iMechanica - dislocation nucleation - Comments

i need your emeil adreess

mohamed ali — Thu, 13 Mar 2008 08:59:12 -0400

i need your emeil adreess

subdivision method to solve integral with weak singularity

mohamed ali — Thu, 13 Mar 2008 08:57:19 -0400

i need this papers about the use of subdivision method to solve integral with weak singularity in bem

- J O Lachat & J O watson " effective numerical treatment of boundary inegral equation " a new formulation for three dimenssional elasostatics Int .Jr.Numer.methods .Eng.211-228(1958)

Continuous Dynamic vs Incremental Dynamic Loading in MD

amleach — Mon, 19 Feb 2007 10:11:45 -0500

Our group has compared these two loading techniques in simulations of nanowire deformation, and found that there is essentially no difference in the observed response. We loaded nanowires in tension using a strain rate of 108 s-1 and also loaded the same nanowires with an incremental displacement of around 0.1 Angstroms followed by a thermal equilibration over 100 ps. The response is near identical for the two loading methods. These results are presented in a paper I posted on my blog a few weeks ago, and may be found here.

Strain Rates and Virial Stress Definition

Mark Tschopp — Mon, 19 Feb 2007 00:32:06 -0500

With regard to the strain rates, we will examine further the influence of strain rate in these calculations. One method would be as described, equilibrating during loading to achieve a quasi-static loading. Another method would be to perform a molecular static calculation with energy minimization after each loading increment. We will try both methods.

As to the choice of using the virial stress definition without the microkinetic portion, this definition is motivated by a few papers by Min Zhou and David McDowell regarding the virial stress as it relates to the continuum Cauchy stress. I believe that they find that the balance of linear momentum is not satisfied with the kinetic portion of the virial stress as the system size decreases; I have not read these papers in a while, instead I forward the interested reader to their publications on the subject.

As you mention, the stress definition does not change the results qualitatively. Even quantitatively, we found that the kinetic portion, for the most part, adds approximately 0.35 GPa to the virial stress at 300 K, with only a small standard deviation from this mean value. At low stresses, this may be a significant, but for the peak stress required to initiate plasticity in these simulations, this is typically a small fraction.

Virial Stress and Continuum Cauchy stress

Arun K. Subramaniyan — Sun, 18 Feb 2007 23:03:18 -0500

Dear Pradeep,

We did a simple thermo elastic MD study. This is part of a paper under review in Applied Physics Letters. I could not attach a write up regarding this in my reply here. I have posted it in my blog instead. You can find it here : http://imechanica.org/node/871

Hope this helps.

atomic stress

Pradeep Sharma — Sun, 18 Feb 2007 21:16:28 -0500

Dear Arun,

Could you elaborate further on your comment on total virial stress being the same as Cauchy stress? You mention that you have shown this recently; is that work published?

Strain rates in MD simulations and definition of stress

Arun K. Subramaniyan — Sun, 18 Feb 2007 21:06:07 -0500

I find it hard to accept the practice of relating the thermal excitation frequency of atoms (usually in THz) to strain rates. MD simulations by nature were developed to calculate statistical average of quantities. An individual snapshop (some particular t) of MD results may not have any meaning by itself. If we equilibrate periodically during loading, then the entire response is similar to quasi-static loading (irrespective of temperature). If the system is not equilibrated periodically after being perturbed (in this case loaded), then it is not clear whether the solution is converged.

Only the potential contribution to the virial stress has been taken. There is some debate whether the total virial stress is the continuum Cauchy stress. Do the authors have any particular reason for not using the total virial stress? Recently we found that from a thermo-elastic standpoint, the total virial stress is indeed the continuum Cauchy stress. However, ignoring the kinetic part of the stress may not change the results qualitatively.

Molecular dynamics simulations and high strain rates

Mark Tschopp — Sun, 18 Feb 2007 11:49:12 -0500

Molecular dynamics (MD) uses an iterative Euler method to calculate the positions of atoms, which requires the magnitude of the timestep to be on the order of atomic vibrations (i.e., picoseconds). Unfortunately, small timesteps mean that MD simulations of material deformation have strain rates that are much higher than experimental strain rates. Due to the computational expense of tracking the positions and velocities of each atom (100,000s-1,000,000s) over each timestep, strain rates on the order of experiments is prohibited.

So, how does strain rate affect the tension-compression asymmetry?

One way to approach this problem is to simulate different strain rates with MD and investigate the response. The typical response that we see is that higher strain rates (10^10) have a significant effect on the mechanical response compared to 10^9, while lower strain rates (10^8) have a very similar response to 10^9. In terms of the yield stress, the MD simulations start to converge around strain rates of 10^9. This is similar to what other groups have found in terms of strain rates. In addition, we run these simulations at 10 K as well, to simulate a quasi-static deformation mode. Thus, we deem our strain rate sufficient to capture the influence of lattice orientation on tension-compression asymmetry.

If this is a more general question of how to get around the problem of high strain rates in MD simulations, there is currently work being done in a number of different areas: e.g., temperature-accelerated molecular dynamics (e.g., Voter), quasicontinuum method (e.g., Curtin), multiscale modelling techniques, nudged elastic band simulations for calculating activation energy, quasi-static calculations, etc. However, these methods are outside the scope of the current research, though.

GigaHertz-level Excitation of Strain

Cetin Cetinkaya — Mon, 12 Feb 2007 17:50:23 -0500

To my knowledge (based on literature), there are, at least, two ways to excite motion at GigaHertz levels: (1) photoacoustic (phonons) excitation by pulsed lasers, and (2) sonochemical reactions (e.g. cavitation).

strain rate too high, impossible to be achieved experimentally

Henry Tan — Mon, 12 Feb 2007 00:08:58 -0500

How does the strain rate of compression and tension affect the tension-compression asymmetry? The strain rate in your MD simulation was too high, 1 billion per second, which is impossible to be achieved in a real test.

3D singularity and oscillatory singularity

Zhen Zhang — Fri, 09 Feb 2007 15:22:11 -0500

Dear Yuval,

Thank you very much for your interest and great comment.

We are also interested in 3D corner singularities since it is such an important issue. May I ask for a copy of your work and the related references about 3D corner singularities? I also suggest you to post your 3D singularity work on iMechanica. There are so many failure modes driven by corner singularities, and 3D corner with large elastic mismatch and CTE mismatch is so common in microelectronics, so I think this is a very interesting topic, at least for the mechanics people working in semiconductor industry.
In order avert the complexity of 3D corner singularities, we use a long stripe instead of a square pad, but the idea and the frame work still apply, as Martijn pointed out in his reply. The mode III is not considered because the stress state is plain strain stress state, there is no out-of-plane shearing.
Beta effect was studied a lot for cracking problems, such as those by He and Hutchinson in late 1980s and early 1990s. All those works showed the beta effect was no big deal, because beta effect is basically Poisson’s ratio effect. So for simplicity, we also adopt this assumption. Of course, as you pointed out, this assumption needs to be verified or studied.
As shown in Figure 2 in our paper, the upper right triangle region is of complex conjugates. In order to avert this complexity, we adopt beta=0, then there is no worry when we use the linear superposition of stronger and weaker singular stress fields. Since for strained silicon structure, such as the typical SiN/silicon system, alpha and beta are rarely in the complex region, therefore we needn’t worry about it.
You raise a very point about the “oscillatory” effect. For a material combination such as silicon/polymer, the singularity exponents are commonly a pair of complex conjugates. In such cases, we have to consider the “oscillatory” effect, just as dealing with the interfacial crack.

Mode III and Dundurs Beta

Martijn Feron — Fri, 09 Feb 2007 12:13:53 -0500

Dear Yuval Freed,

Thank you very much for your interest in this topic, I am pleased to answer your questions.

1. We do not consider the mode III stress intensity factor, since we analyze the "infinite stripe" problem, and therefore the mode III is not of interest.

2. By stating Dundurs beta = 0, we basically assume Poisson's ratio to be equal for both materials, i.e. 0,5. This would mean incompressible material behaviour, which is physically not applicable. However, our goal was to present a framework and method, which could be applied over a wide range of material and geometric properties. Poisson's ratio could be altered and analysis repeated for other values, in order to study the effect of beta effects.

Best regards,

Martijn Feron

Few remarks and questions

Yuval Freed — Thu, 08 Feb 2007 00:39:18 -0500

It is a very interesting topic. I investigated cracks in three-dimensional domains and corner singularities in the past. It is an important issue. However, I have few questions:

What about the mode III stress intensity factor? Is it not considered here since it is decoupled from K1 and K2 and has no 'lambda' singularity?

You assumed beta=0 (Dundur's parameter). Is it a reasonable assumption? It seems as if it degenerates the problem. It is interesting to observe how the 'oscillatory' parameter beta affects on the results.

--
Yuval

Spacing effect

Zhen Zhang — Fri, 08 Dec 2006 14:52:30 -0500

Hi, Xiaohu

As you pointed out before, the SiN are usually patterned periodically. The spacing effect is of more interest to device strain engineering. So we did new calculation and obtained the results. Here is the link of this post or pdf file of the results.

From the curves we plotted, the spacing effect is quite small except that the SiN stripes are quite close to each other, e.g. S/h<1, where S is spacing, h thickness. So for real application, e.g. S/h>1, the spacing effect maybe not a big concern.

Xiaohu, thank you very much

Zhen Zhang — Tue, 21 Nov 2006 18:09:24 -0500

Xiaohu, thank you very much for your comments and ideas.

1. You mention a very important point about the range of curving fitting, i.e. the range of k-annulus. When I did the fitting, I checked range up to 10-3<r/h<10-1, the fitting was good for all the cases when L/h>2. When we fitted the case of L/h=1, the error is too big due to the perturbation of symmetry boundary condition. Hence, for accuracy and validity, I just used the range of 10-3<r/h<10-2 for all the cases. But in reality, L/h>=2, so the validity range can reach r/h<=10-1. Even the film thickness is just about tens of nanometers, the equation of singular stress field still works.

If you consider the full expansion of power-law eigenfunctions, not only stronger singular term and weaker singular term, but also non-singular term, then no worry about the range of validity of fitting. That is, the location r=b for dislocation injection still works. Meanwhile, there are two or more k’s in evaluating the critical condition of dislocation injection. This makes the criterion not neat and simple to use. But, you are absolutely right. You point out a very important issue, so currently we are trying to use both singular terms to consider more generic cases.

For SiN/silicon system, from evaluation of local mode mixity (i.e. stress ratio of weaker mode to stronger mode, please refer to the paper), the contribution from weaker mode is less than 5% when L/h>2. So the stronger singular term is enough with reasonable error.

2. I thought about effect of periodic spacing in my notebook, sketch some figures qualitatively. We did not include this effect in this paper, because we wanted to explain the method clearly rather than put too much in a short paper. The spacing effect can be done very soon. Hopefully, you will like it.