Buckling of residually stressed plates: An asymptotic approach

We consider a residually stressed plate-like body having the shape of a cylinder of cross-section ω and thickness hε, subjected to a system of loads proportional to a positive multiplier λ. 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 Ω_ε =ω ×(−εh/2,+εh/2). We let ε → 0 and we show that if the residual stress and the incremental stress induced by the applied loads scale with ε 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 ω. In the special case of null residual stress, the Euler–Lagrange equations associated to this functional coincide with the classical equations governing plate buckling.