Asymptotic analysis of perforated plates and membranes. Part 1: Static problems for small holes
Asymptotic analysis of perforated plates and membranes. Part 1: Static problems for small holes
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穿孔板和膜的渐近分析第 1 部分：小孔的静态问题
DOI：
10.1016/j.ijsolstr.2011.10.005
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发表时间：
2012
影响因子：
3.6
通讯作者：
Kalamkarov A.L.
中科院分区：
文献类型：
--
作者：
Andrianov I.V;Danishevs’kyy V.V;Kalamkarov A.L.
将渐近方法与多尺度均匀化方法相结合，求解了周期开孔弹性板和杆的静力问题。利用渐近均匀化方法将原边值问题化为两类问题的组合。第一种是周期连续条件下的单位胞元问题的递归系统。第二个问题是整个区域的齐次化边值问题，其特征在于由单胞问题的解得到的常有效系数。将摄动法与逐次逼近法相结合，求解了单胞问题。考虑到孔洞尺寸较小，采用了边界形状摄动法和施瓦茨交替法。连续考虑了带孔截面杆的扭转问题;带孔薄膜在法向载荷作用下的挠度问题;带圆孔和方孔的带孔板的弯曲问题。所应用的渐近技术的误差估计和所获得的解决方案的高精度证明。
Static problems for the elastic plates and rods periodically perforated by small holes of different shapes are solved using the asymptotic approach based on the combination of the asymptotic technique and the multi-scale homogenization method. Using the asymptotic homogenization method the original boundary-value problem is reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. The combination of the perturbation method and the technique of successive approximations is applied for the solution of the unit cell problems. Taking into the account small size of holes the method of perturbation of the shape of the boundary and the Schwarz alternating method are used. The problems of torsion of a rod with perforated cross-section; deflection of the perforated membrane loaded by a normal load; and bending of perforated plates with circular and square holes are considered consecutively. The error of the applied asymptotic techniques is estimated and the high accuracy of the obtained solutions is demonstrated.