Iterative interpretation of Saint-Venant semi-inverse method for construction of composite material thin-walled structural elements equations

Аuthors

Zveryaev E. M.^1*, Olekhova L. V.^2**

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. Keldysh Institute of Applied Mathematics (Russian Academy of Sciences), 4, Miusskaya sq., Moscow, 125047, Russia

*e-mail: zveriaev@gmail.com
**e-mail: olekhova@gmail.com

Abstract

A problem of the construction of the composite material beam and plate state equations similar to the classic ones is considered. The construction process is based on the Saint-Venant semi-invers method treated as the iteration one and subjected to the contraction mapping principle. The theory equations are transformed to be applied the simple iteration method. The small parameter is isolated to analyze the convergence process. In accordance with the Saint-Venant semi-inverse method to extract the wishing solution from general equations it is necessary to ask some of the stresses and displacements, and via the rest equations to calculate all the other unknowns. Because of understanding the semi-inverse method as the iterative one the null approximation values are taken the same as these in the classic theory. The first approximation solution gives the possibility to determine all unknowns expressed via the null approximation values. The boundary conditions verification on the face surfaces gives the state body equations with the effective coefficients to determine the first approximation values taking into account the integral composite properties. In general case 16 effective coefficients are obtained for the strip whereas the plate 48. To determine these simple formula are given. All coefficients have to be taken into account to satisfy all boundary conditions at the end faces and determine the stress-strain state in the domain of application of the local load. In the rest body domain the stress-strain state is described by the slowly varying functions calculated from the theory of bending beam and plate classical equations with only effective coefficient of flexural rigidity. The substitution of the found from these equations values into the expression for first approximation unknowns gives the local stresses and displacements. Thus the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) may be considered as the generalization of Saint-Venant semi-inverse method.

Keywords:

contraction mapping principle, beam, plate theory, small parameter, composite material

References

1. Barzokas D.I., Zobnin A.I. Matematicheskoe modelirovanie fizicheskikh protsessov v kompozitsionnykh materialakh periodicheskoi struktury (Mathematic modeling of physic process for periodic structure material), Moscow, Editorial URSS, 2003, 376 p.
2. Kristensen R. Vvedenie v mekhaniku kompozitov (Mechanics of composite materials), Moscow, 1982, 334 p.
3. Bakhvalov N.S., Panasenko G.P. Osrednenie protsessov v periodicheskikh sredakh (Homogenization: Averaging processes in periodic media), Moscow, Nauka, 1984, 352 p.
4. Sen-Venan B. Memuar o kruchenii prizm, Memuar ob izgibe prizm (Mémoire sur la torsion des prismes avec des considerations sur leur flexion. etc., Mémoire sur la flexion des prismes, etc.) Moscow, GIFML, 1961, 519 p.
5. Vol’mir A.S. Ustoychivost’ deformiruemykh system (Stability of deformable bodies), Moscow, Nauka, 1967, 984 p.
6. Zveryayev Ye.M. Prikladnaya matematika i mekhanika, 2003, vol. 67, no. 3, pp. 472–481.
7. Gol’denveyzer A.L. Teoriya uprugikh tonkikh obolochek (Theory of elastic thin shells), Moscow, Nauka, 1976, 512 p.
8. Zveryayev Ye. M. Mekhanika kompozitsionnykh materialov i konstruktsii, 1997, vol. 3, no. 2, pp. 3-19.
9. Kamke E. Spravochnik po obyknovennym differentsial’nym uravneniyam (Differential equations. Ordinary differential equations), Moscow, Nauka, 1971, 576 p.
10. Vilenkin N.Ya. Metod posledovatel’nykh priblizheniy (Method of successive approximations), Moscow, Nauka, 1963, 108 p.
11. Zveryayev Ye.M., Makarov G.I. Prikladnaya matematika i mekhanika, 2008, vol. 72, no. 2, pp. 308-321.
12. Zveryayev Ye.M., Makarov G.I. Stroitel’naya mekhanika i raschet sooruzheniy, 2013, no. 4, pp. 38-42.
13. Zveryayev Ye.M. Trudy Mezhdunarodnoy molodezhnoy nauchnoy konferentsii «Prochnost’, polzuchest’ i razrushenie stroitel’nykh i mashinostroitel’nykh materialov i konstruktsiy», Moscow, 2014, 2014, pp. 66-81.

Download