各向同性Gent-Thomas材料中空穴生成的突变性

研究了由均匀各向同性不可压缩Gent-Thomas材料组成的球体在给定表面拉伸死载荷作用下的球对称变形问题.得到了描述球体内部空穴生成和增长的空穴分岔方程.证明了方程的平凡解支上存在唯一的分岔点,以及非平凡解在分岔点附近可以局部向左或向右分岔.特别地,当材料参数取某些值时,局部向右分岔的非平凡解对应于拉伸死载荷超过某临界值时,球体内部有空穴生成并连续增长;而局部向左分岔的非平凡解支上还存在一个二次转向分岔点,在这种情形下,当拉伸死载荷还未超过临界值时,球体内部便有一个半径相对较大的空穴生成,这与其他各向同性不可压缩超弹性材料有明显的不同.同时给出了相应的数值算例.

Catastrophe of Cavity Formation in Isotropic Gent-Thomas Materials

The spherical symmetric deformation problem is examined for a sphere composed of the homogeneous isotropic incompressible Gent-Thomas material,in which the sphere is subjected to a prescribed tensile dead load on its surface.A bifurcation equation of cavity that describes cavity formation and growth in the sphere is obtained.It is proved that there exists a unique bifurcation point on the trivial solution branch of the equation,and that the nontrivial solution can bifurcate locally to the left or to the right near the bifurcation point.In particular,corresponding to the nontrivial solution bifurcating locally to the right,a cavity forms in the sphere and then grows continuously once the tensile dead load exceeds a certain critical value.However,there also exists a secondary turning point on the nontrivial solution,in this case,a cavity with relative larger radius forms in the sphere as the tensile dead load does not exceed the certain critical value,which is quite different from other isotropic incompressible hyper-elastic materials.Some numerical examples are carried out simultaneously.