Asymptotic Model of Composite Piezoelectric Beam
I. V. Andrianov1, A. A. Kolpakov2, L. Faella3
1Rhine Westphalian Technical University, Aachen, Germany
2University of Neuchatel, Neuchatel, Switzerland
3University of Cassino and Southern Lazio, Cassino, Italy
Keywords: piezoelectric composite, effective characteristics, asymptotic homogenization method, multiscale method, fast corrector, local (cell) problem, beam
Abstract
A method of transition from the three-dimensional piezoelastic problem for a composite material to the one-dimensional problem for a piezoelastic beam is presented. This is done using an asymptotic method of homogenization based on the separation of fast and slow variables in the solution. A special feature of the problem is the presence of two small parameters, one of which characterizes the microstructure of the composite material, and the other the cross-sectional size. Averaged relations describing the piezoelastic beam and fast correctors were obtained. Their joint use makes it possible to correctly describe the total stress-strain state of the initial three-dimensional body. The proposed method is suitable for solving the three-dimensional problem of deformation of an extended bodies with an arbitrary periodic structure and new problems (e.g., the torsion problem) that have no analogues in the theory of piezoelastic plates.
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