenyuk, “The quasilinearization and WKB methods in connected dynamic problems of thermoviscoelasticity,” in:Thermal Stresses of Structural Elements [in Russian], Issue 15 (1975), pp. 36–44.

  42. V. G. Karnaukhov, I. F. Kirichok, and B. P. Gumenyuk, “Influence of a cyclic load on the thermomechanical behavior of a viscoelastic round plate of variable thickness,” in:Thermal Stresses of Structural Elements [in Russian], Issue 16, (1976), pp. 17–22.

  43. V. G. Karnaukhov, V. V. Mikhailenko, and F. Ts. Franovskii, “Development of the theory of constitutive equations for physically nonlinear viscoelastic bodies under cyclic strain,”Prikl. Mekh.,32, No. 10, 46–51 (1996).

    Google Scholar 

  44. V. G. Karnauhov, I. K. Senchenkov, and B. P. Gumenyuk,The Thermomechanical Behavior of Viscoelastic Bodies under Harmonic Loading [in Russian], Naukova Dumka, Kiev (1985).

    Google Scholar 

  45. V. G. Karnaukhov, G. A. Yakovlev, and L. P. Goncharov, “Investigation of the self-heating of viscoelastic materials under cyclic loads,”Probl. Prochn., No. 2, 26–29 (1975).

    Google Scholar 

  46. A. M. Kiiko and O. B. Rudakova, “The connected problem of thermoviscoelasticity on free nonlinear vibrations of a rectangular plate,”Mekh. Tverd. Tela, No. 5, 145–148 (1989).

    Google Scholar 

  47. I. F. Kirichok, “The thermomechanical behavior of a composite rod with viscoelastic layers,”Prikl. Mekh.,8, No. 4, 107–115 (1982).

    Google Scholar 

  48. I. F. Kirichok, “The linear thermomechanical theory of cyclically deformed viscoelastic shells,”Dokl. Akad. Nauk Ukr. SSR Ser. A, No. 3, 34–38 (1984).

    Google Scholar 

  49. I. F. Kirichok, “The vibrations of prestressed three-layer shells with a viscoelastic filler with allowance for heat generation,”Prikl. Mekh.,21, No. 10, 37–46 (1985).

    Google Scholar 

  50. I. F. Kirichok, “Influence of shear strain and rotary inertia on the thermomechanical behavior of rods subject to flexural vibrations,”Prikl. Mekh.,27, No. 1, 107–115 (1991).

    Google Scholar 

  51. I. F. Kirichok and V. G. Karnaukhov, “The thermomechanical behavior of flexible viscoelastic plates and shells under cyclic loads,”Probl. Prochn., No. 3, 10–14 (1979).

    Google Scholar 

  52. M. Kishkilov, “The finite-element method for investigation of a beam under the effect of continuous vibrational loading,”Teor. Pril. Mekh., No. 2, 74–80 (1990).

    Google Scholar 

  53. A. D. Kovalenko and V. G. Karnaukhov, “Equations and solution of some problems of the theory of viscoelastic shells,” in:Thermal Stresses of Structural Elements [in Russian], Issue 7 (1967), pp. 11–24.

  54. A. D. Kovalenko and V. G. Karnaukhov, “The main features of the behavior of simple material,” in:Thermal Stresses of Structural Elements [in Russian], Issue 12, (1972), pp. 5–18.

  55. A. D. Kovalenko and V. G. Karnaukhov, “The effect of cyclic loading on the temperature of a viscoelastic cylinder,”Dop. Akad. Nauk Ukr. RSR, No. 9, 1135–1140 (1966).

    Google Scholar 

  56. A. D. Kovalenko and V. G. Karnaukhov, “Heat generation in viscoelastic shells of, revolution under periodic actions,” in:Thermal Stresses of Structural Elements [in Russian], Issue 11 (1971), pp. 5–11.

  57. A. D. Kovalenko and V. G. Karnaukhov, “Heat generation in viscoelastic bodies under periodic actions,”Prikl. Mekh.,5, No. 2, 28–35 (1969).

    Google Scholar 

  58. A. D. Kovalenko and V. G. Karnaukhov, “Heat generation in viscoelastic bodies,”Dop. Akad. Nauk Ukr RSR, No. 11, 1029–1033 (1968).

    Google Scholar 

  59. A. D. Kovalenko, V. G. Karnaukhov, G. A. Yakovlev, B. P. Gumenyuk, and L. P. Goncharov, “Heating of a viscoelastic rod in transversal vibrations,” in:Thermal Stresses of Structural Elements [in Russian], Issue 12 (1972), pp. 36–39.

  60. A. D. Kovalenko, V. G. Karnaukhov, G. A. Yakovlev, B. P. Gumenyuk, and L. P. Goncharov, “Heat generation in viscoelastic rods subject to forced transversal vibrations,” in:Thermal Stresses of Structural Elements [in Russian], Issue 13 (1973), pp. 11–14.

  61. V. I. Kozlov, “Vibrations and dissipative heating-up of a multilayer viscoelastic shell of revolution,”Prikl. Mekh.,32, No. 6, 82–89 (1996) (Int. Appl. Mech.,32, No. 6, 480–486 (1996)).

    Google Scholar 

  62. V. V. Vasil'ev, V. D. Protasov, V. V. Bolotin, et al.,Composite Materials: A Reference Book [in Russian] (V. V. Vasil'ev and Yu. M. Tarnopol'skii, general editors) Mashinostroyeniye, Moscow (1990).

    Google Scholar 

  63. B. J. Korites and F. C. Nelson, “The influence of, dissipative heating on the loss factor of a viscoelastically damped beam,”Trans. ASME, Konstr. Tekhn. Mash., Ser. B, No. 4, 56–61 (1969).

    Google Scholar 

  64. A. I. Lukomskaya and V. F. Evstratov,Fundamentals of Predicting the Mechanical Behavior of Rubber [in Russian], Khimiya, Moscow (1975).

    Google Scholar 

  65. G. I. Marchuk and V. I. Agoshkov,An Introduction to Projection-Net Methods [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  66. V. V. Moskvitin,Resistance of Viscoelastic Materials [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  67. M. A. Narbut, “The ray-path method of calculation of the intensity of wave fields in viscoelastic media” in:Problems of the Dynamic Theory of Propagation of Seismic Waves [in Russian], Issue 12, (1974), pp. 61–68.

  68. A. D. Nashif, D. I. G. Jones, and J. P. Henderson,Vibration Damping, John Wiley and Sons. New York (1985).

    Google Scholar 

  69. P. F. Nedorezov, “Determination of the temperature field in forced longitudinal vibrations of a viscoelastic rod,” in:The Mechanics of Deformable Media [in Russian], Issue 5 (1978), pp. 8–17.

  70. P. F. Nedorezov, “Determination of the temeprature field in a polymeric cylindrical shell undercyclic loading: engineering theory,” in:Applied Theory of Elasticity [in Russian], Izd. Saratov. Univ., Saratov (1980), pp. 28–33.

    Google Scholar 

  71. Yu. N. Nemish and Yu. I. Khoma, “The stress-strain state of nonthin shells and plates. A generalized theory (review),”Prikl. Mekh.,29, No. 11, 3–34 (1993) (Int. Appl. Mech.,29, No. 11, 873–902 (1993)).

    MathSciNet  Google Scholar 

  72. W. C. Rheinboldt and J. Ortega,Iterative Solution of Nonlinear Systems in Serveral Variables [Russian translation], Mir, Moscow (1975).

    Google Scholar 

  73. Ya. S. Podstrigach and R. N. Shvets,Thermoelasticity of Thin Shells [in Russian], Naukova Dumka, Kiev (1978).

    MATH  Google Scholar 

  74. V. M. Porksheyan, “Some features of the solution of the heat conduction equation with allowance for the coupling of thermomechanical processes,”Izv. Sev.-Kavkaz. Nauch. Tsentra Vyssh. Shk. Estest. Nauk, No. 2, 29–33 (1984).

    Google Scholar 

  75. V. M. Porksheyan, “Heating-up of thermoviscoelastic rods and circular plate under cyclic strains,”Prikl. Mekh.,20, No. 1, 94–100 (1984).

    Google Scholar 

  76. A. P. Prusakov and Yu. K. Rasteryaev, “Bending of shallow multilayer shells in a high-temperature field,” in:Proc. 8th All-Union Conf. on The Theory of Shells and Plates (Rostov-on-Don, 1971) [in Russian], Nauka, Moscow (1973), pp. 756–672.

    Google Scholar 

  77. S. B. Ratner and V. I. Korobov, “Self-heating of polymers subject to multiple strain,”Dokl. Akad. Nauk Ukr. SSR, No. 4, 161–166 (1965).

    Google Scholar 

  78. S. B. Ratner and V. I. Korobov, “Self-heating of plastic subject to cyclic strain,”Mekh. Polim., No. 3, 93–100 (1965).

    Google Scholar 

  79. O. B. Rudakova, “The problem of thermoviscoelasticity on free nonlinear vibrations of a rectangular plate,”Izv. Akad. Nauk Ukr. SSR Mekh. Tverd. Tela., No. 5, 183–187 (1990).

    Google Scholar 

  80. V. G. Safronenko, “Application of the variation principle to the formulation of connected problems for thermoviscoelastic shells,” in:Design of Plates and Shells [in Russian], Izd. Rostov. Univ., Rostov-on-Don (1980), pp. 12–22.

    Google Scholar 

  81. V. G. Safronenko, “Variation formulation of connected problems for thermoviscoelastic shells,”Izv. Sev.-Kavkaz. Nauch. Tsentr, No. 4, 21–27 (1976).

    Google Scholar 

  82. I. K. Senchenkov, “The dynamic behavior of a rod with allowance for the temperature dependences of the properties of the material,”Prikl. Mekh.,20, No. 2, 85–92 (1984).

    MathSciNet  Google Scholar 

  83. I. K. Senchenkov, “Longitudinal vibrations and heating-up of a rod made from nonlinear viscoelastic material,”Prikl. Mekh.,21, No. 9, 79–85 (1985).

    Google Scholar 

  84. I. K. Senchenkov, “Approximate formulation of connected problems of nonlinear thermoviscoelasticity under harmonic actions,”Prikl. Mekh.,21, No. 2, 91–99 (1985).

    MathSciNet  Google Scholar 

  85. I. K. Senchenkov and B. P. Gumenyuk, “The thermomechanical behavior of a viscoelastic spring-rod with an amplitude-dependent module,”Prikl Mekh.,16, No. 5, 107–112 (1980).

    Google Scholar 

  86. I. K. Senchenkov and V. G. Karnaukhov, “Some general variation principles of thermomechanics,” in:Mechanics of Elastomers [in Russian], Krasnodar. Politekh. Inst., Krasnodar (1981), pp. 43–50.

    Google Scholar 

  87. I. K. Senchenkov, V. G. Karnaukhov, and B. P. Gumenyuk, “Influence of vibrational heating-up on the mechanical stability of a viscoelastic rod,” in:Mechanics and Strength of Machines [in Russian], Issue 34, (1981), pp. 109–113.

  88. I. K. Senchenkov and I. F. Kirichok, “Forced nonlinear vibrations and dissipative heating-up of a viscoelastic beam,”Prikl. Mekh.,23, No. 1, 91–97 (1987).

    Google Scholar 

  89. L. R. Soshevskaya, “Determination of the temeprature field in a plate subject to steady-state transversal vibrations,” in:Mechanics of Deformable Media [in Russian], Issue 6, 34–44 (1979).

  90. I. A. Sprishevskaya, “Two problems of determination of the temperature field of a cyclically deformable medium,”Vestnik Moskovskogo Univ., No. 4, 91–95 (1968).

    Google Scholar 

  91. S. P. Timoshenko and S. Voinovskii-Kriger,Plates and Shells [in Russian], Fizmatgiz, Moscow (1963).

    MATH  Google Scholar 

  92. J. F. Tormey and S. C. Britton, “Effect of cyclic loading on solid propellant grain structures,”Raketn. Tekhn. Kosmon., No. 8, 70–82 (1963).

    Google Scholar 

  93. Yu. N. Shevchenko and I. V. Prokhorenko,The Theory of Elastoplastic Shells under Nonisothermal Loads [in Russian], Naukova Dumka, Kiev (1981).

    Google Scholar 

  94. R. A. Shapery, “The effect of cyclic loading on the temperature in viscoelastic media with variable properties,”Raketn. Tekhn. Kosmon., No. 5, 55–60 (1964).

    ADS  Google Scholar 

  95. R. A. Shapery, “Thermomechanical behavior of viscoelastic media with variable properties subjected to cyclic loading,”Trans. ASME, Prikl. Mekh., Ser. E, No. 3, 150–161 (1965).

    Google Scholar 

  96. R. A. Shapery and D. E. Cantey, “Thermomechanical response studies of solid propellants subjected to cyclic and random loading,”Raketn. Tekhn. Kosmon., No. 2, 255–264 (1966).

    Google Scholar 

  97. J. Constable, J. G. Williams, and D. J. Burns, “Fatigue and cyclic thermal softening of thermoplastics,”J. Mech. Eng. Schi.,12, No. 1, 20–29 (1970).

    Article  Google Scholar 

  98. D. D. Joseph and E. M. Sparrow, “Nonlinear diffusion induced by nonlinear sources,”Quart. Appl. Math.,28, No. 3, 327–341 (1970).

    MATH  MathSciNet  Google Scholar 

  99. D. D. Joseph and T. S. Lundgren, “Quasilinear Dirichlet problems driven by positive sources,”Arch. Ration. Mech. Anal.,43, No. 4, 241–269 (1973).

    MathSciNet  Google Scholar 

  100. N. C. Huang and E. N. Lee, “Thermomechanical coupling behavior of viscoelastic rods subjected to cyclic loading,”Trans. ASME Ser. E,34, No 1, 127–132 (1967).

    Google Scholar 

  101. F. S. Karal Jr. and J. B. Keller, “Elastic wave propagation in homogeneous and inhomogeneous media,”J. Acoust. Soc. Amer.,31, No. 6, 1241–1255 (1959).

    Article  MathSciNet  Google Scholar 

  102. S. Mukherjee, “Variational principle in dynamic thermoviscoelasticity,”Int. J. Sol. Struct.,9, No. 10, 1301–1316 (1973).

    Article  MATH  Google Scholar 

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  1. V. G. Karnaukhov
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S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated, from Prikladnaya Mekhanika, Vol. 36, No. 2, pp. 39–62, February, 2000.

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Karnaukhov, V.G., Kirichok, I.F. Forced harmonic vibrations and dissipative heating-up of viscoelastic thin-walled elements (Review). Int Appl Mech 36, 174–195 (2000). https://doi.org/10.1007/BF02681993

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