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[Solved] Dimensions and Units for FEA


Jorgen
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Finite element programs do not consider the units of given quantities , it is the users responsibility to ensure that the given numbers have consistent units. There are numerous different sets of units that can be used when performing FE simulations. The best set of units will depend on the problem, typically the most accurate results are obtained if the units are chosen such that the values of the input quantities to the FE simulation are close to unity. By having the input quantities close to 1, the influence of round-off errors and truncation errors are reduced.

I have attached a document summarizing some the more common sets of of units and dimensions

.

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Jorgen
Posts: 3941
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------------------------------------------------------------

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 = 1e-3 m/s
[acceleration]         = [length] / [time]^2 = mm/s^2 = 1e-3 m/s^2
[mass]                 = [force] / [acceleration] = Mg = 1e3 kg
[volume]               = [length]^3 = mm^3 = (1e-3)^3 m^3 = 1e-9 m^3
[density]              = [mass] / [volume] = 1e3 kg / (1e-3)^3 m^3 = 1e12 kg/m^3 = Mg/mm^3
[energy]               = [force] * [length] = N * mm = 1e-3 J = mJ
[energy density]       = [energy] / [volume] = 1e6 J/m^3 = MJ/m^3
[effect]               = [energy] / [time] = mW
[moment]               = [force] * [length] = N * mm = 1e-3 Nm = mNm
[thermal conductivity] = [effect] / ([length] * [temp]) = mW / (mm K) = W/(m K)
[specific heat]        = [energy] / ([mass] * [temp]) = 1e-3 J / (1e3 kg K) = 1e-6 J/(kg K)
[heat flux]            = [effect] / [length]^2 = 1e3 W/m^2
[heat transfer coeff]  = [effect] / ([length]^2 * [temp]) = 1e3 W/(m^2 K)

Usage examples:  if the density = 1000 kg/m^3, then in ABAQUS use 1000e-12 (1e12 kg/m^3)
                 if the acceleration = 9.8 m/s^2, then use 9.8e3 mm/s^2 

------------------------------------------------------------

CASE 3: SI Units (micro-scaled parts)

[length]       = 1e-6 m = 1.0 micro m
[force]        = 1e-6 N
[time]         = s
[temperature]  = K

[pressure]             = [force] / [length]^2 = 1e6 Pa = MPa
[stress]               = [pressure] = 1e6 Pa
[velocity]             = [length] / [time] = 1e-6 m/s
[acceleration]         = [length] / [time]^2 = 1e-6 m/s^2
[mass]                 = [force] / [acceleration] = 1 kg
[volume]               = [length]^3 = 1e-18 m^3
[density]              = [mass] / [volume] = 1e18 kg/m^3
[energy]               = [force] * [length] = 1e-6 N * 1e-6 m = 1e-12 J

Usage example:  if the density = 1000 kg/m^3, then in ABAQUS use 1000e-18 (1e18 kg/m^3)

------------------------------------------------------------

CASE 4: Imperial 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

F = m * a

Example: steel: [density] = 7.3e-4 snails/in^3
Example: polymer: [density] = 1/7.85 * 7.3e-4 snails/in^3 = 9.3e-5 snails/in^2

(1 lbf) = [m] * (1 in/s^2)
(4.448 N) = [m] * (25.4 mm/^2) => [m] = 0.175

------------------------------------------------------------

CASE 5: SI Units (nano-scale parts)

[length]       = 1e-9 m = nm
[force]        = 1e-9 N = nN
[time]         = s
[temperature]  = K

[pressure]             = [force] / [length]^2 = 1e9 Pa = GPa
[stress]               = [pressure] = 1e9 Pa
[velocity]             = [length] / [time] = nm/s = 1e-9 m/s
[acceleration]         = [length] / [time]^2 = nm/s^2 = 1.0e-9 m/s^2
[mass]                 = [force] / [acceleration] = (1e-9 N) / (1.0e-9 m/s^2) =
                         (1e-9 kg * m / s^2) / (1.0e-9 m/s^2) = kg
[volume]               = [length]^3 = nm^3 = 1e-27 m^3
[density]              = [mass] / [volume] = kg/nm^3 = (1 kg) / ((1e-9)^3 m^3) = 1e27 kg/m^3
[energy]               = [force] * [length] = (1e-9 N) * (1e-9 m) = 1e-18 J

------------------------------------------------------------

CASE 5b: SI Units (nano-scale parts, second version)

[length]       = 1e-9 m = nm
[force]        = 1e-12 N
[time]         = s
[temperature]  = K

[pressure]             = [force] / [length]^2 = MPa
[stress]               = [pressure] = 1e6 Pa
[velocity]             = [length] / [time] = nm/s = 1e-9 m/s
[acceleration]         = [length] / [time]^2 = nm/s^2 = 1.0e-9 m/s^2
[mass]                 = [force] / [acceleration] = (1e-12 N) / (1.0e-9 m/s^2) =
                         (1e-12 kg * m / s^2) / (1.0e-9 m/s^2) = 1.0e-3 kg = g
[volume]               = [length]^3 = nm^3 = 1e-27 m^3
[density]              = [mass] / [volume] = (1e-3 kg)/nm^3 = (1e-3 kg) / ((1e-9)^3 m^3) =
                         1e24 kg/m^3
[energy]               = [force] * [length] = (1e-9 N) * (1e-9 m) = 1e-18 J

------------------------------------------------------------

CASE 5c: SI Units (nano-scale parts, third version)

[length]       = 1e-9 m = nm
[force]        = 1e-18 N
[time]         = s
[temperature]  = K

[pressure]             = [force] / [length]^2 = Pa
[stress]               = [pressure] = Pa
[velocity]             = [length] / [time] = nm/s = 1e-9 m/s
[acceleration]         = [length] / [time]^2 = nm/s^2 = 1.0e-9 m/s^2
[mass]                 = [force] / [acceleration] = (1e-18 N) / (1.0e-9 m/s^2) =
                         (1e-18 kg * m / s^2) / (1.0e-9 m/s^2) = 1.0e-9 kg
[volume]               = [length]^3 = nm^3 = 1e-27 m^3
[density]              = [mass] / [volume] = (1e-9 kg)/nm^3 = (1e-9 kg) / ((1e-9)^3 m^3)
                         1e18 kg/m^3
[energy]               = [force] * [length] = (1e-18 N) * (1e-9 m) = 1e-27 J

------------------------------------------------------------

CASE 5d: SI Units (nano-scale parts, forth version)

[length]       = 1e-9 m = nm
[force]        = 1e-18 N
[time]         = 1e-6 s
[temperature]  = K

[pressure]             = [force] / [length]^2 = Pa
[stress]               = [pressure] = Pa
[velocity]             = [length] / [time] = nm/micro s = 1e-3 m/s
[acceleration]         = [length] / [time]^2 = nm/(micro s)^2 = 1.0e3 m/s^2
[mass]                 = [force] / [acceleration] = (1e-18 N) / (1.0e3 m/s^2) =
                         (1e-18 kg * m / s^2) / (1.0e3 m/s^2) = 1.0e-21 kg
[volume]               = [length]^3 = nm^3 = 1e-27 m^3
[density]              = [mass] / [volume] = (1e-21 kg)/nm^3 = (1e-21 kg) / ((1e-9)^3 m^3)
                         1e6 kg/m^3
[energy]               = [force] * [length] = (1e-18 N) * (1e-9 m) = 1e-27 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) / ((1e-3)^3 m^3) = 1e9 kg/m^3
[energy]               = [force] * [length] = mN * mm = 1e-6 J = micro J

Usage example:  if the density = 1000 kg/m^3, then in ABAQUS use 1000e-9 (1e9 kg/m^3)


------------------------------------------------------------

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) / ((1e-3)^3 m^3) = 1e9 kg/m^3
[energy]               = [force] * [length] = kN * mm = J

Usage example:  if the density = 1000 kg/m^3, then in ABAQUS use 1000e-9 (1e9 kg/m^3)

------------------------------------------------------------

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] = 1e-3 kg = g
[volume]               = [length]^3 = mm^3
[density]              = [mass] / [volume] = g/mm^3 = (1e-3 kg) / ((1e-3)^3 m^3) = 1e6 kg/m^3
[energy]               = [force] * [length] = N * mm = mJ
[strain rate]          = 1 / [time] = 1 / ms = 1e3 /s

Usage example:  if the density = 1000 kg/m^3, then in Abaqus use 1000e-6 (1e6 kg/m^3)
Usage example:  if the strain rate is x/s, then in Abaqus use x 1e-3 (1e3/s)

------------------------------------------------------------

CASE 10: SI Units

[length]       = mm
[force]        = micro N
[time]         = s
[temperature]  = C

[pressure]             = [force] / [length]^2 = (1e-6 Pa) / (1e-3 m) / (1e-3 m) = Pa
[stress]               = [pressure] = Pa
[velocity]             = [length] / [time] = mm/s
[acceleration]         = [length] / [time]^2 = mm/s^2
[mass]                 = [force] / [acceleration] = (1e-6 N) / (1e-3 m/s^2) = 1e-3 kg = g
[volume]               = [length]^3 = mm^3
[density]              = [mass] / [volume] = 1e-3 kg/mm^3 = (1e-3 kg) / ((1e-3)^3 m^3) = 1e6 kg/m^3
[energy]               = [force] * [length] = (1e-6 N) * (1e-3 m) = 1e-9 J = nano J

Usage example:  if the density = 1000 kg/m^3, then in Abaqus use 1000e-6 (1e6 kg/m^3) = 0.001 (1e6 kg/m^3)

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