Ideal Gas Material

Constitutive Laws

This MPM material models either an ideal gas or a non-ideal van der Waals gas. Note the MPM is not in ideal method for odeling gas dynamics. The main use of this material type is when combine gas material points interact with other material types. TO optimize such modeling, the gas pressure can be is tracked by one of two methods:

1. Actual Pressure: Set reference pressure to [math]\displaystyle{ P_0 }[/math] and track total pressure, [math]\displaystyle{ P }[/math], on the each particle from gas laws.
2. Gauge Pressure: Omitting reference pressure material property, P0, tells the calculation track a gauge pressure, [math]\displaystyle{ p = P - P_0 }[/math] where [math]\displaystyle{ P_0 }[/math] is the gauge pressure set to 1 atmosphere (= 0.101325 MPa), on each particle. In other words 1 atmosphere pressure is a new baseline that correspond to zero gauge pressure on the particle.

The "Actual Pressure" works best when modeling just an ideal gas. The "Gauge Pressure" method appears to work better when gas particles are interacting with other materials (all of which have zero stress under starting conditions).

Ideal Gas Law

The ideal gas law is:

      [math]\displaystyle{ PV = nRT }[/math]

where P is pressure, V is volume, n is number of moles, R is the gas constant, and T is absolute temperture. By making use of a Reference State, this equation is recast in a more convenient form for modeling as a hyperelastic material in MPM code. The resulting constitutive laws for pressure and gauge pressure are:

      [math]\displaystyle{ P = P_0 {T\over T_0} {1\over J} \quad{\rm and}\quad p = P_0 \left({T\over T_0} {1\over J}-1\right) }[/math]

where J is determinant of the deformation tensor (J = V/V₀) and T is temperature. P₀, T₀, and V₀ (the initial particle value) define a gas Reference State]. Note that P₀ is always 1 atmosphere when using gauge pressure.

The pressure is stored in the normal stresses or σ_xx =σ_yy = σ_zz = -P (or -p). All shear stresses are zero. This material is equivalent to a hyperelastic material with strain energy function of

      [math]\displaystyle{ P = -\frac{dW(J)}{dJ} \quad {\rm or}\quad W(J) = -P_0{T\over T_0} \ln J }[/math]

where [math]\displaystyle{ C }[/math] is any constant. This energy function is equivalent to the energy per unit initial volume for isothermal compression or expansion of an ideal gas.

When using isothermal mode, this material models isothermal compression and expansion, which implies all work results in heating or cooling. The amount of heat generated is tracked in the particle's heat energy. The problem may include heat input (in thermal boundary conditions), which may cause temperture rises. In other words, this mode only means the gas itself will not cause temperature changes. The internal energy will not change unless there is external heating.

To model adiabatic compression and expansion, activate adiabatic mode. This mode will convert work to heat resulting in heating during compression or cooling during expansion. You need to enter heat capacity (which can only pick monotonic or diatomic gas) and thermal conductivity. The current implementation uses a temperature independent conductivity (which may change in the future).

Stability

Ideal gas particles are fairly stable, but can be made unstable by certain boundary conditions on constraining walls. If stability problems arise, try different boundary conditions. They also do not work for irreversibale processes such as free expansion into empty space. They are intended to always be bounded by stable pressure.

Non-Ideal van der Waals Gas

The van der Waals gas law is

     [math]\displaystyle{ \left(P + {an^2\over V^2}\right)\left({V\over n}-b\right) = RT }[/math]

where a and b are van der Waals gas properties. Using reference state calculations to eliminate n, actual and gauge pressure constitutive laws can be written as:

     [math]\displaystyle{ P = P_0'{T\over T_0}\left({1\over J-b'}\right) - {a'\over J^2} \quad{\rm and}\quad p = P_0'{T\over T_0}\left({1\over J-b'}\right) -\left(P_0 + {a'\over J^2}\right) }[/math]

where

     [math]\displaystyle{ a' = \frac{a\rho_0^2}{M_g},\ b'=\frac{b\rho_0}{M_g},\ {\rm and}\ P_0'=(P_0+a')(1-b') }[/math]

Note that these law revert to ideal gas law when [math]\displaystyle{ a'=b'=0 }[/math]. The constant [math]\displaystyle{ b' }[/math] can also be written as [math]\displaystyle{ b'=bn/V_0 }[/math] that corresponds to fraction of thw references volume that is excluded by gas molecules (an ideal gas ignores volume of gas molecules).

Reference State

Gas material properties are set by defining a reference state with pressure, [math]\displaystyle{ P_0 }[/math], temperature, [math]\displaystyle{ T_0 }[/math], and initial particle volume, [math]\displaystyle{ V_0 }[/math]. When using gauge pressure method, [math]\displaystyle{ P_0 }[/math] is set to 1 atmosphere. The reference state is used eliminate number of moles in the gas law. For an ideal gas the analysis is easy:

     [math]\displaystyle{ n = \frac{P_0V_0}{RT} }[/math]

Substitution into ideal gas law gives the constitutive laws used in code. The calculation is the same for a van der Waals gas but is more complicated (it needs to solve a cubic equation). The details are provided in the materials.pdf file in the technical notes.

MPM coding also needs the initial density. For an ideal gas, initial density can be found from

     [math]\displaystyle{ \rho_0 = \frac{nM_g}{V_0} = {P_0 M_g\over R T_0} }[/math]

where [math]\displaystyle{ M_g }[/math] is molecular weight of the gas molecule. The second form uses reference state to eliminate [math]\displaystyle{ V_0/n }[/math]. For example, air might have reference pressure of 1 atm = 0.101325 MPa, [math]\displaystyle{ M_g=28.97 }[/math] g/mol at room temperature T₀=300K with R = 8.3144621 J/k/mol leading to [math]\displaystyle{ \rho_0=0.1176 }[/math] g/cm². The initial density for a van der Waals gas is slightly different (and found by revised [math]\displaystyle{ V_0/n }[/math] in reference conditions). The difference in calculated density compared to an ideal gas calculated density is usually small.

Material Properties

The ideal gas properties are set with

Property	Description	Units	Default
P0	Reference pressure at reference temperature and reference density. This must be a positive value greather than zero.	pressure units	none
T0	Reference temperature	K	none
rho	Density at reference conditions.	density units	none
Cv	Instead of heat capacity, this parameter determines if the gas is monatomic or diatomic. Enter 1 (or any number smaller than 1) for monatomic or 2 (or any number larger than 2) for diatomic. The resulting heat capacity is CV = (3/2)R for a monotonic gas and CV = (5/2)R for a diatomic gas.	none	1
(other)	Properties common to all materials	varies	varies

When using gas particles, the stress free temperature must always be set to a desired temperature in Kelvin.

History Variables

None

Examples

Note that you must always set the stress free temperature when use ideal gas law for sample particles.

Material "air","Air","IdealGas"
  P0 0.101325
  T0 300
  rho 0.001176
  Cv 2
Done
StressFreeTemp 300