Uniform approximations and effective boundary conditions for a high-contrast elastic interface

Chapman, C. J.; Mogilevskaya, S. G.

Abstract

A difficulty in the theory of a thin elastic interface is that series expansions in its thickness become disordered in the high-contrast limit, i.e. when the interface is much softer or much stiffer than the media on either side. We provide a mathematical analysis of such series for an annular coating around a cylindrical fibre embedded in an elastic matrix subject to biaxial forcing. We determine the order of magnitude of successive terms in the series, and hence the terms which need to be retained in order to ensure that every neglected term is smaller in order of magnitude than at least one retained term. In this way, we obtain uniform approximations for quantities such as the jump in the displacement and stress across the coating, and explain some peculiarities which have been observed in numerical work. A key finding is that it is essential to distinguish three types of boundary-value problem, corresponding to 'distant forcing', 'localized forcing' and 'the homogeneous problem', since they give different patterns of disorder in the corresponding series expansions. This provides a meaningful correspondence between physical principles and our mathematical results.

• Journal Article Type: Article
• Acceptance Date: Aug 4, 2023
• Online Publication Date: Sep 6, 2023
• Publication Date: 2023-09
• Deposit Date: Aug 24, 2023
• Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Print ISSN: 1364-5021
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 479
• Issue: 2277
• Article Number: 0140 A20230140
• Pages: 1 - 22
• DOI: https://doi.org/10.1098/rspa.2023.0140
• Keywords: General Physics and Astronomy, General Engineering, General Mathematics