Impact of ridge cracking on the morphology of buckle-delamination

Highlights

• Nonlinear solutions for the morphology of straigh-sided ridg-cracked blisters and TC-like ridge-cracked blisters are obtained.

• The presence of ridge crack drastically influences the morphology of straight-sided ridge-cracked blisters and TC-like ridge-cracked blisters.

• The asymmetry of the cross-sectional profile of the TC-like ridge-cracked blister is mainly determined by the deviation of the position of the ridge crack and the asymmetry enhanced by the amplitude of the compressive stress is suppressed.

Abstract

Ridge-cracked blisters form when ridge cracking occurs in a biaxially compressed film on a relatively rigid substrate during buckle-delamination. The observation of telephone-cord (TC) blister with wavy ridge crack indicates that the ridge-cracked blister is not straight-sided as it used to be. How ridge cracking influences the profile of buckle-delamination remains unclear. Based on Föppl–von Kármán (FvK) nonlinear plate theory, we derive an improved analytical solution for a better description of the cross-sectional profile of the straight-sided blister with straight-sided ridge crack through considering the effect of hinge position caused by the ridge crack. Following a similar procedure in the case of the straight-sided ridge-cracked blister, an approximate solution of FvK plate in curvilinear coordinates is further obtained for the cross-sectional profile of the TC ridge-cracked blister. The results show that the position of the ridge crack significantly modifies the asymmetry, the shape, and the maximum deflection of the cross-sectional profile of the resulting blister.

Introduction

Residual stress inevitably appears in film–substrate structures during their processing and lifetimes [1]. The residual compressive stress in a film on a relatively rigid substrate usually drives the film to buckle and delaminate away from the substrate [2], leading to the formation of various buckle-delamination patterns, including but not limited to straight-sided, circular, telephone-cord (TC), and network-like shape [3], [4]. Investigation of buckle-delamination morphology becomes a topic of considerable research interest because it not only provides excellent examples of mechanical instability driven self-assembly but also helps us improve the reliability of such structures by suppressing buckle-delamination instability [5]. However, due to the strong nonlinearity of post-buckling analysis based on Föppl–von Kármán (FvK) theory with fixed delamination boundary, only some closed forms for critical conditions at the onset of buckling instability are possible [6], [7], [8] and quite a few analytical solutions for the post-buckling profile can be derived. It is well known that the shape of the straight-sided buckle-delamination has been well described by an elegant solution of a clamped FvK plate [2], [6], while there are only different approximate solutions for the cases of circular [2], [8], and telephone cord buckle-delamination [9], [10], [11], [12]. At the same time, various numerical simulations techniques such as finite element method (FEM) [13], [14], [15], [16], [17], gradient-flow kinetic approach [18], and triangular lattice model [19] have also been successfully developed to probe growth and equilibrium profile of complex buckling-delamination patterns [13], [14], [15], [16], [17], [18], [19].

The strain energy in compressed film–substrate structures can be released not only by buckle-delamination but also by other activated stress relaxation modes. Plastic folding of gold thin films on silicon substrates significantly modifies the profile of a circular blister [20]. The step structure due to substrate plasticity leads to an asymmetric profile of blisters in a Nickel film on LiF single crystal substrate [21], [22], [23]. Interface sliding is found to increase the maximum buckle height and facilitates the stable growth of the buckle-delamination [24], [25]. Atomistic simulations combined continuum model clearly show that the interface sliding may be induced by interface plasticity via emission of dislocations [26], [27], [28]. A theoretical effort shows that sliding at interface suppresses the transition of a straight-sided blister to the telephone cord buckle [29]. In particular, when the brittle film is under compression, experimental observations demonstrate that a local bending induced tensile strain in the compressive film can cause the formation of ridge cracking [30], [31], [32], [33], [34], [35]. Ridge cracking has also been found to occur in the transverse direction in brittle coating on a polymer substrate [36], [37], [38], [39], [40], [41], [42], [43]. Up to now, most frequently observed ridge-cracked buckles are straight-sided. The energy releasing rate of the straight-sided ridge-crackled blister on a flat [44], [45], [46] or curved substrate [47] is estimated respectively. An analytical investigation evidenced by experimental observations demonstrates that ridge cracking promotes symmetric secondary buckling instability mode of the straight-sided ridge-crackled blister and favors bubble-like blisters [48]. However, both experiments and FEM simulations indicate that the ridge-cracked buckles are unnecessarily straight-sided as it used to be in the previous study [47], the telephone cord buckle with wavy ridge cracking is possible [34], [49]. It remains elusive how ridge cracking impacts on the morphology of buckle-delamination, especially when the ridge crack is not straight. One of the main purposes in this study is to elucidate the influence of the ridge crack on the profiles of the straight-sided blisters and the TC blisters. We revisit an analytical solution for the profile of the straight-sided blister with straight-sided ridge crack based on FvK nonlinear plate theory after considering the effect of hinge position caused by the ridge crack. We also try to derive an approximate solution of the FvK plate in curvilinear coordinates for the cross-sectional profile of the TC ridge-cracked blister by using the ring-like blister approximation.

In Section 2 we first present our experimental observations of the ridge-cracked TC blister. In Section 3 we present formulations to derive solutions of the FvK plate in Cartesian coordinates and curvilinear coordinates for the cross-sectional profiles of the straight-sided ridge-cracked blister and the TC ridge-cracked blister, respectively. In Section 4 we discuss the effect of ridge crack on the cross-sectional profile of the resulting blister. In Section 5 we summarize the main conclusion.