# Cracking and Toughening Mechanisms in Nanoscale Metallic Multilayer Films: A Brief Review
## 4 Other Fracture Properties
### Adhesion Energy  
The performance of thin films all depend on adhesion between the substrate and the film [118; 119; 120; 121]. Estimation of interfacial adhesion is also critically important in NMMFs [52; 122]. In measuring NMMF adhesion, uniaxial tensile and indentation tests are generally preferred to initiate and drive delamination between film and substrate. The specific value of the interfacial adhesion energy \(\Gamma\) is related to the cracking and buckling processes as [52]:  
\[\sqrt{\frac{\delta^{*}}{2h}}=(2\alpha)^{1/4}\frac{1}{2h}[1+\sqrt{1+\frac{3}{4} \alpha(\frac{l}{2h})}^{4}]^{-1/4} \tag{21}\]  
\[\alpha=2\Gamma(\frac{2}{\pi})^{4}=\frac{4\Gamma_{cri}(1-v_{f}^{2})}{2 hE_{f}}( \frac{2}{\pi})^{4} \tag{22}\]  
where \(\delta^{*}\) is the buckle height, \(l\) is the half of the buckle width, \(\alpha\) is an experimentally calibrated parameter and \(h\) is the layer thickness. For nanoindentation, the strain energy release rate for a circular buckle due to residual and indentation stresses was calculated as [122]:  
Figure 17: Evolution of fracture toughness with \(h\) for fcc/hcp multilayers with structure transformation and fcc/bcc without structure transformation. The figure demarcates regimes that exhibit distinct processes governing the crack and fracture phenomenon in this two multilayer systems. The sketches of the micro-mechanical fracture model, which shows the interface shielding effect, is presented in each region for clarity [15].  
\[G=\frac{(1-v)H\sigma_{r}^{2}}{E}c[(1-\frac{\sigma_{c}^{2}}{\sigma_{r}^{2}})+( \frac{\sigma_{v}^{2}}{\sigma_{r}^{2}})(\frac{2\sigma_{c}}{\sigma_{v}}+\frac{1+v }{2c}-1)] \tag{23}\]  
where:  
\[\sigma_{v}=\frac{\beta EV}{a^{2}h_{f}},\sigma_{c}=\gamma E(h^{*}/a)^{2},\beta= \frac{1}{2\pi(1-v)},\gamma=\frac{k}{12(1-v^{2})} \tag{24}\]  
here, \(c=[1+0.902(1-v)]^{-1}\)[123], \(a^{*}\) is the crack radius, \(h_{f}\) is the film thickness, \(V\) is the plastic volume displaced by the indenter, \(\sigma_{r}\) is the residual stress in the film, \(E\) and \(v\) is the weighted elastic modulus and Poisson's ratio respectively.  
Attention should be paid that the adhesion energy of NMMFs is related to the length scale when the modulation period \(\lambda\) is larger than 50 nm, as seen in Figure 18. Because the curvature of buckles inevitably causes a local deformation gradient, geometrically dislocations are required to accommodate the deformation discrepancy. At \(\lambda<\) 50 nm, dislocations slide individually in-plane within the constrained layers based on the discussion in Section 3.2. The buckles have smaller dimensions and mainly produce elastic deformation. As a result, the determined adhesion energy exhibits \(\lambda\)-independence (I regime in Figure 18) and is close to the intrinsic value. At \(\lambda>\) 50 nm, dislocations can move loosely both in-plane and out-of-plane within the layers. Buckles with greater dimensions can be formed since a larger deformation gradient can be accommodated, which leads to an overestimate of the adhesion energy due to the plastic deformation. The thicker \(\lambda\) the greater the plastic deformation, and vice versa. Therefore, a remarkable \(\lambda\)-dependence of adhesion energy ismeasured, shown as regime II in Figure 18.