PolymerFEM - Constitutive Models FAQ

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Conversion Between Elastic Constants

The following table shows how to convert between elastic constants (Young's modulus, Poisson's ratio, Lame's constants) in different formats:

FEA Unit Systems

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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)

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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)

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

Running ANSYS from command line

I often run ANSYS from the command line. One way to do this is to use the command:

ansys140 -i simulation_file.dat -j res_file -o res_file.log -b -np 4

where
• -i specifies the input file
• -j specifies the base name of the job files
• -o specifies the name of the log file
• -b specifies that the simulation should run in batch mode
• -np specifies the number of CPUs to use for the calculation

Filler particle concentration - definitions

Definitions:
• mf = mass of filler particles
• mr = mass of rubber (resin)

PHR = parts per hundred of filler particles by mass

That is, if PHR=10, then 10 kg of filler particles is added to 100 kg of raw rubber resin.

PHR = 100 * mf / mr

MASS FRACTION OF FILLERS = mf / (mf + mr) = PHR / (100 + PHR)

To get the volume fraction of fillers it is necessary to know the ratio of the density of the fillers to the density of the rubber resin:

VOL. FRAC. FILLERS = rhof/rhor * PHR / (100 + rhof/rhor * PHR)

Often, rho_fillers / rho_rubber = 0.5, giving
VOL. FRAC. FILLERS approx = 0.5 * PHR / (100 + 0.5 * PHR)

Neo-Hookean parameters in ABAQUS

ABAQUS requires that the neo-Hookean material parameters be expressed in terms of C10 and D1. These parameters can be obtained from the shear modulus (mu) and the bulk modulus (kappa) by the following expressions:

C10 = mu / 2

D1 = 2 / kappa

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