关于超弹性材料球体中空穴分岔问题的研究(Ⅱ)

研究了一类可压缩超弹性材料组成的球体的空穴分岔问题.建立了球体在给定的表面均匀径向拉伸作用下的球对称变形问题的数学模型.证明了当给定的表面伸长超过某临界值时,球体内部可以发生空穴分岔,并且得到了描述球体的径向变形函数的参数型空穴分岔解;指出了临界伸长随泊松比的增加而减少,讨论了空穴分岔方程的解的稳定性.证明了在空穴生成之前,整个球体的变形是膨胀的;空穴生成后,空穴附近的变形是收缩的.最后给出了球体内部有空穴生成时的应力分布.

Cavitated Bifurcation Problems for Spheres Composed of Hyper-elastic Materials (Ⅱ)

A spherical cavitated bifurcation problem is examined for a solid sphere composed of a class of compressible hyper-elastic materials. The mathematical model that describes spherical symmetric deformation of the sphere under a uniform homogeneous radial stretch is proposed. It is proved that cavitated bifurcation can occur for this class of compressible hyper-elastic spheres as the prescribed stretch exceeds the critical stretch. A group of parameter-type solutions for the cavitated deformation for a solid sphere are obtained;on the other hand, the critical stretch decreases monotonously with the increasing Poisson ratio. Stability of the solutions is discussed. It is proved that the deformation of the whole sphere is extension before the cavity forms and the deformation near the cavity becomes compressive after the cavity forms. Finally, analyses of distribution of radial and circumference stresses as a cavity forms in the sphere are carried out.