Asymptotic theory of elastic plate buckling under lateral compression

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Homogeneous solutions obtained in /1/ are used for investigating in three-dimensional formulation the stability of a thick plate of arbitrary shape of neo-Hookean material and free of constraints, /2/. Boundary conditions at lateral surfaces are obtained using the variational principle of superposition of a small deformation on a finite one, as proposed in /3/. As the result, the problem of critical pressure determination is reduced to the general problem of eigenvalues for an infinite homogeneous system of operator equations whose dependence (explicit as well as implicit) on the initial deformation parameters is essentially nonlinear. The asymptotic method proposed in /4/ is extended so as to make possible the determination of critical load asymptotics, as the plate thickness ε approaches zero. The effect of potential solutions that correspond to irregular (with the initial deformation eliminated) roots of the characteristic equation /1/, and have no analogs in the linear theory of elasticity /5/, on the plate stress-strain state and on the magnitude of critical pressure is determined. It is shown that the two-dimensional theory of plate buckling based on the Kirchhoff hypotheses /6/ makes possible the correct determination of the principal terms of the critical load asymptotics as ε → 0. As an example, the axisymmetric buckling of a circular plate is considered. Five terms of the asymptotic expansion of the critical load are obtained. It is established that the classic theory yields a deficient critical force, with a relative error of the order of ε 2.