The uniqueness of solution in the small sense of tasks of equally-stressed reinforcement of composite

A. P. Yankovskii

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 4 (2014), pp. 121-132.

Résumé

The uniqueness of a solution in the small sense (in the sense of lack of infinitely close solution) is proved for the boundary-value problem of equally-stressed reinforcing metal composite plates in conditions of steady creep of materials of all phases of the composition, when in addition to static and kinematic boundary conditions and boundary conditions for the densities of the reinforcement, which is natural in such problems, on the contour of the plates an additional boundary conditions are specified for angles of reinforcement. In a large sense (in the sense of significant differences solution) this problem can have multiple, but not infinitely close, alternative solutions because of the nonlinearity of the static boundary conditions and equally-stressed of reinforcement. The study of the problem of uniqueness of the solution of this problem is necessary when examining the issue of correctness setting of problems of equally-stressed of reinforcement.

Détail

Keywords: equal-stressed reinforcement, metal-composite plate, steady-state creep, uniqueness of solution, correctness of boundary value problem.

Bibliographie

[1] Banichuk N. V., Kobelev V. V., Rikards R. B., Optimizatsiia elementov konstruktsii iz kompozitsionnykh materialov [Optimization of Structural Elements Made of Composite Materials], Mashinostroenie, Moscow, 1988, 224 pp. (In Russian)

[2] Nemirovskii Yu. V., Yankovskii A. P., Ratsional'noe proektirovanie armirovannykh konstruktsii [Rational Design of Reinforced Structures], Nauka, Novosibirsk, 2002, 488 pp. (In Russian)

[3] Khazhinskii G. M., Modeli deformirovaniia i razrusheniia metallov [The Models of Metal Deformation and Destruction], Nauchnyi mir, Moscow, 2011, 231 pp. (In Russian)

[4] Nemirovskii Yu. V., Yankovskii A. P., “Equal-stressed reinforcement of metal-composite plates with fibers of constant cross section in steady-state creep”, Mechanics of Composite Materials, 44:1 (2008), 9–24 | DOI

[5] Yankovskii A. P., “On some properties of equal-stress problem solution reinforcement bending the metal-composite plates working in steady creep conditions”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2011, no. 2(23), 62–73 (In Russian) | DOI

[6] Yankovskii A. P., “Application of methods of the theory of perturbations in flat problem of equally-stressed reinforcing of metal-composite plates at steady creep”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2009, no. 2(19), 53–71 (In Russian) | DOI

[7] Yankovskii A. P., “Equal-Stress Reinforcement the Ring Bending Metal-Composites Plates Working in Conditions of Steady Creep”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2010, no. 5(21), 42–54 (In Russian) | DOI

[8] Yankovskii A. P., “Application of methods of the perturbation theory to problem of equally-stressed reinfocing of bending metal-composite plates in conditions of steady-state creep”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2013, no. 2(31), 17–35 (In Russian) | DOI

[9] Yankovskii A. P., “Numerical-analytical method of solving the plane problem of equal-stressed reinforcement of metal-composite plates under steady creep conditions”, Computational Continuum Mechanics, 2:2 (2009), 108–120 (In Russian) | DOI

[10] Novatskii V., Teoriia uprugosti [Elasticity Theory], Mir, Moscow, 1975, 872 pp. (In Russian)

[11] Gorin V. V., “Existence and uniqueness of solutions of the nonlocal equation for the ionization source in a glow discharge and a hollow cathode”, Trudy MFTI, 2:3 (2010), 71–80 (In Russian)

[12] Golichev I. I., “On uniqueness and iteration method of solving a non-linear non-stationary problem with non-local boundary conditions of “radiation heat transfer” type”, Ufimsk. Mat. Zh., 2:4 (2010), 27–38 (In Russian) | Zbl

[13] Volynskaya M. G., “Uniqueness of the solution of a nonlocal problem for a degenerate hyperbolic equation”, Vestn. Samar. Gos. Univ., Estestvennonauchn. Ser., 2008, no. 2(61), 43–51 (In Russian) | Zbl

[14] Zhernovyi Yu. V., “Uniqueness of the solution of some boundary value problems for ordinary differential equations with a nonlinear occurrence of the higher derivative”, Differ. Equ., 36:4 (2000), 497–503 | DOI

[15] Dzhuraev T. D., Kraevye zadachi dlya uravnenii smeshannogo i smeshanno-sostavnogo tipov [Boundary value problems for equations of mixed and mixed-composite types], Fan, Tashkent, 1979, 238 pp. (In Russian) | Zbl

[16] Nayfeh A. H., Introduction to perturbation techniques, John Wiley Sons, New York, 1981, xiv+519 pp. | Zbl

[17] Andrianov I., Awrejcewicz J., Metody asimptoticheskogo analiza i sinteza v nelineinoi dinamike i mekhanike deformiruemogo tverdogo tela [Asymptotic Analysis and Synthesis in Nonlinear Dynamics and Mechanics of Deformable Bodies], Institut komp'iuternykh issledovanii, Moscow, Izhevsk, 2013, 276 pp. (In Russian)

[18] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii [Systems of quasi-linear equations], Nauka, Moscow, 1969, 592 pp. (In Russian)

[19] Bitsadze A. V., Boundary value problems for second order elliptic equations, North-Holland Series in Applied Mathematics and Mechanics, 5, North-Holland Publishing Company, Amsterdam, 1968, 211 pp. | Zbl | Zbl

[20] Kachanov L. M., Teoriia polzuchesti [Creep Theory], Fizmatgiz, Moscow, 1960, 456 pp. (In Russian)