Gamma收敛的非线性弱曲杆
Nonlinear weakly curved rod by Gamma convergence
摘要：
本文提出了一种弱曲杆的非线性模型，即曲率为截面直径量级的曲杆。我们使用类似于杆和曲杆的方法，从三维非线性弹性的应变能泛函开始，并且不假设任何结构行为。为了推导出该模型，通过$Γ$ -收敛，我们需要提出应变能的阶数如何与主体$h$的厚度相关。我们分析的情况下，应变能（除以体积的数量级）的顺序为$h^4$。这与获得板的Föppl-von Kármán模型和杆的类似模型时的方法相同。所得到的模型类似于Marguerre-von Kármán的扁壳模型，其线性化是线性浅拱模型，可以在文献中找到。
英文摘要：
We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use the approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity and do not presuppose any constitutional behavior. To derive the model, by means of $Γ$-convergence, we need to propose how is the order of strain energy related to the thickness of the body $h$. We analyze the situation when the strain energy (divided by the order of volume) is of the order $h^4$. That is the same approach as the one when Föppl-von Kármán model for plates and the analogous model for rods are obtained. The obtained model is analogous to Marguerre-von Kármán for shallow shells and its linearization is the linear shallow arch model which can be found in the literature.
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