A Study of the Catastrophe and the Cavitation for a Spherical Cavity in Hooke’s Material with 1/2 Poisson’s Ratio
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摘要: 研究Poisson比为1/2的Hooke材料中,空穴的突变和萌生现象.求解一个球对称几何非线性弹性力学的移动边界(moving boundary)问题,空穴为球形,远离空穴处为三向均匀拉伸应力状态,在当前构形上列控制方程;在当前构形边界上列边界条件.找到了这个自由边界问题的封闭解并得到空穴半径趋于零时的叉型分岔解.计算结果显示,在位移-载荷曲线上存在一个切分岔型分岔点(或鞍结点型分岔点、极值型分岔点),这个分岔点说明在外力作用下空穴会发生突变,即突然“长大”;当球腔半径趋于零时,这个切分岔转化为叉型分岔(或分枝型分岔),这个叉型分岔可以解释实心球中的空穴萌生现象.Abstract: In this paper, the catastrophe of a spherical cavity and the cavitation of a spherical cavity for Hooke. s material with 1/2 Poisson's ratio are studied. A nonlinear problem, which is a moving boundary problem for the geometrically nonlinear elasticity in radial symmetric, was solved analytically. The governing equations were written on the deformed region or on the present configuration. And the conditions were described on moving boundary. A closed form solution was found. Furthermore, a bifurcation solution in closed form was given from the trivial homogeneous solution of a solid sphere. The results indicate that there is a tangent bifurcation on the displacement-load curve for a sphere with a cavity. On the tangent bifurcation point, the cavity grows up suddenly, which is a kind of catastrophe. And there is a pit chfork bifurcation on the displacement-load curve for a solid sphere. On the pitchfork bifurcation point, there is a cavitation in the solid sphere.
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Key words:
- moving boundary /
- tangent bifurcation /
- pitchfork bifurcation /
- cavitation
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