Broadening Global Families of Anti-Plane Shear Equilibria
Broadening Global Families of Anti-Plane Shear Equilibria
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DOI：
10.1137/21m1392838
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发表时间：
2021-01
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影响因子：
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通讯作者：
T. Hogancamp
中科院分区：
文献类型：
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作者：
T. Hogancamp
本文发展了两类非线性弹性材料的全局分叉理论。假设它们受到反平面剪切变形，并在参考配置中占据无限长圆柱体。利用解析全局分歧理论构造了相应弹性静力问题解的曲线。与第一类相关的曲线显示出加宽行为，而对于第二类，我们发现控制方程在极限下经历了损失椭圆性。当有效支集无限增长时，解的序列发生展宽。这种现象在孤立水波的背景下受到了相当大的关注;它已经被数值预测，但仍有待严格证明。椭圆度的破坏与裂纹和失稳有关，使其成为破坏力学理论的一个重要方面。
We develop a global bifurcation theory for two classes of nonlinear elastic materials. It is supposed that they are subjected to anti-plane shear deformation and occupy an infinite cylinder in the reference configuration. Curves of solutions to the corresponding elastostatic problem are constructed using analytic global bifurcation theory. The curve associated with first class is shown to exhibit broadening behavior, while for the second we find that the governing equation undergoes a loss ellipticity in the limit. A sequence of solutions undergoes broadening when their effective supports grow without bound. This phenomena has received considerable attention in the context of solitary water waves; it has been predicted numerically, yet it remains to be proven rigorously. The breakdown of ellipticity is related to cracks and instability making it an important aspect of the theory of failure mechanics.