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Journal Club Theme of July 15 2008: Plasticity at Sub-Micron Scales

Cai Wei's picture

Our topic is a continuation of the May 15 discussion led by Professor Julia Greer on “Experimental Mechanics at Nano-scale”.  The whole story about the “micro-pillars” started in 2004, when Mike Uchic et al. used focused ion beams (FIB) to make micro-pillars from pure Ni and Ni alloys that can then be uni-axially compressed by a flattened AFM tip [Science 305, 986-989, 2004].  The flow stress is found to increase with decreasing sample diameter even though there is no imposed strain gradient as in micro-indentation, bending or torsion experiments.  This finding generated a lot of excitement worldwide.  Many experimental groups are now performing similar tests for a wide range of materials, and many computational groups are doing atomistic and dislocation dynamics (DD) simulations with similar geometries.  The excitement can be traced to two reasons.  First, this is a new way to measure local properties of materials under a simple stress state.  It is (presumably) easier to associate the data with the intrinsic properties of the material, as compared with the data from other tests.  Second, it is now possible to quantitatively compare computational models of defect dynamics with experimental measurements on mechanical strength.  Our sacred theory of dislocations and our favorite numerical toolboxes are now under a stringent test.  This certainly makes me feel excited, perhaps a little anxious, with my fingers crossed.  

Many theoretical models have been proposed to explain the observed size dependence in these micro-compression experiments.  Much of the recent debate in the community has been around the question of whether or not the micro-pillar is “dislocation-starved” during deformation.  An active debate is an exciting thing to have in the scientific community (the most famous is probably the one between Niels Bohr and Albert Einstein about the indeterminism of quantum mechanics).  To contribute constructively to this debate, I would like to bring to your attention some of the latest findings both from the experimental side and from the computational side, as well as a few thoughts for you to consider. 

Dislocation starvation model.  

The main idea here is that the dislocation is easy to leave the micro-pillar.  We now have a dislocation depletion rate that can be greater than the dislocation multiplication rate (depending on the pillar diameter).  Because we still believe plastic deformation in metals require dislocations, we can imagine lots of dislocations nucleated from one side of the cylindrical surface, rushing through the pillar and escape from the other side of the cylindrical surface.  The “dislocation starvation” model then predicts the dominance of dislocation nucleation (probably from surface) on micro-pillar plasticity.  The derivation in the dislocation starvation model is quite interesting, but it was given in an appendix that is difficult to find.  So I am attaching it here for you to take a look [PRB 73, 245410, 2006, see below].  A recent evidence for the starvation model was the in situ experiment by Shan et al. [Nature Materials 7, 116, 2008] already discussed in the May 15 issue.  On the other hand, Uchic has found dislocation networks in micro-pillars similar to that in bulk metals [Acta Mater. 2008 see below].  It is worth noting that in the in situ experiment (supporting the starvation model) the pillar is very thin (D < 300 nm) while in the latter experiment (where dislocation network is observed) the pillar is much thicker (D > 1 micron, but still within the diameter range where the size effect exists).   
Effect of existing dislcoations and pre-strain.

A very different model from “dislocation starvation” maybe called the “source truncation model” [Scripta Materialia 56, 313-316, 2007].  To appreciate the main idea, we can imagine a pre-existing dislocation network inside the pillar.  DD simulations show that strain can be carried out by “single-arm” Frank-Read sources, which are dislocation lines with one end pinned inside the pillar and the other end free to rotate around the cylindrical surface.   The stress to activate these sources is higher if the length of these “single-arm” sources is shorter.  At the same time, it is reasonable to expect that, statistically, the length of dislocation arms in a thinner pillar must be shorter, hence the observed size effect.  But what is the physical origin of the pinning points that create the “single-arm” sources?  One possibility is the junction formed between the dislocation line and other dislocations.  In this context, the recent paper by Tang, Schwarz and Espinosa [PRL 100, 185503, 2008, see below] is especially interesting.  In this paper, they performed DD simulations in a micro-pillar starting with a pre-existing network of dislocations.  Many aspects of the simulation result resembles experimental observation, such as alternating periods of elastic loading and sudden bursts of plastic strain. 

Perhaps the recent paper by Bei et al. [Acta Mater. 2008, see below] can be considered as an “experimental counterpart” of this simulation.  In this paper, Bei et al. performed micro-compression experiments on (BCC) Mo alloy pillars that were fabricated without FIB damage, but were subjected to different amount of pre-strain, which introduce a dislocation network into the pillar.  Interestingly the observed yield stress is found to decrease with the amount of pre-strain.  Here I do want to point out that this experiment is done on a BCC crystal (whereas much of the dislocation starvation model was based on dislocation behavior in FCC crystals).  The fact that we have an alloy, instead of pure Mo, here may also influence dislocation behavior not accounted for in existing models.  One more point: these pillars have a rectangular cross section while the other FIB-machined pillars have a circular cross section. 

Is it an artifact after all?  

I’d like to end my introduction to this journal club discussion with a frightening thought.  Is it possible that the original report of the size-dependent flow stress in FIB machined micro-pillars is an experimental artifact after all?  I hope not, but it is interesting to consider this possibility and realize that the original experimental set up does have artifacts.  First of all, there is the FIB damage, which may exist in the form of dislocation loops near the cylindrical surface.  (There is a debate here that I do not intend to get into, but you can look into Appl. Phys. Lett. 91, 111915, 2007; 92, 096101, 2008.)  Recently, micro-tensile experiments become possible (thanks to the creativity of our fellow experimentalists), which points to the possible artifacts of the micro-compression tests [Acta Mater. 56, 580-592, 2008, see below]. 

As a theorist, I am not in a position to assess the experimental artifacts.  All I know is that we have a full bag of computational artifacts to worry about.  Now, a positive note: since we have multiple theories and computer simulations all consistent with a size-dependent flow stress at the sub-micron scale, the experimentalists better not conclude that the size-effect is not really there in the end.

Articles for discussion:


Mike Ciavarella's picture

... that it is well know that materials are stronger at small scales, withouth the need of gradients!

Even Griffith in 1921 has essentially said that.

So what is really new here?  An increase towards theoretical strength is entirely predictable and expected.

Speaking of nanoindentation, but only to start with, it is easy to play with equations that fit very easily what dislocation dynamics simulates, see e.g.


Formato file: PDF/Adobe Acrobat - Versione HTML
FOR NANOINDENTATION, BUT NOT ONLY. Nicola M. Pugno. Department of Structural Engineering, Politecnico di Torino,. Corso Duca degli Abruzzi 24, Torino, ...

Title: A general shape/size-effect law for nanoindentation

Author(s): Pugno, NM
Source: ACTA MATERIALIA   Volume: 55   Issue: 6   Pages: 1947-1953   Published: APR 2007

Times Cited: 4 

Title: Microindentation of titanium: Dependence of plastic energy on the indentation depth and time-dependent plastic deformation

Author(s): Chen R, Yang FQ, Imam AA, et al.
Source: JOURNAL OF MATERIALS RESEARCH   Volume: 23   Issue: 4   Pages: 1068-1075   Published: APR 2008


 So please reformulate so that perhaps I also get excited!  ;)


Cai Wei's picture

Mike, thanks for your comment.  Let me give it another try.  The key factor that is new in this type of experiment is that there is no imposed strain gradient.  A strain gradient introduces geometrically necessary dislocations, from which a size effect may naturally arise.  It is therefore interesting to see whether free-standing micro-pillars can have size-dependent flow stress or not.

Indeed, we need to discuss the micro-pillar plasticity size effects in light of Griffith, where the essential idea is that when the sample is free of any defect, it may reach ideal strength before it collapses.  This seems to be what is observed in experiments of Script. Mater. 57, 397-400 (2007).

Even though (we believe) there is no imposed strain gradient in the micro-pillar compression experiments, it is possible that a local strain gradient may exist near the cylindrical surface (APL 89, 151905, 2006).  If this is the case, the scenario becomes similar to that in a grain of nano-crystals.  That is why people are trying to assess the amount of FIB damage or to elimiate it all together.

Wei Cai

Mike Ciavarella's picture

However, I think there was some general misunderstanding since the times when strain gradient plasticity was introduced (was it Gao, Hutchinson, Fleck, Aifantis or all of them at the same time?).

SGP was a nice thing, but size effect were already there much before, with NO strain gradient there --- Weibull, Griffith.

Indeed, what my friend Pugno shows is that SGP has a hard time to predict the deviation up to the theoretical strength, because essentially is an attempt like a first order correction to the standard plasticity, but you need many more terms to reach the plateau...

So Pugno's equation works better ALSO with strain effects, see the paper, and is very simple!

This is not to say that SGP, Gao, Hutchinson, Fleck, Aifantis did not make interesting contributions.

Moving back to the surface effect, indeed it is interesting to see more explanation of that, this is why I referred in my further comment to the work of Lu from Hong Kong.  Please give a look at that.  Including, please, remind me if it is correct that, instead of plateau, you can see a "decrease" of strength again.  Is that a mistake by some authors -- indeed it does NOT fit the simple model, unless something new is going on.

Thanks again.


Mike Ciavarella's picture

At Ecole Polytechnique we recently had a nice seminar from Jian Lu head of mech eng in Hong Kong.  See abstract at

He seemed to suggest his nanocrystalline structures could also see a plateau of highest strength and then decrease with an "inversion".  Did I understand correctly?

More reference:



, LU K. Surface Nanocrystallization (SNC)
of Materials and its effect on Mechanical
Behavior, Comprehensive Structural
Editors Milne, Ritchie, Karihaloo, Ed. Elsevier,
vol. 8, 2003, p. 495-528.


Y.P., LU J., A new method to extract the
plastic properties of metal materials
from an instrumented spherical indentation
loading curve, Acta Materialia, 52 (13):
4023-4032 AUG 2 2004.


Nanostructure Formation Mechanism of
using SMAT, Acta Materialia,
52 (14): 4101-4110 AUG 16 2004.


, Residual
stresses in laser welded aluminium
plate by use of ultrasonic and optical methods.
Materials science and engineering
, 2004,
vol.382, p. 257-264.


Y.P., LU J., Size-dependent sharp indentation-I:
A closed-form expression of the indentation
loading curve and II: A reverse algorithm
to identify plastic properties of metallic
materials”, Journal of the Mechanics
and Physics of Solids
, Jan. 2005, vol. 53, pp33-62.


Z.B., LU J., LU K., Chromizing behaviors
of a low carbon steel processed by means,
ACTA MATERIALIA 53 (7): 2081-2089 APR 2005.


X.H., LU J., LU L., LU K., Tensile properties
of a nanocrystalline 316L austenitic stainless
steel, Scripta Materialia, 52 (2005) 1039–1044,
May 2005.


Julia R. Greer's picture

Wei, thank you very much for this very interesting to  me (of course) discussion! I wanted to second your sentiment that this is really an engaged debate that is happening in the mechanics/materials science community - and I think it goes beyond just the "size effect" part - I think it is forcing us to return to the fundamentals and basics. In other words - to the main mantra of Materials Science: "Structure-Processing-Properties." We find ourselves in the regime (nano-scale) where these relationships, which we had comfortably been using without the use of scale, no longer hold.

 One very important (and controversial) question is determining the role of surfaces - whether the inscrease in strength continues all the way up to theoretical strength as the size is reduced to nearly 0 or whether there is a softening that occurs at smaller sizes. Contrary to the dislocation starvation theory, some atomistic simulations using empirical potentials suggest that the self-similar hardening behavior may not hold below a critical diameter (typically 100 nm) where surface effects become dominant [J. Marian, J. Knap, Breakdown of self-similar hardening behavior in Au nanopillar microplasticity. International journal for multi-scale computational engineering, 5, 287 (2007).]. Experiments at that regime have not been possible yet, but a lot of us in the experimental community are working on it...


Julia Rosolovsky Greer

Pradeep Sharma's picture

Julia and Wei, I have a simple question---related to a comment made by Julia. I understand how surfaces play a role in elasticity behavior and generally the size-scale where they become relevant are sub-10 nm. Exactly what do plasticity researchers expect the surfaces to contribute in (relatively) large sizes as 50-100 nm? Does it have to do with dislocation image forces? It is true that even in absence of external strain gradients, near surfaces, we can expect some but those decay exponentially will a decay length of the order of lattice spacing so I expect those would be irrelevant, right?

grahamcross's picture

Julia -

 Interesting thread, thanks for your comments and comments of others.

I would just like to point out that there is experimental work in the sub-100 nm regime.  

For indentation, we published a study of surface plasticity of atomically flat Au(111) terraces under nanoindentation with ultra-sharp (3 nm radius) tips, the shape of which was rigorously characterized by field ion microscopy (custom experimental setup).  These experiments were conducted fully under proper UHV surface science conditions at approximately 100 K temperature, using a sensitive differential interferometer to measure displacements and forces. 

In this system, we discovered numerous interesting phenomena, the most strking of which was a reverse plastcity or, in other words, a rapid surface healing effect that nevertheless could be extinguished by cyclic working of the pristine surface.  We attributed this qualitative change in plasticity to be due to an enhanced role of (surface) point defect currents at the length scale of a few nm indentation depth.  The indentatios appeared to exhibit a hardening effect, although we were reluctant to precisely estmate stresses without direct knowledge of contact areas (worried about adhesion etc. influences.)  When compared to computational MD studies, the hardening we observed was mirrored in simulation but the reverse plasticity was not, which emphasized the point that constraint on materials models by experiments must match conditons in both space *and* time.  That is to say, point defect mediated surface plasticity effects probably occur on time scales still not accessiblt to conventional atomistic simulation, and therefore require more sophisticated approaches in either models or experiment to make them truly comparable.

Graham L. W. Cross, André Schirmeisen, Peter Grütter and Urs T. Dürig, Nature Materials 5, 370 - 376 (2006) doi:10.1038/nmat1632


 In a similar vein, reverse plasticity in compression of sub-100 nm Si spheres has been inferred, eg.

  1. Gerberich, W. W. et al. Reverse plasticity in single crystal silicon nanospheres. Int. J. Plasticity 21, 2391–2405 (2005).
  2. Kramer, D. E., Yoder, K. B. & Gerberich, W. W. Surface constrained plasticity: oxide rupture and the yield point process. Phil. Mag. A 81, 2033 (2001).


 There is also work on tethered nanowire bending via AFM for wires down to 40 nm diameter, eg. for gold nanowires.

Bin Wu, Andreas Heidelberg, & John J. Boland, Nature Materials 4, 525 - 529 (2005), doi:10.1038/nmat1403

which finds a pronounced hardening effect.











Mike Ciavarella's picture

Hello guys I am interested by this reversed flow mechanisms and shakedown too. In these respect, does anyone know the old KL Johnson paper in Nature (I ask because it has an incredibly low citation number, only 7!, so either people refer only to his book, admitting all the details are there, or this paper is not well known!)

Reversed Plastic Flow during the Unloading of a Spherical Indenter - all 2 versions »
KL Johnson - Nature, 1963 -
Nature 199, 1282 (28 September 1963); doi:10.1038/1991282a0. Reversed Plastic
Flow during the Unloading of a Spherical Indenter. KL JOHNSON. ...

Cited by 7 - Related Articles - Web Search - ACNP Holdings


Speaking of shakedown, does anyone here know the Melan's theorem to find out the shakedown limit?  In this from 1936, very well known in some communities, but not all of them...

If I find time, I will read the papers above from Graham they look interesting.  But please check yourself if you forget some classics.

Dear Wei,

Thank you for bringing this up. It is an exciting subject.

Regarding Mike's comparison of the size effect observed in micro/nanoplasticity to size dependence in brittle fracture, I am not sure these are phenomena of the same nature. The size effect observed first by Griffith in his experiments with glass fibers is usually explained from the probabilistic point of view (Weibull statistics), where larger solids have a higher chance to contain a critical flaw, and hence have lower mean strength.

In plasticity at micro/nanoscales, however, the size effect (which is expressed in size-dependant strain-stress curve) is explained from a deterministic point of view, which eventually refers to collective behaviour of dislocations.

We have also recently been looking at this problem, trying to accurately (re-) derive the equations for arrays of dislocation. We actually showed, that there are two sources for size effects: one is the correlation within a dislocation array, which explicitly leads to gradient terms. The second is the self-image stress on a dislocation due to the presence of boundaries, which might be even more significant for the size effects than the strain gradient corrections. (I am attaching two recent papers about the correlation functions and the strain gradient corrections, which might be of interest)

Thanks again

Vinogradov, V., Willis, J.R. 2008 The pair distribution function for an array of screw dislocations. International Journal of Solids and Structures. 45(13), 3726-3738.

Vinogradov, V., Willis, J.R. 2008 The pair distribution function for an array of screw dislocations: implications for gradient plasticity. Mathematics and Mechanics of Solids, (accepted).


Mike Ciavarella's picture

to see I seem to raise some animated "discussion".

Some attempts to reply (remember that I am ignorant in this, as in many other areas --- all I proposed usually is NOT to follow the croud, so it has the advantage of originality, but could be crap!).

Clearly, you folks come from material science point of view, and I come more from solid mech eng, but this is good.

1) It is true that Griffith is for brittle materials (no dislocation motions grossly speaking), but remember also that what defines "brittle" is the intrinsic material "flaw tolerance".   So metals or standard ductile materials at the macro scale, behave ductile as their flaw tolerance is high.  You can of course measure this by defining "intrinsic crack lengh scale"

a0 = 1/Pi (KIc / sigma_Y) ^2

and this comes out meters in some cases!  See nice maps by Ashby in his books and papers (but be warned that for brittle materials he takes sigma_Y as compressive strength which I find confusing).  Here KIc is thoghness and sigma_Y the yield stress

2) Now, for brittle materials, either you follow Griffith for a deterministic crack size in the LEFM size effect (which is the strongest), or Weibull (which is milder, which applies Griffith to a distribution of cracks and recnognizes the larger cracks are more likely in larger structures.

3) Now, move to smaller scales.  We can agree for the time being that KIc stays constant, but sigma_Y we also agree that increases.  So what happens to a0?  It gets smaller and smaller!   When sigma_Y reaches theoretical strength, we all know it is an increase of maybe 100 times, in terms of a0 this is a decrease of 10^4 ---- so now what you considered ductile, becomes brittle!  The dominant mechanism now returns to be Griffith!

4) all of this, has been published in many versions, and to return to Pugno, I think he and Ruoff have applied Weibull to carbon nanotubes.  It doesn't matter. What it matters is that you realize that you have a hard time to explain "ductile behaviour at nanoscale" using dislocations, simply because the material becomes more and more brittle and there is a competition and interaction between two failure modes.   As often the case, an empirical equation (you can say elegantly as Bazant would say an "asymptotic matching" betweeen the two mechanisms) gives a simple solution.  Dislocation simulations may in some cases provide the brute force long answer.

5) Thanks for Julia to speak about softening.  I have heard Lu talk about it, but I don't have references.  I don't know if it can be seen as from picture of 4) above, but I suspect it is another transition to another mechanism perhaps as you say dislocation facilitated to move near surfaces?


Hope this helps to excite you even more, for me I am only excited when I find something relatively simple.  Usually this contrasts with Editors and Reviewers, and the contrast between my excitement and their rejection is hard!  I should better get excited with something not too new and not too simple, publish nice figures and this may more successfull with editors...


Dear Wei,

Thanks for this interesting introduction. Acutually, in my opinion, the strength also becomes sensitive to factors other than the sample size per se, when the characteristic dimensions (such as the diameter of a pillar) of the metallic object are on the scale of a few nanometers to a few tens of nanometers (or larger). In other words, on this scale, the ‘‘size effects” can be derived not only directly from the sample dimension itself, but also from accompanying consequences that may not be immediately apparent. Our recent paper has demonstrated and explained such effects: unusually large influence is derived from the sample shape and temperature on the apparent strength of nanoscale objects. This work underscores the importance of acquiring detailed information about the sample geometry, including local curvature and atomic-scale features, before predicting and explaining the mechanical properties in measurements.

Cao A, Ma E, Sample shape and temperature strongly influence the yield strength of metallic nanopillars, Acta Mater (2008), doi:10.1016/j.actamat.2008.05.044

Ajing Cao




Mike Ciavarella's picture


Science 23 May 2008:
Vol. 320. no. 5879, pp. 1057 - 1060
DOI: 10.1126/science.1156084

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Inverse Temperature Dependence of Toughness in an Ultrafine Grain-Structure Steel

Yuuji Kimura,*
Tadanobu Inoue,
Fuxing Yin,
Kaneaki Tsuzaki

Materials are typically ductile at higher temperatures and become brittle at lower temperatures. In contrast to the typical ductile-to-brittle transition behavior of body-centered cubic (bcc) steels, we observed an inverse temperature dependence of toughness in an ultrahigh-strength bcc steel with an ultrafine elongated ferrite grain structure that was processed by a thermomechanical treatment without the addition of a large amount of an alloying element. The enhanced toughness is attributed to a delamination that was a result of crack branching on the aligned {100} cleavage planes in the bundles of the ultrafine elongated ferrite grains strengthened by nanometer-sized carbides. In the temperature range from 60° to –60°C, the yield strength was greater, leading to the enhancement of the toughness.

National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan.

* To whom correspondence should be addressed. E-mail:' + u + '@' + d + '<\/a>'//-->


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The editors suggest the following Related Resources on Science sites:
In Science Magazine

MATERIALS SCIENCE: Stronger, Tougher Steels
J. W. Morris Jr. (23 May 2008)
320 (5879), 1022.
[DOI: 10.1126/science.1158994]
Mike Ciavarella's picture



Science 18 July 2008:
Vol. 321. no. 5887, pp. 382 - 385
DOI: 10.1126/science.1156211

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Observations of Intergranular Stress Corrosion Cracking in a Grain-Mapped Polycrystal

A. King,1,2*
G. Johnson,1,2
D. Engelberg,1
W. Ludwig,2,3
J. Marrow1

Nondestructive three-dimensional mapping of grain shape, crystallographic orientation, and grain boundary geometry by diffraction contrast tomography (DCT) provides opportunities for the study of the interaction between intergranular stress corrosion cracking and microstructure. A stress corrosion crack was grown through a volume of sensitized austenitic stainless steel mapped with DCT and observed in situ by synchrotron tomography. Several sensitization-resistant crack-bridging boundaries were identified, and although they have special geometric properties, they are not the twin variant boundaries usually maximized during grain boundary engineering.

1 School of Materials, University of Manchester, Manchester M13 9PL, UK.
2 European Synchrotron Radiation Facility, Grenoble, France.
3 Matériaux: Ingénierie et Science, Institut Nacional des Sciences Appliquées de Lyon, Villeurbanne, France.

* To whom correspondence should be addressed. E-mail:' + u + '@' + d + '<\/a>'//-->


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MATERIALS SCIENCE: Tracking Corrosion Cracking
Andreas Stierle (18 July 2008)
321 (5887), 349.
[DOI: 10.1126/science.1160939]
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Rashid K. Abu Al-Rub's picture

Dear Cai,

Thank you very much for the interesting topic.

There is no doubt that these micron and submicron pillar experiments have greatly renewed the interest in scale-dependent plasticity through the development of various theoretical, computational, and experimental techniques to address this type of size-scale effect. However, it is very early to conclude that nonlocal or gradient-dependent plasticity cannot explain this type of size effect. Many of these experiments argue that gradient-dependent plasticity fails to explain size effect under macroscopically homogeneous state of stress or strain but without characterizing the initial or local microscopic presence or evolution of geometrically necessary dislocations (which scales with strain gradients) due to synthesis, preparation, and deformation processes of the tested samples. Therefore, to my best knowledge, there are several issues that these experiments did not address sufficiently:


(1)   The role of the initial dislocation density on the observed size effect. One can show through three-dimensional discrete dislocation dynamics that the presence of one dislocation loop due to sample preparation can lead to significant increase in the yield strength due to the microstructural heterogeneous evolution of dislocations that yield a non-zero net Burgers vector which indicates the presence of geometrically necessary dislocations. The reason for mentioning three-dimensional as opposed to two-dimensional dislocation dynamics is the consideration of the effect of line tensions and dislocation loop curvatures, which are very important in explaining this type of size effect.

(2)   There is no doubt that the Focus Ion Beam (FIB) testing methodology is a powerful tool that will give many physical insights on the causes of size effects, but at this stage damage due to sample preparation through FIB is unavoidable. More work is needed to improve the sample preparation techniques through FIB to minimize or even eliminate the presence of initial defects, which as outlined above, may be the source of the observed size effect.

(3)   Many of the tested specimens are not perfectly aligned. One can easily see some imperfections in the geometry and alignment of the compressed pillars. Such imperfections, even very small ones, will trigger the evolution of plastic slip gradients on preferred slip systems, which also can explain the observed size effect.

(4)   Many of the tested specimens are attached to a substrate, which also contribute to the observed size effect and its role in the interpretation of this type of size effect is often ignored. The specimen-substrate interface can be sources of geometrically necessary dislocations.

In my opinion, the reason for not being able to use the current gradient-dependent plasticity theories by those experimentalists (who in many cases did not work before on gradient plasticity) to explain this type of size effect, is due to the lack of the physical understanding of the higher-order boundary conditions that result from the mathematical framework of gradient theories. If you are using a lower-order gradient theory, which ignores the application of these non-classical boundary conditions, then you will not be able to capture the effect of boundary layer thickness at free surfaces and interfaces. These higher-order boundary conditions can be directly related to the surface/interfacial energy at free surfaces and interfaces, which increases with the surface area-to-volume ratio, such that one can with such understanding interpret the increase in yield strength and strain-hardening rates with decreasing size. See this recent paper on this very issue.

Abu Al-Rub, R.K., "Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals,” International Journal of Plasticity, Vol. 24, No. 8, pp. 1277-1306, 2008.

Rashid K. Abu Al-Rub, Ph.D. Texas A&M University

Xiaodong Li's picture

Thanks a lot. This theme is of great interest. I think there is still a lot we need to do. The so called size effect is related to surface lattice. This needs a team work from many disciplies. The discussion here is very useful.

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