Post-buckling of an elastic half-space coated by double layers

Abstract

We investigate the buckling and post-buckling properties of a hyperelastic half-space coated by two hyperelastic layers when the composite structure is subjected to a uniaxial compression. In the case of a half-space coated with a single layer, it is known that when the shear modulus [Formula: see text] of the layer is larger than the shear modulus [Formula: see text] of the half-space, a linear analysis predicts the existence of a critical stretch and wave number, whereas a weakly nonlinear analysis predicts the existence of a threshold value of the modulus ratio [Formula: see text] below which the buckling is super-critical and above which the buckling is sub-critical. It is shown that when another layer is added, a larger variety of behaviour can be observed. For instance, buckling can occur at a preferred wavenumber super-critically even if both layers are softer than the half-space although the top layer would need to be harder than the bottom layer. When the shear modulus of the bottom layer lies in a certain interval, the super-critical to sub-critical transition can happen a number of times as the shear modulus of the top layer is increased gradually. Thus, an extra layer imparts more flexibility in producing wrinkling patterns with desired properties, and our weakly nonlinear analysis provides a road map on the parameter regimes where this can be achieved.

A refined model for the buckling of film/substrate bilayers

“…As the next step in deriving the reduced model, we need to express the traction vector t(x 1 , h) in terms of the displacement components at x 2 = h which, with the use of equation (22), are given by…”

“…Under the framework of nonlinear elasticity, early studies include the linear analyses by Dorris and Nemat-Nasser [10], Shield et al [11], Ogden and Sotiropoulos [12], Bigoni et al [13], Steigmann and Ogden [14], and the non-linear post-buckling analysis by Cai and Fu [15]. More recent studies have addressed the phenomenon of period doubling [16], effects of pre-stretching in the half-space [17], compressibility [18,19], multi-layering [20][21][22], growth [23][24][25], and magnetorheology [26]. There also exists a large body of literature based on various approximate models for the layer and substrate (see, for example, [27][28][29][30] and the references therein).…”

Understanding the role of interfacial mechanics on the wrinkling behavior of compressible bilayer structures under large plane deformations

“…When two materials are bonded at an interface, unlike what is assumed in the great majority of the works on bilayer structures thus far, the bonding is subject to various imperfections, hence creating an interphase layer between the two [30]. Stafford et al [4] accounted for this interphase layer, motivated by the different properties a film on a substrate displays along its depth, and have paved the way for many contributions on trilayer systems [9,31–35], also finding applications serving as an adhesive layer [36] or for strain isolation [37]. Bifurcation of elastic solids with sliding interfaces has been recently investigated by Bigoni et al [38].…”

A refined model for the buckling of film/substrate bilayers

“…Under the framework of nonlinear elasticity, early studies include the linear analyses by Dorris & Nemat-Nasser (1980), Shield et al (1994), Ogden & Sotiropoulos (1996), Bigoni et al (1997), Steigmann & Ogden (1997), and the nonlinear post-buckling analysis by Cai & Fu (1999). More recent studies have addressed the phenomenon of period doubling (Fu & Cai, 2015), effects of pre-stretching in the half-space (Hutchinson, 2013), compressibility (Liu & Dai, 2014;Cai & Fu, 2019), multi-layering (Cheng et al, 2014;Wang et al, 2020;Zhou et al, 2022), growth (Alawiye et al, 2019(Alawiye et al, , 2020Liu et al, 2020b), and magnetorheology (Rambausek & Danas, 2021). There also exists a large body of literature based on various approximate models for the layer and substrate; see, e.g., Chen & Hutchinson (2004), Huang et al (2005), Audoly & Boudaoud (2008), Zhao et al (2015), and the references therein.…”