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Paper by Ellad Tadmor & Nikhil Admal---Atomistic Definition of Stress

Pradeep Sharma's picture

For both mechanicians and physicists, linking continuum concepts to the underlying microscopic characteristics of materials has remained a major preoccupation. Certainly, with the recent advent of the so-called "multiscale" modeling approaches, this topic has been brought to a sharp focus. Cauchy stress is one such continuum concept that has courted a fair amount of controversy.

I personally became interested in this topic a few years ago when I initiated research on a quantum definition of stress. This became troublesome quite quickly as I found that (let alone quantum perspective) I was bewildered and confused by even the literature on classical definition of atomistic stress. Papers by Professor Ian Murdoch went a long way in contributing to my understanding of this subject.

Last week, I had the pleasure of reading a pre-print by Ellad Tadmor and his graduate student (Nikhil Admal). This paper (honestly more of a treatise given its scope and length!) provides one of the clearest and most comprehensive yet exposition on this topic---at least for me. Apart from a fresh approach to the derivations, the paper clarifies several mis-notions and establishes connections between the various definitions of atomistic stress that abound in the literature.

The readers can make up their own mind. The pre-print is attached below (with permission from the authors). It will appear in a special issue of Journal of Elasticity honoring Professor Walter Noll. Incidentally, it is worth mentioning that a translation is now available of Professor Noll's relatively lesser known paper on atomistic stress----see also a discussion by Richard LeHoucq . Noll's original paper appeared in 1955 and was in German. Ellad, Ian (Murdoch) and I, all use the results derived by Professor Noll in some form or the other.

PDF icon admal_tadmor_final.pdf918.99 KB


Mike Ciavarella's picture


I don't want to sound offensive, or too steryle polemic, as I understand this is high class mathematics... but really, if Leonardo da Vinci, was able to work out beam theory without Chauchy stress, what has this to do with Engineering?  Especially where is the connection with  Aerospace and Mechanics?

I suggest instead to read Leonardo of 500 years ago... today we're going too far with atomistic... and academic discussions about stress, see the other guy who made a lot of fuss a few months ago about a "new theory of stress" -- I forget the name ... he was german I think.

Better to read Leonardo if we are to learn something from a genius.


ME Magazine Print

copy is for your personal, non-commercial use only.  This article may
not be reprinted for commercial purposes without the written permission
of Mechanical Engineering magazine and ASME.  © 2008
Mechanical Engineering magazine

The Da Vinci-Euler-Bernoulli Beam Theory?


by Roberto Ballarini

Many mechanical engineers and educators are not aware of some of
Leonardo da Vinci's fundamental contributions to solid mechanics, fluid
mechanics and mechanical design. These contributions appear in Codex
Madrid I, one of two remarkable notebooks that were discovered in 1967
in the National Library of Spain (Madrid), after being misplaced for
nearly 500 years. Selected scientific and artistic works that appear in
Codex Madrid I are summarized, translated, and discussed by Vincian
scholars in The Unknown Leonardo (edited by L. Reti, McGraw
Hill Co., New York, 1974), a book that should be of significant
historical interest to the mechanical engineering community.

Mechanics textbooks and instructors often credit Galileo's analytical
attempt to determine the load carrying capacity of a transversely
loaded beam as the beginning of beam theory (see for example Mechanics
of Materials
, J.M. Gere and S.P. Timoshenko, Fourth Edition, PWS
Publishing Company, 1997). This is consistent with Timoshenko's
influential History of Strength of Materials (Dover
Publications, Inc., New York, 1983), which summarizes the individual
contributions made to beam theory by Galileo, Mariotte, (Jacob)
Bernoulli, Euler, Parent, and Saint-Venant.

However, while his name does not appear in any textbook discussion of
beam theory, the mechanics education community should be aware that da
Vinci made a fundamental contribution to what is commonly referred to as
Euler-Bernoulli (engineering) beam theory 100 years before Galileo.
Historians of mechanics did not cheat Leonardo; they simply were not
aware that he made the fundamental hypothesis upon which Euler-Bernoulli
beam theory rests in Codex Madrid I.

The normal stress and the moment-curvature formulas for slender
linear elastic beams,
The Da Vinci-Euler-Bernoulli Beam<br />
Theory? - Formula

constitute what is universally referred to as Euler-Bernoulli beam
theory. In these formulas
The Da Vinci-Euler-Bernoulli Beam<br />
Theory? - Figures

represent, respectively, the normal stress, bending moment, distance
from the neutral axis, moment of inertia, deflection of the neutral
axis, and Young's modulus. The fact that the theory is not called
Galileo-Euler-Bernoulli beam theory is understandable, since Galileo
incorrectly assumed that under transverse loading a beam's cross-section
develops a uniform stress distribution. Parent, on the other hand, who
was the first to obtain the correct stress distribution and to relate
the stress to the bending moment, was arguably short-changed.

As everyone well versed in mechanics knows the key to the development
of the above referenced formulas is the assumption made by Bernoulli in
the 17th century that the strain is proportional to the distance from
the neutral surface, the constant of proportionality being the
curvature. It is remarkable that da Vinci hypothesized this form of
strain distribution two centuries earlier. As explained by Carlo
Zammattio in his article "Mechanics of Water and Stone" (in The Unknown
Leonardo), da Vinci established all of the essential features of the
strain distribution in a beam while pondering the deformation of
springs. For the specific case he considered of a rectangular
cross-section, da Vinci argues for equal tensile and compressive strains
at the outer fibers, the existence of a neutral surface, and a linear
strain distribution. Of course, da Vinci did not have available to him
Hooke's law and the calculus. If he did, it is conceivable that he would
have derived the formulas listed above, so that beam theory would be
referred to as Da Vinci Beam Theory.

As discussed by Zammattio, the Codex includes other remarkable
contributions, including the partitioning of energy into potential and
kinetic components, within the context of what is commonly referred to
as the Bernoulli Equation for fluids. In fact, da Vinci introduces (in
words and not equations) this fundamental theorem of fluid mechanics,
which states that for frictionless flow along a stream line the total
head is constant. Contributions to conceptual mechanical design include
the gear drive used in a bicycle.

The following image is a scanned version of the reproduction of folio
84 of Codex Madrid I that appears in Zammattio's article, together with
Zammattio's translation of da Vinci's notes. Ignoring the fact that da
Vinci wrote right-to-left in mirror-image script, this discussion may as
well have been copied from a modern mechanics of materials text.

The Da Vinci-Euler-Bernoulli Beam Theory?Da Vinci's discussion of
the deformation of a beam/spring with rectangular cross-section.
taken from the book "The Unknown Leonardo," McGraw Hill Co., New York,

Translation: "Of bending of the springs: If a straight spring is
bent, it is necessary that its convex part become thinner and its
concave part, thicker. This modification is pyramidal, and consequently,
there will never be a change in the middle of the spring. You shall
discover, if you consider all of the aforementioned modifications, that
by taking part 'ab' in the middle of its length and then bending the
spring in a way that the two parallel lines, 'a' and 'b' touch a the
bottom, the distance between the parallel lines has grown as much at the
top as it has diminished at the bottom. Therefore, the center of its
height has become much like a balance for the sides. And the ends of
those lines draw as close at the bottom as much as they draw away at the
top. From this you will understand why the center of the height of the
parallels never increases in 'ab' nor diminishes in the bent spring at

In conclusion, I suggest that the mechanics community remove the
question mark in the title of this paper, that future discussions on the
mechanics of beams acknowledge da Vinci's fundamental contribution, and
that everyone look for a copy of The Unknown Leonardo.


ASME member Roberto Ballarini is a professor of civil
engineering at Case Western Reserve University in Cleveland. In August
of 2003, he will be F.W. Olin Professor of Mechanical Engineering at
F.W. Olin College of Engineering.



Pradeep Sharma's picture

There are many aspects to "mechanics". One of them is its use to understand materials behavior. Although I cannot claim to be very familiar with your work, a cursory glance suggests that to some degree, your own research is in the same vein.

Atomistic methods; by themselves, or as part of a "multiscale modeling" approach where they are connected with a continuum analysis, have emerged as a powerful tool to understand the basic science underpinning materials phenomena. I can provide many examples. Here is a particularly challenging one that is related to both mechanics and aerospace.

Mechanical deformation and damage of ceramics that can reliably operate at high temperature (> 2000 C) environment is an active area of research and currently even the best materials candidates are incapable of doing so. Metal alloys are simply unusable. What kind of microstructural engineering must be performed to design new materials that resist creep and oxidation effectively at such high temperatures? Creep research has been around for several decades but many questions still remain open and conventional mechanics approaches (already exhausted to some degree) are unlikely to provide further insights. Atomistic methods (although not easy to use for such problems) are already showing possible paths to understanding fundamental mechanisms underlying creep damage. For atomistic methods to provide relevant insights of interest to experimentalists or continuum mechanicians in this, and many other problems I did not mention, we must establish a common language. How will you compare an experimental stress-strain rate curve to predictions from atomistics if we cannot agree on how to calculate/define stress in a discrete ensemble? Admal and Tadmor, in their introductory paragraphs, also provide some additional examples.

Who should work on defining mechanical stress within atomistics? Mechanicians or Physicists? Would you (or would Da Vinci) draw a sharp boundary between the two classes of scientists? If I recall correctly, since you do invoke him constantly, Da Vinci pursued with unabashed curiosity all aspects of natural phenomena that interested him.

Finally, in closing, I am not inclined to comment on the bulk of your post which appears to be "cut & paste" of an article on Da Vinci. I am unclear how it is relevant to Tadmor & Admal's paper. I am glad that Da Vinci was a genius. I would be however more curious about your technical comments on the posted paper (if the topic interests you). You may consider taking a look at this helpful post created by iMechanica administrators and moderators (see the section on "Elements of Style"). It provides instructions on how to include links in your posts and comments obviating the need to cut & paste large, entire articles in a comment body.


Mike Ciavarella's picture

Pradeep, since you sounded like taking this personally, I did read the (not easy) 81 pages of the paper.

It is certainly highly mathematics as I had suspected from a glance and this is not an offense.

The paper is extremely nice and well written.  Just it is mathematics, no doubt about it, applied to engineering, not pure and abstract of course.

There is no reference in 81 papers, to development of new materials, or to applications in Aerospace or Mechanics engineering.   I would see this more naturally from Mathematics department.  However, I have nothing against their authors, and I know some Schools (including a similar one in Elasticity) exist in Italy too.  Actually, the School of Elasticity is dominant in Italy, with names such as Villaggio, Podio-Guidugli, Gianpietro Del Piero.   You know, incidentally, how strongly Gianpietro Del Piero was resisted by his students and the entire university and the entire city of Ferrara, since he was considered too mathematical and too difficult to pass his exam at Engineering?  It went to the press.  And Del Piero was "suggested" to take many years sabbatical leave.  Some people suggest in Italy it was a disaster to have such pure Elasticity in standard Engineering course...   But I don't take side on this.


I am glad you do add some loose connections to materials engineering.

I do understand there are various definitions of stress possible, and this gives rise to confusion.   But in the end, except for the unifying framework, I see the authors return to 3 of them in the conclusion, and are not able to suggest a "best one".   So?

Finally, numerical experiments involving molecular dynamics and lattice statics
simulations are conducted to study the various stress definitions derived in this paper.
It is generally observed that the Hardy stress definition appears to be most accurate
and converges most quickly with the averaging domain size. In situations where a
more localized measure of stress is needed, such as near surfaces or defects, the Tsai
traction can be used instead. The virial stress is less accurate than the other two definitions
and converges most slowly with averaging domain size. Its main advantage
is its simple form and low computational cost. One of the most interesting results,
which requires further study, comes from Experiment 2 of a crystalline system under
uniform hydrostatic stress. Fig. 6.2(c) shows that although the potential and kinetic
parts of the Tsai traction largely depend on the position of the Tsai plane between two
adjacent lattice planes, the total stress remains constant. This calculation provides a
striking demonstration of the interplay between the kinetic and potential parts of the stress tensor.



With reference to Leonardo, yes, I am an admirer (and I notice also the Elasticians, if the Doctoral School in Pisa is named after Leonardo, as many other things).  In his time, and for his genius, he was able to keep all branches of science, or art, and of human knowledge in one person, and despite he had no formal education (or maybe because of it), as he was, in his words, "omo sanza lettere" --- illiterate.  The scientists of his age had to speak latin, as the scientists of our time have to specialize in a little and narrow field, to make their names.

In these respects, maybe useful to quote

Human subtlety will never devise an invention more
beautiful, more simple or more direct than does nature because in her
inventions nothing is lacking, and nothing is superfluous.

With respect not to post an entire paper, I could not resist not to attach the original Leonardo text.  Sorry!

Mike Ciavarella's picture

New theory
of elasticity & deformation

Starting with a few questions which I asked in my
introductory class 30 years ago at UC ... found enough reasons over the
years to reject the current theory of
elasticity, stress and continuum mechanics entirely.
... viscous and plastic.
However, asking new questions resulted in the
stoniest of silence; if you folks think you are ...

Blog entry - Falk
H. Koenemann
- 2009-03-13 22:02 - 117 comments - 0 attachments

Prof. Michele Ciavarella
Politecnico di BARI
V.le Gentile
70125 BARI, Italy
tel+390805962811 fax+390805962777

personal blog


Rui Huang's picture


Thanks for suggesting this paper. I have always had questions about how stresses are calculated in various atomistic models, including first-principle quantum mechanics calculations. In my understanding stress is a continuum concept, which is only meaningful if defined in a continuum framework. I wish I will soon have time to read this paper in details. After a quick glance, however, I am surprised by the discrepancies in the three stress calculations (Hardy, Tsai, and virial). I must admit I am only slightly more familiar with virial stress calculation and essentially unaware of the other two. I have also noticed that there has been some debates over the contribution of kinetic energy in the virial stress calculation. In the case of molecular mechanics (statics) models, no kinetic energy is involved and thus such debate does not apply. To further simplify the situation, under a macroscopically homogeneous deformation, the stress can also be calculated from the derivatives of the potential energy, which can be calculated directly from the atomistic model (Cauchy-Born). Sure, this is rather special case. But at least in such simple cases, to my experiences (very limited, MM only), the stress calculation is unambiguous: the potential method, the virial calculation (without the kinetic term), and direct traction calculations (interatomic forces) all yield essentially the same stress (true or nominal). Of course, a self-consistent definition of stress in a continuum framework must be followed in all calculations. Now, my question is: Would the Hardy or Tsai stress calculations be any different even in these simple cases? If so, what would be the fundamental physics (or mathematical process) that leads to the discrepancies?



Pradeep Sharma's picture

Dear Rui,

Sorry for the delay in my response. I am glad you got a chance to go through the paper. Regarding your second and third sentences, reconciliation of what we understand to be stress in the continuum sense and the closest quantity to that which can be derived from a discrete setting is precisely the objective of the posted paper.

Indeed, the discrepancies between various stress measures are quite striking and I have to give credit to the authors for choosing nice illustrative problems to emphasize this.

I think as far as the necessity of kinetic energy is concerned, that debate has been settled a little while back. Yes, there was (and perhaps still is) confusion in the literature, but kinetic part of the stress tensor cannot be discarded. The need for the kinetic part is clear not only from the derivations in this paper but also in previous works (see the paper by Murdoch that I provided a link to). A physical argument is more easily digested however, and Tadmor/Admal provide one on page 58 where they discuss the so-called "Experiment 1". In an ideal gas, the potential contribution is zero and the trasmitted pressure is entirely due to the kinetic term of the definition of stress.

Hardy, Tsai and Virial stress measures will all give different results even at zero kelvin. In fact, there is no ambiguity or difference in the kinetic parts. The difference stems from how the potential interaction term is considered. Among the many other contributions of this paper, the illustration that all three definitions of stress can be derived from a common setting flushes out those differences quite clearly. Rather than write a lengthy note on the specific cause to why they are so different, I will simply point to page 61 (Experiment 3/4). Here the authors analyze sample boundary value problems at zero kelvin. The discrepancy you see between the various measures (e.g. figures on page 65) all stem from the different ways the potential term is handled.

Rui Huang's picture

Hi Pradeep,

Following your suggestion, I read from page 61 of the paper and stopped at page 63. I have several questions about Experiment 3. As the authors stated, this is a zero-temperature (0 K) and homogeneous deformation problem. In such a case I expect the stress to be homogeneous as well and thus am surprised to see oscillations (or variations) of all the stress calculations in Fig. 6.3. At the equilibrium state, should all the unit cells deform identically? At least for the virial stress that I am somewhat familiar with, I do not see how it could vary with the domain size in such a homogeneous equilibrium state, unless the homogeneous equilibrium state was never reached.

The second question I have is regarding the material model by the modified L-J potential. The authors list the constant elastic modulus in Eqs. (6.6)-(6.8). However, is the material really linear elastic? The potential function in Eq. (6.5) is definitely nonlinear. The stress value used in Experiment 3 is 1, corresponding to about 2% in strain. Is the material still linear elastic at this strain level? If not, the boundary conditions in Eqs. (6.9) and (6.10) do not give a uniaxial stress state. In other words, the real stress in the atomistic model is not 1.

Another question is: how were the interatomic forces calculated in this model? The authors did not mention (perhaps they did elsewhere) whether multibody interactions are accounted for in this experiment. Apparently the L-J potential is a pair potential. But the cut-off radius (2.5) is long enouth to include interactions between second and even third nearest neighbors. Are all these interactions included in the calculations of potential and interatimic forces? I am also curious if these interactions contribute to the predicted elastic constants.



admal's picture

Dear Rui,

Following are my answers to your questions:

1. From a continuum mechanics perspective, you are right in saying that one would expect the stress to be homogeneous for the conditions assumed in experiment 3. But this is hardly the case in an atomistic system because a continuum model is obtained as a local thermodynamic limit of an atomistic system. For example, in experiment 3, I can get to a homogeneous stress state as predicted by the continuum model, from an atomistic system with the same boundary conditions only in a limit, in the sense that if I started off with a much larger system (relative to what I consider in the experiment) so that I can have an averaging domain (a ball for the virial and Hardy calculation and a plane for Tsai) with larger number of particles, then the fluctuations observed in the experiment will decrease. So fluctuations in the fields obtained from an atomistic system have their source also from the spatial averaging that is being used in addition to the dynamic effects resulting from non-zero temperature.

2. I now realize that probably the first question that comes up is: why did the authors push the linear elastic model to its limit by taking such high strains. This is because I had to do a balancing act taking into consideration the limited computational capcacity. In order to obtain a reasonable magnitue of stress well within the scope of the linear elastic model and which has small fluctuations relative to its magnitude (the reason for which is explained above), I had to choose a much larger model than what I had considered in the experiment and much larger averaging domain, which means higher computation. Therefore to keep the fluctuations small relative to the average stress I had to increase the stress in the system. So to answer to question - is the system linearly elasic? - I performed an experiment and observed that the system is behaving in a linear fashion even at such high strains.

3. The interatomic forces are calculated using the expression given in (3.41). The L-J potential is a pair potential, so there are only pairwise interactions in the model. The cut-off radius only dictates that pairs of particles which are within the cut-off radius from each other interact. Therefore the cut-off radius does not dictate if there are any multi-body interactions in the system. The elastic constants are calculated using the pair potential being used. A reference I used for this - Crystals, Defects and Microstuctures by Rob Phillips.

Finally, in addition to my first answer, I was curious if it is possible to define the thermodynamic limit in a more rigourous way, so that one can get to the continuum stress (which would predict homogeneous stress in experiment 3) from the stress fields for the atomisitic systems derived in the paper. One solution is to take the averaging domain as the whole system, so the stress obtained is just a single value, but this fails to capture variations of stress observed in experiment 4. Another procedure is to actually take the number of particles and the volume to infinity so that their ratio remains constant. In such a limiting procedure one gets to a thermodynamic equilibrium (defined in footnote 1) but not a local thermodynamic equlibrium and under this limiting procedure the system cannot have a non-zero shear stress (Statistical mechanics of nonlinear elasticity by Oliver Penrose is an excellent paper for reference).

In addition to what Pradeep suggested, and based on your questions, I would like to suggest that section 2 can be overlooked in the first reading as most of the theory for the understading of various stress definitions begins from section 3 and and it is independent of section 2. Section 2 talks about how geometric ideas can be used to interpret stress in an equlibrium setting. Also section 3.6 can also be over-looked in the first reading as this is a generalization of the stress definition, in order to quickly get to the main point. Let me know if there are any other question you have.

Nikhil Admal

Rui Huang's picture

Dear Nikhil,

Thank you very much for answering my questions. I take this as an opportunity to learn more about MD and atomistic simulations in general. I will follow your suggestions to read other parts of your paper. Hopefully, with some time I will be able to finish reading the entire paper.

I think you missed the point in my first question. I thought Experiment 3 is a zero-temperature problem according to the paper. Otherwise I would understand fluctuations due to the dynamic effects at a non-zero temperature. For example, in molecular mechanics (statics) calculations, no temperature or dynamic effect is involved. Would you still expect fluctuations in the stress calculations? I think the fluctuation due to spatial averaging alone can be easily fixed.

For the third question, you did not answer directly if the interactions between second and third nearest neighbors are included in the calculations of potential and interatomic forces. Your answer seems to imply this is the case. I will check out Phillips' paper for more details on this.



The question you raise is very interesting. I think the fluctuation can be explained by several reasons:

+ The simulation domain is rather small so that the size effect becomes important. 10 unit cells is not somthing very big. According to my experiences, the surface effect is only negligible after 3 or 4 atoms layers from each sides.

+ Because the simulation domain is small, the average domain is also small. Furthermore the average domain is a circle not square. The convergence also depends on the geometry of the average domain. A square domain is better for highly ordered materials (ex: crystal) whereas a circle one is prefered for fluid materials.

admal's picture

Dear Rui,

There are multiple ways fluctuations can be observed. So let me write all possible ways:

Before that, note that stress in atomistic systems depends on the point of interest and the averaging domain.

1. Fix the point of interest at which you want to calculate the stress and also fix the averaging domain. Now, we can observe the fluctuations in stress w.r.t time only at a non-zero temperature because of dynamic effects. So this is ruled out in experiment 3.

For the following two cases consider a system at 0K (for example, experiment 3 and 4).

2. Fix the point of interest at which you want to calculate the stress but not the averaging domain size. Plot the stress as a function of the averaging domain size. In this case, we observe fluctuations w.r.t averaging domain size. These fluctuations are observed because of the dependence of stress on the averaging domain size. This case applies to experiment 3 and Fig. 6.3. The only possible way I can see to decrease these fluctuations is in the limit as the averaging domain size goes to infinity, i.e., the thermodynamic limit, as is observed in Fig. 6.3. In this limit the stress is made independent of the averaging domain size and there are no fluctuations, and this is exactly what continuum stress refers to.

3. Fix the averaging domain size. Plot the stress as a function of the point of interest. In this case we observe fluctuations w.r.t space. This case applies to experiment 4 and Fig. 6.4.

You mentioned -  I think the fluctuation due to spatial averaging alone can be easily fixed. I am not sure at this point how this can be done other than taking a thermodynamic limit as mentioned in 2. Let me know if I am still missing the point you are trying to raise.

Regarding the final question, when you said multibody interactions in your earlier post, I thought you were referring to multibody interactions that arise in multibody potentials. Since we are using a pair potential such a situation does not arise. Since all possible pair interactions are included, interaction between second and third neighbor is also included by definition of a pair interaction.

 Nikhil Admal

Rui Huang's picture

Dear Nikhil,

Thank you again for answering my questions. I believe I now understand what caused variation in your calculations of virial stresses in Experiment 3. As pointed out above by TO Quy Dong, the domain you used to calculate the virial stress is a sphere, which is different (in shape) from the FCC unit cell. If you instead use domains of the same shape as the unit cell, containing one, two, or any integer number of unit cells, would you still have such fluctuation or variation with the domain size? In the case of homogeneous deformation with periodic boundary conditions, the forces across the boundary of the selected domain can and should be included in the virial stress calculations. Otherwise the volume of the domain should be corrected. Again, for Experiment 3, using a fixed domain shape and size, the stress is homogeneous everywhere. The smallest domain one can use for a correct stress calculation is one unit cell.

For Experiment 4, since the deformation and stress are expected to be inhomogeneous, it is not obvious to me what domain shape and size shall be used to calculate the true stress at one particular point. Intuitively, it cannot be too small or too large. Depending on the inhomogeneous distribution of stress, both the shape and size of the domain could have effects, and there may not exist an unique answer to this question for a generally inhomogeneous (and anisotropic) stress field.



admal's picture

Dear Rui,

Before I answer your questions, I would like to clarify (if there is any confusion) that the non-uniqueness of the stress tensor that is addressed in the paper does not deal with the question of what is the ideal shape of the averaging domain so that the stress is unique. The non-uniqueness that is addressed is realated to different functional forms of the stress tensor that many people have proposed in the past which seem (to me) to be completely unrelated. So in this sense, I do not differentiate, say, two hardy stresses with different averaging domain.

The question that you and TO Quy Dong are very interesting, but these are not raised in the paper, because of the following reason:

You asked "If you instead use domains of the same shape as the unit cell, containing one, two, or any integer number of unit cells, would you still have such fluctuation or variation with the domain size?"

Even if the averaging domain has the same shape as the unit cell, the fluctuation would, in general, be non-zero and it is zero only under a very special condition, which I think is too much to ask for. The fluctuations are zero only when the point of interest (at which your averaging domain is centered) along with other positions of the crystal satisfy some symmetry conditions. For example, in cubic crystals if my point of interest is at the position of an atom then a cubic averaging domain would probably work and give you zero fluctuations. But instead, if my point of interest is at some arbitrary point inside the cube, due to which the symmetry in the crystal is broken, i.e., the positions of the atoms and the position of interest all taken into account do not satisfy any symmetry, then I think that the cubic domain also will give you some fluctuation, with a different signature compared to that resulting from a spherical avg domain.

So the non-uniqueness of the stress tensor in atomistic systems that we refer to (see the expressions for Hardy, virial, Tsai and DA stress - they all look very different) is different from the non-uniqueness resulting from different averaging domain, and I think the former one is much more serious in the sense that it gives a more qualitative picture of how these stress definitions are different from each other. Moreover the former question is more tractable, as I think there is no definite answer on the existence of an averaging domain that gives a unique stress tensor (in functional form and value).

"In the case of homogeneous deformation with periodic boundary conditions, the forces across the boundary of the selected domain can and should be included in the virial stress calculations"

The virial stress, by definition, does not include the forces across the boundary and this is the reason is falls short (see Fig.6.3) (But in the thermodynamic (TD) limit it converges because the contribution due to forces across the boundary tends to zero as volume tends of infinity). These forces are taken in to account in the Hardy stress. Therefore the Hardy stress converges much faster to the continuum stress in the TD limit.



Dear Nikhi,

From my point of view, stress tensor makes little sense for small systems, especially for crystal systems where the potential forces play an important role.

Let us look at the virial stress formula. The potential term depends on the volume and the number of molecules. Roughly speaking, it depends on the number density. For small systems the number density varies a lot whereas for large systems it is stable.

For example, for small crystal systems, when averaging, there must be a big jump (or discontuity) in number density and in stress whenever the average domain boundary meets an atom.

In your figure, the virial stress is not stable, it can still increase. It would be interesting to see if all the stress formulae converge to the same line when the domain is larger than the one studied in the paper.

Quy Dong

admal's picture

Dear  TO Quy Dong,

You said - "From my point of view, stress tensor makes little sense for small systems"

I like to take the point of view that the stress tensor in atomistic system depends on the length scale and one is free to choose this length scale by choosing an averaging domain. I think it is very important to remember that the stress tensor that we try to define for atomistic system strictly corresponds to the Cauchy stress tensor in continuum mechanics only in the thermodynamic (TD) limit. By this reasoning, I am not worried about the fluctuations, as long as these tend to zero in the TD limit.

I agree with your example and the reasoning for the fluctuation due to the size of the averaging domain. 

In your figure, the virial stress is not stable, it can still increase.

Based on the derivation of the virial stress, I can say that the virial stress tends to other stress definitions (Hardy, Tsai) and the continuum stress in the thermodynamic limit. I agree that it would have been much better if we had taken a larger domain to show that the virial stress actually converges. Based on the steps involved in the derivation, I think it does.


Nikhil Admal




admal's picture

Dear Pradeep,

Thank you for starting this discussion and clarifying various question posted by others. I will be glad to particiate in the discussion and answer any questions.

 Nikhil Admal

Mike Ciavarella's picture

Of the questions.  Which is:  given 500 years ago Leonardo da Vinci was able to invent many things we have built only today, and without notion of stress -- he did invent beam theory incidentally, and not in the wrong way as did much later on, Galileo Galilei --- and for centuries we have anyway built cathedrals, etc. etc.

What do you think these details on atomistic simulations should bring?

Of course, my question is a little provoquative.  I like "debates".  But, even in terms of speculation, I am sure sometimes even purely curiosity driven research is useful perhaps 50 years later.  But a lot of the curiosity driven, remains unused.  

So are you hoping your curiosity will be, at least in 50 years if not, now, of any use?

Thanks Mike

admal's picture

Dear Mike,

The usage of the results in the paper are highlighted in the paper and to some extent described by Pradeep in an earlier post to your question. Let us know if you have any technical question/comment related to the paper.

Regarding the question that you have raised, I think this is just not mathematical abstraction or curiousity (which, by the way, I think is very important), but it is more than that. There is an urgent need (not 50 years from now, but now!) to understand the various stress definitions and this saves many researcher's time in sorting out the confustion regarding stress in atomistic systems.


Mike Ciavarella's picture


my technical questions are the following


1) can you mention to me a list of 4-5 applications (like where do we develop this atomistic simulation in order to get some use in, say, areonautical engineering)

2) can you mention to me a list of 4-5 materials where this has been used -- with examples

3) can you mention to me a list of 4-5 challenging problems where you see this WILL be used?




admal's picture

Dear Mike,

Following are my brief answers to your questions:

1. Since I come from a structural engineering background, the simplest and most studied problem is the study of crack propagation. Although there are many good continuum theories (somewhat related to atomistics are nonlocal continuum theories such as Peridynamics), atomisitic simulations offer a good alternative to study these problems. Any problem where there are multiple length scales involved, say in spatial dimension, such as study of reentry vehicle from space. This problem involves multiple length scales at the gas surface interface and as far as I can see, the only tractable option is to use a coupled atomistic-continuum simulation.

2. Again I list a few: Carbon nanotubes, Silicon wafers are studied using atomisitic simulations.

3.Challenging problem related to academics: This study can be used to further study plasticity, in particular dislocations. Something that I think is challenging is to relate this work to the work by Ortiz and Ariza - Discrete Crystal Elasticity and Discrete Dislocations in Crystals.

Nikhil Admal

Mike Ciavarella's picture

Dear Nikhil

  thanks for answer. In these qualitative terms, I had also seen something.  A good review paper for each topic, would be useful here.  However, I wonder if we really need the notion of stress for these things.  Particularly, for fracture, we need different concepts.  Perhaps, while we make so much effort to "return" to familiar concepts, there are easier ones out there which we are NOT looking for?  Just a thougth provoquing idea...


Bin Liu's picture

I think my previous blog and two attached papers might also attract the attention of readers here. Several examples are discussed to demonstrate the validness of various atomic stress definitions.

Mike Ciavarella's picture


your references are also useful, and elegant. But I do see exactly the same problem.  I don't see where is the application!   Take you intro below.  You say that atomistic tend to confirm continuum mechanics.  Not only, but "atomistic are a powerful tool".   But can you be more specific?    Powerful for what?   You guys do not make a single example, and seem obsessed to return to a definition like Cauchy stress.   But what for??! Especially if, as you say, most continuum mechanics results are confirmed.   So when they are not confirmed?  In which case, why do you think we need Cauchy stress?  If the results are different, maybe we need to interpret them with a different semantic....

Maybe we need to take some time to step back and think about it...




Continuum mechanics has been very successful in predicting
material behaviors at macroscopic scale.Moreo ver,
with the emergence of nanotechnology and nanoscience,
many recent researches demonstrated that, in many cases,
the concepts of continuum mechanics can still be applied
to discrete atom systems at microscopic scale (e.g.,
Refs.[1–5]).On the other hand, atomistic simulation has
been a powerful study tool in mechanics research.In the
theoretical framework of continuum mechanics, Cauchy
stress is one of the most important quantities.Ho w to
correctly extract stress information from atomistic simulations
is a key to link atomistic and continuum studies.
F or discrete atom systems, the virial stress and its
modified editions,

Mike Ciavarella's picture


In fact, I
have maybe "resisted" to atomistic simulations for long enough.... Although some
times they are published in top journals like Nature (see Luan and Robbins
below), I find continuum mechanics explains almost everything (see also my post
to imechanica of few years ago see discussion
Some notes on Luan and
Robbin's papers on contact and adhesion at atomic scale
).  But to try
on my own, do you have atomistic surfaces in some simple format that I can play
with?   Something like


Hybrid Atomistic/Continuum Study of Contact and Friction
Between Rough Solids

B Luan, MO Robbins - Tribology Letters, 2009 - Springer.
Abstract A hybrid simulation method is used to study the effect of atomic
structure and self-affine roughness on non- adhesive contact and friction between two-dimensional
Rough-on-flat and rough-on-rough contact are compared as a function of system
size up ...




The breakdown of continuum models for mechanical contacts

B Luan, MO Robbins - Nature, 2005 -
Forces acting within the area of atomic contact between surfaces play a central
role in friction
and adhesion. Such forces are traditionally calculated using continuum contact
mechanics, which is known to break down as the contact radius approaches atomic
dimensions. Yet ...




S Medina, D Dini - ... Recently, Luan and Robbins have modelled the contact
of a spherical tip, of the size found in atomic force microscopes, on a flat substrate using atomistic simulations based
upon Lennard-Jones potentials [4]. They demonstrated the significant errors that are


Any suggestions?  Thanks in advance.


Mike Ciavarella's picture


Surface identification, meshing and analysis during
large molecular dynamics simulations


Laurent M Dupuy
Robert E Rudd


Lawrence Livermore National Laboratory, University of
California, L-415, Livermore, CA 94551 USA



Modelling and
Simulation in Materials Science and Engineering

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Volume 14, Number 2


Laurent M Dupuy and Robert E Rudd 2006 Modelling
Simul. Mater. Sci. Eng.
14 229

doi: 10.1088/0965-0393/14/2/008

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Techniques are presented
for the identification and analysis of surfaces and interfaces in
atomistic simulations of solids. Atomistic and other particle-based
simulations have no inherent notion of a surface, only atomic positions
and interactions. The algorithms we develop here provide an unambiguous
means to determine which atoms constitute the surface, and the list of
surface atoms and a tessellation (meshing) of the surface are determined
simultaneously. The tessellation is then used to calculate various
surface integrals such as volume, area and shape (multiple moment). The
principle of surface identification and tessellation is closely related
to that used in the generation of the r-reduced surface, a step in the
visualization of molecular surfaces used in biology. The algorithms have
been implemented and demonstrated to run automatically (on the fly) in a
large-scale parallel molecular dynamics (MD) code on a supercomputer.
We demonstrate the validity of the method in three applications in which
the surfaces and interfaces evolve: void surfaces in ductile fracture,
the surface morphology due to significant plastic deformation of a
nanoscale metal plate, and the interfaces (grain boundaries) and void
surfaces in a nanoscale polycrystalline system undergoing ductile
failure. The technique is found to be quite robust, even when the
topology of the surfaces changes as in the case of void coalescence
where two surfaces merge into one. It is found to add negligible
computational overhead to an MD code.


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