Geodesic
Conditions for convexity of the isotropic function of the second-rank tensor
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 4 (2011) no. 2, pp. 265-272.

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

For a scalar function, depending on the invariants of the second-rank tensor, condition of convexity and strong convexity are obtained with respect to the components of this tensor in an arbitrary Cartesian coordinate system. It is shown that if a function depends only on the four invariants: three principal values of the symmetric part of a tensor and modulus of pseudovector of the antisymmetric part, these conditions are necessary and sufficient. A special system of convex invariants is suggested to construct potentials for the stresses and strains in the mechanics of structurally inhomogeneous elastic media, exhibiting moment properties.
Keywords: isotropic tensor function, convexity, nonlinear elasticity, plasticity.
Mots-clés : invariants 
@article{JSFU_2011_4_2_a13,
     author = {Vladimir M. Sadovskii},
     title = {Conditions for convexity of the isotropic function of the second-rank tensor},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {265--272},
     publisher = {mathdoc},
     volume = {4},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2011_4_2_a13/}
}
TY  - JOUR
AU  - Vladimir M. Sadovskii
TI  - Conditions for convexity of the isotropic function of the second-rank tensor
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2011
SP  - 265
EP  - 272
VL  - 4
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2011_4_2_a13/
LA  - ru
ID  - JSFU_2011_4_2_a13
ER  - 
%0 Journal Article
%A Vladimir M. Sadovskii
%T Conditions for convexity of the isotropic function of the second-rank tensor
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2011
%P 265-272
%V 4
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2011_4_2_a13/
%G ru
%F JSFU_2011_4_2_a13
Vladimir M. Sadovskii. Conditions for convexity of the isotropic function of the second-rank tensor. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 4 (2011) no. 2, pp. 265-272. http://geodesic.mathdoc.fr/item/JSFU_2011_4_2_a13/

[1] L. I. Sedov, Mekhanika sploshnoi sredy, v. 2, Nauka, M., 1994

[2] Yu. N. Rabotnov, Mekhanika deformiruemogo tverdogo tela, Nauka, M., 1979

[3] S. K. Godunov, E. I. Romenskii, Elementy mekhaniki sploshnykh sred i zakony sokhraneniya, Nauchnaya kniga, Novosibirsk, 1998 | Zbl

[4] P. P. Mosolov, V. P. Myasnikov, Mekhanika zhestkoplasticheskikh sred, Nauka, M., 1981 | MR | Zbl

[5] G. I. Bykovtsev, D. D. Ivlev, Teoriya plastichnosti, Dalnauka, Vladivostok, 1998

[6] V. A. Palmov, “Osnovnye uravneniya teorii nesimmetrichnoi uprugosti”, Prikl. matem. i mekh., 28:3 (1964), 401–408 | MR

[7] O. V. Sadovskaya, V. M. Sadovskii, Matematicheskoe modelirovanie v zadachakh mekhaniki sypuchikh sred, Fizmatlit, M., 2008 | Zbl

[8] E. Spenser, Teoriya invariantov, Mir, M., 1974

[9] P. A. Zhilin, “Modifitsirovannaya teoriya simmetrii tenzorov i tenzornykh invariantov”, Izvestiya VUZov, Severo-Kavkazskii region. Estestvennye nauki. Nelineinye problemy mekhaniki sploshnykh sred, 2003, Spetsvypusk, 176–195

[10] R. T. Rokafellar, Vypuklyi analiz, Mir, M., 1973

[11] V. G. Karmanov, Matematicheskoe programmirovanie, Nauka, M., 1980 | MR | Zbl

[12] W. H. Yang, “A useful theorem for constructing convex yield functions”, Trans. ASME. J. Appl. Mech., 47:2 (1980), 301–305 | DOI | MR