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Optimal Design of Thin Plates by a Dimension Reduction for Linear Constrained Problems

The goal of this paper is to give a rigorous justification for the Hessian-constrained problems introduced in [G. Bouchitte and I. Fragala, Arch. Ration. Mech. Anal., 184 (2007), pp. 257-284] and to show how they are linked to the optimal design of thin plates. To that aim, we study the asymptotic behavior of a sequence of optimal elastic compliance problems in the double limit when both the maximal height of the design region and the total volume of the material tend to zero. In the vanishing volume limit, a sequence of linear constrained first order vector problems is obtained, which in turn—in the vanishing thickness limit—produces a new linear constrained problem where both first and second order gradients appear. When the load is orthogonal to the plate, only the Hessian constraint is active, and we recover as a particular case the optimization problem studied in [G. Bouchitte and I. Fragala, Arch. Ration. Mech. Anal., 184 (2007), pp. 257-284] (see also [T. Lewinski and J. J. Telega, Arch. Mech. (Arch. Mech. Stos.), 53 (2001), pp. 457-485]).