Buckling modes of a compressed plate on an elastic substrate

The buckling modes of a homogeneously compressed elastic plate on a soft elastic substrate are studied. The critical compression is uniquely determined by the bifurcation equation, but this compression is associated with a wide set of buckling modes. It was proved that any solution of the Helmholtz equation satisfies the bifurcation equation. At the same time, in microelectronics, it is required to know which buckling mode is realized. Experimental and theoretical investigations show that the chessboard-like buckling mode should be expected. In what follows, this problem is discussed theoretically. The expected buckling mode can be found by analyzing the energy of the initial postcritical deformation, and the desired mode is determined from the condition of its minimum. The analytic expression of this energy is obtained. Its minimization results in the chessboard-like buckling mode.