13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- On propagation of interface cracks parallel to free boundaries in relation to delamination of multilayered coatings Konstantin B. Ustinov*, Robert Goldstein IPMech RAS, 119526, Russia * Corresponding author: ustinov@ipmnet.ru Abstract. The problem of a crack propagation in multilayered coating either between the individual layers, or between the whole package forming the coating and substrate (possibly also non-uniform) is considered. According to the model the delaminated part of the coating is treated as a plate with a special attention given to the boundary conditions, which are supposed to be of the elastic clamping type, which means that the angle of rotation at the clamping point is related to the acting bending moment at that point. The generalized model of elastic clamping is also considered in the frame of which it is supposed that components of displacements and angle of rotation at the clamping point are related to the acting total force and total moment by means of 3x3 matrix of elastic coefficients. Several variants of analytical models to determine the coefficients of this 3x3 matrix were considered and discussed. The case of non-uniform (multi-layered) coating is considered with a special attention paid to analysis of the influence of the relative dimensions involved, such as the individual layer and the multilayer coating thicknesses, distance from the crack to the free surface, crack length. Using the obtained solution the problem on buckling of the delaminated coatings is addressed. It has been shown that for a wide range of parameters even for anisotropic and multilayered structures the ratio of the obtained critical stress and the critical stress of a rigidly clamped plate is determined by a single nondimensional parameter, which is a combination of the elastic constants of the coating and substrate, and the ratio of the delamination length to the coating thickness. The analytical results correlate well with the results obtained using the numerical calculations. Keywords. Interface crack, Multilayered coatings, Delamination, Elastic clamping, Buckling 1. Introduction Problems related to propagation of interface cracks have been investigated widely due to importance for industry. Lately they appealed additional interest owing to applications in micro- and nano-electronics, biology, medicine. The problems were addressed by many authors by using both analytical and numerical approaches e.g. [1-10]. In [2], [3], in particular, the problem of stability loss of coating delaminated from rectilinear rigid substrate was solved. The critical stress, magnitude of deflection [2], [3], and energy release rates along and across the delamination front were found. The delaminated part of coating was considered as a clamped plate, which corresponds to rigid substrates. However, detailed studies yield that, from the one hand, the condition of rigid clamping are not satisfied exactly even for absolutely rigid substrates [5], and, from the other hand, on the base of numerical calculations it is said [6], [7] that even for substrates tree times softer then coating the condition of rigid clamping yields acceptable errors. In our opinion the last statement has to be accepted with cautions, because the influence of delamination size, importance of which being confirmed by other studies [5], [8], was not investigated in [6]. Influence of the substrate rigidity was studied by [4-6], however without paying attention to the influence of the delamination width to the critical stress. The analysis of influence of both substrate compliance and delamination width on the critical stress was performed in [5]. The delamination was modeled by a plate, with boundary condition corresponding to generalized elastic clamping, i.e. magnitudes of tangential displacements and gradient of deflection of the clamped edges were supposed to be proportional to tangential force and bending moment acting at the points of clamping. The coefficients of elastic clamping depending on two mentioned parameters were
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