My main topic of research over the past 5 years has been the application of the variational approach to the fracture of thin films. Thin-film fracture is a multi-dimensional problem by nature that involves a vast number of fracture phenomena such as the creation of cells and patterns, kinking and coalescing of cracks, etc. The successful simulation of these phenomena has been impossible until recently. Classical fracture mechanics, though powerful in predicting the breaking loads in simple fracture topologies, lacks the necessary generalization to simulate all these effects.
The curse of thickness!
When dealing with thin films, (even before considering fracture) one has to decide how to represent the thin film:
- Use a brute force method to simulate the 3-D domain using very fine discretization to resolve the film thickness.
- Use a 2-D slice of the 3-D domain; this would introduce many ad-hoc assumptions.
- Perform a dimension reduction finding an asymptotic expression for the thin-film with respect to its thickness.
My research’s focus has been mainly on the third approach. Starting from 3-D elasticity
$$\mathcal{P}(u):=\frac{1}{2}\int_{\Omega\setminus\Gamma}Ae(u):e(u)\,dx$$
and using the right scaling hypothesis, one can derive different models for thin films. Two of these models are:
- in-plane model: $$\mathcal{P}(u):=\frac{1}{2}\int_{\Omega}Ae(u):e(u)\,dx+\frac{K}{2}\int_{\Omega}(u-u_{0})^{2}\,dx$$
- out-of-plane model: $$\mathcal{P}(u):=\frac{D}{2}\int_{\Omega}((\partial_{xx}u+\partial_{yy}u)^{2}-2(1-\nu)(\partial_{xx}u\partial_{yy}u-(\partial_{xy}u)^{2}))\,dx+\frac{K}{2}\int_{\Omega}(u-u_{0})^{2}\,dx$$
the details of which you can read in Mesgarnejad et al. 2013 and Leon-Baldelli et al.
The Variational Approach to the Fracture of Thin Films
Use of the variational approach to fracture mechanics comes naturally when the elastic potential is known. If we define the potential energy of the system as:
$$\mathcal{E}(u):=\mathcal{P}(u)+G_{c}\mathcal{H}^{1}(\Gamma)$$
then one can easily follow Ambrosio and Tortorelli’s approximation. For a smeared fracture field $$\alpha$$ and characteristic length $$\epsilon$$ we can write:
$$\mathcal{E}_{\epsilon}(u):=\mathcal{P}_{\epsilon}(u,\alpha)+\mathcal{S}_{\epsilon}(\alpha)$$
where:
$$\mathcal{P}_{\epsilon}(u,\alpha):=\frac{1}{2}\int_{\Omega}\alpha^{2}Ae(u):e(u)\,dx$$
and
$$\mathcal{S}_{\epsilon}(\alpha):=\frac{3G_{c}}{8}\int_{\Omega} \frac{1-\alpha}{\epsilon}+\epsilon |\bigtriangledown \alpha|^{2}\,dx$$
my FD code and FE code both implement the above formulation, while my FE code also includes many more elastic potentials. Both codes are private, but you can always contact me to obtain access to them.