Shape optimization of elasto-plastic axisymmetric bodies. (English) Zbl 0756.73094

Hlaváček, Ivan

Appl. Math., Praha 36, No. 6, 469-491 (1991).

An optimal design problem for elasto-plastic axisymmetric bodies is studied. It consists in the following data: given body forces, surface loads and material characteristics, find the shape of the meridian section such that the cost functional defined by the integral of the yield function is minimal under the conditions that either zero displacements or zero surface tractions are prescribed on the unknown part of the boundary. Using, for the studied bodies, the Hencky’s law of state and the Haar-Kármán’s principle the author reduces the problem to the solution of the variational inequality \((\sigma,\tau-\sigma)\geq 0\), \(\forall\tau\in P(D)\cap E(D)\) where \(\sigma\), \(\tau\) and \(D\) have known significances and \(P(D)\), \(E(D)\) are the set of plastically admissible stress fields and the set of statically admissible stress fields respectively, defined by the author in the paper. For this variational problem, an approximate optimal design problem can be formulated by means of a piecewise linear approximation of the stress field and of the unknown boundary. For the exact and the approximate problems the author proves the existence and the uniqueness of the solution, which constitutes the main original contribution of the author to the studied problem.

Reviewer: S.Zanfir (Craiova)

MSC:

74S30	Other numerical methods in solid mechanics (MSC2010)
74P10	Optimization of other properties in solid mechanics
74B99	Elastic materials
74C99	Plastic materials, materials of stress-rate and internal-variable type
65K10	Numerical optimization and variational techniques

Keywords:

domain optimization; Hencky’s law; Haar-Kármán’s principle; variational inequality; approximate optimal design problem; piecewise linear approximation; existence; uniqueness

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