Week - 1 - Consistency of Units

Consistency of Units

The SI unit system (Internation System of Units) defines the units for length in meters(m), time is seconds(s) and mass in kilograms(kg). Additionally, the units for force in the SI system is Newton(N).

The unit 1PA (pascals) can be expressed in terms of SI units as: 1 N/m^2.

Simillarly,

1 MPA (Megapascals) = 1 x 10^6 N/m^2

1 GPA (Gigapascals) = 1 x 10^9 N/m^2

The conversion highlighted in red is used as a conversion factor for defining the Elastic Modulus of Steel in SI units (kg/m/s) in LS-DYNA. But when a kg/mm/ms  system is chosen, 10^-9 is used as a conversion factor.

Explanation:

SI units: E = 210 x 10^9 N/m^2 ......(1)

(kg/mm/ms) system: 

The N/m^2 units will be converted to N/mm^2 as follows,

N/m^2 --> N/(10^3)^2 mm^2 --> 10^-6 N/mm^2 (Since 1m = 1000mm)......(2)

Now, since force (1N) is expressed in kg.m/s^2 in SI system, a similar conversion will be applied to the force unit Newton (N) as follows,

1N --> kg.m/s^2 --> kg.10^3 mm/(10^3)^2 ms^2 --> 10^-3 kg.mm/ms^2 --> 10^-3 N ......(3)

Substituting eq(3) in eq(2)

N/m^2 --> 10^-6 x 10^-3 N/mm^2 --> 10^-9 N/mm^2......(4) (Factor of conversion!)

Substituting eq(4) in eq(1)

(kg/mm/ms): E = 210 x 10^9 x 10^-9 N/mm^2 --> 210 N/mm^2

Hence, in (kg/mm/ms) system a factor of conversion:10^-9 is used to define the elastic modulus of steel.

Conversion from SI unit system to other systems (including their conversion factors)

	Mass 1kg	Length 1m	Time 1s	Force 1N	Stress 1Pa	Energy 1Nm	Density 1kg/m3	E 210GPa	Velocity 56.33kmph	Gravity 9.8m/s2
kg cm ms	1	10^2	10^3	10^-4	10^-8	10^-2	10^-6	210x10	56.33 x 1/36	9.8 x 10^-4
kg cm s	1	10^2	1	10^2	10^-2	10^4	10^-6	210x10^7	56.33 x 10^3/36	9.8 x 10^2
gm mm ms	10^3	10^3	10^3	1	10^-6	10^3	10^-6	210x10^3	56.33 x 1/3.6	9.8 x 10^-3
kg m s	1	1	1	1	1	1	1	210x10^9	56.33 x 1/3.6	9.8
gm cm s	10^3	10^2	1	10^5	10	10^7	10^-3	210x10^10	56.33 x 10^3/36	9.8 x 10^2
gm cm us	10^3	10^2	10^6	10^-7	10^-11	10^-5	10^-3	210x10^-2	56.33 x 1/36x10^3	9.8 x 10^-10
gm mm s	10^3	10^3	1	10^6	1	10^9	10^-6	210x10^9	56.33 x 278	9.8 x 10^3
ton mm s	10^-3	10^-3	1	1	10^-6	10^-3	10^6	210x10^3	56.33 x 278	9.8 x 10^-3
slug ft s	1/14.594	3.281	1	3.281/14.594	1/(14.594x3.281)	1/14.594	1.94 x 10^-3	210x(10^9/(14.594x3.281))	56.33 x 1/1.097	9.8 x 3.281
kg mm s	1	10^3	1	10^3	10^-3	10^6	10^-9	210x10^6	56.33 x 278	9.8 x 10^3
g cm ms	10^3	10^2	10^3	10^-1	10^-5	10	10^-3	210x10^4	56.33 x 1/36	9.8 x 10^-4
kg mm ms	1	10^3	10^3	10^-3	10^-9	1	10^-9	210x1	56.33 x 1/3.6	9.8 x 10^-3

As given in the table above, the mass, length and time in gm/mm/ms system is given as,

M = 10^3 gm

L = 10^3 mm

T = 10^3 ms

The conversions for rest of the measurements are as follows:

Force = kg.m/s^2 --> 10^3 . 10^3/(10^3)^2 --> 1 gm.mm/ms^2

Stress = N/m^2 --> 1/(10^3)^2 --> 10^-6 N/mm^2

Energy = Nm --> 1 . 10^3 Nmm

Density = kg/m^3 --> 10^3/(10^3)^3 --> 10^-6

E = 210 x 10^9 N/m^2 --> 210 x 10^9 x 10^-6 N/mm^2 --> 210 x 10^3 N/mm^2

Velocity = 56.33 kmph --> 56.33 x (10^6/3.6 x 10^6) --> 56.33/3.6 mmpms

Gravity = 9.8 m/s^2 --> 9.8 x 10^3/(10^3)^2 --> 9.8 x 10^-3 mm/ms^2