Impact buckling of orthotropic shells of revolution with allowance for geometric nonlinearity

Abstract

Numerous experiments have established the presence of axisymmetric and nonaxisymmetrie stages in the buckling of elastic cylindrical and conical shells subjected to axial impact. It has been shown [8, 10, 14] that metallic, glassplastic, and polymer shells become unstable due to the intensive development of nonaxisymmetric strains caused by geometric shape flaws. Until recently, calculations of the nonaxisymmetric buckling of cylindrical shells in a geometrically nonlinear formulation have been performed only by the Bubnov--Galerkin method in lower-level approximations [5]. In [6], a numerical solution was obtained for the problem of the dynamic axial compression of a metallic cylindrical shell while retaining a large number of terms in the expansion for nonaxisymmetric deflection. The authors of [2, 4] formulated and numerically solved a problem concerning the impact buckling of flawed isotropic cylindrical and conical shells within the framework of the Kirchhoff--Love hypothesis with allowance for boundary, wave, and nonlinear effects associated with the coupling of the axisymmetric mode of instability and one of the nonaxisymmetric modes. The authors of [3] solved the problem of the impact buckling of an isotropic cylindrical shell with allowance for the interaction of the axisymmetric mode and a finite number of nonaxisymmetric modes. The 9 method in [2] was used in [7] to study the stability and strength of laminated orthotropic shells. The goal of the present investigation is to use the findings in [2-4] as a basis for developing a method of numerically analyzing the impact buckling of orthotropic shells of revolution with initial shape flaws. We also want to study the interaction of nonaxisymmetric modes of buckling. We will examine the axial impact buckling of thin orthotropie shells of revolution of constant thickness h referred to orthogonal curvilinear coordinates ,~ and/3 which coincide with the lines of curvature of the middle surface of the shell. The equations of motion of the shell will be written in the form [13]