热超弹性薄板非对称拉伸的分岔和稳定性

Bifurcation and Stability of Asymmetric Stretch for Thermo-Hyperelastic Sheet

本文应用超弹性材料的有限变形理论分析了在面内等双向拉伸死载荷作用下不可压热超弹性方形薄板发生非对称变形的分岔及其稳定性问题。给出了方板变形的分岔曲线和临界载荷,发现对受面内等双向拉伸载荷作用的均匀方板,当拉伸载荷值较小时,方板双向等伸长变形,发生对称的拉伸变形;但当此载荷值大于某一临界值时,从方板的对称拉伸变形中分岔出非对称的变形,方板在两个方向的变形不再相等。通过变形发生分岔前后的能量比较发现,分岔后的对称变形是不稳定的,而非对称变形是稳定的。同时,给出了板中的应力分布曲线,并由不同温度下变形的分岔曲线和应力分布曲线讨论了温度对方板变形和板中的应力分布的影响。

Bifurcation and stability of asymmetric deformations of equal-biaxial stretching of a thin elastic square sheet was examined within the framework of finite elasticity. The sheet is composed of an incompressible thermo-hyperelastic material and subjected to the equal tensile dead loads on its edges. Bifurcation curves and critical load were given. It shows that, when the loads are small, the two principal stretches of sheet are equal to each other; but when the loads are larger than a certain critical value, the solutions corresponding to the asymmetric stretching bifurcate from the symmetric stretching solution. Through the comparison of energy, the asymmetric solution is stable while the symmetric one is revealed unstable. Furthermore, the distribution of tensile stresses was given. The temperature effect on the bifurcation and the distribution of stresses were discussed.