Influence of local wrinkling on membrane behaviour: A new approach by the technique of slowly variable Fourier coefficients

A new Fourier-related double scale analysis for wrinkling analysis of thin films on compliant substrates

“…But at least two functions U 0 ðx; zÞ and U 1 ðx; zÞ are needed to describe nearly periodic patterns: U 0 ðx; zÞ can be identified to the mean value while U 1 ðx; zÞ represents the envelope or the amplitude of the spatial oscillations, see Fig. (2). These two harmonics of the Fourier series have been used to describe instability phenomena in beams on Winkler foundation [21], sandwich beams [31,32] and thin metal films [23,33,34], and rather good estimations in wrinkling patterns and critical loads have been observed.…”

“…Based on an efficient multiscale approach established by Damil and Potier-Ferry [20,21] that exploits such periodic nature and the effective CUF model of Yang http://dx.doi.org/10.1016/j.compstruct.2016. 10.062 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.…”

“…In the framework of the multi-scale approach (often referred to as ''Fourier series with slowly varying coefficients"), the unknowns to be solved in Yang et al [1] are expanded by the Fourier series, which leads to the Fourier coefficients as the new unknowns varying much more slowly than the former unknowns (we address the former unknowns as ''microscopic" and the latter ''macroscopic" in this paper). Usually using rather coarse spatial meshes is sufficient to describe the slowly varying envelops, therefore computational efficiency can be improved significantly [21][22][23]. This approach is firstly applied to study the instability phenomena in beams on Winkler foundation [20,21] with a most simplified macroscopic model, in which only the zero harmonic (mean field) and the first order harmonics (envelops) are taken into account.…”

“…Usually using rather coarse spatial meshes is sufficient to describe the slowly varying envelops, therefore computational efficiency can be improved significantly [21][22][23]. This approach is firstly applied to study the instability phenomena in beams on Winkler foundation [20,21] with a most simplified macroscopic model, in which only the zero harmonic (mean field) and the first order harmonics (envelops) are taken into account. Results showed that the macroscopic model has two advantages compared to Landau-Ginzburg technique: (1) not only the bifurcation point but also the post-buckling path can be captured and (2) the coupled global and local instability patterns can be incorporated and characterised, see Liu et al [22].…”

The interactive bending wrinkling behaviour of inflated beams

“…Consequently, there is a need to develop theoretical and experimental methods for describing the instability phenomena that couples global and local wrinkling characteristics. A Fourier series method, which corresponds to the periodic and sinusoidal characteristics of wrinkles correctly, is able to account for instability phenomena, such as bending instabilities of circular tubes [22,23], buckling of sandwich structures [24] or of a Winkler foundation beam [25].…”