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# Delamination of stiff islands patterned on stretchable substrates

As another celebration of March Journal Club of Mechanics of Flexible Electronics, this paper has just been submitted.

**Abstract **

In one design of flexible electronics, thin-film islands of a stiff material are fabricated on a polymeric substrate, and functional materials are grown on these islands. When the substrate is stretched, the deformation is mainly accommodated by the substrate, and the islands and functional materials experience relatively small strains. Experiments have shown that, however, for a given amount of stretch, the islands exceeding a certain size may delaminate from the substrate. We calculate the energy release rate using a combination of finite element method and complex variable method. Our results show that the energy release rate diminishes as the island size or substrate stiffness decreases. Consequently, the critical island size is large when the substrate is compliant. We also obtain an analytical expression for the energy release rate of debonding islands from a very compliant substrate.

**Keywords:** flexible electronics, island size, delamination, complex potential

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## Comments

## nice work!

nice work!

## Some suggestions

Some suggestions:

above equation (3): change H->infinity to H/h -> infinity

below equation (3): change Ef->infinity to Ef/Es -> infinity

## how to go from Hutchinson and Suo's paper to equation (1)?

Dear Nanshu,

How was the equation (1) derived from the Hutchinson and Suo’s paper (1992)?

The expression was not explicitly given in their paper for a steady state crack on the interface between the film and the substrate.

Can you help me to reach to equation (1)? Thanks.

## Hint for Eq. (1)

Dear Henry,

Thanks for your attention. Regarding to Eq. (1) in this paper, please focus on Eq. (3.22) in Hutch & Suo's paper. According to our loading conditions shown in Fig. 1(b), we know P1=0, M1=0, M2=0 P2=P3=E_sεH and M3=(hΔ-H/2)P, where hΔ is the neutral axis of the structure and Δ is given in Eq. (3.18). Also you can find expressions for A and I in Eq. (3.21). Plug in everything and one will be able to get Eq. (1) in our paper.

## Thanks.

Thanks.

## effect of Poisson's ratios

Should the result also depend on the Poisson's ratios of the film and the substrate in general?

RH

## Good paper

hi Nanshu,

Thanks, and i have learnt very much from your paper! But it seems wrong in Page 8 the third line---" Es=200GPa ...." should be reversed, Es-- Ef, Ef -- Es .

I am a third year Phd in Xi'an Jiaotong university in china, and have done some experimental work in Cu films on PI substrates. Accordingly, i am very interesting to your group's research.

## Thanks very much

Dear Rongmei,

Thanks very much to point out the mistake. Nice to meet you :)

Actually I am also doing some experiments of metal films on Kapton substrate in Vlassak's group. I have read your previous discussions carefully and I am wondering is there any paper published on this topic by your group? Or any crucial references you can recommend to me?

Thanks.

Nanshu

## pleasure to discuss with you!

Dear Nanshu,

I am very pleasure to recommend excellent references to you! but you will find that most of them are from your group. Another paper in Journal of the Mechanics and Physics of Solids48 (2000) P1107――Crack patterns in thin films by Z. Cedric Xia, John W. Hutchinson, may be useful.

I admire and congratulate you that you are so fortunate in a great group, and conveniently communicate and discuss with great people. My major is materials science and engineering, and i only did some experiments, the attatchment in the following is part of the experiments, i sincerely hope it helpful to you! To be frankly, i still did not completely understand your paper. Why use "island" instead of "film or coating"? In the actual application, the films are continuous. Regards

## My pleasure to answer your question

Dear Rongmei,

Thanks for your recommendation very much. Now with the platform of imechanica, mechanics people can communicate freely all over the world. We are able to find speeches, lecture notes, manuscripts and many other resources here. Actually all of us are fortunate, right?

Regarding to your question about "island", it is actually a very nice design in flexible electronics. In order to be flexible, we need to fabricate inorganic brittle material onto flexible PI substrate. Since the brittle film can only sustain little strain, the film cannot continuously cover the whole substrate surface. Instead, we fabricate a lattice of brittle islands with metallic interconnections. When the substrate is stretched, the deformation is mainly accommodated by the substrate, so that the island and functional materials on it experience relative small strains.

Regarding to "coating", we use it to refer an additional continuous layer covering all over the islands and substrate. It can significantly reduce corner singularity and suppress delamination. A recent study from our group is here.

Regards

## attatchment

Dear Nanshu,

I am sorrry for missing the attachment. This paper has been accept, and may be referenced to your experiments.

Regards!

Rongmei

## Nice job

Dear Rongmei,

Congratulations to your new paper and thanks so much for sharing with us. It is very nice work and exactly what I am looking for. I just had a glance and have two questions needing your help,

1. How do you define and measure the crack density?

2. To obtain Cu yield strength did you take the tensile system compliance into account? What Young's modulus for film Cu did you get?

I'll read it more carefully later.

Nanshu

## Response to your question

Hi Nanshu,

Cracks density is the total cracks length per unit area in this paper. in case of parallel channel cracks , the average crack spacing is often used. Howerver, as to the zigzag cracks, i thinkit is reasonable to use cracks length per unit area.

I donot understand what is you meaning of the second question. And the modulus of Cu from my test is about 80~100GPa. How about you?

Regards!

Rongmei

## questions and suggestions

Hi Nanshu:

With interest I read your paper. Nice job for a first-year graduate student! If you don't mind, I have a few questions as well as some suggestions as follows.

Question first (one above already). The analytical solutions are always desirable. In addition to the limits of steady state cracking and convergent debonding you provided, is it possible to develop an analytical solution for a short emanating crack from the island edge? In this case, both the island size and thickness can be considered infinity. Thus the problem becomes a finite crack between two half spaces. It seems to be possible to get an analytical solution.

Another question: how did you calculate G by ABAQUS? J-intgeral or other method? What kind of elements and mesh did you use (especially near the crack tip)? How did you split the mode I and mode II stress intensity factors? Computationally, very short cracks (a->0) can be challenging. What was your minimum crack length?

Last question: Eq. (10) was obtained by setting a = 0 in Eq. (7). Conceptually, how do we understand the energy release rate for zero crack length? This is troubling me because in a plot like Fig. 4(a) I always wonder what should be the limit as a -> 0?

Enough of the questions. Some suggestions: Eq. (4) is also true for the limit h/H -> infinity, which can be shown in Fig. 2 (a log-log scale would give you a straight line of slope -1 toward the right side).

Another suggestion: you can get an asymptoic solution to Eq. (10) for the limit of small spacing, i.e., S << b (the length scale determined by the interface toughness). It is L = S(1-2S/b(pi)). You may plot this in Fig. 8 for comparison.

RH

## Response to your questions

Hi, Prof. Huang,

Thanks very much for your interests and helpful suggestions. It's my pleasure to answer these questions.

1. It is well known that for linear elastic bimaterial system the elastic mismatch is taken care by two Dundurs parameters α and β, or correspondingly E and ν. However, in the paper I just fix the Poisson's ratio of the film and substrate both to be 0.3 because otherwise there would be too many variables blurring our main focuses.

2. It is very important to point out what's the behavior of the very starting period of an edge crack. Actually it is my next-step research.

3. I calculated G by J integral in ABAQUS v6.5. It also gives you the K1 and K2 stress intensity factors. To do this, we create a seam crack beginning at the film edge and propagating along the film/substrate interface, use 0.25 for the midside node parameter and collapsed element side, single node to create a r1/2 singularity in strain and make use of “swept meshing” technique around the crack tip. My minimum crack length is a=0.01h, where h is the film thickness.

4. Notice that Eq. (7) is our analytical result valid for periodic rigid islands. Even though we took a=0 to arrive at Eq. (10), what really happens is that at the edge of a rigid island there's no difference for the wedge angle to be 90 degree or 0 degree. This means one can imagine the free surface of the elastic substrate as a crack surface and take the surface covered by rigid island as the bonded ligament. Our results show that the energy release rate is only related to the bonded length, but has nothing to do with the crack length.

5. You are absolutely right about Eq. (4). But here let's just name the "film" as the thinner layer and take the "substrate" to be the thicker one.

6. For S<<b, compared to your formula the critical island size can be expressed even simpler, namely L=S. From Fig. 8 this point is evident. That's why we just give one curve here.

Thanks for your attention and I hope these are helpful.

Regards,

Nanshu

## two points about your response

Nanshu:

Thanks for your responses to my questions. I think you probably can include some of the explanations in the paper after you receive the reviewers' comments. Here are two points regarding what you mentioned.

First, for item 6 in your response, remember that S must be greater than L in Fig. 1. Therefore, you must include the second term as I suggested, even though it is small.

Second, you apparently used the singular element (quarter nodes) at the crack tip to get the sqaure root singularity. However, at the interface between two dissimilar materials, the crack-tip singularity is not necessarily square root. Disregarding the oscillatory singularity, for a rigid island on a compliant substrate as shown in Fig. 3, the singularity can be as strong as r^(-1) (see Zak and Williams, JAM, vol. 30, pp. 142, 1963). Zhigang and I had a paper (Publication 138) a few years ago about channeling cracks, which showed the challenge for ABAQUS (with the singular elements) in modeling a similar crack problem. It is not a trivial exercise, I think.

RH

## Singularity for interface crack

Prof. Huang,

I am not quite sure what is your definition of interface crack but from my understanding interface crack means a crack lying on the bilayer interface and also propagate along the interface. If both layers are elastic, the interface crack tip stress field singularity is 1/2+iε referring to the paper "Interface crack between two elastic layers" by Suo & Hutch 1990 (#5 in Suo group publication)

However, when the crack is imbeded in one layer and perpendicular to the interface and when the layer without crack is very soft, it is possible that the singularity goes to 1. Please refer to Fig. 4 in Zhen's paper

Regards,

Nanshu

## interface crack or corner

As Zhigang mentioned below, it does not matter whether the crack is along the interface or perpendicular to the interface when the island is rigid.

RH

## The singularity of a rigid island on an elastic substrate

Dear Rui:

For a rigid island on an elastic substrate, the singularity is the same as that of a crack on a bimaterial interface:

-1/2+iε

This is because when a material is rigid, all it does is to prescribe a fixed displacement on the interface. The shape of the rigid material makes no difference to the elastic field in the elastic substrate.

## Williams singularity

Zhigang:

I would agree with you, from what I know about interface cracks. However, I seem to have trouble to get the same from the Zak and Williams's solution (Eq. 2.7 in our paper, Publication 138). Even when beta = 0, the singularity factor s does not equal to 0.5 when alpha approaches 1 (for a rigid film). Anything wrong?

RH

## Zak-Williams vs. islands.

Dear Rui: In the Zak-Williams problem, a crack impinges upon an interface. This would correspond to a limiting case in our problem when two islands approach each other. For us, we need to consider a corner singularity.

When the islands are rigid, the Zak-Williams problem will prescribe displacement on the entire interface. However, at the edge of an island (our problem), we will only prescribe displacement underneath the island, leaving the other part of the substrate surface traction-free.

## wedge singularity and Williams singularity

Dear Rui,

I am not sure that what you mean by the so-called Williams singularity. Is that a crack perpendicular to the interface?

Here are my two cents about the wedge singularity and crack singularity.

1. for the wedge singularity of rigid film on substrate, alpha=1, the singularity exponents are -1/2+i*epsilon, the same as interfacial crack.

2. for Zak-William singularity, i.e. a crack in mat#2 perpendicular to the interface, the singularity exponents can vary from 0 to 1 for the whole alpha-beta plane.

Zhen

## This solves my problem!

Thank you again, Zhigang.

It is always a pleasure to read your paper. Now with iMechanica, I can learn more from the discussions, just like those days back in Princeton.

RH

## mode mixity

Nanshu,

Nice work. Just a bit confused how you got K2 and K1 in Abaqus if you opted to use J-integral, and what is the characteristic length for you to calc mode mixity unless you neglected the ossicillatory index?

Charlie Zhai, AMD

## Reply to mode mixity

Thanks, Charlie.

When you request history output for the domain of contour integral there are several choices of the type. To obtain energy release rate only, we just choose "J-integral". To obtain both ERR and Ks, we choose "Stress intensity factors", with prescribing a crack initiation criterion. After calculation is completed, from the history output we can get both K1 and K2 as well as ERR estimated from Ks.

You are right, to present the ABAQUS results in the paper I just assume ε=0 and get my mode mixity simply from tanΨ=K2/K1. This is because our main purpose is to study the behavior of energy release rate.

## Nice work, Nanshu.

Nanshu,

Very nice work. Thanks for this a wonderful addition to the March Journal Club issue.

I was away in the last several days, and it seems I missed the instructive discussion here. I've added a link in the March issue of Journal Club to this active discussion.

-Teng

## Corner Effects.

Hi Nanshu,

Very interesting work. A question. I noticed that you used plane strain assumption for your calculation, because the island's width is usually much higher than its thickness. Have you ever considered the effects of the island's corners, which in my opinion are the highest stress concentration points in this kind of structure, and therefore maybe the crack initiations? In this way, do you you need a 3D simulation to analyse the problem?

-Xuanhe

## Justification for 2D modeling of island delamination

Xuanhe, very interesting question.

You're right, from engineering point of view the ideal modeling should be 3D, multilayer structure, true flaws... as real as possible. Intuitively, the corners of a square shape island should have higher driving force so that these areas are more susceptible for debonding. However, here I can give three justifications for our 2D modeling of island delamination.

First, from some experimental pictures of island delamination (Fig. 7 in Lacour et al, JAP

100014913, 2006 and Fig. 11 in Bhttacharya et al, JES153(3) G259, 2006) we can see debonding occurs at both corners and laterals of the square island. Thus our 2D modeling is reasonable for the later one.Second, even if the crack front is a 2D curve (many people modeling the 3D corner crack with quarter circular front) our delamination criterion says the energy release rate of each point on the front should reach interface toughness Γ(Ψ), where Ψ is the mode mix angle. So sometimes this 3D problem can be reduced to a 2D problem.

Last but not least, the main purpose of our scientific research is to try to understand the natural laws and intrinsic mechanisms. We make as many as possible simplifications so that the main object gets emphasized. Starting from 2D study we can find out how elastic mismatch, island size and crack size can influence the energy release rate, which is of great guidance for 3D problem. Meanwhile, 3D modeling is of high calculation and time cost. Comparing the gain and cost we just adopted 2D modeling.