Geodesic
Development of the decomposition method in mechanics of solids
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 10 (2010) no. 3, pp. 54-59 
V. A. Lokhov; Yu. I. Nyashin; V. S. Tuktamishev. Development of the decomposition method in mechanics of solids. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 10 (2010) no. 3, pp. 54-59. http://geodesic.mathdoc.fr/item/ISU_2010_10_3_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The orthogonal projection method for solution of boundary value problem of theory of elasticity with eigenstrain is presented. The main feature of the method is that the orthogonal decomposition is performed in the Hilbert function space of eigenstrains instead of function space of stresses, which is commonly accepted. As a result, the opportunities to create the desired stress field by eigenstrain keeping strain unchanged (strain-free stress control) and vice versa to create the desired strain distribution keeping stress unchanged (stress-free shape control) are shown. The developed approach is applied to control of residual stress in thermoplasticity.

[1] Zaremba S., “Sur le principe de minimum”, Bull. intern. l'Acad. d. sciences de Cracovie. Cl. des sciences math. et natur., 1909, no. 7, 197–264 | Zbl

[2] Weil H., “The method of orthogonal projections in potential theory”, Duke Math. J., 7 (1940), 411–444 | DOI | MR

[3] Mikhlin S. G., Variatsionnye metody v matematicheskoi fizike, 2-e pererab. i dop., Nauka, M., 1970, 512 pp. | MR | Zbl

[4] Reissner H., “Eigenspannungen und Eigenspannungsquellen”, Z. Angew. Math. Mech., 11:1 (1931), 1–8 | DOI | MR | Zbl

[5] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1968, 543 pp. | MR | Zbl

[6] Nyashin Y., Lokhov V., Ziegler F., “Decomposition method in linear elastic problems with eigenstrain”, Z. Angew. Math. Mech., 85 (2005), 557–570 | DOI | MR | Zbl

[7] Pozdeev A. A., Nyashin Yu. I., Trusov P. V., Ostatochnye napryazheniya: teoriya i prilozheniya, Nauka, M., 1982, 112 pp.

[8] Tall L., Huber A., Beedle L., “Residual stress and column instability under axial loads”, XII Annual Assembly (Liege, June 13–19, 1960), Intern. Institute of Welding, Liege, 1960, 381–396