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FEA Unit Systems

CASE 1: SI Units
Choose the following base dimensions:
Length in meters (m)
Force in Newtons (N)
Time in seconds (s)
Temperature in Kelvin (K)
Then the following dimensions need to be used:
[pressure] = [force] / [length]^2 = N/m^2 = Pa
[stress] = [pressure] = N/m^2 = Pa
[velocity] = [length] / [time] = m/s
[acceleration] = [length] / [time]^2 = m/s^2
[mass] = [force] / [acceleration] = kg
[volume] = [length]^3 = m^3
[density] = [mass] / [volume] = kg / m^3
[energy] = [force] * [length] = N * m = J
[energy density] = [energy] / [volume] = J/m^3
[effect] = [energy] / [time] = J/s = W
[thermal conductivity] = [effect] / ([length] * [temp]) = W / (m K)
[specific heat] = [energy] / ([mass] * [temp]) = J / (kg K)
[heat flux] = [effect] / [length]^2 = W/m^2
[heat transfer coeff] = [effect] / ([length]^2 * [temp]) = W/(m^2 K)

CASE 2: SI Units (small parts)
Length in millimeters (mm)
Force in Newton (N)
Time is seconds (s)
Temperature in Kelvin (K)
[pressure] = [force] / [length]^2 = N/mm^2 = 1e6 Pa = MPa
[stress] = [pressure] = 1e6 Pa = N/mm^2 = MPa
[velocity] = [length] / [time] = mm/s = 1e3 m/s
[acceleration] = [length] / [time]^2 = mm/s^2 = 1e3 m/s^2
[mass] = [force] / [acceleration] = Mg = 1e3 kg
[volume] = [length]^3 = mm^3 = (1e3)^3 m^3 = 1e9 m^3
[density] = [mass] / [volume] = 1e3 kg / (1e3)^3 m^3 = 1e12 kg/m^3 = Mg/mm^3
[energy] = [force] * [length] = N * mm = 1e3 J = mJ
[energy density] = [energy] / [volume] = 1e6 J/m^3 = MJ/m^3
[effect] = [energy] / [time] = mW
[moment] = [force] * [length] = N * mm = 1e3 Nm = mNm
[thermal conductivity] = [effect] / ([length] * [temp]) = mW / (mm K) = W/(m K)
[specific heat] = [energy] / ([mass] * [temp]) = 1e3 J / (1e3 kg K) = 1e6 J/(kg K)
[heat flux] = [effect] / [length]^2 = 1e3 W/m^2
[heat transfer coeff] = [effect] / ([length]^2 * [temp]) = 1e3 W/(m^2 K)

CASE 3: SI Units (microscaled parts)
[length] = 1e6 m = 1.0 micro m
[force] = 1e6 N
[time] = s
[temperature] = K
[pressure] = [force] / [length]^2 = 1e6 Pa = MPa
[stress] = [pressure] = 1e6 Pa
[velocity] = [length] / [time] = 1e6 m/s
[acceleration] = [length] / [time]^2 = 1e6 m/s^2
[mass] = [force] / [acceleration] = 1 kg
[volume] = [length]^3 = 1e18 m^3
[density] = [mass] / [volume] = 1e18 kg/m^3
[energy] = [force] * [length] = 1e6 N * 1e6 m = 1e12 J

CASE 4: English Units
[length] = in
[force] = lbf
[time] = s
[temperature] = K
[pressure] = [force] / [length]^2 = lbf/in^2 = psi
[stress] = [pressure] = psi
[velocity] = [length] / [time] = in/s
[acceleration] = [length] / [time]^2 = in/s^2
[mass] = [force] / [acceleration] = 1 snail (about 386 lbf on earth)
[volume] = [length]^3 = in^3
[density] = [mass] / [volume] =
[energy] = [force] * [length] = lbf * in
[energy density] = [energy] / [volume] = lbf / in^2 = psi

CASE 5: SI Units (nanoscale parts)
[length] = 1e9 m = nm
[force] = 1e9 N = nN
[time] = s
[temperature] = K
[pressure] = [force] / [length]^2 = 1e9 Pa = GPa
[stress] = [pressure] = 1e9 Pa
[velocity] = [length] / [time] = nm/s = 1e9 m/s
[acceleration] = [length] / [time]^2 = nm/s^2 = 1.0e9 m/s^2
[mass] = [force] / [acceleration] = (1e9 N) / (1.0e9 m/s^2) =
(1e9 kg * m / s^2) / (1.0e9 m/s^2) = kg
[volume] = [length]^3 = nm^3 = 1e27 m^3
[density] = [mass] / [volume] = kg/nm^3 = (1 kg) / ((1e9)^3 m^3) = 1e27 kg/m^3
[energy] = [force] * [length] = (1e9 N) * (1e9 m) = 1e18 J

CASE 5b: SI Units (nanoscale parts, second version)
[length] = 1e9 m = nm
[force] = 1e12 N
[time] = s
[temperature] = K
[pressure] = [force] / [length]^2 = MPa
[stress] = [pressure] = 1e6 Pa
[velocity] = [length] / [time] = nm/s = 1e9 m/s
[acceleration] = [length] / [time]^2 = nm/s^2 = 1.0e9 m/s^2
[mass] = [force] / [acceleration] = (1e12 N) / (1.0e9 m/s^2) =
(1e12 kg * m / s^2) / (1.0e9 m/s^2) = 1.0e3 kg = g
[volume] = [length]^3 = nm^3 = 1e27 m^3
[density] = [mass] / [volume] = (1e3 kg)/nm^3 = (1e3 kg) / ((1e9)^3 m^3) =
1e24 kg/m^3
[energy] = [force] * [length] = (1e9 N) * (1e9 m) = 1e18 J

CASE 5c: SI Units (nanoscale parts, third version)
[length] = 1e9 m = nm
[force] = 1e18 N
[time] = s
[temperature] = K
[pressure] = [force] / [length]^2 = Pa
[stress] = [pressure] = Pa
[velocity] = [length] / [time] = nm/s = 1e9 m/s
[acceleration] = [length] / [time]^2 = nm/s^2 = 1.0e9 m/s^2
[mass] = [force] / [acceleration] = (1e18 N) / (1.0e9 m/s^2) =
(1e18 kg * m / s^2) / (1.0e9 m/s^2) = 1.0e9 kg
[volume] = [length]^3 = nm^3 = 1e27 m^3
[density] = [mass] / [volume] = (1e9 kg)/nm^3 = (1e9 kg) / ((1e9)^3 m^3)
1e18 kg/m^3
[energy] = [force] * [length] = (1e18 N) * (1e9 m) = 1e27 J

CASE 5d: SI Units (nanoscale parts, forth version)
[length] = 1e9 m = nm
[force] = 1e18 N
[time] = 1e6 s
[temperature] = K
[pressure] = [force] / [length]^2 = Pa
[stress] = [pressure] = Pa
[velocity] = [length] / [time] = nm/micro s = 1e3 m/s
[acceleration] = [length] / [time]^2 = nm/(micro s)^2 = 1.0e3 m/s^2
[mass] = [force] / [acceleration] = (1e18 N) / (1.0e3 m/s^2) =
(1e18 kg * m / s^2) / (1.0e3 m/s^2) = 1.0e21 kg
[volume] = [length]^3 = nm^3 = 1e27 m^3
[density] = [mass] / [volume] = (1e21 kg)/nm^3 = (1e21 kg) / ((1e9)^3 m^3)
1e6 kg/m^3
[energy] = [force] * [length] = (1e18 N) * (1e9 m) = 1e27 J

CASE 6: SI Units (small parts, second version)
[length] = mm
[force] = mN
[time] = s
[temperature] = K
[pressure] = [force] / [length]^2 = 1e3 Pa = kPa
[stress] = [pressure] = 1e3 Pa
[velocity] = [length] / [time] = mm/s
[acceleration] = [length] / [time]^2 = mm/s^2
[mass] = [force] / [acceleration] = 1 kg
[volume] = [length]^3 = mm^3
[density] = [mass] / [volume] = kg/mm^3 = (1 kg) / ((1e3)^3 m^3) = 1e9 kg/m^3
[energy] = [force] * [length] = mN * mm = 1e6 J = micro J

CASE 7: SI Units but with long times
Choose the following base dimensions:
Length in meters (m)
Force in Newtons (N)
Time in seconds (days)
[temperature = K
Then the following dimensions need to be used:
[pressure] = [force] / [length]^2 = N/m^2 = Pa
[stress] = [pressure] = N/m^2 = Pa
[velocity] = [length] / [time] = m/days = (1/86400) m/s
[acceleration] = [length] / [time]^2 = m/days^2 = (1/86400^2) m/s^2
[mass] = [force] / [acceleration] = (86400^2) kg
[volume] = [length]^3 = m^3
[density] = [mass] / [volume] = (86400^2) kg / m^3
[energy] = [force] * [length] = N * m = J

CASE 8: SI Units (large forces, short times)
[length] = mm
[force] = kN
[time] = ms
[temperature] = K
[pressure] = [force] / [length]^2 = 1e9 Pa = GPa
[stress] = [pressure] = 1e9 Pa
[velocity] = [length] / [time] = m/s
[acceleration] = [length] / [time]^2 = km/s^2
[mass] = [force] / [acceleration] = 1 kg
[volume] = [length]^3 = mm^3
[density] = [mass] / [volume] = kg/mm^3 = (1 kg) / ((1e3)^3 m^3) = 1e9 kg/m^3
[energy] = [force] * [length] = kN * mm = J

CASE 9: SI Units (small parts, short times)
[length] = mm
[force] = N
[time] = ms
[temperature] = K
[pressure] = [force] / [length]^2 = 1e6 Pa = MPa
[stress] = [pressure] = 1e6 Pa
[velocity] = [length] / [time] = m/s
[acceleration] = [length] / [time]^2 = km/s^2
[mass] = [force] / [acceleration] = 1e3 kg = g
[volume] = [length]^3 = mm^3
[density] = [mass] / [volume] = g/mm^3 = (1e3 kg) / ((1e3)^3 m^3) = 1e6 kg/m^3
[energy] = [force] * [length] = N * mm = mJ
[strain rate] = 1 / [time] = 1 / ms = 1e3 /s