Issue 48

Y. Sun et alii, Frattura ed Integrità Strutturale, 48 (2019) 648-665; DOI: 10.3221/IGF-ESIS.48.62 650 F ORMATION MECHANISMS OF THE WRINKLES AND CRACKS hen a metal film is deposited onto a soft substrate, tensile stress and compressive stress are often stored in the film during the deposition and cooling processes, respectively. In order to release the stored stresses, the film and the substrate are usually deformed at the same time. When the deformation is beyond a critical value, the cracks and wrinkles will form in the film to reduce the elastic strain energy of the system. The cracks and wrinkles are widely found in nature and engineering materials. For example, dehydrated fruits and aging skin can cause the formation of wrinkles [18,76,77], and fatigue failure of metal parts can produce progressive cracks [78-91]. Understanding the formation mechanisms of the wrinkles and cracks has a great significance for predicting the material failure. When the film/substrate is subjected to the compressive stress, the film is placed in an unstable state with high energy. In order to balance the system, the film and the substrate will deform to release the elastic strain energy. Generally, the film has a larger Young's modulus relative to the substrate and is less prone to in-plane deformation. Furthermore, because the film has a small thickness and bending stiffness, it usually generates out-of-plane deformation. When the deformation is beyond a critical value, the wrinkles will be formed to release the compressive stress. Many previous studies have shown that the critical strain  c for the outset of wrinkling obeys the following scaling law [14,76,92-94]:                   2 3 2 2 3 1 1 = 4 1 s f c f s E E (1) In the formula, E is the Young’s modulus,  is the Poisson’s ratio, the subscripts f and s refer to the film and substrate respectively. The two most important parameters wavelength  and amplitude A for charactering the wrinkles are [14,76,92-95]:                    1 3 2 2 1 2 3 1 f s s f E h E and           1 2 1 c A h (2) h is the film thickness and  is the applied strain. Eqns. (1) and (2) are classical critical wrinkling condition and characteristic scale of the film under the low deformation regime, respectively. Their application condition is that the stiff film adhered to an isotropically elastic substrate with infinite thickness cannot debond from the substrate. It can find from the Eqn. (1) that the critical wrinkling strain depends on the Young’s modulus of the film and substrate. In the case that the substrate is softer and the film is harder, the critical wrinkling strain is smaller, and the film is more prone to form wrinkles. From the Eqn. (2), it is clear that the characteristic wavelength of the wrinkles is dependent on the film thickness and the Young’s modulus ratio of the film and the substrate. By energy analysis, Cerda et al . [76] showed that small wavelength will increase the bending energy of the film significantly, large wavelength significantly increase the deformation energy of the substrate, and the competition between the two energies will result in an equilibrium wrinkle wavelength. It can also find that the wrinkle amplitude depends on the film thickness and the applied strain. If the applied strain increases, the amplitude will increase. When the amplitude increases to a critical value, the wrinkles will collapse into the folds [22]. For the metal film, it is difficult to withstand tensile stress because it is quite brittle. According to the Griffith classical criterion, crack will propagate when the elastic energy stored in the film released per unit length overcomes the fracture energy [11],   2 f he G h with       2 1 t f f e h E (3) where e is the elastic energy in the film per unit surface,  t is the tensile stress, f G is the fracture energy per unit area and  is a constant that depends on the mismatch in elastic properties between the film and the substrate. Eqn. (3) shows that the crack propagation depends not only on the tensile stress but also on the film thickness. Below a critical thickness       2 2 1 f f t f h G E the film is free of cracks [10,11,96,97]. Above the critical thickness, the cracks appear. When the W