LIMIT STATE COMPRESSED PLATE FROM THE FIRST-ORDER MATERIAL

Abstract

The post-buckling of a uniformly compressed plate was considered. An analytical-numerical solution was obtained using the Rayleigh-Ritz method. Legendre polynomials and their linear combinations were used as basis functions. The resolving equations were obtained from the principle of the stationarity of the total potential energy, in which the elastic potential is a strain energy density function of the semilinear material. This hyperelastic material means a linear binding between the right stretch tensor (right Biot-stretch tensors) and the conjugate symmetric tensor of stresses Biot (Jaumann stress) pertaining to the Seth-Hill family. To reduce the problem to a two-dimensional (plane stressed state), Kirchhoff-Love's classical hypotheses were used. The components of the right stretch tensor in the reference Cartesian coordinate system were written explicitly in terms of the derivatives of the three displacement functions. To do this, the deformation gradient tensor of the deformed middle surface, the right Cauchy–Green deformation tensor, the right stretch tensor and the proper orthogonal rotation tensor were first constructed. After this, the right tensor of the multiplicity of elongations for an equidistant surface was similarly constructed. After this, the right tensor of elongations was constructed similarly for a surface that is located from a deformed middle surface to a small distance z. The components of this tensor were expanded in a Taylor series about the neglected coordinate z = 0 and its linear part were singled out. The algorithm for solving the problem, which includes the implementation of the Rayleigh-Ritz method and Newton's method, was performed using the C ++ programming language. Diagrams of equilibrium states were constructed and graphs of the change in the total potential energy were presented. Two branches of equilibrium states, which begin with the first and second critical Euler loads, were traced, and these branches, obtained by the Föppl–von Kármán plate theory, are also given. It has been shown that Föppl–von Kármán plate theory gives a satisfactory result only near the bifurcation point and can lead to a qualitatively incorrect result. It was demonstrated that post-buckling for uniformly compressed plates, jumps from one form of equilibrium to another (as for shells) are observed. To determine the limiting value of the compressive load from the condition of a minimum of the total potential energy was proposed.