The ratio between the combined mode I/II energy release rate and the mode III energy release rate changes as the shear to normal stress ratio varies along the crack front. Denoting by ω the angle, which has been tabulated as a function of the elastic mismatch between the plate and the substrate in Suo and Hutchinson [9], then the parameters k and σc are given by ( ) ( ) 3 2 2 1/2 1c c 2 2 2 2 k = (1- ) 1+( 1)sin 2EG = (1- )h 1+( 1)sin λ ν λ − ω σ ν λ − ω (4) In the following, it is assumed that ν = 1/3 so that the usual mode independent Griffith fracture criterion corresponds to k = 3 , while k = 0 corresponds to a fracture criterion independent of mode 3. In [6] a value close to k = 1 gave best agreement with experimental results for a polyamide/glass system, while the criterion applied in [8] and [3] for spot welded sheet metal corresponded to a value close to k = 2 . NUMERICAL RESULTS FOR BOND STRENGTH In the calculations below, the bond shape (sketched in Fig. 1) is taken to be circular with diameter d , and its centre is located at the centre of a square plate of in-plane dimension αd with α specified below. The plate is loaded by a constant normal traction, σ, in the x1-direction along a side parallel to the x2axis, and it is free along the other side parallel to the x2-axis. The plate sides parallel to the x1-axis are both free in the calculations below or symmetry conditions are specified on both sides so that the interaction between bonds in an infinite periodic array of identical bonds is modelled. Symmetry with respect to the x1-axis is assumed and the results below are presented for the lower half of the crack front, only. The stress state in the plate is solved numerically by the finite element method using typically 288 x 50 planar elements each consisting of two linear displacement triangles. The mesh is graduated with decreasing element sizes towards the interface crack for the best possible accuracy along the crack front. In Fig. 2, the variation along the bond edge of F defined in (3) is shown in the case k = 2 with k defined in (4). The plate edges parallel to the x1-axis are free in these calculations. By the fracture criterion (3), the most critical location along the bond edge, Θ = Θc, is at the peak value of F , which is denoted Fp so that F(Θc) = Fp. The value of the applied traction, σ = σ0, required to initiate crack propagation is c 0 p = F σ σ (5) with σc given in (4). The angle Θ and the aspect ratio, α, defined as the ratio between the in-plane dimensions of the square plate and the bond diameter, are indicated in the inset in Fig. 2. By Fig. 2 it is seen that Fp is significantly increased due to the presence of the free edge as in particular this increases the shear stress along the crack front close to the free edges. The applied traction is scaled by the aspect ratio, α, in Fig. 2 so that the resulting force acting on the bond is independent of the size of the plate. By (5) an increase in the peak value of F is equivalent to a decrease in the bond strength. Furthermore by (3) in the case k = 2 it is seen that the fracture criterion is more sensitive to shear stresses along the crack front than to normal stresses. Calculations similar to those in Fig. 2 are presented in Fig. 3 but now with a value k = 1 so shear stresses contribute relatively less to the fracture criterion. It can be seen, as one would expect, that the strength of the bond in cases where the fracture criterion (3) applies with k = 1 is higher than bonds with
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