Ideal Gas Material

Constitutive Law

This MPM material models an ideal gas implemented as a large-deformation, isotropic, hyperelastic material at finite deformations. Its contitutive law for pressure is:

      [math]\displaystyle{ P = P_0 {T\over T_0} {1\over J} }[/math]

where J is determinant of the deformation tensor (J = V/V₀), T is temperature, and P₀ and T₀ are reference conditions. The pressure P is stored in the normal stresses or σ_xx =σ_yy = σ_zz = -P. All shear stresses are zero. This material is equivalent to a hyperelastic material with strain energy function of

      [math]\displaystyle{ W = -P_0{T\over T_0} \ln J }[/math]

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 br bounded by stable pressure.

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.	MPa	none
T0	Reference temperature	K	none
rho	Density at reference conditinos.	g/cm^3	none
Cv	Instead of head capacity, this parameter just determined is the gas is monatomic or diatomic. If this term is omitted or is less than or equal 1, CV is set to (3/2)R for a monotonic gas. If the entered value is greater than 1, CV is set to (5/2)R for a diatomic gas.	none	0
(other)	Properties common to all materials	varies	varies

When using gas particles, the stress free temperature must be set to the 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 MPa at any 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 M_g 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

Material "air","Air","Ideal Gas"
  P0 0.101
  T0 288.15
  rho 0.001163
Done
StressFreeTemp 300