Gamma收敛的扁壳模型

摘要：

本文从三维非线性弹性力学出发，用$Γ$ -收敛方法导出扁壳模型。我们使用类似于壳和板的方法。我们从一般三维弹性体的极小化公式出发，该弹性体受法向体积力和自由边界条件的作用，并且不假定任何本构行为。为了推导出模型，我们需要提出外部载荷的量级如何与主体$h$的厚度以及浅壳的“几何形状”的量级相关。我们分析了当外部法向力为$h^α$阶时的情况，其中$α>2$。对于$α=3$，我们得到了Marguerre-von Kármán模型，对于$α>3$，我们得到了线性化的Marguerre-von Kármán模型。对于$α\in (2,3)$，我们只能得到$Γ$ -极限的下界。这类似于最近的结果为普通壳模型。

英文摘要：

In this paper we derive, by means of $Γ$-convergence, the shallow shell models starting from non linear three dimensional elasticity. We use the approach analogous to the one for shells and plates. We start from the minimization formulation of the general three dimensional elastic body which is subjected to normal volume forces and free boundary conditions and do not presuppose any constitutional behavior. To derive the model we need to propose how is the order of magnitudes of the external loads related to the thickness of the body $h$ as well as the order of the "geometry" of the shallow shell. We analyze the situation when the external normal forces are of order $h^α$, where $α>2$. For $α=3$ we obtain the Marguerre-von Kármán model and for $α>3$ the linearized Marguerre-von Kármán model. For $α\in (2,3)$ we are able to obtain only the lower bound for the $Γ$-limit. This is analogous to the recent results for the ordinary shell models.

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