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# Mixed-Mode Fracture. Curved Crack Path

These notes belong to a course on fracture mechanics

A crack pre-exists in a body. When the body is loaded, the two faces of the crack may simultaneously open and slide relative to each other. The crack is said to be under a mixed-mode condition. When the load reaches a critical level, the crack starts to grow, and usually kinks into a new direction. Subsequently the crack often grows along a curved path.

This lecture discusses the critical condition to initiate the growth, the direction of the kink, and the method to predict the curved path.

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## Comments

## Energy release rate

Dear Professor Suo,

I read with great pleasure your posts on fracture mechanics. Thank you for offering us your fm class, I feel it very valuable.

I may be wrong but the energy release rate formula, in the form you posted, appears to be valid only if the crack, when subjected to mixed moad loading, is moving straightforward. This was also stated in Broberg, Cracks and Fracture, Academic Press at page 88. In a more general fashion, it should be written in terms of the SIFs right after a crack kinking, as proved in Ichikawa M., Tanaka S., "A critical analysis of the relationship between the energy release rate and the SIFs for non-coplanar crack extension under combined mode loading", International Journal of Fracture,18,pp. 19-28,(1982). The expressions of SIFs have been provided in Amestoy M. and Leblond J.B., "Crack paths in plane situations - II. Detailed form of the expansion of the stress intensity factors", International Journal of Solids and Structures, 29, pp.465-501, (1992).

Best regards

Alberto Salvadori

## Re: Energy release rate

Dear Alberto: Thank you for your note. Indeed, the stress intensity factors and the energy release rate on pages 1-3 are for a stationary crack.

## SIFs and Energy release rate

Dear professor Suo,

thanks. Again, I can definitely be wrong, but I am not convinced that the energy release rate formula, in the form you posted, is valid even for a stationary crack. Let me set an example. Assume that, at a given time t, we are in the presence of a crack under mixed-mode condition, and bot SIFs are non zero. Than, mode mixity factor is clearly given. At least for proportional loading, at each mode mixity uniquely corresponds a crack propagation angle, whatever criteria one prefers. In other terms, the crack does not propagate, is stationary at time t for any t smaller than a suitable time T that I am not discussing here. But when it will propagate, the angle has been set by the criteria and is already known at time t. Therefore, at time t the SIFs that would appear right after an eventual crack kinking (at time t) are known: they are "in potential" up to time T, but are known. They have been provided in Amestoy M. and Leblond J.B., "Crack paths in plane situations - II. Detailed form of the expansion of the stress intensity factors", International Journal of Solids and Structures, 29, pp.465-501, (1992).

The energy release rate is related to such a value of SIFs, i.e. the "potential" ones, and not to the usual SIFs, the ones measured in the Frenet frame at time t. As a consequence, if one aims at measuring the energy released by a potential crack growth at any time $t <=T$, it differs from the one you posted. This is because the matrix $F$ in the paper above is not orthonormal.

Of course, for mode 1 crack growth the two notions of SIFs do coincide and the posted formula is correct.

In conclusion, the problem of crack kinking is a tough one. According to many scientists, it has not been solved yet.

Best regards, and again, many thanks

Alberto

## Re: SIFs and Energy release rate

Dear Alberto and Han: Thank you very much for the comments.

Significance of KI and KII. In the notes for the lecture, KI and KII are the stress intensity factors for a stationary crack. They represents the amplitudes of the loads of the two modes. When discussing a kink from a pre-existing crack (the main crack), so long as the length of the kink is small compared to that of the main crack, we assume that an annulus exists, in which the stress field is the same as that of the main crack, characterized by KI and KII. Thus, KI and KII are used as loading parameters to drive any small-scale event at the tip of the main crack, including a small kink.Definition of G. In this context, G is still defined as the reduction in the potential energy associated with the main crack advancing per unit area, in the direction directly ahead. G represents the amplitude of the remote load. This usuage is consistent with the previous idea of using KI and KII as loading parameters to describe any small-scale event at the tip of the main crack.Thus, KI, KII, and G are

notthe stress intensity factors and the energy release rate of the kink.One can use either (KI,KII) or (G,psi) to represent the remote loads. Under these loads, one can describe how a kink happen.

Also, Roy has reminded us that, under certain conditions, one needs to use additional parameters (such as the T-stress) to specify the remote load.

## Kink angle, growth path and energy criterion

Hi Zhigang,thanks for the clarification. Two questions in reading the notes, I think we touched them in the lectures, just want to be clear1. Kink angle vs growth path. We introduced two criteria that crack kinks to the plane with the maximum hoop stress, and a crack growths along a pure mode I path. Can I think the kink is only crack initiation, while crack growth is a stationary process?

2. For "crack growths along pure mode I path", in my understanding, what we do is to find the stress intensity factor as a function of the path angle, and set the local mode II SIFs as 0, it gives us the growth angle. In this way, the SIFs refer to the local SIFs, instead of aheading of the crack tip, right?

Hi LiHan,regarding the energy release criterion, I read (Anderson, P81) "a propogating crack seeks the path of least resistance or the path of maximum driving force". If the material is isotropic and homogeneous, the path will seek a path with maximum energy release rate. Therefore, we can find G as a function of the path angle, maximize it and it gives the angle. However this is consistent with the "pure mode I" criterion.Kejie

## Re: Kink angle, growth path and energy criterion

Kink angle vs. growth path. By kinking we mean a crack changes its direction of extension abruptly by a finite angle. Kinking is often observed when a pre-cut crack under mixed-mode loading starts to grow. Kinking may also happen during the growth of a crack when the external load suddenly changes proportion.When the external load varies smoothly with time, a growing crack often grows along a smooth-curved path.

Crack grows along a mode I path. As explained in the notes, a well known hypothesis is that, in a homogeneous, isotropic and brittle material, the growing crack grows along a path such that the tip of the crack is locally under the mode I condition. The word "locally" means that we talk about the stress intensity factors at the tip of the crack.## Dear Professor Suo, I

Dear Professor Suo,

I recently considered the issue under debate in a publication:

Salvadori, A., “Crack kinking in brittle materials”, Journal of the Mechanics and Physics of Solids, 58 (2010) 1835–1846

The conclusions seem to be interesting. In particular it seems that the local symmetry as well as any other criterion is not compatible with Griffith theory in the presence of kink. Furthermore, the onset of crack propagation seems to be always related to a prediction of the kinking angle in the eventuality of a crack elongation. In other words, the safety of a stationary crack, regardless of how far away it is from the critical state, depends on the angle the crack is going to kink at the time it grows.Therefore, SIFs or T seem to be unsufficient to fully characterize the safety of a stationary crack.

looking forward to your remarks,

very best

Alberto

## Hi Alberto, thanks for

Hi Alberto, thanks for bringing up this question. A related question is: how is energy release rate, G, is defined for a crack under mixed mode loading, when the direction of crack propagation is not known a priori. My understanding is that in this case, the energy release rate is no longer a number, but a function of the kink direction. The crack will propagated along a direction where G first surpass the fracture resistance of the material. Would the direction predicted this way agree with the Mode-I criterion? Not sure.

lh

## Hi Li, you may find a

Hi Li,

you may find a recent paper:

Salvadori, A., “Crack kinking in brittle materials”, Journal of the Mechanics and Physics of Solids, 58 (2010) 1835–1846

of interest for your question. I'm glad to send it if you are interested. Best!

Alberto

## T-stress and the mix-mode fracture education experiment

Dear Zhigang,

This

is a very important topic. I'd add a few comments based on my experimental

observation:

1. In

your first illustration, a mixed-mode crack suddenly kinks from its

initial path. In my previous experiments using high-speed photography, a

mixed-mode crack always kinks very smoothly as shown in my figure 1 (load-induced

kink). A sharp kink only occurs at the interface due to the material

resistance change.

2. In

your modeling, I didn't see the T-stress in the mode-I component. Indeed,

the T-stress has significant influence on the crack path if this value is

large based on JR Rice's early work in 1980. In 2007, we published a systematic

investigation on the T-stress change across static crack kinking in

Journal of Applied Mechanics (web link here) . We hope

someone can model the T-stress change for a dynamically kinked crack.

3. Your

second illustration on a mixed-mode fracture problem is very interesting.

I designed one education experiment using copy paper as shown in my figure

2. This paper was published in International Journal of Engineering

Education (web link here) . It is easy for every student to

try, but two persons should use their 20 fingers to apply uniform tensile

load at the paper edges (don't follow my figure 2(b)).

4. Erdogan

and Sih's criterion was supported by their famous PMMA plate experiment

conducted in 1963. I believe there was no EDM to make a sharp crack tip for a plate in 1960s. In 2009, I

met Prof. Erdogan and asked him- your "crack" was indeed a notch, so their

stress singular orders were very different...

Roy

## Are SIFs and T-stress independent loading parameters?

Dear Prof.Xu,

Very nice to see you here. I have been interested in the T-stress effect but not read much literature yet. One basic question in my mind is, are the K and T-stress independent loading parameters? Or in other words, for a particular boundary value problem, can they vary independently? Thanks

Kejie

## Dear Kejie, Very glad to

Dear Kejie,

Very glad to hear from you again. You can find previous T-stress papers in our JAM T-stress paper in 2007. Yes, K and T-stress are independent mechanics parameters. K can be solved using the asymptotic solutions at the crack tip, but the T-stress needs a full-field stress analysis (including these high-order terms). For some cases, a so-called ‘‘biaxial ratio'' was used to correlate K and the T-stress, as shown in our paper on dynamic crack propagation and kinking in 2006 (web link to download the full paper).

## Re: T-stress and the mix-mode fracture education experiment

Dear Roy: Thank you for your note.

## Curved crack path due to mixed-mode load

Dear Zhigang,

We published a paper on dynamic crack propagation in homogenous materials with interfaces (click to download the paper). As shown in Figure 11, these dynamic cracks kinked very smoothly due to stress wave interaction. These curved crack paths are similar to racing-car tracks!

On the cracks kinked from a notch (experiments and modeling), I recommend a very good paper but I cannot upload this paper here. The citation is "A failure criterion for brittle elastic materials under mixed-mode loading by Zohar Yosibash • Elad Priel • Dominique Leguillon, Int J Fract (2006) 141:291-312, DOI 10.1007/s10704-006-0083-6". We are conducting systematic experiments on this topics, and hope to post our results soon.

Roy

## Re: Curved crack path due to mixed-mode load

Dear Roy: Thank you so much for these references. I'll follow them up when updating this lecture. I appreciate your kindness.

## KI, KII and T-stress

Hi, Zhigang, Roy and everyone else:

I cover the theoretical part of my Fracture Mechanics class by answering three questions:

(a) when does a crack grow?

(b) in what direction does it grow?

(c) how sensitive is its trajectory to perturbation?

We all know question (a) is answered by setting effective fracture parameter (K, G, J, C...) => threshold, and question (b) answered by imposing criterion of max G, tangential stress, shear stress, S-parameter or local symmetry. Seldom question (c) is addressed, which is about crack path stability/instability, in a textbook. In the linear limit (save K-T and J-Q approaches), the first two questions are answered essentially with KI and KII. The third question is answered with T-stress in addition, which is the Cotterell-Rice T-stress criterion (1980, if my memory serves). I teach the class like this. However, I also note that the crack path stability/instability problem has not been understood/solved. A dozen years ago, K. Ravi-Chandar and I tested on the T-stress criterion with the thermal-stress driven crack pattern bifurcation (cited in Zhigang's note) but found it not to work. After these many years, I am still scratching my head and wonder if there exists a local parameter like K and T that can describe this phenomenon.

All the best,

Bo

## Zhigang, I think you need to add an important update,helic crack

I guess you have avoided as too complicated the case of Mode I + Mode III which is seeing many recent developments, see here .

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

http://poliba.academia.edu/micheleciavarella

Editor, Italian Science Debate, www.sciencedebate.it

Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

## Hi,Prof. Zhigang,

Hi,Prof. Zhigang, recently I read all your lecture notes about fracture. Maybe I just perform an introductory study on this topic, but some

confusions really make me troublesome. As long as I pick up a textbook or lecture notes about fracture mechanics, I see a body (exactly may

say a plate) with a pre-existing crack and following the authour's thinking, I seem to understand many issues. However it doubts me that if a body,

say a pillar sample under compression not tension, without pre-existing crack, how will it fracture and how to predict the crack path? It seems so

complicated for me. Could you give some tips? Just as an example, if compressive load is exerted on a pillar specimen (we assume that the body

has no or litle flaws, or pre-existing crack) and shear bands inclied to the load axis by some degree appear in the pillar surface, then the shear band

may become instability and followed fracture happens. So can the fracture mechanics about pre-existing crack analysis this process? I choose to take

the intial microcrack induced by the shear bands with intense deformation as the 'fictive pre-existing crack' and utilize the knowledge I gain from the

textbook and your notes to analyze this issue. How do you think this dicussion? Maybe the confusions for me is silly or naive for you! Thanks!

Yours Sincerely

Min Yi

## fracture of concrete under compression

Dear Min: Thank you for this perceptive comment. You have not really missed anything. A sample with a pre-cut crack is used to measure the toughness of a material. How to use the measured toughness to answer any real question is an art all by itself.

The least troublesome use of toughness is to apply it to a brittle material under tension. In such a brittle material, there are always small cracks. For example, in a glass, a crack of length 100 microns will lead to a tensile strength on the order of 100 MPa. You can check these numbers with the Griffith formula.

For a concrete under compression, the application of fracture mechanics becomes less straightforward. One picture involves small cracks at an angle to the axis of compression, so that the sliding of the faces of the crack induces large tension at the tips of the crack, and causes the crack to extend. Each of such a crack may be stable, and stops after some extension. When many such small cracks join together, a shear band forms. I have never studied this phenomenon myself, but vaguely recall seeing this picture described in papers by Mike Ashby, and by Sia Nemat-Nasser. You may wish to look for their papers or contact them.

## Dear Zhigang

Dear Zhigang, your profound viewpoint about fracture mechanics impressed me strongly and maybe it can be thought that the toughness measured by a plate with 'pre-existing crack' is more related to engineering application within faracture mechanics. It seems that the present fracture mechanics is very limited in predicting, analyze fracture or fracture path in nonstandard body.

You refer to Mike Ashby and Sia Nemat-Nasser when discussing the concrete's fracture, and I will search their papers about this topic in Google firstly, though it'll be lucky for me to gain your recommending papers or books list by them. As far as know, the inclined small cracks in concrete under compression is extended by the tension in the crack tip, while the tension may be induced by shear dilatation in the crack tip, i.e. plastic flow in the tip under remote compression is induced by shear, but this shear will contribute to volume dilatation, thus bringing out tension. But it's unclear for me that why the crack will be stable.

At last, I have some confusions about your lecture notes in htis topic. When you discuss 'Direction of the kink' ,you write 'We will We will only consider an opening crack, so that KI>0. When KII>0 , the crack kinks down . When KII<0, the crack kinks down. When the crack is pure mode I, this criterion predicts that the crack extends straight ahead. ....' . Does it means that if we consider an closure crack (under compression), KI<0 ? And I have search and read may textbooks,but fail to find the negitive or positive definition about KII. But in the book 'Fracture Mechanics:With an Introduction to Micromechanics, Springe 2006', when discussing mixed-mode fracture, Dietmar Gross and Thomas Seelig write '...In order to ensure traction-free crack surfaces, a certain minimum crack opening hence should always exist (KI > 0). That means that all mentioned fracture criteria basically are physically meaningful only for KI ≥ 0. If crack closure occurs, a mode-I crack-tip ﬁeld no longer exists and a pure mode-II crack-tip loading (KI = 0) then is present.' This seems that KI only can be >=0. It upsets me very much.

## Tensile stresses could develop perpendicular to compression

Dear Min Yi,

I have heard civil engineers saying "cracks propagate in the direction of compression ...".

What they really intend to convey is, when there is uni-axial compression as you have considered, tensile stresses will generally develop in the direction perpendicular to the direction of the compressive loading. Cracks propagate in the direction perpendicular to tensile stresses in brittle materials like concrete, thus effectively in the direction of compression (though there are assumptions like uni-axial loading, it serves the purpose in most cases!).

Regarding the case of pre-existing cracks, I believe that in a highly heterogenous "material" like concrete, it will always be present. Estimating the crack-length may be the significant consideration...

Regards,

Jayadeep

## Dear Jayadeep, It's nice

Dear Jayadeep,

It's nice to hear the words 'cracks propagate in the direction of compression ..'. Thanks for your tips about local tension under global compression. From my viewpoint, as long as shear dilatation exsits in a material such as concrete, the material may generate tension under plastic flow, but it's hard for me to predicted and analyze the fracture behavior under this circumstance.

Yours Sincerely

Min Yi

## key=>local tensile stress around a crack tip

Dear colleagues:

In order to understand the fracture under (uniaxial/biaxial) compression, the key is to see the development of local tensile stress at a crack tip. It occurs when a crack is inclined to the principal directions (of loading). The (initially straight) crack would kink and propagate asymptotically approaching to one of the principal directions--sometimes called "wing crack development". A crack parallel to either one of the principal directions doesn't propagate. Thus, isolated cracks/flaws may grow but don't grow far. Only after populating, they interact and coalesce leading to the macroscopic fracture of a brittle solid under compression...

All the best,

Bo

## Dear Bo Yang,

Dear Bo Yang,

Thanks for your information about 'wing crack development' which is very intresting and important for me to understand this issue. Can you recommand papers or books coping with this problem?

Also, I have confusions about your comments. If we consider a plane stress problem, i.e. a tow-dimensional plate under compression and a inclined crack initiating in the plate edge, from the Mor-Circle we can only get the compressive stress at any plane, so how the tension happens in the crack tip? Maybe there exist some micromechnisms. As far as I know, some material such as soil,rock,amorphous alloy et.al, always suffer plastic dilatation or shear dilatation, thus inducing tension. Can there be other micromechanisms? Or my stress analysis based Mor-Circle is wrong?

Yours Sincerely Min Yi

## consider a crackED specimen under compression

Min:

After a crack is introduced, tensile stress would appear in certain planes at the crack tip. Even if the crack is all closed...

Please tell me your email address. Or, directly email me at boyang@fit.edu. I can then send you an old paper of mine with a brief discussion on this problem. Therein you may find a few classical references as well.

Best wishes,

Bo

## Thanks for your help! I

Thanks for your help! I have emailed you.

Yours Sincerely

Min Yi