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strain evaluation in relaxed nanowires

Due to the existence of surface stress, the interior of a free standing nanowire is in non-zero stress state especially near free surface (see attached). This stress state corresponds to strain field in nanowires. A reference configuration would then be necessary to define the strain. Any good suggestions on the selection of the reference configuration?


Xiaodong Li's picture

Thanks a lot for posting this interesting topic. Surface stress plays a very important role in the elastic modulus of a nanowire. The key is to understand both the mechanics and the surface structure reconstruction. The following two papers may help. 

Guofeng Wang and Xiaodong Li, "Size Dependency of the Elastic Modulus of ZnO Nanowires: Surface Stress Effect," Applied Physics Letters, 9 (2007) 231912.

Guofeng Wang and Xiaodong Li, "Predicting Young's Modulus of Nanowires from Firstprinciples Calculations on Their Surface and Bulk Materials," Journal of Applied Physics, 104 (2008) 113517.


Hi Dr. Li,

Thanks for your reply and your impressive refereence. Your model is very helpful to the problem we are working on. I here have a question about the surface strain in the APL paper. By equilibrate the contribution to total energy from the bulk strain energy and from the surface energy, a critical strain is defined as the surface strain. Since the bulk strain energy is used, the critical strain is equivalent to the nominal strain respect to the length to bulk lattice at the point of minimum total energy. Is it right?  

What we are trying to do here is to evaluate the local strain developed during the relaxation. By doing so a reference configuration is needed to calculate the displacement field directly from the atomic coordination (if doable). Our purpose is to get a distributed elastic moduli from the surface. However, the configuration in bulk lattice sames not very proper for this purpose. 

Thanks a lot,



Pradeep Sharma's picture


Your questions really pertains to defining a Cauchy-Born rule for nanostructures. This is to some degree, still an active area of research. I can suggest 3 papers which should prove helpful to you:

(1) A recent review article by Jerry Ericksen on Cauchy-Born rule. This paper is a rich source of references. 

(2) Work by Harold Park and co-workers who in fact have proposed the so-called "surface Cauchy-Born rule" and address directly the topic of your interest (surface energy effects) 

(3) A paper by Sunyk and Steinmann.



Thanks a lot Dr Sharma, I will look into the papers you recommend.



Harold S. Park's picture

Dear Yongfeng:

My apologies for the late response.  I think the problem you are interested in (strain relaxation in nanowires due to surface stresses) can be addressed in multiple ways.  The first is obviously to conduct MD simulations of nanowires by setting up nanowires assuming bulk lattice spacings, and then minimizing the energy to see how the nanowire either expands or contracts due to the surface stresses.  It is very easy to generate a strain field through these MD simulations as the initial (reference) configuration you desire will be the initial (bulk) lattice positions of the atoms before deformation.

If you desire a continuum approach to calculating the strain relaxation due to surface stresses, which might be necessary for larger cross section nanowires, then Pradeep is right - I have worked on a surface Cauchy-Born model, which is a continuum model to capture surface stress effects on nanostructures, and which I have validated for both FCC metallic and semiconducting (silicon) nanowires.  The method works in a similar form to atomistics in that you initially assume bulk lattice positions for bulk and surface atoms; however, due to surface stresses and differing elastic constants of the bulk and surface, you generate strain relaxation which is compressive for FCC metals, and tensile for silicon.  You can find the relevant papers here: (  Specifically, you can look at Figure 3 in the 2007 PRB paper for a comparison between relaxation strain as computed using both SCB and MD for FCC metal nanowires, and Figures 4-6 in the 2008 CMAME paper of Park and Klein for comparisons of relaxation strain as computed using SCB and MD for silicon nanowires.  Other papers investigate surface stress effects on resonant frequencies, bending modulus, and other mechanical properties of interest of nanowires using the SCB model and incorporating strain relaxation and surface stress effects. 

Also, I have also written, along with Hanchen Huang, Wei Cai and Horacio Espniosa a review article in the MRS Bulletin which discusses at length the effect that strain relaxation and surface stresses have on the mechanical properties of metallic and semiconducting nanowires - this may be of interest to you as well (

I'd be happy to discuss anything with you in further detail.





Thanks a lot for the detailed response. We are now doing the MD approach on this problem. My main concern were (1) the definition of the reference configuration and (2) the definition of deformation (strain) field using atomic data. Now it seems that the bulk (lattice) configuration is good as a reference state. The method proposed by Dr. Jon Zimmerman will be used to calculate the deformation field (International Journal of Solids and Structures 46 (2009) 238–253). By the way, your papers on nanowire mechanics are great references to us.

Thanks again for all the replies, this is really a good place to learn.


Dear Yongfeng,

The issues ofyour concern are interesting. The key point is the choice of the referenceconfiguration. Due to the existence of the surface tension, in the absenceof external loadings, the bulk materials of nano structures will be in a stressedstate. This state is usually chosen as the reference configuration. Hence, theelasticity with residual stresses should be used to describe the mechanicalresponses of the bulk materials. Here is a recent paper which concerns thisproblem.

ZhiqiaoWang, Yapu Zhao and Zhuping Huang. "Theeffects of surface tension on the elastic properties of nano structures ",InternationalJournal of Engineering Science, doi:10.1016/j.ijengsci.2009.07.007 . 



Pradeep Sharma's picture

Zhi-Qiao, thanks for brining this paper to our attention.....It is very interesting.....congratulations on a nice work......I expect it to be quite useful for some of our own ongoing work.

A while back (perhaps more than a year or so), I had some discussion with Professor ZP Huang regarding impact of residual surface stress on elastic behavior (--this was in the context of a paper he published on composites). While I (now) understand this assertion, a few aspects still bother me (which hopefully I or rather my students can resolve in the near future). Anecdotal evidence obtained from atomistic calculations suggest that the computed surface stress invariably turns out to be symmetric---in contrast to what we expect from Gurtin-Murdoch constitutive law. Your derivations in the present paper as well as the constitutive law of Gurtin-Murdoch clearly suggests that this should not be case in general. 


Dear Pradeep,

       Thanks for your attention and appreciation.

       The Cauchy stress and the second Piola-Kirchhoff stress of the surface are symmetric tensors. However, the first Piola-Kirchhoff stress of the surface is a two-point tensor, which is unsymmetric in general.


           Zhi qiao

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