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Evaluation of theoretical strength of porous materials according to catastrophe theory

НазваEvaluation of theoretical strength of porous materials according to catastrophe theory
Назва англійськоюEvaluation of theoretical strength of porous materials according to catastrophe theory
АвториMykola Stashchuk; Zinoviy Nytrebych; Roman Hromyak
ПринадлежністьKarpenko Physico-Mechanical Institute of the National Academy of Sciences of Ukraine, Lviv, Ukraine Lviv Polytechnic National University, Lviv, Ukraine Ternopil Ivan Puluj National Technical University, Ternopil, Ukraine
Бібліографічний описEvaluation of theoretical strength of porous materials according to catastrophe theory / Mykola Stashchuk; Zinoviy Nytrebych; Roman Hromyak // Scientific Journal of TNTU. — Tern. : TNTU, 2020. — Vol 99. — No 3. — P. 44–54.
Bibliographic description:Stashchuk M.; Nytrebych Z.; Hromyak R. (2020) Evaluation of theoretical strength of porous materials according to catastrophe theory. Scientific Journal of TNTU (Tern.), vol 99, no 3, pp. 44–54.
DOI: https://doi.org/10.33108/visnyk_tntu2020.03.044
УДК

539.4 

Ключові слова

strength, catastrophe theory, Morse lemma, catastrophe fold, catastrophe function, volume concentration of pores, critical nominal stress, porous composite, effective electrical conductivity, engineering formulas.

 

With the rapid development of modern science, in particular, applied mechanics, the catastrophe theory proved to be quite effective in the analysis of classical results and the development of modern ones. This theory has developed significantly in the study of a number of issues in the theory of elastic stability, which studies the response of elastic bodies and structures to existing mechanical loads. Catastrophe theory predictions have important technical applications for estimating the critical forces that initiate the loss of stability of elastic bodies and engineering structures. The main basics of the research are analysed in this paper; based on the catastrophe theory, the problems are set; the main types of catastrophes’ functions are described; and the simplest of them, in particular the fold catastrophe, is applied. Based on the set analytical relations for the calculations of effective electrical conductivities and elastic modules by the pore concentration of the electrically conductive material, the estimation of the element strength of the composite sample is simulated in the form of a rod.

 

ISSN:2522-4433
Перелік літератури
  1. Stashchuk M. H., Irza E. M. Thermal Stressed States of the Bodies of Revolution made of Functionally Graded Materials. Materials Science. 2019. Volume 55, pр. 311–319.
  2. Staschuk M. G., Irza E. M. Optimizatsiya rezhimiv termoobrobki elementiv konstruktsiy z funktsionalno-gradientnih materialiv. Fiz.-him. mehanika materialiv. 2020. 56. No. 1. Р. 101–105.
  3. Thom R., Stabilite Structurelle et Morpho-genese, New York, Benjamin, 1972; transl. Structural Stability and Morphogenesis, Reading, Benjamin, 1975.
  4. Arnol`d V. I., Critical Points of Smooth Functions, Proc. Int. Cong. Math., Vancouver, 1974, pp. 19–75.
  5. Arnold V. I. Kriticheskie tochki gladkih funktsiy i ih normalnyie formyi. UMN, 1975, 30:5, 3–65.
  6. Poston T., Styuart I. Teoriya katastrof i ee prilozheniya. M.: Mir, 1980. 608 р.
  7. A`Campo N. A., Le Groupe de Monodromie de Deploiement des Singularites Isolees de Courbes Planes, I, Math. Ann., 213, 1–32 (1975).
  8. A`Campo N. A., Le Groupe de Monodromie de Deploiement des Singularites Isolees de Courbes Planes, II, Proc. Int. Cong. Math., Vancouver, 1974, pp. 395–404.
  9. Zeeman E. S. Catastrophe Theory, Sci. American, 234 (4), 65–83 (1976). Published in original from in: E. C. Zeeman, Catastrophe Theory, Selected Papers, 1972–1977, Reading: Addition–Wesley, 1977. P. 18
  10. Gilmor R. Prikladnaya teoriya katastrof: v 2-h kn. M.: 1984.
  11. Tompson Dzh. M. T. Neustoychivosti i katastrofyi v nauke i tehnike. M.: Mir, 1985. 254 р.
  12. Eksperementalnaya mehanika: v 2-h knigah / per.s angl. pod red. A. Kobayasi. M.: Mir, 1990. Kniga 1, 2. 552 р.
  13. Morse M., The Critical Points of a Function of n Variables. Trans. Am. Math. Soc. 33. 1931. Р. 72–91
  14. Gromoll D., Meyer W., On Differentiable Functions with Isolated Critical Points. Topology. 1969. 8. Р. 361–370.
  15. Mushelishvili N. I. Nekotoryie osnovnyie zadachi matematicheskoy teorii uprugosti. M.: “Nauka”, 1966.
  16. Emets Yu. P. O provodimosti sredyi s neodnorodnyimi vklyucheniyami v magnitnom pole. Zhurn.tehn.fiziki. 1974. 44. No. 5. Р. 916–921.
  17. Emets Yu. P. Elektricheskie harakteristiki kompozitsionnyih materialov s regulyarnoy strukturoy. Kiev: Nauk. dumka, 1986. 192 р
  18. Vanin G. A. Mikromehanika kompozitsionnyih materialov. Kiev: Nauk. dumka, 1977. 264 р.
  19. Kristensen R. Vvedenie v mehaniku kompozitov. M.: Mir, 1982. 334 р.
  20. Staschuk M. G. Vpliv kontsentratsiyi vodnyu na napruzhennya u sutsilnomu metalevomu tsilindri. Fiz.-him. mehanika materialiv. 2017. 53. No. 6. P. 73–79.
  21. Tkachev V. I., Levina I. M., Ivas'kevych L. M. Distinctive features of hydrogen degradation of heat-resistant alloys based on nickel. Mater Sci. 33. 1997. No. 4. Р. 524–531.
  22. Maksimovich G., Kholodnyi V., Belov V., Tretyak I., Ivas'kevich L., Slipchenko T. Influence of gaseous hydrogen on the strength and plasticity of high-temperature strength nickel alloys. Soviet Materials Science. 1984. 20. No. 3. Р. 252–255.
  23. Stashchuk M. H. Determination of the Distribution of Hydrogen Near Cracklike Defects. Materials Science. 2017. Vol. 52. No 6. P. 803–810.
  24. Stashchuk M., Boiko V., Hromyak R. Determination of hydrogen concentration influence on stresses in structures. Scientific Journal of TNTU. Tern.: TNTU, 2019. Vol. 94. No. 2. P. 134–144.
  25. Hromyak R., Stashchuk M., Stashchuk N.Calculation of the deformed state of the cable pipeline with circular surfaces. Scientific Journal of TNTU. Tern.: TNTU, 2018. Vol. 92. No. 4. P. 42–52.
References:
  1. Stashchuk M. H., Irza E. M. Thermal Stressed States of the Bodies of Revolution made of Functionally Graded Materials. Materials Science. 2019. Volume 55, pр. 311–319.
  2. Staschuk M. G., Irza E. M. Optimizatsiya rezhimiv termoobrobki elementiv konstruktsiy z funktsionalno-gradientnih materialiv. Fiz.-him. mehanika materialiv. 2020. 56. No. 1. Р. 101–105.
  3. Thom R., Stabilite Structurelle et Morpho-genese, New York, Benjamin, 1972; transl. Structural Stability and Morphogenesis, Reading, Benjamin, 1975.
  4. Arnol`d V. I., Critical Points of Smooth Functions, Proc. Int. Cong. Math., Vancouver, 1974, pp. 19–75.
  5. Arnold V. I. Kriticheskie tochki gladkih funktsiy i ih normalnyie formyi. UMN, 1975, 30:5, 3–65.
  6. Poston T., Styuart I. Teoriya katastrof i ee prilozheniya. M.: Mir, 1980. 608 р.
  7. A`Campo N. A., Le Groupe de Monodromie de Deploiement des Singularites Isolees de Courbes Planes, I, Math. Ann., 213, 1–32 (1975).
  8. A`Campo N. A., Le Groupe de Monodromie de Deploiement des Singularites Isolees de Courbes Planes, II, Proc. Int. Cong. Math., Vancouver, 1974, pp. 395–404.
  9. Zeeman E. S. Catastrophe Theory, Sci. American, 234 (4), 65–83 (1976). Published in original from in: E. C. Zeeman, Catastrophe Theory, Selected Papers, 1972–1977, Reading: Addition–Wesley, 1977. P. 18
  10. Gilmor R. Prikladnaya teoriya katastrof: v 2-h kn. M.: 1984.
  11. Tompson Dzh. M. T. Neustoychivosti i katastrofyi v nauke i tehnike. M.: Mir, 1985. 254 р.
  12. Eksperementalnaya mehanika: v 2-h knigah / per.s angl. pod red. A. Kobayasi. M.: Mir, 1990. Kniga 1, 2. 552 р.
  13. Morse M., The Critical Points of a Function of n Variables. Trans. Am. Math. Soc. 33. 1931. Р. 72–91
  14. Gromoll D., Meyer W., On Differentiable Functions with Isolated Critical Points. Topology. 1969. 8. Р. 361–370.
  15. Mushelishvili N. I. Nekotoryie osnovnyie zadachi matematicheskoy teorii uprugosti. M.: “Nauka”, 1966.
  16. Emets Yu. P. O provodimosti sredyi s neodnorodnyimi vklyucheniyami v magnitnom pole. Zhurn.tehn.fiziki. 1974. 44. No. 5. Р. 916–921.
  17. Emets Yu. P. Elektricheskie harakteristiki kompozitsionnyih materialov s regulyarnoy strukturoy. Kiev: Nauk. dumka, 1986. 192 р
  18. Vanin G. A. Mikromehanika kompozitsionnyih materialov. Kiev: Nauk. dumka, 1977. 264 р.
  19. Kristensen R. Vvedenie v mehaniku kompozitov. M.: Mir, 1982. 334 р.
  20. Staschuk M. G. Vpliv kontsentratsiyi vodnyu na napruzhennya u sutsilnomu metalevomu tsilindri. Fiz.-him. mehanika materialiv. 2017. 53. No. 6. P. 73–79.
  21. Tkachev V. I., Levina I. M., Ivas'kevych L. M. Distinctive features of hydrogen degradation of heat-resistant alloys based on nickel. Mater Sci. 33. 1997. No. 4. Р. 524–531.
  22. Maksimovich G., Kholodnyi V., Belov V., Tretyak I., Ivas'kevich L., Slipchenko T. Influence of gaseous hydrogen on the strength and plasticity of high-temperature strength nickel alloys. Soviet Materials Science. 1984. 20. No. 3. Р. 252–255.
  23. Stashchuk M. H. Determination of the Distribution of Hydrogen Near Cracklike Defects. Materials Science. 2017. Vol. 52. No 6. P. 803–810.
  24. Stashchuk M., Boiko V., Hromyak R. Determination of hydrogen concentration influence on stresses in structures. Scientific Journal of TNTU. Tern.: TNTU, 2019. Vol. 94. No. 2. P. 134–144.
  25. Hromyak R., Stashchuk M., Stashchuk N.Calculation of the deformed state of the cable pipeline with circular surfaces. Scientific Journal of TNTU. Tern.: TNTU, 2018. Vol. 92. No. 4. P. 42–52.
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