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# Discussion of fracture paper #27 - Phase-field modelling of cracks and interfaces

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*^(1-2*s*). 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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## Does ESIS have a twitter account?

I have tweeted a link to these ESIS entires on fracture: https://twitter.com/zhigangsuo/status/1319060444758376453. Does ESIS have a twitter account?

As an experiment, I have tweeted a thread called

One Hundred Years of Toughness

https://twitter.com/zhigangsuo/status/1290570290657996800

## Dear Zhigang,

Dear Zhigang,

Yes, you can find ESIS on Twitter: @ESIS_web

Dear Per,

It is remarkable how phase field methods can revolutionise many areas, including fracture mechanics but also others such as corrosion modelling - as you have pioneeringly demonstrated. Its ability to (naturally) predict features such as crack deflection and branching make it a very suitable technique to gain insight into problems such the case of a crack impinging on an interface. There have been a number of papers using phase field to gain complementary insight to that of He and Hutchinson [1-3].

In my group, we have also embraced this success with relish, and tried to contribute to the community. For example, by extending phase field formulations to deal with fracture in hydrogen-embrittled solids [4-6], functionally graded materials [7] and shape memory alloys [8].

Some aspects of the formulation might require further research. There has been some discussion on the lack of a explicit connection with the material strength and the implications for crack nucleation - an interesting paper by Oscar Lopez-Pamies was posted in iMechanica recently (https://imechanica.org/node/24284).

Regarding your question (implications of the length scale), I would argue that all fracture/damage models that are based on a toughness or critical energy release rate have embedded a material length scale: L~GcE/sigma_c^2=Kic^2/sigma_c^2. Analogous to what Rice calls the material's characteristic length or Zhigang refers to as the fractocohesive length. I found the paper by Tanné et al. a nice piece of work in the subject [9].

Best,

Emilio Martínez-Pañeda

PS: If someone is interested in playing around with the method, I have released all my phase field codes for Abaqus and FEniCS, including phase field fatigue [10]: https://www.imperial.ac.uk/mechanics-materials/codes/

[1] https://www.sciencedirect.com/science/article/pii/S0045782516317066

[2] https://www.sciencedirect.com/science/article/pii/S0045782518305772

[3] https://www.sciencedirect.com/science/article/pii/S0045782519300192

[4] https://www.sciencedirect.com/science/article/pii/S0045782518303529

[5] https://www.sciencedirect.com/science/article/pii/S0010938X19316464

[6] https://www.sciencedirect.com/science/article/pii/S0022509620303276

[7] https://www.sciencedirect.com/science/article/pii/S135983681930229X

[8] https://arxiv.org/pdf/2010.04390.pdf (just accepted in CMAME)

[9] https://www.sciencedirect.com/science/article/pii/S0022509617306543

[10] https://www.sciencedirect.com/science/article/pii/S0167844219305580

## Dear Emilio,

Dear Emilio,

Thanks for the well-informed comment. I appreciate your mentioning of Oscar Lopez-Pamies's blog. From there I learned about the 1998 paper by Francfort and Marigo. When we wrote our first paper on the subject in 2011 we thought we were first but that was soon revised. I didn't know about the early Francfort and Marigo paper until now, though.

I like your papers and not the least the most recent "just accepted", where you use phase field modelling to understand the evolving concentration of martensite and the propagation of cracks. If I understand it correctly the martensite/austenite composite is treated as homogeneous, but I guess it wouldn't be a big theoretical step to include a third phase to separate the martensite and austenite phases. I guess the numerical treatment will provide some challenges.

You are right regarding the material length scale connected to the fracture toughness. With the stress level at which the fracture processes operates, the toughness gives a characteristic length, but it doesn't help until it is allowed to play a role in the model. If the fracture processes are referred to a point, a sharp crack tip, I don't think the crack can penetrate an interface to a stiffer material. The required remote load becomes unlimited for any non-zero fracture toughness. A somewhat paradoxical result.

The phase-field model has a length scale, given by the square root ratio of a molecular mobility coefficient vs. a coefficient defining the energy barrier of a surface energy potential. The length scales with the thickness of the transition region between the phases. The square root product of the mentioned two coefficients gives the surface energy.

Replacing the crack tip stress singularity with a cohesive zone, or a box model, allows the crack to penetrate the interface to a stiffer material. I know from own experience that phase field models of corrosion pitting, does the trick. Also we see that the blur crack tip in the present paper works. I could not find any comments in the present paper regarding the expected decrease of the crack energy release rate or necessity to increase the remote load to maintain a constant crack growth rate. If there were a dip of the crack tip driving force versus the remote load it would give a possible comparison with the length scale L, as defined by you.

Best regards, Per

## Dear Per,

Dear Per,

Thank you for your kind reply. In addition to the 1998 paper by Francfort and Marigo I would recommend the 2000 work by Bourdin, Francfort and Marigo ("Numerical experiments in revisited brittle fracture"), where details of the phase field framework, including Gamma convergence and the Ambrosio and Tortorelli functional, are introduced.

Regarding our recent paper on phase field fracture (and fatigue) of Shape Memory Alloys; an internal variable - the martensite volume fraction - is used to determine which regions correspond to martensite, austenite or the transformation region.

I agree with your statement, the fracture energy-related length scale only plays a role if incorporated into the model. In that sense, I was thinking of the role of the phase field length scale (which can be tought as proportional to this fracture energy-related length scale) versus other models that include such a fracture length scale: cohesive zone models, non-local continuum damage models, etc.

Kind regards,

Emilio Martínez-Pañeda

## More references

Dear Per, Emilio, and others,

The 2001 paper is a bit dated now. Our 2008 long article / book "The variational approach to fracture" (DOI: 10.1007/s10659-007-9107-3), although also dated by today's standard, provides a good view of the foundations and theory of a field that is about 25 years old, even though many have discovered it recently.

Blaise

## Since we are in the business

Since we are in the business of advertising codes, let me also pitch in mef90/vDef, the open source parallel implementation that I have used for almost all of my work in the field in the last 25 years or so.

Blaise

## Dear Professor Ståhle,

Dear Professor Ståhle,

Thank you for considering our paper [1] in your blog entry and your

interesting thoughts!

At first, it seems not trivial to compare our simulations to the

considerations of Zak and Williams [2]. In phase-field models for

fracture, there is no real stress singularity anymore due to the

regularization of the crack surface with the length scale l_c.

Furthermore, the phase-field method comprises the specification of the

fracture toughness, which is a key ingredient in controlling the onset

of fracture. In other words, the value of the energy release rate at

which crack growth takes places is à priori prescribed. In addition to

this, we investigate cases with heterogeneous fracture toughnesses

compared to the publication of He and Hutchinson [3].

Nevertheless, we try to answer your expectation

"The expected outcome I think would be that the crack growth energy

release rate increases for a crack in a stiffer material and decreases

for a crack in a weaker material."

Let us reformulate your expectation a bit and replace "crack growth

energy release rate" with "tendency". Now it reads "[...] the tendency

towards the interface increases for a crack in a stiffer material and

decreases for a crack in a weaker material." Now, looking at Fig. 13 in

[1], we exactly observe what is stated! This is also in line with your

first considerations, where the crack is further away from the

interface than (any) of the introduced length scales. Therefore, an

influence of the heterogeneity of the fracture toughness can be

excluded. This is good news, because the phase-field method does, what

LEFM predicts :-)

Investigating heterogeneities and their manifold influence on crack

propagation, crack delay or increase of overall cracking resistance is

indeed a topic of great interest and subject to ongoing research of us

[4,5] but also of many other research groups such as [6-10]. The list

is not exhaustive and further work of these and other research groups

can be found in the references of the cited literature.

Warm regards,

Arne Hansen-Dörr and Franz Dammaß

[1]: Hansen-Dörr, Dammaß, de Borst and Kästner, Phase-field modeling of

crack branching and deflection in heterogeneous media, Eng Fract Mech

232: no. 107004, 2020.

[2]: Zak and Williams, Crack point stress singularities at a bi-

material interface, GALCIT SM 62-1, 1962.

[3]: He and Hutchinson, Crack deflection at an interface between

dissimilar elastic materials, Int J Solids Struct 25(9): pp. 1053-1067,

1989.

[4]: Hansen-Dörr, Hennig, de Borst and Kästner, Phase-field modelling

of interface failure in brittle materials, Comput Method Appl M 346:

pp. 25-42, 2019.

[5]: Hansen-Dörr, Brummund and Kästner, Phase-field modeling of

fracture in heterogeneous materials: jump conditions, convergence and

crack propagation, Arch Appl M, 2020.

[6]: Hossain, Jsueh, Bourdin and Bhattacharya, Effective toughness of

heterogeneous media, J Mech Phys Solids 71: pp. 15-32, 2014.

[7]: Kuhn and Müller, Phase field modeling of interface effects on

cracks in heterogeneous materials, PAMM 19(1), 2019.

[8]: Nguyen, Yvonnet, Zhu, Bornert and Chateau, A phase field method to

simulate crack nucleation and propagation in strongly heterogeneous

materials from direct imaging of their microstructure, Eng Fract Mech

139: pp. 18-39, 2015.

[9]: Schneider, Schoof, Huang, Selzer and Nestler, Phase-field modeling

of crack propagation in multiphase systems, Comput Method Appl M 312:

pp. 186-195, 2016.

[10]: Lebihain, Leblond and Ponson, Effective toughness of periodic

heterogeneous materials: the effect of out-of-plane excursions of

cracks, J Mech Phys Solids 137: no. 103876, 2020

-----------

Posted by ESIS on behalf of Drs. Hansen-Dörr and Dammaß / PS

## Dear Arne Hansen-Dörr and

Dear Arne Hansen-Dörr and Franz Dammaß,

You are right when you say your problem does not have a stress singularity. That, you have in common with everyone who is interested in the interaction between fracture processes and anything that compromises the process region autonomy. Point shaped process regions cannot be used for cracks that are short, initiate from notches, or as in your case interacts with bi-material interfaces.

In the paper you have a case, where the ratio of the phase-field layer thickness,

lc, is only 1.5% of the width of the layer with the initial crack. The ratios crack l. vs. distance crack tip to interface and the same distance vs. phase-field layer both simultaneously become around 8 at its best. If it is enough to reproduce Zak and William's result I am not sure, but some correlation I guess should be expected. Never the less, the phase-field thickness,lc, is the key parameter and as you pointed out, it is important to get right.Thanks again for an interesting paper and best regards,

Per