PARADISCYL:Scale-Rule

Manual 03 for ParaDiS Cylinder Codes

How Units Are Scaled

This document describes the extension of the ParaDiS program that implements the cylindrical free surface boundary condition. This extension is based on Weinberger and Cai, J. Mech. Phys. Solids, 55, 2027 (2007)(PDF). The code is written by Chris Weinberger. It is owned by Chris and is not included in the standard distribution of ParaDiS. Keonwook Kang wrote the first draft of this document, which was later revised by Chris Weinberger and Wei Cai. Please read the manual of ParaDiS before reading this document.

Motivation

For simplicity, the ParaDiS Cylinder program requires the radius of the cylinder to be 1. Because elasticity model does not have an intrinsic length scale, we can model a cylinder with arbitrary radius by appropriately scaling all lengths in the problem. Accordingly, other parameters, such as stress may also change. In this manual, we will describe how all quantities scale with the cylinder radius R.

Rule of Scale

In our test case concentric_loop_test.ctrl in M02 Test Run, we can find the following lines.

 
 burgMag = 1.0e0

 #Elastic constants 
 shearModulus = 1.0e+0
 pois = 3.050000e-01

 #Core cut-off radius
 rc = 1.0e-3

 #Applied stress in Pa (xx,yy,zz,yz,zx,xy) 
 appliedStress = [ 0 0 0 0 0 0 ]

Let's find out what each line means in real physical units, if we want to model to represent a cylinder with a radius R that is not 1.

Non-dimensional quantity will be notified with the asterisk (*). For example, the dimensionless radius is expressed as $R^{*}=R/R_{c}$ and is fixed to be 1 in ParaDiS Cylinder codes, which means all the length units are scaled by the cylinder radius (R) since $R_{c}=R$. The cut-off radius rc in the script is the scaled cut-off radius (r_c^*) and is .375 nm in real unit if the cylinder radius is given as 375 nm, because

$r_{c}=r_{c}^{*}\times R_{c}=r_{c}^{*}\times R.$

In addition, according to the script (burgMag = 1.0e0 and shearModulus = 1.0e+0), both shear modulus and Bergers vector magnitude are scaled to be 1 or μ^* = 1 and b^* = 1. In other words,

μc = μ and bc = b.

In the table below, listed are four key physical quantities which will be used to scale other physical quantity.

Scaling Parameters e.g. Shear Modulus, μ 23e9 (Pa) Burgers vector magnitude, b 3e-10 (meter) Cylinder radius, R .375e-6 (meter) Mobility, m 1 /(Pa*sec)

You might think that shear modulus could be a good scaler for stress because they share same unit. However, the reference stress is, in fact, μb/R or

$\sigma _{c}={\frac {\mu _{c}b_{c}}{R_{c}}}={\frac {\mu b}{R}}$

and hence nondimensional stress (σ^*) is

$\sigma ^{*}=\sigma /\sigma _{c}=\sigma \times (\mu b/R)^{-1}$

You would understand the choice of reference stress considering that the stress due to a dislocation is proportional to

$\sigma \sim {\frac {\mu b}{L}},$

where L is in the unit of distance, according to the elasticity solution. Thus, one of the stress components -3 in the script, though commented out, becomes 55.2 (MPa) in compression in real unit if you multiply μb/R (=18.4 MPa when μ = 23e9Pa, b = 3Å and R = 375 nm )

Energy stored per unit length E' in the elastic media due to a dislocation is proportional to μb², which is natural reference choice for energy per unit length. (or effectively force.)

The next table lists multiplcation factors to convert the scaled quantity to the real one.

Physical quantity	Multiplication factor	e.g.
Length, L	R	.375e-6 (meter)
Stress, σ	μb/R	18.4e6 (Pa)
Energy per length, E' (effectively, force)	μb2	2.07e-9 (Newton or Pa$\cdot$meter2)
Force per length, F'	μb2/R	5.52e-3 (Newton/meter or Pa$\cdot$meter)
Velocity, v	μb2m/R	5.52e-3 (meter/sec)
Strain rate, ${\dot {\epsilon }}$	μb2m/R2	1.472e+4 (1/sec)
Time, t	(μb2m/R2)-1	6.7935e-5 (sec)

The Poisson number pois is already dimensionless and it doesn't need to be scaled additionaly.