Homogenization in open sets with holes
Homogenization in open sets with holes
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DOI：
10.1016/0022-247x(79)90211-7
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发表时间：
1979-10
影响因子：
1.3
通讯作者：
D. Cioranescu;J. Paulin
中科院分区：
文献类型：
--
作者：
D. Cioranescu;J. Paulin
设Qr是一个有r个圆柱形空腔的圆柱杆，空腔的生成元与Qr的生成元平行。设Ω为杆的横截面，Ω为材料所占据的畴的横截面，Ω i（i= 1，.，r）为空腔的横截面：Ω <$i <$Ω <$i <$Ω <$k= φ，i <$k。对这种杆的弹性扭转的研究导致了以下问题[见2.，3., 267-320）]：Δ r+ 2μα= 0 in Ω r Ω= 0（1）r= Ω i上的常数; i= 1，.，r其中μ是材料的剪切模量，α是扭转角，r代表应力函数。本文考虑了周期分布的孔数增加的问题（1）。人们想知道当r→+∞时，是否有一个极限<$∞，如果有，这个极限满足的方程。这是一个“均匀化”问题--用一个均匀杆代替非均匀杆Qr，其扭转响应尽可能接近Qr。我们将研究一个更一般的问题，并作为应用得到弹性扭转的情况。证明使用能量方法[见狮子（法兰西学院，1975年至1977年），鞑靼（法兰西学院，1977年）]和扩展定理。一个相关的问题是多孔板的均匀化[cf.（Duvaut）。
Let Q r be a cylindrical bar with r cylindrical cavities having generators parallel to those of Q r. Let Ω be the cross-section of the bar, Ω∗ the cross-section of the domain occupied by the material and Ω i (i= 1,…, r) the cross-sections of the cavities: Ω ̄ i⊂ Ω Ω ̄ i∩ Ω ̄ k= φ, i≠ k. The study of the elastic torsion of this bar leads to the following problem [see 2., 3., 267–320)]: Δƒ r+ 2μα= 0 in Ω∗ ƒ r¦∂ Ω= 0 (1) ƒ r= constant on∂ Ω i; i= 1,…, r where μ is the shear modulus of the material, α is the angle of twist and ƒ r represents the stress function. In this paper the problem (1) with an increasing number of holes which are distributed periodically is considered. One would like to know if ƒ r has a limit ƒ∞ as r→+∞, and if so, the equation satisfied by this limit. This is an “homogenization” problem—the heterogeneous bar Q r is replaced by a homogeneous one, the response of which under torsion approximates as closely as possible that of Q r. A more general problem will be studied and the case of elastic torsion will be obtained as an application. The proof uses the energy method [see Lions (Collège de France, 1975–1977), Tartar (Collège de France, 1977)] and extension theorems. A related problem is the homogenization of a perforated plate [cf. Duvaut (to appear)].