Landau and Ginzburg formulated a theory that includes the free energy of phases, with the purpose to derive coupled PDEs describing the dynamics of phase transformations. Their model with focus on the phase transition process itself also found many other applications, not the least because many exact solutions can be obtained. During the last few decades, with focus on the bulk material rather than the phase transition, the theory has been used as a convenient tool in numerical analyses to keep track of cracks and other moving boundaries. As a Swede I can’t help myself from noting that both of them received Nobel prizes, Landau in 1962 and Ginzburg in 2003. At least Ginzburg lived long enough to see their model used in connection with formation and growth of cracks.
The Ginzburg-Landau equation assumes, as virtually all free energy based models do, that the state follows the direction of steepest descent towards a minimum free energy. Sooner or later a local minimum is reached. It doesn’t necessarily have to be the global minimum and may depend on the starting point. Often more than one form of energy, such as elastic, heat, electric, concentration and more energies are interacting along the path. Should there be only a single form of energy the result becomes Navier’s, Fourier’s, Ohm’s or Fick’s law. If more than one form of energy is involved, all coupling terms between the different physical phenomena are readily obtained. By including chemical energy of phases Ginzburg and Landau were able to explain the physics leading to superfluid and superconducting materials. Later by mimicking vanished matter as a second phase with virtually no free energy we end up with a model suitable for studies of growing cracks, corrosion, dissolution of matter, electroplating or similar phenomena. The present paper
describes a usable benchmarked numerical model for computing crack growth based on a phase field model inspired by the Ginzburg-Landau’s pioneering work. The paper gives a nice background to the usage of the phase field model with many intriguing modelling details thoroughly described. Unlike in Paper #11 here the application is on cracks penetrating interfaces. Both mono- and bi-material interfaces at different angles are covered. It has been seen before e.g. in the works by He and Hutchinson 1989, but with the phase field model results are obtained without requiring any specific criterium for neither growth nor branching nor path. The cracking becomes the product of a continuous phase transformation.
According to the work by Zak and Williams 1962, the stress singularity of a crack perpendicular to, and with its tip at, a bimaterial interface possesses a singularity r^-s that is weaker than r^-1/2 if the half space containing the crack is stiffer than the unbroken half space. In the absence of any other length scale than the distance, d, between the interface and the crack tip of an approaching crack, the stress intensity factor have to scale with d^(1/2-s). The consequence is that the energy release rate either becomes unlimited or vanishes. At least that latter scenario is surprisingly foolish whereas it means that it becomes impossible to make the crack reach the interface no matter how large the applied remote load is.
In the present paper the phase field provides an additional length parameter, the width of the crack surfaces. That changes the scene. Assume that the crack grows towards the interface and the distance to the interface is large compared with the width of the surface layer. The expected outcome I think would be that the crack growth energy release rate increases for a crack in a stiffer material and decreases it for a crack in a weaker material. As the surface layer width and the distance to the interface is of similar length the changes of the energy release rate does no more change as rapid as d^(1-2s). What happens then, I am not sure, but it seems reasonable that the tip penetrates the interface under neither infinite nor vanishing load.
I could not find any observation of this mentioned in the paper so this becomes just pure speculation. It could be of more general interest though, since it could provide a hint of the possibilities to determine the critical load that might lead to crack arrest.
Comments, opinions or thoughts regarding the paper, the method, or anything related are encouraged.
Per Ståhle
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