Ideal Gas Material
Constitutive Laws
This material can model either an ideal gas or a non-ideal van der Waals gas. The gas is modeled one of two ways:
- Set P0 to a reference pressure and track total pressure, [math]\displaystyle{ P }[/math], on the each particle from gas laws.
- Omit P0 and track a gauge pressure, [math]\displaystyle{ p = P - 1 }[/math] atmosphere (= 0.101325 MPa), on each particles
Version 1 works best when modeling just an ideal gas. Version 2 appears to work better when gas is interacting with other materials (all of which have zero stress under 1 atmosphere of pressure).
Ideal Gas Law
This MPM material models an ideal gas implemented as a large-deformation, isotropic, hyperelastic material at finite deformations. Its contitutive law for pressure and gauge pressure are:
[math]\displaystyle{ P = P_0 {T\over T_0} {1\over J} \quad{\rm and}\quad p = P_1 \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. When using actual pressure, P0 and T0 are reference pressure and temperature at the reference density. When using gauge pressure, [math]\displaystyle{ P_1 }[/math] is set equal to 1 atmosphere and T0 is reference temperature for 1 atmosphere of pressure at the reference density.
The pressure is stored in the normal stresses or σxx =σyy = σzz = -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_1\left({T\over T_0} \ln J - J + C\right) }[/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 laws for actual pressure and gauge pressure are
[math]\displaystyle{ \left(P + {an^2\over V^2}\right)\left({V\over n}-b\right) = RT \quad{\rm and}\quad \left(p+P_0 + {an^2\over V^2}\right)\left({V\over n}-b\right) = RT }[/math]
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. The material properties refer to gas state at any reference conditions. For example, air might have reference pressure of 1 atm = 0.101325 pressure units at some reference temperature T0. The density of air (or any gas) is then given by
[math]\displaystyle{ \rho = {P_0 M_g\over R T_0} }[/math]
where Mg is the molecular weight of the gas (such as 28.97 g/mol for air) and R is the gas constant, R = 8.3144621 J/k/mol.
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 288.15 rho 0.001163 Cv 2 Done StressFreeTemp 300