We consider a residually stressed plate-like body having the shape of a cylinder of cross-section Ï and thickness hÎµ, subjected to a system of loads proportional to a positive multiplier Î». We look for the smallest value of the multiplier such that the plate buckles, the so-called critical multiplier. The critical multiplier is computed by minimizing a functional whose domain of definition is a collection of vector fields defined in the three-dimensional region Î©<inf>Îµ</inf> =Ï Ã(-Îµh/2,+Îµh/2). We let Îµ â 0 and we show that if the residual stress and the incremental stress induced by the applied loads scale with Îµ in a suitable manner, then the critical multiplier converges to a limit that can be computed by minimizing a functional whose domain is a collection of scalar fields defined on the two-dimensional region Ï. In the special case of null residual stress, the Euler-Lagrange equations associated to this functional coincide with the classical equations governing plate buckling.

### Buckling of residually stressed plates: An asymptotic approach

#### Abstract

We consider a residually stressed plate-like body having the shape of a cylinder of cross-section Ï and thickness hÎµ, subjected to a system of loads proportional to a positive multiplier Î». We look for the smallest value of the multiplier such that the plate buckles, the so-called critical multiplier. The critical multiplier is computed by minimizing a functional whose domain of definition is a collection of vector fields defined in the three-dimensional region Î©Îµ =Ï Ã(-Îµh/2,+Îµh/2). We let Îµ â 0 and we show that if the residual stress and the incremental stress induced by the applied loads scale with Îµ in a suitable manner, then the critical multiplier converges to a limit that can be computed by minimizing a functional whose domain is a collection of scalar fields defined on the two-dimensional region Ï. In the special case of null residual stress, the Euler-Lagrange equations associated to this functional coincide with the classical equations governing plate buckling.
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2015
Paroni, Roberto; Tomassetti, Giuseppe
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11568/885738`
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