Singular global bifurcation problems for the buckling of anisotropic plates

P. V. Negrón-Marrero, S. Antman
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引用次数: 13

Abstract

This paper treats a variety of unexpected pathologies that arise in the global bifurcation analysis of axisymmetrie buckled states of anisotropic plates. The geometrically exact plate theory used accounts for flexure, extension and shear. The nonlinear constitutive functions have very general form. As a consequence of the anisotropy the trivial solution may depend discontinuously on the load parameter. Accordingly, the equations for the bifurcation problem have the same character, so that bifurcating branches of solutions become disconnected as the load parameter crosses values at which discontinuities occur. The anisotropy furthermore implies that the governing equations have a singular behaviour much worse than that for isotropic plates. Consequently, a variety of novel constructions are required to demonstrate the validity of the essential results upon which global bifurcation theory stands. (These results include the compactness of certain operators and the uniqueness of solutions of initial value problems for singular ordinary differential equations.) It is shown that in regions of solution-parameter space in which the equations depend continuously on the load parameter there exist connected global branches of solution pairs that have detailed nodal properties inherited from eigenfunctions of the linearized problem. Moreover, these nodal properties are preserved across gaps occurring where discontinuities occur. The methodology used to show this result actually supports constructive methods for finding disconnected branches.
各向异性板屈曲的奇异全局分岔问题
本文讨论了各向异性板轴对称屈曲状态全局分岔分析中出现的各种意想不到的病态。使用的几何精确板理论考虑了弯曲、伸展和剪切。非线性本构函数具有非常一般的形式。由于各向异性,平凡解可能不连续地依赖于荷载参数。因此,分岔问题的方程具有相同的性质,当荷载参数越过发生不连续的值时,解的分岔分支就会断开。各向异性进一步表明,控制方程的奇异性比各向同性板差得多。因此,需要各种新颖的结构来证明全球分岔理论所依据的基本结果的有效性。(这些结果包括某些算子的紧性和奇异常微分方程初值问题解的唯一性。)结果表明,在方程连续依赖于荷载参数的解-参数空间区域中,存在着具有继承线性化问题特征函数的详细节点性质的解对的连通全局分支。此外,这些节点属性在发生不连续的间隙中保持不变。用于显示此结果的方法实际上支持查找断开分支的建设性方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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