Geodesic
Mathematical modeling of the deformation of composite plane with interface crack for semi-linear material
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 2, pp. 89-102 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The exact analytical solutions have been obtained for the nonlinear problems (plane-strain and plane-stress) for the bi-material plane with an interface crack. The plane is formed by joining of two half-planes made from different materials. Mechanical properties of halfplanes are described with the model of semi-linear material. The application of this model has allowed using the methods of the complex functions in the nonlinear boundary value problems. For this particular case the problem is solved for the plane with a free interface crack at given constant nominal (Piola) stresses at infinity. The expressions for nominal stresses, Cauchy stresses and displacements are obtained. From the general solutions the asymptotic expansions of these functions have been constructed in vicinities of crack tips. It is established that in the nonlinear problem of uniaxial extension of a plane with a free crack the formulas which give the crack disclosing differ by a constant factor from the formulas of linear elasticity. The stress intensity factors (SIF) of nonlinear and linear problems coincide. The nominal stresses have the root singularity at the tips of a crack; the Cauchy stresses have no singularity.
Keywords: bi-material plane, plane-strain problem, plane-stress problem, method of complex functions, interface crack, semi-linear material. 
@article{VSPUI_2018_14_2_a1,
     author = {T. O. Domanskaya and V. M. Malkov and Yu. V. Malkova},
     title = {Mathematical modeling of the deformation of composite plane with interface crack for semi-linear material},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {89--102},
     year = {2018},
     volume = {14},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2018_14_2_a1/}
}
TY  - JOUR
AU  - T. O. Domanskaya
AU  - V. M. Malkov
AU  - Yu. V. Malkova
TI  - Mathematical modeling of the deformation of composite plane with interface crack for semi-linear material
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2018
SP  - 89
EP  - 102
VL  - 14
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2018_14_2_a1/
LA  - ru
ID  - VSPUI_2018_14_2_a1
ER  - 
%0 Journal Article
%A T. O. Domanskaya
%A V. M. Malkov
%A Yu. V. Malkova
%T Mathematical modeling of the deformation of composite plane with interface crack for semi-linear material
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2018
%P 89-102
%V 14
%N 2
%U http://geodesic.mathdoc.fr/item/VSPUI_2018_14_2_a1/
%G ru
%F VSPUI_2018_14_2_a1
T. O. Domanskaya; V. M. Malkov; Yu. V. Malkova. Mathematical modeling of the deformation of composite plane with interface crack for semi-linear material. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 2, pp. 89-102. http://geodesic.mathdoc.fr/item/VSPUI_2018_14_2_a1/

[1] John F., “Plane strain problems for a perfectly elastic material of harmonic type”, Commun. Pure and Applied Mathematics, 13:2 (1960), 239–296 | DOI | MR | Zbl

[2] Lurie A. I., Non-linear elasticity, Nauka Publ., M., 1980, 512 pp. (In Russian)

[3] Chernykh K. F., Litvinenkova Z. N., Theory of large elastic deformations, Leningrad State University Publ., L., 1988, 256 pp. (In Russian)

[4] Zubov L. M., Nonlinear theory of dislocations and disclinations in elastic bodies, Springer, Berlin, 1997, 205 pp. | MR | Zbl

[5] Malkov V. M., Malkova Yu. V., “Plane problems of elasticity for semi-linear material”, Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy, 2012, no. 3, 93–106 (In Russian)

[6] Malkov V. M., Malkova Yu. V., “Plane problems on concentrated forces for semi-linear material”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2013, no. 3, 83–96 (In Russian)

[7] Malkov V. M., Malkova Yu. V., Domanskaya T. O., “Analysis of stresses of bi-material plane and half-plane at action of a point force for two models of harmonic materials”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2016, no. 1, 38–52 (In Russian)

[8] Varley E., Cumberbatch E., “Finite deformation of elastic materials surrounding cylindrical holes”, Journal of Elasticity, 10:4 (1980), 341–405 | DOI | Zbl

[9] Ru C. Q., “On complex-variable formulation for finite plane elastostatics of harmonic materials”, Acta Mechanica, 156:3–4 (2002), 219–234 | Zbl

[10] Ru C. Q., Schiavone P., Sudak L. J., Mioduchowski A., “Uniformity of stresses inside an elliptic inclusion in finite plane elastostatics”, Intern. Journal of Non-linear Mechanics, 38:2–3 (2005), 281–287 | MR

[11] Malkov V. M., Malkova Yu. V., “Plane problem of non-linear elasticity for harmonic material”, Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy, 2008, no. 3, 114–126 (In Russian)

[12] Malkov V. M., Malkova Yu. V., “Modeling nonlinear deformation of a plate with an elliptic inclusion by John's harmonic material”, Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy, 4(62):1 (2017), 121–130 (In Russian) | DOI

[13] Mal'kov V. M., Mal'kova Yu. V., “Modeling nonlinear deformation of a plate with an elliptic inclusion by John's harmonic material”, Vestnik of Saint Petersburg University. Mathematics, 50:1 (2017), 74–81 | DOI | MR | Zbl

[14] Malkov V. M., Malkova Yu. V., Stepanova V. A., “Bi-material plane of John's material with interface crack loaded by pressure”, Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy, 2013, no. 3, 113–125 (In Russian)

[15] Domanskaia T., Malkov V., Malkova Yu., “Bi-material plane of John's harmonic material with a point force at interface”, XXIV Intern. Congress of Theoretical and Applied Mechanics (ICTAM) (21–26 August 2016, Montreal, Canada), 2016, 1958–1959

[16] Domanskaya T. O., Malkov V. M., Malkova Yu. V., “Nonlinear problem for bi-material plate with interface crack for harmonic John's material”, 2nd International conference “Deformation and Failure of Composite Materials and Structures” (DFCMS-2016) (October 2016), Institute of Engineering RAS Publ., M., 2016, 33–35 (In Russian)

[17] Malkov V. M., Introduction to non-linear elasticity, Saint Petersburg State University Publ., Saint Petersburg, 2010, 276 pp. (In Russian)

[18] Malkova Yu. V., Some problems for bi-material plane with curvilinear cracks, Saint Petersburg State University Publ., Saint Petersburg, 2008, 160 pp. (In Russian); Ито Ю. и др., Справочник по коэффициентам интенсивности напряжений, в 2-х т., ред. Ю. Мураками, ред. Р. В. Гольдштейн, Н. А. Махутов, Мир, М., 1990, 1014 с.

[19] Ito Yu. et all., Stress intensity factors handbook, In 2 vol., Oxford University Press, Oxford etc., 1985–1986