# Length Scales in Fracture

Some material scientists and experimentalists are generally sceptical of simulations and reproach the theoreticians with lacking knowledge of real materials. Sometimes they may just be ignorant of the mathematics behind the models but sometimes they appear to be right. An example: Introducing “damage” into a constitutive equation simply as an internal variable which obeys some evolution law, not having the foggiest notion about the specific nature of damage - whether brittle, ductile, creep, fatigue - and its micromechanical mechanisms, promotes scepticism about the benefits of modelling in general, and deservedly so.

This is not my particular point today, however, but it is related. My problem today is the handling of length scales in

H. Krull and H. Yuan: Suggestions to the cohesive traction–separation law from atomistic simulationsEngineering Fracture Mechanics, Vol. 78, 2011, pp. 525-533.

Ductile tearing is governed by the initiation, growth and coalescence of voids in an elasto-plastic or viscoplastic material. Koplik and Needleman (1981) have been the first to perform unit-cell calculations of void growth to analyse this mechanism, and numerous studies by other authors followed varying the void shape, accounting for inclusions etc.. They helped improving and generalising constitutive equations of ductile damage like the Gurson model of porous metal plasticity. Unit-cell calculations have also been used to derive traction-separation laws for cohesive zones. The physical processes take place at length scales of micrometers to millimetres, accordant to the dimensions of the microstructure. Continuum models still apply at this length scale.

Atomistic simulations and molecular dynamics are based on models that relate binding energies or forces to spatial configurations in order to calculate accelerations of particles via Newton’s law. They describe interactions at a length scale of nanometers, which is at least three orders of magnitude below the relevant length scale of ductile tearing. What occurs at this length scale has absolutely nothing in common with plastic deformations due to dislocation motion and ductile tearing of metals, and hence the authors’ conclusions are apocryphal and unsubstantiated:

>     The computations under mode I conditions show that crack growth even in the nano-scale single-crystal aluminum is in the form of void nucleation, growth and coalescence, which is similar to ductile fracture at meso-scale.”

What do the authors actually mean by vague formulations like “in the form of void nucleation ...” and “similar to ductile fracture”? Voids nucleate at particles, for instance. Which particles are of atomic or sub-atomic size? What is their criterion for a process being “similar to” another?

>   Understanding the failure process based on atomistic simulation can provide detailed information for the cohesive law.”

This is correct, of course, provided the correct failure process is modelled.

>   The relationship between atomic traction and atomic separation including elastic deformations confirms the exponential function form suggested by Needleman.”

This is neither a miracle nor striking news, as the cohesive law proposed by Needleman in Int. J. Fracture 42 (1990) is based on the universal atomistic binding energy function proposed by Rose et al. in Phys. Rev. Letters 47 (1981), and he actually analysed “tensile decohesion along an interface” in an elastic medium.

>  The computations show that void nucleation and growth are controlled by tensile stress and hydrostatic stress. The Mises stress is not involved in material failure.”

This is the final death sentence to any simulation: that it yields results which are contrary to everything that is known about the real process. Apart from this, the hydrostatic stress and the maximum tensile stress depend on each other under fully plastic conditions of plane strain. But plasticity is out of the scope of the MD simulations, anyway.

After all, the simulations could indeed provide useful information on a cohesive law for a process which is not void growth and coalescence in ductile metals, however. It is the authors’ business to present an actual decohesion mechanism following their model.

### On the use of molecular dynamics for CZMs

Dear Wolfgang,

Dear Colleagues,

thanks for your post. I was recently involved in a close research, trying to interpret and use the results of MD simulations to have a better understanding of the physics underlying cohesive zone models (CZMs).

We published these two papers:

M. Paggi, P. Wriggers: "A nonlocal
cohesive zone model for finite thickness interfaces – Part I:
mathematical formulation and validation with molecular dynamics", Computational
Materials Science
, Vol. 50 (5), 1625-1633, 2011. doi:10.1016/j.commatsci.2010.12.024

M. Paggi, P. Wriggers: "A nonlocal
cohesive zone model for finite thickness interfaces – Part II: FE
implementation and application to polycrystalline materials", Computational
Materials Science
, Vol. 50 (5), 1634-1643, 2011. doi:10.1016/j.commatsci.2010.12.021

In the firtst one we found that, REMOVING the elastic deformation from the total MD traction-displacement curves, a unique inelastic traction-separation relation can be obtained, regardless of the size of the MD simulation. It seems that the authors of the present paper did not remove such an elastic component. This would lead to results dependent on the size of the sample. I would kindly ask the authors a clarification on this point.

Depending on the type of nonlinearities and on the material, we found that MD simulations provide different shapes of the CZM (in spite of the limited number of interatomic potentials that can be used). Therefore, I do not think that a single CZM can be reproduced by MD. Rather, different shapes of the CZM can be obtained dependig on the actual evolution of damage (and nonlinear dissipative mechanisms) occurring at that scale.

Thanks in advance and best regards,

Marco Paggi

___________________________________________
Dr. Ing. Marco
Paggi, Ph.D.
Assistant Professor of Structural Mechanics at Politecnico di Torino
Alexander von
Humboldt Research Fellow 2010-2011
Member of the Executive Board of the
Italian Group of Fracture 2009-2013

### length scale

Dear Marco,

thank you for your comment! This is what my blogs actually intend to promote: scientific discussions on relevant topics. But the response to my contributions was “moderate”, so far, and not even the authors of the papers felt addressed to respond. They did not react to your request for clarification, either. The idea of publishing has (once) been to face up to the scientific community and accept the response of the fellow scientists. It appears however that publishing has partly degenerated to increasing the individual number of references in journals. As soon as a contribution has been accepted there seems to be no more interest in discussing it.

Anyway, I agree with your conclusion that “different shapes of the CZM can be obtained depending on the actual evolution of damage ... occurring at that scale“, of course. My point was “that scale” and my argument is that the scale of MD simulations differs by some orders of magnitude to the damage mechanism of ductile tearing. I would still appreciate further comments to this issue.

Wolfgang

### Cohesive zone models for different scales

As the corresponding author of the paper you commented, I would like to address our opinion to your comments. For many years the molecular dynamics (MD) has been introduced to model deformations and damage mechanisms of single crystals and polycrystals, with many known limitations. One of the common big issues is the dimension of the MD model, the dimension of such computational models is by far below 1 micon3. The small dimension causes other problems into modeling, e.g. reflection of dislocations, interactions with boundaries, etc., so that these simulations cannot predict the property of the engineering metals. Bridging different length scales is still a difficult task in materials simulation and quantitative studies of materials damage in nano-scale are still an open issue. In this sense, it is absolutely correct to be skeptic, to directly compare MD simulations with macroscopic, even microscopic behavior of materials.

However, it is often not the goal of these works to catch the real metal property, but to find interdependence of different field variables. With help of this kind of computations, one may find new ideas which cannot be measured in laboratory and observed under modern SEM/TEM. Computational experiments in a nano-scale model could provide new evidences for description of micro-scale even meso-scale material behavior. In this sense these scientists are not ignorant of physics behind the reality, but they are trying to consider the problem in different way.

In the paper (H. Krull and H. Yuan: Suggestions to the cohesive traction–separation law from atomistic simulations. Engineering Fracture Mechanics, Vol. 78, 2011, pp. 525-533) we have studied the fracture mechanism in nano-scale to provide qualitative implication for formulating a cohesive law. The values from MD like cohesive strength and critical separation, of course, are much too high and low, respectively, for continuum models. But the internal correlation among different variables is of interest and can provide fresh ideas for formulating a cohesive zone model.

The cohesive zone is a strong simplification of the damage zone around a crack tip, which can neither be measured nor observed in real materials. However, it should contain nano-, micro and meso-scopic processes happened around the crack tip. It is certainly a meaningful way to study the failure process based on debonding and continuum mechanics models. If a fine grained material, e.g. fine grained Nickel superalloy, with grain size between 1 to 10 microns, the applicability of the conventional continuum plasticity becomes an open issue. In nano-srystalline the crystal plasticity could even be too coarse. The MD simulation cannot reach the dimension of whole grains, but can provide the characters of subgrains. At least, it will generate new information from the small scales. Additionally, the simulation of debonding is based on an additional model, which may affect the final results significantly. The agreement with the exponential law of Needleman cannot simply be taken as the direct consequence of the MD modeling. The predicted damage process in nanoscale substantially depends on MD model. The model of Needleman is directly derived from simple atomistic calculations of interfacial separation, whereas the present MD computations considered effects of deformations and degradations around the crack tip. The cohesive zone should contain influence of stress gradients, stress triaxiality and discrete damage process around the crack tip. In this sense, the cohesive law from MD simulation is various and sensitive to evaluation of numerical results. Our work developed a reasonable averaging method and has shown that the cohesive law has a similar form as the exponential function suggested by Needleman, but only for steady crack growth. Crack initiation is somehow different. Moreover, it is shown that this exponential law is also valid for single-crystal materials without interface, which is beyond assumption of Needleman.

Regarding to nucleation of crack, Professor Brocks focuses on void nucleation and coalescence in ferretic alloys, described in the Gurson-Tvergaard-Needleman model. This consideration is certainly not sole mechanism for metal failure and not contradictory to our computation for a cracked fcc crystal. In the paper the MD was used for studying crack growth in a single crystal. The crack under mode I shows that the void may be nucleated due to high stress triaixiality and coalescence each other. Phenomenologically the nano-event is similar to meso-observation, but from different origins. Whether or not the nano-scale failure runs in this form has to be verified in futural experiments.

If one uses cohesive zone model, he is trying to separate plastic deformations from material damage. That is, the plastic deformations of the material should be described by the plasticity model, and the material failure should be considered in the cohesive zone, representing the simplified damage zone, which is additionally described by the cohesive law (or the traction-separation-law). Under mode I loading condition, the traction responsible for void nucleation is the tensile stress ahead of the crack tip, as predicted in the MD computations, while the metallic plastification is characterized by the Mises stress. This consideration is confirmed in the MD simulations. This is the significant difference from continuum damage mechanics, in which damage and deformation should be described in a set of evolution equations. This is not the final death sentence to any other simulation, but could be a new starting point for further investigations of the influence of stress triaxiality to the cohesive law.

In MD simulation the plasticity is considered in the form of nucleation and motion of dislocations and cannot be excluded in computations. In the present work, however, the large amount of dislocations is suppressed by the rigid boundaries of the specimen, so that the crack initiation and propagation are along in the symmetric plane. f the crack is located in a (100) plane of the fcc lattice, crack propagates without significant dislocations. However, if the crack is located in a (111) plane, dislocations are observed and plasticity plays a role. Due to the reflection of dislocations at the boundaries of the simulation model, however, the stress field at the crack tip is influenced by these dislocations, so that an explicit investigation becomes more difficult.

Additionally, the cohesive zone models can not only be applied for meso-scopic problem, but also in micro- and even nano-scale to predict, e.g., separations of materials in various scales. One common problem is in formulating the cohesive law. From this point of view, the MD simulation could even be directly used for identifying the material property.

_______________________________________

Prof. Dr. H. Yuan

Department of Mechanical Engineering

University of Wuppertal

Email: h.yuan@uni-wuppertal.de