Packaging waste contributes significantly to resource depletion and pollution. Despite the crucial role of packaging in product preservation, its environmental impact has become a major issue. Addressing the circularity and sustainability (C&S) of packaging by design offers a route to mitigate these impacts and reduce waste. However, integrating C&S into the current packaging design process presents significant challenges, such as conflicts between C&S and functional requirements and inadequate tools to provide packaging-specific practical solutions. To address these challenges, this study proposes a novel packaging design framework developed through literature review, brainstorming sessions, and field visits. By incorporating iterative design strategies and leveraging past design knowledge, the framework empowers designers to create packaging solutions that meet C&S requirements.

In this paper, exact vibration frequencies of circular, annular and sector membranes with a radial power law density are presented for the first time. It is found that in general, the sequence of modes may not correspond to increasing az-imuthal mode number n. The normalized frequency increases with the absolute value of the power index |ν|. For a circular membrane, the fundamental frequency occurs at n = 0 where n is the number of nodal diameters. For an annular membrane, the frequency increases with respect to the inner radius b. When b is close to one, the width 1 – b is the dominant factor and the differences in frequencies are small. For a sector membrane, n – 1 is the number of internal radial nodes and the fundamental frequency occurs at n = 1. Increased opening angle β increases the frequency.