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.Pa

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

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 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

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_c b_c}{ R_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 due to a dislocation 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. 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 (or reference value)	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 PaFailed 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 \cdot} meter2)
Force per length, F'	μb2/R	5.52e-3 (Newton/meter or PaFailed 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 \cdot} meter)
Velocity, v	μb2m/R	5.52e-3 (meter/sec)
Strain rate, 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 \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.