Mathematical modeling of the pure bending for the aviation material beam at creep conditions
Mathematics. Physics. Mechanics
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
The paper deals with the solution of pure bending of a rectangular beam of D16T material at constant temperature loaded with a constant bending moment. Research of the construction for creep and long-term strength are considered. A comparison is provided of the two methods for the problem solution based on the numerical results, as well as a comparison of the numerical solution with the experimental data.
The numerical calculation problem is solved using the equations of the energy variant of the creep theory, as well as the method of solution continuation with respect to a specific parameter and the best parametrization. The problem is being reduced to the solution of ordinary differential equations. The fourth-order Runge–Kutta method was chosen as the numerical solution method. By using these two methods of solution, creep curves were obtained, as well as the value for the long-term strength of the material.
According to the results, the following conclusions were made:
- A quite satisfactory similarity was found between the calculated and experimental data for the value of the curvature of the beam, which confirms the choice of defining equations in the energy form to describe the creep of beams until the destruction;
- The method of solution continuation with respect to a specific parameter and the best parametrization can be used for research long-term creep strength. The method of solution continuation with respect to a specific parameter and the best parametrization allows the integration step to be increased, reduces the number of steps to find the independent variable and the computation time, compared with non parametrized equations.
The method of solution continuation with respect to a specific parameter is first shown in this work, as well as the best parametrization of the problems of the creep theory. Obtained results and considered methods can be widely used in mechanical engineering, aviation and other high-tech industries.
Keywords:creep, destruction, specific energy scattering, damage parameter, beam bending value, solution continuation with respect to a parameter, parameterization, the best parameter, ordinary differential equations
- Rabotnov Ju.N. Polzuchest' jelementov konstrukcij (Creep Problems in Structural Members), Moscow, Nauka,1966, 752 p.
- Sosnin O.V., Gorev B.V., Nikitenko A.F. Jenergeticheskij variant teoriipolzuchesti (Energy Variant of the Creep Theory), Novosibirsk, Institut gidrodinamiki SO AN SSSR, 1986, 95 p.
- Gorev B.V., Panamarev V.A., Peretyat'ko V.N. Izvestija Vuzov. Chernaja metallurgija, 2011, no. 6, pp. 16-18.
- Kachanov L.M. Teorija polzuchesti (The Theory of Creep), Moscow, Fizmatgiz, 1960, 455 p.
- Lepin G.F., BondarenkoJu.D. Problemy prochnosti, 1970, no. 7, pp. 68-70.
- Nikitenko A.F., Sosnin O.V. Problemy prochnosti, 1971, no. 6, pp. 67-70.
- Gorev B.V. Dinamika sploshnoy sredy, 1973, vol. 14, pp. 44-51.
- Gorev B.V., Klopotov I.D. Prikladnaja mehanika i tehnicheskaja fizika, 1999, vol. 40, no. 6, pp. 157-162.
- Sosnin O.V., Gorev B.V., Rubanov V.V. Problemy prochnosti,1976, no. 11, pp. 9-13.
- Sosnin O.V., Torshenov N.G. Zavodskaja laboratorija, 1969, no. 10, pp. 1273-1274.
- Shalashilin V.I., Kuznetsov E.B. Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics, Dordrecht / Boston / London: Kluwer Academic Publishers, 2003, 236 p.