Effect of substrate compliance on the global unilateral post-buckling of coatings: AFM observations and finite element calculations

The Mexican hat effect on the delamination buckling of a compressed thin film

“…The studies on a ruck in a rug [4,5] and a blister buckling [6] fall into this category, in which the substrate is (implicitly) assumed rigid and the clampedclamped boundary conditions apply. However, recent studies [1][2][3] show that the thin film delamination buckling load can be much lower than what Eq. (1) predicts, especially for the case of a hard film on a soft substrate.…”

“…When the elastic deformation of the substrate is taken into account, the film/substrate bump-like deformation profile, as shown in Fig. 1b, resembles a Mexican hat and is often analyzed by the so-called Mexican hat wavelet decomposition [2]. The boundary conditions at the delamination edges are essential for the buckling patterns [6].…”

“…In contrast, Yu and Hutchinson's analysis [3] is based on (the assumption of) nonzero displacement and rotation at the delamination edge; the clamped-clamped boundary conditions are thus violated, which leads to a lower buckling load. For a Mexican hat deformation shape, the atomic force microscope (AFM) measurement and finite element analysis (FEA) [2] show that both the displacement and the rotation are nonzero at the delamination edge. Yu and Hutchinson's free body diagram analysis is on the delamination edge and only an implicit solution can be obtained [3].…”

“…where E * f is the effective Young's modulus of the film; E * f = E f for a beam structure and E * f = E f /(1 − ν 2 f ) for a thin plate structure [1,2]; E f and ν f are the film Young's modulus and Poisson's ratio. I = ct 3 /12 (c and t are the film width and thickness, respectively).…”

“…For a freestanding thin film with clamped-clamped boundary conditions and a length of 2b, the buckling load (P c−c ) is given as follows [1][2][3] P c−c = 4π 2 E * f I (2b) 2 …”