On chessboard buckling modes in compressed materials

Abstract

The article is concerned with a transversely isotropic homogeneous elastic medium subjected to uniform compression in the isotropy plane. The medium becomes unstable in the sense of Hadamard at a certain level of initial strain. The critical strain is established to be uniquely determined from the system of equations of the equilibrium bifurcation; however, there are many modes of buckling corresponding to this strain. A solution of the system of the bifurcation equations is considered in the form of double periodic functions of the kind sin r 1 x 1 sin r 2 x 2. The uncertainty in the buckling mode implies that the wave numbers r 1 and r 2 remain arbitrary. In order to determine the relationship between the wave numbers, we examine the initial supercritical behavior of the material. Only two types of buckling modes (the shear type and the volume type) are possible. It is established that the buckling mode of the volume type is a chessboard-like one, and the mode of the shear type is not chessboard like. The stability of the supercritical equilibrium state is discussed.