可压缩球膜的膨胀和分叉

INFLATION AND BIFURCATION OF A COMPRESSIBLE SPHERICAL MEMBRANE

利用大变形迭加小变形的一般方法详细考察了可压缩球膜的膨胀和分叉问题.得到了可压缩球膜分叉的控制微分方程组,并通过求解非线性偏微分方程组,给出了不同情形的分叉模式及相应的分叉判据,但为了与Alexander的实验对比,只认为01模式的分叉解在物理上是可行的.结果表明:可压缩球膜分叉解的控制微分方程组与不可压缩时非常相似,都只有3个独立的弹性系数,但弹性系数的定义是不同的;从理论上证明了可压缩球膜的分叉也是在内压达到极大值之后发生的,且在球膜的膨胀过程中,当内压达到极大值后,球膜的形状不再是标准的球形,而是上半球的厚度变大,下半球厚度变小,此时球膜分叉了,这与实验结果是一致的.

With the help of small deformations superposed on large ones,the inflating and bifurcation of a compressible spherical membrane were studied. The controlling partial differential equations were obtained. According to seeking the solution of the equations, all kinds of the bifurcation models and criteria were given. In order to compare with the Alexander＇s test,only the 01 model was assumed to be possible. The results indicated that the controlling differential equations of the compressible situation were very sim- ilar with the incompressible case, both are with three independent elastic coefficients, although their definitions are different. It was proved theoretically that bifurcation happens after the internal pressure reached the critercal value. In the inflating process of a spherical membrane, the spherical shape was no longer kept after the internal pressure reached the critercal value,which implies that the spherical membrane bifurcated. This phenomenon is same as the experimental result.