z-TRANSFORM METHODS FOR THE OPTIMAL DESIGN OF ONE-DIMENSIONAL LAYERED ELASTIC MEDIA

In this work, we develop a finite trigonometric series representation for the stress in a multilayered Goupillaud-type elastic strip subjected to transient Heaviside loading on one end while the other end is held fixed. This representation is achieved by means of the z-transform method and involves the so-called base angles. Generally, different layered designs could share the same set of base angles, and the more layers the design has, the more base angles are expected. Necessary conditions for the base angles and design parameters for any given design are described. As a result of the stress representation, we are able to identify optimal layered designs which provide the smallest stress amplitude. For two- and three-layered designs, for which the coefficients of the stress representation are easy to find, the optimization results are achieved using a custom-made discrete optimization technique applied in [A. P. Velo and G. A. Gazonas, Int. J. Solids Structures, 40 (2003), pp. 6417–6428]. For other layered designs, the optimality conditions are predicted heuristically using pattern recognition and the necessary conditions for the base angles and design parameters. Applications of these optimization results include design improvement in making a nonoptimal design optimal. They are also extended to non-Goupillaud-type layered media with integer layer length ratios. Our results are supported by numerical experiments and provide means to validate numerical optimization codes.