The Mie–Grüneisen equation of state is an
equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal ...
that relates the
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
and
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of a solid at a given temperature.
[Roberts, J. K., & Miller, A. R. (1954). Heat and thermodynamics (Vol. 4). Interscience Publishers.][Burshtein, A. I. (2008). Introduction to thermodynamics and kinetic theory of matter. Wiley-VCH.] It is used to determine the pressure in a
shock
Shock may refer to:
Common uses Collective noun
*Shock, a historic commercial term for a group of 60, see English numerals#Special names
* Stook, or shock of grain, stacked sheaves
Healthcare
* Shock (circulatory), circulatory medical emerge ...
-compressed solid. The Mie–Grüneisen relation is a special form of the
Grüneisen model which describes the effect that changing the volume of a crystal lattice has on its vibrational properties. Several variations of the Mie–Grüneisen equation of state are in use.
The Grüneisen model can be expressed in the form
:
where is the volume, is the pressure, is the
internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
, and is the Grüneisen parameter which represents the thermal pressure from a set of vibrating atoms. If we assume that is independent of and , we can integrate Grüneisen's model to get
:
where
and
are the pressure and internal energy at a reference state usually assumed to be the state at which the temperature is 0K. In that case ''p''
0 and ''e''
0 are independent of temperature and the values of these quantities can be estimated from the
Hugoniot equations. The Mie–Grüneisen equation of state is a special form of the above equation.
History
Gustav Mie
Gustav Adolf Feodor Wilhelm Ludwig Mie (; 29 September 1868 – 13 February 1957) was a German physicist.
Life
Mie was born in Rostock, Mecklenburg-Schwerin, Germany in 1868. From 1886 he studied mathematics and physics at the University of ...
, in 1903, developed an intermolecular potential for deriving high-temperature equations of state of solids.
[Mie, G. (1903) "Zur kinetischen Theorie der einatomigen Körper." Annalen der Physik 316.8, p. 657-697.] In 1912,
Eduard Grüneisen
Eduard Grüneisen (26 May 1877 – 5 April 1949) was a German physicist and the co-eponym of Mie–Grüneisen equation of state.
Grüneisen was born in Giebichenstein, near Halle (Saale).
The Grüneisen parameter was named after him.
Since ...
extended Mie's model to temperatures below the
Debye temperature
In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (Heat capacity) in a solid. It treats the vibrations of the atomic lattice (hea ...
at which quantum effects become important.
[Grüneisen, E. (1912). Theorie des festen Zustandes einatomiger Elemente. Annalen der Physik, 344(12), 257-306.] Grüneisen's form of the equations is more convenient and has become the usual starting point for deriving Mie–Grüneisen equations of state.
[Lemons, D. S., & Lund, C. M. (1999). Thermodynamics of high temperature, Mie–Gruneisen solids. American Journal of Physics, 67, 1105.]
Expressions for the Mie–Grüneisen equation of state
A temperature-corrected version that is used in computational mechanics has the form
:
where
is the bulk speed of sound,
is the initial density,
is the current density,
is Grüneisen's gamma at the reference state,
is a linear Hugoniot slope coefficient,
is the shock wave velocity,
is the particle velocity, and
is the internal energy per unit reference volume. An alternative form is
:
A rough estimate of the internal energy can be computed using
:
where
is the reference volume at temperature
,
is the
heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
and
is the specific heat capacity at constant volume. In many simulations, it is assumed that
and
are equal.
Parameters for various materials
Derivation of the equation of state
From Grüneisen's model we have
where
and
are the pressure and internal energy at a reference state. The
Hugoniot equations for the conservation of mass, momentum, and energy are
:
where ''ρ''
0 is the reference density, ''ρ'' is the density due to shock compression, ''p''
H is the pressure on the Hugoniot, ''E''
H is the internal energy per unit mass on the Hugoniot, ''U''
s is the shock velocity, and ''U''
p is the particle velocity. From the conservation of mass, we have
:
Where we defined
, the specific volume (volume per unit mass).
:For many materials ''U''
s and ''U''
p are linearly related, i.e., where ''C''
0 and ''s'' depend on the material. In that case, we have
:
The momentum equation can then be written (for the principal Hugoniot where ''p''
H0 is zero) as
:
Similarly, from the energy equation we have
:
Solving for ''e''
H, we have
:
With these expressions for ''p''
H and ''E''
H, the Grüneisen model on the Hugoniot becomes
:
If we assume that and note that
, we get
The above ordinary differential equation can be solved for ''e''
0 with the initial condition ''e''
0 = 0 when ''V'' = ''V''
0 (''χ'' = 0). The exact solution is
:
where Ei[''z''] is the exponential integral. The expression for ''p''
0 is
:
For commonly encountered compression problems, an approximation to the exact solution is a power series solution of the form
:
and
:
Substitution into the Grüneisen model gives us the Mie–Grüneisen equation of state
:
If we assume that the internal energy ''e''
0 = 0 when ''V = V''
0 () we have ''A'' = 0. Similarly, if we assume ''p''
0 = 0 when ''V = V''
0 we have ''B'' = 0. The Mie–Grüneisen equation of state can then be written as
:
where ''E'' is the internal energy per unit reference volume. Several forms of this equation of state are possible.
If we take the first-order term and substitute it into equation (), we can solve for ''C'' to get
:
Then we get the following expression for ''p'':
:
This is the commonly used first-order Mie–Grüneisen equation of state.
See also
*
Impact (mechanics)
In mechanics, an impact is a high force or shock applied over a short time period when two or more bodies collide. Such a force or acceleration usually has a greater effect than a lower force applied over a proportionally longer period. The ...
*
Shock wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...
*
Shock (mechanics)
A mechanical or physical shock is a sudden acceleration caused, for example, by impact, drop, kick, earthquake, or explosion. Shock is a transient physical excitation.
Shock describes matter subject to extreme rates of force with respect to ti ...
*
Shock tube
: ''For the pyrotechnic initiator, see Shock tube detonator''
The shock tube is an instrument used to replicate and direct blast waves at a sensor or a model in order to simulate actual explosions and their effects, usually on a smaller scale. ...
*
Hydrostatic shock
Hydrostatic shock is the controversial concept that a penetrating projectile (such as a bullet) can produce a pressure wave that causes "remote neural damage", "subtle damage in neural tissues" and/or "rapid incapacitating effects" in living ta ...
*
Viscoplasticity
Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The ine ...
References
{{DEFAULTSORT:Mie-Gruneisen equation of state
Continuum mechanics
Solid mechanics
Equations of state