# Buckling and Wrinkling of a Thin Solid Film with Non-uniform Thickness

## DOI:

https://doi.org/10.13052/ejcm2642-2085.29234## Keywords:

Thin solid film, structural stability, Winkler foundation, finite difference method## Abstract

The instability of a strip as a free-standing film and also a deposited film on a substrate is studied in this work. The non-uniform thickness of the film is assumed with a quadratic profile. The problem is categorized under the topic of structural stability and the eigenvalue problem corresponding with the ODE of the system is solved. For the free-standing film, the buckling loads and mode shapes are derived analytically through a closed-form solution. For the substrate-bonded film with a finite length, the uniaxial wrinkling of the film is investigated by using a series solution and a finite difference method and the wrinkling load and wrinkling pattern are characterized. Unlike the wrinkling of thin films with uniform thickness in which the wrinkles propagate along the entire span, it is shown that for the non-uniform film wrinkles are localized near the location with a minimum thickness along the length span; and the wrinkling accumulation is very sensitive to the thickness variations. Therefore, this work is expected to increase the insight into the localization of the wrinkles in thin film-substrate systems in engineering, industry and medical science.

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