# Tension-Compression Asymmetry in Homogeneous Dislocation Nucleation

**Abstract.** This letter addresses the dependence of homogeneous dislocation nucleation on the crystallographic orientation of pure copper under uniaxial tension and compression. Molecular dynamics simulation results with an embedded-atom method potential show that the stress required for homogeneous dislocation nucleation is highly dependent on the crystallographic orientation and the uniaxial loading conditions; certain orientations require a higher stress in compression (*e.g.*, <110> and <111>) and other orientations require a higher stress in tension (<100>). Furthermore, the resolved shear stress in the slip direction is unable to completely capture the dependence of homogeneous dislocation nucleation on crystal orientation and uniaxial loading conditions.

This manuscript was recently accepted in Applied Physics Letters. I have converted this to the journal format (3-pages) using LaTex. A few points that I have found interesting:

1. The tension-compression asymmetry observed in the dislocation nucleation stress in single crystal copper may help explain the tension-compression asymmetry that exists in the presence of heterogeneities (i.e., grain boundaries in nanocrystalline materials, free surfaces in nanowires). The tension-compression contour plot shows that most orientations require a larger dislocation nucleation stress in compression than tension. Therefore, a nanocrystalline material with random crystallographic texture should require a larger yield stress in compression than tension based *solely* on the effect of the crystal orientation; previous work confirms that this is the case.

2. The stress required for dislocation nucleation in single crystals depends on both the resolved shear stress on the slip plane in the direction of slip and the stress normal to the slip plane. While not explicitly shown in this letter, we find that there are certain regions of the stereographic triangle in tension where the dislocation nucleation stress correlates best with the resolved stress normal to the slip plane. In other words, while the resolved shear stress is important for the ** motion** of dislocations (Schmid Law), the resolved stress normal to the slip plane can be important for the

**of dislocations in a pure fcc material.**

*nucleation*I welcome any comments. Thank you.

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## strain rate too high, impossible to be achieved experimentally

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

impossibleto be achieved in a real test.## Molecular dynamics simulations and high strain rates

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

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 rates in MD simulations and definition of stress

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.

## Strain Rates and Virial Stress Definition

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.

## atomic stress

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?

## Virial Stress and Continuum Cauchy stress

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.

## Continuous Dynamic vs Incremental Dynamic Loading in MD

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.