Geodesic
One approach to determination of the ultimate load-bearing capacity of mechanical systems with softening elements
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 4, pp. 762-773.

Voir la notice de l'article provenant de la source Math-Net.Ru

The fundamental provisions of the limiting load calculation theory are presented for a discrete mechanical system with softening elements. The method is based on the numerical determination of degenerate critical points for the potential function of the system. At these points there is a transition from the stability of the loading process to instability such as a catastrophe or a failure. This approach helps to avoid solving a large number of nonlinear equilibrium equations. The problem of determining the limiting internal pressure in a thin walled cylindrical tank is solved as an example. A unified potential specially defined for a flat square element of material in biaxial tension is used in developing a potential function of the system. It describes all stages of deformation including the softening stage.
Keywords: potential function, degenerate critical points, unified potential, thin walled reservoir, limiting pressure.
Mots-clés : Hesse matrix 
@article{VSGTU_2018_22_4_a8,
     author = {V. V. Struzhanov and A. V. Korkin and A. E. Chaykin},
     title = {One approach to determination of the ultimate load-bearing capacity of mechanical systems with softening elements},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {762--773},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2018_22_4_a8/}
}
TY  - JOUR
AU  - V. V. Struzhanov
AU  - A. V. Korkin
AU  - A. E. Chaykin
TI  - One approach to determination of the ultimate load-bearing capacity of mechanical systems with softening elements
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2018
SP  - 762
EP  - 773
VL  - 22
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2018_22_4_a8/
LA  - ru
ID  - VSGTU_2018_22_4_a8
ER  - 
%0 Journal Article
%A V. V. Struzhanov
%A A. V. Korkin
%A A. E. Chaykin
%T One approach to determination of the ultimate load-bearing capacity of mechanical systems with softening elements
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2018
%P 762-773
%V 22
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2018_22_4_a8/
%G ru
%F VSGTU_2018_22_4_a8
V. V. Struzhanov; A. V. Korkin; A. E. Chaykin. One approach to determination of the ultimate load-bearing capacity of mechanical systems with softening elements. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 4, pp. 762-773. http://geodesic.mathdoc.fr/item/VSGTU_2018_22_4_a8/

[1] Sedov L. I., Mekhanika sploshnoi sredy [Continuum Mechanics], v. 1, Nauka, Moscow, 1970, 492 pp. (In Russian)

[2] Struzhanov V. V., Mironov V. I., Deformatsionnoe razuprochnenie materiala v elementakh konstruktsii [Deformational Softening of Material in Structural Elements], UrO RAN, Ekaterinburg, 1995, 190 pp. (In Russian)

[3] Vil'deman V. E., Chausov N. G., “Conditions of strain softening upon stretching of the specimen of special configuration”, Zavodskaia laboratoriia. Diagnostika materialov, 73:10 (2007), 55–59 (In Russian)

[4] Vil'deman V. E., Tretyakov M. P., “Material testing by plotting total deformation curves”, J. Mach. Manuf. Reliab., 42:2 (2013), 166–170 | DOI

[5] Struzhanov V. V., Korkin A. V., “Regarding stretching process stability of one bar system with softening elements”, Vestnik Ural'skogo gosudarstvennogo universiteta putei soobshcheniia, 2016, no. 3(31), 4–17 (In Russian) | DOI

[6] Andreeva E. A., “Solution of one-dimensional softening materials plasticity problems”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2008, no. 2(17), 152–160 (In Russian) | DOI

[7] Arsenin V. Ya., Metody matematicheskoi fiziki i spetsial'nye funktsii [Methods of mathematical physics and special functions], Nauka, Moscow, 1974, 431 pp. (In Russian)

[8] Struzhanov V. V., Korkin A. V., “On the solution of non-linear equations of trim of a single expandable frame structure with softening elements by the method of simple iterations”, Vestnik Ural'skogo gosudarstvennogo universiteta putei soobshcheniia, 2017, no. 2(34), 4–16 (In Russian) | DOI

[9] Poston T., Stewart I., Catastrophe Theory and Its Applications, Surveys and Reference Works in Mathematics, 2, Pitman, London, San Francisco, Melbourne, 1978, xviii+491 pp. | MR | Zbl

[10] Gilmore R., Catastrophe theory for scientists and engineers, A Wiley-Interscience Publication, John Wiley Sons, New York, 1981, xvii+666 pp. | MR | Zbl

[11] Pars L. A., A treatise on analytical dynamics, Heinemann Educational Books, London, 1965, xxi+641 pp. | MR | Zbl

[12] Ilyushin A. A., “On the foundations of the general mathematical theory of plasticity”, Vestn. Mosk. Univ., Ser. I, 1961, no. 3, 31–36 (In Russian) | MR | Zbl

[13] Kachanov L. M., Foundations of the theory of plasticity, North-Holland Series in Applied Mathematics and Mechanics, 12, North-Holland Publ., Amsterdam, 1971, xiii+482 pp. | MR | Zbl

[14] Struzhanov V. V., “The determination of the deformation diagram of a material with a falling branch using the torsion diagram of a cylindrical sample”, Sib. Zh. Ind. Mat., 15:1 (2012), 138–144

[15] Radchenko V. P., Vvedenie v mekhaniku deformiruemykh sistem [Introduction to the mechanics of deformable systems], Samara State Technical Univ., Samara, 2009, 241 pp. (In Russian)

[16] Southwell R. V., An introduction to the theory of elasticity. For engineers and physicists, Oxford Engineering Science Series, Clarendon Press, Oxford, 1936, ix+510 pp. | Zbl