Buckling And Post-Buckling Analysis Of An Elastic Substrate Coated By Double Layers

In this paper,we have investigated the buckling and post-buckling properties of an elastic substrate coated with two layers.At macro-scales,buckling usually undermines a structure’ s integrity and should be avoided.When a structure is sensitive to imperfections,any imperfection,material or geometrical,will significantly reduce the critical load at which bifurcation takes place.Thus,from a practical point of view,it is important to find the parameter regime in which the structure is imperfection sensitive.At micrometre and submicrometre scales,robust wrinkling patterns can be harnessed to serve useful purposes.Since super-critical bifurcation can be observed or realized in practice,results from our weakly nonlinear analysis provide a road map on how to choose a variety of combinations of material parameters to achieve robust wrinkling patterns.The two-layers/elastic substrate structure is composed of two hyperelastic layers bonded to a hyperelastic half-space,and the composite structure is subjected to a uniaxial compression.In the case of a half-space coated with a single layer,a linear stability analysis shows that the composite structure is less stable than the half-space if r<1 and more stable if r>1,where r is the ratio of the shear modulus of the half-space to that of the layer.When the layer is stiffer than the half-space(r<1),there exists a critical buckling mode number corresponding to a minimum(critical)compression.Compared with the single-layer film/elastic substrate structure,the addition of the second film makes the bifurcation condition exhibit two interesting features.First,even if both layers are softer than the half-space,the stretch can still exhibit a maximum at a finite wavenumber.Second,there is a range of thickness ratios for which multiple stretch maxima exist and mode switching becomes possible as a material parameter is varied.This is different from the situation of a single layer where a stretch maximum is only possible when the layer is stiffer than the half-space and no mode switching is possible.Through a weakly nonlinear analysis,we derive the amplitude equation for a single near-critical mode,and the coefficient of the cubic nonlinear term in the amplitude equation determines whether the bonded structure is sensitive to imperfections.In the case of a half-space coated with a single layer,a weakly nonlinear analysis predicts the existence of a threshold value of the modulus ratio r≈0.57,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 supercritical 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 the results of our weakly nonlinear analysis can be used to achieve this goal within a suitable parameter domain.