Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Lokhov V. A., Nyashin Y. I., Tuktamishev V. S. Development of the Decomposition Method in Mechanics of Solids. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2010, vol. 10, iss. 3, pp. 54-59. DOI: 10.18500/1816-9791-2010-10-3-54-59

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Development of the Decomposition Method in Mechanics of Solids

Lokhov V. A., Perm National Research Polytechnic University
Nyashin Y I, Perm National Research Polytechnic University
Tuktamishev V. S., Perm National Research Polytechnic University

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 / S. Zaremba // Bull. intern. l’Acad. d. sciences de Cracovie. Cl. des sciences math. et natur. – 1909. – № 7. – P. 197–264.
  2. Weil, H. The method of orthogonal projections in potential theory / H. Weil // Duke Math. J. – 1940. – V. 7. – P. 411–444.
  3. Михлин, С.Г. Вариационные методы в математической физике / С.Г. Михлин. – 2-е изд., перераб. и доп. – М.: Наука, 1970. – 512 с.
  4. Reissner, H. Eigenspannungen und Eigenspannungsquellen / H. Reissner // Z. Angew. Math. Mech. – 1931. – V. 11, № 1. – P. 1–8.
  5. Колмогоров, А.Н. Элементы теории функций и функционального анализа / А.Н. Колмогоров, С.В Фомин. – М.: Наука, 1968. – 543 с.
  6. Nyashin, Y. Decomposition method in linear elastic problems with eigenstrain / Y. Nyashin, V. Lokhov, F. Ziegler // Z. Angew. Math. Mech. – 2005. – V. 85. – P. 557–570.
  7. Поздеев, А.А. Остаточные напряжения: теория и приложения / А.А. Поздеев, Ю.И. Няшин, П.В. Трусов. – М.: Наука, 1982. – 112 с.
  8. Tall, L. Residual stress and column instability under axial loads / L. Tall, A. Huber, Beedle // XII Annual Assembly, Liege, June 13–19, 1960. – Liege: Intern. Institute of Welding, 1960. – P. 381–396.