# Traction separation laws in Cohesive zone models - Some Questions

Hello!

As a student who has spent a lot of time studying cohesive zone models in fracture mechanics, I have several questions that have bothered me over the past year or so, and I have not been able to find suitable answers to them. I am limiting myself here to questions related to the traction-separation law, which invariably forms the basis of CZM as it is implemented today. I am raising these questions in the hope that I can receive some response here, even if it means my question is invalid (as I suspect a few may be). So here is my list:

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1. Basis: It seems that the only basis cited is for the exponential one, when it is tied to the Rose-Smith-Ferrante traction separation law. Is this a justified basis? Have any attempts been made to DERIVE the form of the traction separation law as opposed to propose it on a phenomenological basis? Would this even be possible?

2. The "separation": Why do CZMs operate with separation, and is it correct to do so? Fracture mechanisms involve, in my understanding, complex phenomena such as strengthening and bridging. By lumping all these phenomena into the traction-separation law, are we making erroneous assumptions? Is there truly a "critical separation"? Can this be expected to be a material parameter? Would not separation suffer from the similar problems that make us prefer stress-strain plots to stress-displacement ones?

3. Statistical origins: Could the traction-separation law be nothing more than a probability distribution, similar to that proposed by Weibull for the distribution of flaws? Is the exponential form used by Needleman and others just a manifestation of this probabilistic distribution of critical strengths? Can/has this been proved?

4. Mixed mode: Is there a consensus on the treatment of mixed mode? Can shear and normal separations be combined to form a resultant separation?

5. Parameters: The maximum stress and critical separation? Is there any reason to believe they are/are not material constants?

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I hope you find my questions appropriate. Thanks for your time. There are other questions I have regarding singularity removal, mesh sensitivity etc., but that is for another post. :-)

Thanks,

Dhruv

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## Re: Traction separation laws in Cohesive zone models

Hello, Mr. Dhruv Bhate. I am trying to answer only some part of your questions with my limited knowledge and information.

CZL (cohesive-zone law) is a phenomenological model but not a physical law characterizing a fracture process zone, while K-field in LEFM (linear elastic fracture mechanics) has solid foundation, since elasticity gives square-root singularity around a sharp crack-tip. One of assumptions of CZM is that all the fracture processes are squeezed into a cohesive-zone line. Derivation of CZL only from known fundamental laws seems to be very difficult. In my opinion, the exponential CZL is widely used since it can usually best fit the existing experimental results. You can propose any functional form of CZL to reflect special characteristics of fracture processes, for example, void formation, bridging, crazing, etc. But, a proposed CZL can only be justified by experiments. I attempted to obtain CZL of FCC crystal from atomistic simulations without any assumption on the form of CZL (S. T. Choi and K.-S. Kim, Phil. Mag., Vol 87, No12, 1889-1919, 2007, http://www.imechanica.org/node/1257#attachments). However, atomistic simulation can also be regarded as a numerical ‘experiment’, and its interatomic potential, EAM (embedded atom potential), is based on the fitting of bulk properties of FCC crystals. And even though there are various interesting phenomena and new findings, the obtained CZL is similar to the exponential model of CZL.

I think energy approach is a better way to appreciate CZL. As K-field in LEFM is related to energy release rate, the area of the traction-separation curve gives the critical energy flow from external loading. That is, traction and separation in CZL is work conjugate, as stress and strain are strain energy conjugate. Strain in cohesive zone is irrelevant. And I think maximum stress or critical separation is not an appropriate material property alone. As you can see in the above mentioned reference, the maximum stresses in CZL of FCC crystals obtained by ‘rigid separation potential’ and ‘field projection method’ give quite different values, but, the energy release rates at the cohesive zone from both methods have similar values, of which difference mainly comes from different phenomena.

I hope these comments partly answer to your questions, but do not bother you more.

Thank you.

Seungtae Choi

## Very helpful

Dear Seungtae,

Thanks for your reply. I think your comments are very helpful, especially about the displacement formulation leading to a fracture work (as opposed to a strain energy term). For some reason, I always prefer to think in terms of volumetric quantities such as strain energy density or dissipation, hence my prior curiosity about Sih's strain energy density method . Also, I was not aware of attempts to use atomistic simulations to develop CZ laws and will read your paper with interest.

Thanks again,

Dhruv

## Atomic simulations

Dear Seungtae,

Am I right? Atomic simulations can only be used to get the nano scale cohesive law?

Can this type of simulations be applied to get a micrometer-scaled cohesive law?

## Hi Dhruv, Here are

Hi Dhruv,

Here are some comments on your questions. Hope they are helpful.

It should be mentioned that the cohesive zone law is not universal, and takes very different forms for different materials/interface. Most existing cohesive zone laws are phenomenological, in which a relation between normal (and shear) tractions and opening (and sliding) displacement is assumed. There are also some experimental studies to measure the microscale cohesive law (for example,PHILOSOPHICAL MAGAZINE,87:2221-2232, 2007). In our work, we found that the cohesive properties, such as the maximum stress (cohesive strength) and critical separation where the cohesive stress reaches the maximum value are all material constants, and can be expressed in terms of the area density and volume density of the atoms at the interface and the constants in the potential representing the van der Waals force, such as Lennard-Jones potential. It is also found that for a CNT in an infinite polymer, the tensile cohesive stress depends only on the opening displacement, and the shear cohesive stress vanishes. However, for a CNT in a finite polymer matrix, the tensile cohesive stress remains the same, but the shear cohesive stress depends on both opening displacement and sliding displacement, i.e., the tension/shear coupling.

The cohesive law for the particle/matrix interfaces in high explosives, Tan et al., JOURNAL OF MECHANICS AND PHYSICS OF SOLIDS, 53: 1892-1917, 2006). We recently have two papers published about the cohesive law. Our work avoids any assumed phenomenological cohesive laws, but accurately accounts for the van der Waals interactions between carbon nanotubes and matrix (A cohesive law for carbon nanotube/polymer interfaces based on the van der Waals force, Jiang et al., JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 54 (11): 2436-2452 NOV 2006), or between carbon nanotubes walls (A cohesive law for multi-wall carbon nanotubes,Lu et al.,Liying Jiang

## 1000 times difference in interface cohesive law

As Liying said, interface cohesive law takes very different forms for different materials/interface. Here I want to emphasise that interface cohesive law is also scale dependent.

The interface cohesive law at micro-structural scale (micrometers) can be 1000 times different from that at the atomic scale (nanometers) for some crystal/polymer interfaces.

## interface cohesive law is scale dependent

Dhruv,

You raised some very tough questions about cohesive zone model.

I would write something to your second comments: what is separation? I would say that separation is a measurement that depends on what is the ruler you are using.

The interface cohesive law at micrometer scale is totally different from the law for the same material but at nano scale!

## scale dependent material parameters

Therefore, to your (Dhruv's) fifth comments:

The maximum stress and critical separation are scale dependent material parameters.

## Thanks

Thanks a lot, Liying and Henry. I am fascinated by the work you'll have done in these papers, and will read it in more detail. At first glance, they look both very interesting: I particularly appreciate the clarity of Fig 11 in Tan et. al. (JMPS).

Regards,

Dhruv

## Some remarks about cohesive zone modelling

I have followed the discussion on cohesive modelling and would like to give a few additional remarks,

1) The cohesive zone is basically a

modelconcept that can be useful in certain cases. It can be used for instance when the fracture process zone is too large and a point-sized crack tip model is not adequate. It can also be used when modelling the initiation of a crack from a medium without cracks. The cohesive zone may or may not be a simulacrum of an actual physical process. It can be used to model different types of separation processes such as void growth and coalescence, fibre bridging, atomic separation, separation of adhesive layers such as glue etc. Once the cohesive law has been set the problem formulation is complete and no other fracture criterion is necessary.2) There are different ways to derive a cohesive law.

a) By experimental measurements on special types of specimens (cf. T. Andersson and U. Stigh, (2004),

Int. J. of Solids and Structures,41, 413-434, B. F. Sørensen and E. K. Jacobsen, (1998),Composites Part A,29A, 1442-1451. and several others).b) By modelling (numerical or analytical) of the process that is to be replaced by the cohesive zone model.

c) By using a predefined functional assumption for the cohesive law, for instance as predefined in a numerical code (cf. ABAQUS). The parameters are estimated from experiments or by reasonable guesswork. This is probably the most common way.

3) It is often stated that a cohcsive zone model is equivalent to assuming that crack growth is governed by constant fracture energy. This is not true in general. Here are some situations when the fracture energy is non-constant and problem dependent.

a) A crack tip that is extending under conditions that are not steady-state (with or without inertia effects) (cf. L. B. Freund (1990),

Dynamic Fracture Mechanics, pp. 237-238).b) When large deformation effects are significant, depending on how the cohesive zone law is formulated (cf. F. Nilsson (2005),

Int. J. Fract.,136, 133-142).c) When the cohesive law depends on other quantities from the problem such as constraint, displacement rate etc.

When, as often is the case, setting up a cohesive law using fracture energy measured from experiments, it is thus important that this is a problem independent quantity.

4) Some caution must be observed when using cohesive models in conjunction with numerical models such as FEM. The minimum size of the cohesive zone will be the size of an element near the crack tip. Should this be larger than the size of the process zone of the physical problem, a length scale has been introduced that does not exist in the physical problem. This is frequently the case when using cohesive zone models for analysis of fatigue crack growth.

## cohesive zone will be the size of an element near the crack tip

Fred,

Thank you very much for your interesting comments. I am a newbie in this area and couldn't completely understood the last comment "Some caution must be observed

when using cohesive models in conjunction with numerical models such as

FEM. The minimum size of the cohesive zone will be the size of an

element near the crack tip. Should this be larger than the size of the

process zone of the physical problem, a length scale has been

introduced that does not exist in the physical problem. This is

frequently the case when using cohesive zone models for analysis of

fatigue crack growth." I would really appreciate if you can elaborate on this or point me to some article where this is not taken into account or has been discussed.

## Length of cohesive zone

Rahul,

I know of one paper that may help you appreciate this a bit better (Fred, do correct me if I have misunderstood your statement)

A. Turon, C.G. Da´vila, P.P. Camanho, J. Costa, "An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models," Engineering Fracture Mechanics, 74, 2007, pp. 1665-1682

The paper is relatively easy to grasp, and collects all the various interpretations of length of cohesive zone...the process zone is essentially a highly nonlinear damaging region and several elements are needed to adequately describe the behavior in that zone.

I hope this helps.

Dhruv

## Element size and cohesive zones

Rahul,

What I meant with my last comment is that the cohesive zone must be at least of the size of several elements to accurately resolve it. If the parameters of the cohesive law are such that the zone size for a specific problem is only one or a few elements, there is a risk that a length parameter is introduced that bears no relation to the physical problem. Thus, the physical problem may demand (through the appropriate cohesive law) elements that are so small that the computations may be very expensive or even impossible to perform. It may then happen that the cohesive parameters (bearing in mind that in general very little is known about the appropriate values of the parameters) are adjusted so that a solution with an artificial length scale is obtained.

Fred

## How to define the crack tip

Dear colleagues,

Many thanks for these very useful comments. Using cohesive law in Abaqus, I'm doing a peeling test simulation. Within the cohesive layer, the position of crack tip is defined as the element with the maximum S22 (Normal stresses). Is that reasonable?

Thanks,

Dong

## Crack tip definition

Dong,

In a cohesive zone model there is no clearly defined crack tip. Sometimes the front edge is used as reference, sometimes the trailing edge. Thus you can choose any definition that you prefer. I suppose that with the definition you suggest the crack tip will be at or near the front end.

Fred

## Modeling R-Curve behaviour using CZM

Hi All,

Thanks for your valuable comments and explanations about this topic. I am relatively new to this (Cohesive Zone Model) field. From a limited amount of reading and discussions with collegues I understood this topic to certain extent. From what I understand, CZM requires a traction separation law for modeling, which is obtained based on fracture energy.

My question is, because fracture energy is not a contstant value i.e., in case of ductile materials we have phenomenon called "R-Curve" behaviour (fracture energy changes with crack length initially), is it justifiable to use only one value of fracture energy (i.e., steady state value) to derive the traction separation law?

Please point me towards any article(s) if exist about modeling "R-Curve" behaviour using CZM.

Thanks a lot for your valuable time and suggestions.

--Siva

## Re: Modeling R-Curve behaviour using CZM

Here is one such paper:

Tvergaard, V., Hutchinson, J.W.,"

The relation between crack growth resistance and fracture process parameters in elastic-plastic solids." J. Mech. Phys. Solids 40, 1377-1397(1992).

## The convergence of cohesive models

Thanks very much,

Using Abaqus default cohesive element, I model the peeling test (the height of the adhesive is 150

umand the length is very long). The interaction between the adhesive and the soft (E=3MPa) is Van De Waels force. The length scale of Van De Waels force is very short (10 nm), compared with the adhesive and substrate. The compution cann't be finished even for some increments. But if the length scale of cohesive stress (200um) becomes very large,it works well.## Cohesive elemet propeties

Dear all:

I am working on grack propagation problem using Cohesive elements in ABAQUS CAE.

I really don't know how to choose the properties of chohesive elements. Can any body tell me how to choose it??

Vikas

## Cohesive Element Properties in Abaqus

I wanted to raise some issues with regards to Vikas' question on implementing cohesive elements in Abaqus CAE.

I have used Abaqus CAE for some simple modeling of my shear specimens. I used Damage for Traction Seperation Law (under the Mechanical Properties tab). Here, the nominal stress for 3 (or 2 in my case) should be entered. This would generally be the tensile and shear failure strengths of the material. Also, using Suboptions, one can enter the displacement at failure value. Now, I am not really sure, if Abaqus uses this value as the ultimate failure value or for the initial drop in the cohesive law. Now, I work more on experimental mechanics, and so I confess that some (or all!) of what I have written may not be correct. It will be great if someone can throw more light on this approcah.

The other issue I wanted to discuss was the phenomenon of snap back, where depending on certain input values, the simulation fails to converge. I guess this issue arises when displacements are imposed in the analysis and I ran into this problem for my simulations. Again, can someone kindly comment on this and say whether they have faced the same problem?

Thanks,

Arun

## a mutiscale cohesive zone model

Recently a mutiscale cohesive zone model was proposed by X. Zeng and S. Li [

A multiscale cohesive zone model and simulations of fractures. Computer Methods in Applied Mechanics and Engineering, 2010, 199(9-12):547-556. ]. With the aid of coarse-graining procedure, the prosperities of the bulk material are determined by the atomistic potential and theCohesive Zone Lawwas acquired based on the same coarse-graining procedure as in the bulk material.## Beginner

Hey all,

I'm new to this whole field and i was wondering what traction separation laws were. I want to read up on it because its going to be relevant to my university studies.

Cheers