Journal of Applied Mechanics and Technical Physics

, Volume 59, Issue 7, pp 1197–1210

# Calculation of the Unsteady Thermal Stresses in Elastoplastic Solids

• A. A. Burenin
• A. V. Tkacheva
• G. A. Shcherbatyuk
Article

## Abstract

The one-dimensional boundary problem of the theory of thermal stresses that simulates the shrink fit assembly of cylindrical parts is used to discuss a computational approach to predicting the evolution of the thermal stresses under piecewise-linear plasticity conditions. The solution of the problem is based on the classical maximum shear stress criterion (Tresca–Saint Venant yield criterion), and the maximum reduced shear stress criterion (Ishlinsky–Ivlev yield criterion) is only used to compare the calculation results. The application of piecewise-linear potentials in the theory of plastic flow is shown to allow equilibrium equations to be integrated in both the region of reversible deformation and various parts of a plastic flow region. The dependences thus obtained are important for a time-step calculation algorithm. This algorithm can trace the site and time of both nucleation and completion of plastic flows at each time step. The calculations demonstrate that, following temperature, the stresses in the assembly element materials can pass from correspondence to a certain face of a loading surface to correspondence to its edge and, then, to another face. This circumstance implies the division of the irreversible deformation region into parts, where a plastic flow obeys different sets of equations, which take into account the assignment of the state of stress to various faces and edges of the loading surface. The computational algorithm also makes it possible to trace the beginning of division of a flow region into parts and the motion of the part boundaries through irreversibly deformed materials, including the times of their coincidence (i.e., the disappearance of the parts of the calculation region). A repeated plastic flow is shown to appear. This flow nucleates when an assembly is cooled, i.e., when the assembly element materials return to reversible deformation conditions due to the evolution of the state of stress. Taking into account the change in plastic flow conditions that is caused by the use of piecewise-linear plastic potentials is found to substantially affect the level and the distribution of residual stresses and the final tightness of the assembly.

## Key words

elasticity plasticity thermal stresses shrink fit interference fit assembly

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## References

1. 1.
Berniker, E.I., Posadka s natyagom v mashinostroenii (Interference Fit in Mechanical Engineering), Leningrad: Mashinostroenie, 1966.Google Scholar
2. 2.
Parkus, H., Instationäre Wärmespannungen (Instationary Thermal Stresses), Wien: Springer, 1959.
3. 3.
Boley, B.A. and Weiner, J.H., The Theory of Thermal Stresses, New York: Wiley, 1960.
4. 4.
Gokhfeld, D.A. and Cherniavsky, O.F., Limit Analysis of Structures at Thermal Cycling, The Netherlands: Sijthoff & Noordhoff, 1980.Google Scholar
5. 5.
Dopuski i posadki: Spravochnik (Tolerances and Landing, A Handbook), Myagkov, V.D., Palej, M.A., Romanov, A.B., and Braginskii, V.A., Leningrad: Mashinostroenie, 1982, vol. 1.Google Scholar
6. 6.
Bland, D.R., Elastoplastic thick-walled tubes of work-hardening material subject to internal and external pressures and to temperature gradients, J. Mech. Phys. Solids, 1956, vol. 4, no. 4, pp. 209–229.
7. 7.
Ishlinskii, A.Yu. and Ivlev, D.D., Matematicheskaya teoriya plastichnosti (Mathematical Theory of Plasticity), Moscow: Fizmatlit, 2001.
8. 8.
Perzyna, P. and Sawczuk, A., Problems of thermoplasticity, Nucl. Eng. Des., 1973, vol. 24, no. 1, pp. 1–55.
9. 9.
Ohno, N. and Wang, J.D., Transformation of a nonlinear kinematic hardening rule to a multisurface form under isothermal and nonisothermal conditions, Int. J. Plast., 1991, vol. 7, no. 8, pp. 879–891.
10. 10.
Orçan, Y. and Gamer, U., Elastic-plastic deformation of a centrally heated cylinder, Acta Mech., 1991, vol. 90, no. 1, pp. 61–80.
11. 11.
Chaboche, J.L., Thermodynamically based viscoplastic constitutive equation: theory versus experiment, in Proceedings of the ASME Winter Annual Meeting, Atlanta, GA, 1991, pp. 1–20.Google Scholar
12. 12.
Lippmann, H., The effect of a temperature cycle on the stress distribution in a shrink fit, Int. J. Plast., 1992, vol. 8, no. 5, pp. 567–582.
13. 13.
Gamer, U., A concise treatment of the shrink fit with elastic-plastic hub, Int. J. Solids Struct., 1992, vol. 29, no. 20, pp. 2463–2469.
14. 14.
Mack, W., Thermal assembly of an elastic-plastic hub and a solid shaft, Arch. Appl. Mech., 1993, vol. 63, no. 1, pp. 42–50.
15. 15.
Knyazeva, A.G., Teplofizicheskie osnovy sovremennykh vysokotemperaturnykh tekhnologii (Thermophysical Fundamentals of Modern High-Temperature Technologies), Tomsk: Tomsk Politekh. Univ., 2009.Google Scholar
16. 16.
Bondar, V.S., Danshin, V.V., and Kondratenko, A.A., Version of the thermoplasticity theory, Vestn. PNIPU, Mekh., 2015, no. 2, pp. 21–35. Google Scholar
17. 17.
Kovtanuk, L.V., The modelling of finite elastic-plastic deformation in non-isothermal case, Dal’nevost. Mat. Zh., 2004, vol. 5, no. 1, pp. 110–120.Google Scholar
18. 18.
Rogovoi, A.A., Constitutive relations for finite elastic-plastic strains, J. Appl. Mech. Tech. Phys., 2005, vol. 46, no. 5, pp. 730–739.
19. 19.
Burenin, A.A. and Kovtanyuk, L.V., Bol’shie neobratimye deformatsii i uprugoe posledejstvie (Large Irreversible Deformations and Elastic Aftereffects), Vladivostok: Dalnauka, 2013.Google Scholar
20. 20.
Alexandrov, S.E. and Chikanova, N.N., Elastic-plastic stress-strain state in a plate with a pressed insertion under the action of a temperature field, Mech. Solids, 2000, vol. 35, no. 4, pp. 125–132.Google Scholar
21. 21.
Alexandrov, S. and Alexandrova, N., Thermal effects on the development of plastic zones in thin axisummetric plates, J. Strain Anal. Eng., 2001, vol. 36, no. 2, pp. 169–175.
22. 22.
Shevchenko, Yu.N. and Steblyanko, P.A., Computing methods in stationary and non-stationary problems of theory thermal-plasticity, Probl. Obchisl. Mekh. Mitsnosti Konstrukts., 2012, no. 18, pp. 211–226.Google Scholar
23. 23.
Shevchenko, Yu.N., Steblyanko, P.A., and Petrov, A.D., Computing methods in non-stationary problems of theory thermal-plasticity, Probl. Obchisl. Mekh. Mitsnosti Konstrukts., 2014, no. 22, pp. 251–264.Google Scholar
24. 24.
Gorshkov, S.A., Dats, E.P., and Murashkin, E.V., Calculation of plane stress field under plastic flow and unloading, Vestn. ChGPU Yakovleva, Ser.: Mekh. Predel’n. Sost., 2014, no. 3 (21), pp. 169–175.Google Scholar
25. 25.
Burenin, A.A., Kovtanyuk, L.V., and Panchenko, G.L., Nonisothermal motion of an elastoviscoplastic medium through a pipe under a changing pressure, Dokl. Phys., 2015, vol. 60, no. 9, pp. 419–422.
26. 26.
Aleksandrov, S.E., Lomakin, E.V., and Dzeng, Y.-R., Solution of the thermoelasticoplastic problem for a thin disc of plastically compressible material subjected to thermal loading, Dokl. Phys., 2012, vol. 57, no. 3, pp. 136–139.
27. 27.
Aleksandrov, S.E., Lyamina, E.A., and Novozhilova, O.V., The influence of the relationship between yield strength and temperature on the stress state in a thin hollow disk, J. Mach. Manuf. Reliab., 2013, vol. 42, no. 3, pp. 214–218.
28. 28.
Burenin, A.A., Dats, E.P., and Murashkin, E.V., Formation of the residual stress field under local thermal actions, Mech. Solids, 2014, vol. 49, no. 2, pp. 218–224.
29. 29.
Pozdeev, A.A., Nyashin, Yu.I., and Trusov, P.V., Ostatochnye napryazheniya: teoriya i prilozheniya (Residual Stresses: Theory and Applications), Moscow: Nauka, 1982.Google Scholar
30. 30.
Bengeri, M. and Mack, W., The influence of the temperature dependence of the yield stress on the stress distribution in a thermally assembled elastic-plastic shrink fit, Acta Mech., 1994, vol. 103, no. 1, pp. 243–257.
31. 31.
Kovács, Á., Residual stresses in thermally loaded shrink fits, Period. Polytech., Ser.: Mech. Eng., 1996, vol. 40, no. 2, pp. 103–112.Google Scholar
32. 32.
Dats, E.P., Tkacheva, A.V., and Shport, R.V., The assemblage of “ring in ring” constructions with shrink fit method, Vestn. ChGPU Yakovleva, Ser.: Mekh. Predel’n. Sost., 2014, no. 4 (22), pp. 225–235.Google Scholar
33. 33.
Burenin, A.A., Dats, E.P., and Tkacheva, A.V., On the modeling of the shrink fit technology, J. Appl. Ind. Math., 2014, vol. 8, no. 4, pp. 493–499.
34. 34.
Bykovtsev, G.I. and Ivlev, D.D., Teoriya plastichnosti (Theory of Plasticity), Vladivostok: Dalnauka, 1998.Google Scholar
35. 35.
Loginov, Yu.N., Med’ i deformiruemye mednye splavy (Copper and Deformable Copper Alloys, The School-Book), Yekaterinburg: Ural’sk. Gos. Politekh. Univ., 2006.Google Scholar
36. 36.
Marochnik stalei i splavov (Database of Steels and Alloys), Zubchenko, A.S., Ed., Moscow: Mashinostroenie, 2003.Google Scholar

## Authors and Affiliations

• A. A. Burenin
• 1
• 2
Email author
• A. V. Tkacheva
• 1
• G. A. Shcherbatyuk
• 2
1. 1.Institute of Machinery and Metallurgy, Far East BranchRussian Academy of SciencesKomsomolsk-on-AmurRussia
2. 2.Komsomolsk-on-Amur State Technical UniversityKomsomolsk-on-AmurRussia