Asymptotic analysis of perforated shallow shells

Static and dynamic problems for the elastic shallow shells periodically perforated by a large number of 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 boundary condition of periodic continuation. 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 unit cell problems are solved using the combination of the perturbation method and the technique of successive approximations. The method of perturbation of the shape of the boundary and the Schwarz alternating method are applied taking into account small size of holes. Simple engineering formulae for the effective stiffnesses of perforated shells are derived on the basis of the obtained asymptotic solutions. The error of the applied asymptotic techniques is estimated and the high accuracy of the obtained solutions is demonstrated.