PARADISCYL:Scale-Rule

Manual 03 for ParaDiS Cylinder Codes
How Units Are Scaled

Keonwook Kang, Chris Weinberger and Wei Cai

Original date : Oct 23 , 2008

Latest update on Oct 23 , 2008

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Rule of Scale

ParaDiS Cylinder program is hard-coded such that the radius of a cylinder be 1. Accordingly, the input numbers such as cut-off radius need to be scaled appropriately. In the test script concentric_loop_test.ctrl in M02 Test Run, you see

 
 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.  -9e+0  0.  0.  0.  ]
 #appliedStress = [ 0 0 0 0 -3e0 0 ]
 appliedStress = [ 0 0 0 0 0 0 ]

Let's figure out what each line means in real physical unit.

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

Scaling Parameters	e.x.
Shear Modulus, μ	23e9 Pa
Burgers vector magnitude, b	3e-10 meter
Cylinder radius, R	.35e-6 meter
Bobility, 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_c = \frac{\mu b}{ R} }

and hence nondimensional stress (σ^*) is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^* = \sigma/\sigma_c = \sigma \times (\mu b/R)^{-1}}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma \sim \frac{\mu b}{ L },}

where L is in the unit of distance, according to the elasticity solution.

physical quantities	scale parameter	e.x.
Stress, σ	αβ/γ	105e6 Pa