13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 1 = − c A h ε ε . (5) Finite element analysis (FEA) of wrinkling was performed using a two-dimensional (2D) plane-strain model in ABAQUS [26]. By a linear eigenvalue analysis, the finite element model predicted the critical strain for onset of wrinkling and the corresponding eigenmode, in good agreement with the analytical solutions in (3) and (4) when the substrate thickness is sufficiently large. To simulate wrinkle growth beyond the critical strain, a nonlinear post-buckling analysis was performed with the finite element model. Figure 2(a) shows the wrinkle amplitude as a function of the nominal strain for 1000 = s f E E and 1 3 = = s f ν ν . For comparison, the approximate analytical solution in (5) is plotted as a continuous solid line. The numerical results vary slightly as the amplitude of the initial imperfection (A0/h) varies. Using a small initial imperfection (A0/h = 10 -4 ), the numerical results compare closely with the analytical solution, with an abrupt transition at the critical strain. It is thus concluded that the analytical solution (5) is a good approximation for the wrinkle amplitude. 0 0.005 0.01 0.015 0.02 0 0.5 1 1.5 Nominal strain, ε Wrinkle amplitude, A W /h Analytical solution FEA, A 0 /h = 10-4 FEA, A 0 /h = 0.01 FEA, A 0 /h = 0.05 Figure 2. (a) wrinkle amplitude; (b) wrinkling-induced normal and shear tractions. As the wrinkle amplitude grows, the normal and shear tractions acting on the film/substrate interface increase, which may cause delamination [5, 18, 19]. An approximate formula was derived to estimate the maximum normal traction at the interface [26]: c s s s m E q ε ε ν ν − − − = 3 4 ) 4(1 2 . (6) The maximum shear traction is linearly related to the maximum normal traction, namely m s s m q ) 2(1 1 2 ν ν τ − − = . (7) By comparing to the FEA results, as shown in Fig. 2(b), it is found that the maximum normal traction by FEA follows (6) remarkably well, while the formula in (7) underestimates the maximum shear traction. The close agreement for the maximum normal traction suggests that the formula (6) may be used to estimate the critical strain for initiation of wrinkle-induced interfacial delamination. By setting the maximum normal traction equal the interfacial strength ( int ˆσ ), the critical strain for initiation of wrinkle-induced delamination is obtained [26]: 2 int 2 ˆ ) 4(1 3 4 − − = + s s s c WID E σ ν ν ε ε . (8) (a) (b)
RkJQdWJsaXNoZXIy MjM0NDE=