Collections

• Collections
• • Usage
  • • Construct a Parameters instance
    • Evaluate energy
    • Evaluate pressure
    • Evaluate bulk modulus
  • Public interfaces

The current Parameterss contain

EquationsOfState.EquationOfStateOfSolidsParameters
 ├─ EquationsOfStateOfSolids.AntonSchmidt
 ├─ EquationsOfStateOfSolids.FiniteStrainParameters
 │   ├─ BirchMurnaghan
 │   │   ├─ BirchMurnaghan2nd
 │   │   ├─ BirchMurnaghan3rd
 │   │   └─ BirchMurnaghan4th
 │   └─ PoirierTarantola
 │       ├─ EquationsOfStateOfSolids.PoirierTarantola4th
 │       ├─ PoirierTarantola2nd
 │       └─ PoirierTarantola3rd
 ├─ EquationsOfStateOfSolids.Holzapfel
 ├─ Murnaghan
 │   ├─ EquationsOfStateOfSolids.Murnaghan2nd
 │   └─ Murnaghan1st
 └─ Vinet

Here the leaves of the type tree are concrete types and can be constructed.

Usage

Construct a Parameters instance

We will use BirchMurnaghan3rd as an example.

A BirchMurnaghan3rd can be constructed from scratch, as shown above. It can also be constructed from an existing BirchMurnaghan3rd, with Setfield.jl @set! macro:

julia> using Setfield

julia> eos = Murnaghan(1, 2, 3.0)
Murnaghan{Float64}(1.0, 2.0, 3.0, 0.0)

julia> @set! eos.v0 = 4
Murnaghan{Float64}(4.0, 2.0, 3.0, 0.0)

julia> eos
Murnaghan{Float64}(4.0, 2.0, 3.0, 0.0)

To modify multiple fields (say, :v0, :b′0, :b′′0, :e0) at a time, use @batchlens from Kaleido.jl:

julia> using Setfield, Kaleido

julia> lens = @batchlens(begin
           _.v0
           _.b′0
           _.b″0
           _.e0
       end)
IndexBatchLens(:v0, :b′0, :b″0, :e0)

julia> eos = BirchMurnaghan4th(1, 2.0, 3, 4)
BirchMurnaghan4th{Float64}(1.0, 2.0, 3.0, 4.0, 0.0)

julia> set(eos, lens, (5, 6, 7, 8))
BirchMurnaghan4th{Float64}(5.0, 2.0, 6.0, 7.0, 8.0)

Users can access BirchMurnaghan3rd's elements by "dot notation":

julia> eos = BirchMurnaghan3rd(1, 2, 3, 4.0)
4-element BirchMurnaghan3rd{Float64}:
 1.0
 2.0
 3.0
 4.0

julia> eos.v0
1.0

Evaluate energy

The $E(V)$ relation of equations of state are listed as below:

1. Murnaghan:

   \[E(V) = E_{0}+K_{0} V_{0}\left[\frac{1}{K_{0}^{\prime}\left(K_{0}^{\prime}-1\right)}\left(\frac{V}{V_{0}}\right)^{1-K_{0}^{\prime}}+\frac{1}{K_{0}^{\prime}} \frac{V}{V_{0}}-\frac{1}{K_{0}^{\prime}-1}\right].\]

2. BirchMurnaghan2nd:

   \[E(V) = E_{0} + \frac{9}{8} B_{0} V_{0} \left(\left( V / V_0 \right)^{-2 / 3}-1\right)^{2}.\]

3. BirchMurnaghan3rd:

   \[E(V) = E_{0}+\frac{9}{16} V_{0} B_{0} \frac{\left(x^{2 / 3}-1\right)^{2}}{x^{7 / 3}}\left\{x^{1 / 3}\left(B_{0}^{\prime}-4\right)-x\left(B_{0}^{\prime}-6\right)\right\}.\]

   where x = V / V_0, and f = \frac{ 1 }{ 2 } \bigg[ \bigg( \frac{ V_0 }{ V } \bigg)^{2/3} - 1 \bigg].

4. BirchMurnaghan4th:

   \[E(V) = E_{0}+\frac{3}{8} V_{0} B_{0} f^{2}\left[\left(9 H-63 B_{0}^{\prime}+143\right) f^{2}+12\left(B_{0}^{\prime}-4\right) f+12\right].\]

   where H = B_0 B_0'' + (B_0')^2.

5. PoirierTarantola2nd:

   \[E(V) = E_{0}+\frac{1}{2} B_{0} V_{0} \ln ^{2} x.\]

6. PoirierTarantola3rd:

   \[E(V) = E_{0}+\frac{1}{6} B_{0} V_{0} \ln ^{2} x\left[\left(B_{0}^{\prime}+2\right) \ln x+3\right].\]

7. PoirierTarantola4th:

   \[E(V) = E_{0}+\frac{1}{24} B_{0} V_{0} \ln ^{2} x\left\{\left(H+3 B_{0}^{\prime}+3\right) \ln ^{2} x\right. \left.+4\left(B_{0}^{\prime}+2\right) \ln x+12\right\}.\]

   where H = B_0 B_0'' + (B_0')^2.

8. Vinet:

   \[E(V) = E_{0}+\frac{9}{16} V_{0} B_{0} \frac{\left(x^{2 / 3}-1\right)^{2}}{x^{7 / 3}}\left\{x^{1 / 3}\left(B_{0}^{\prime}-4\right)-x\left(B_{0}^{\prime}-6\right)\right\}.\]

9. AntonSchmidt:

   \[E(V)=\frac{\beta V_{0}}{n+1}\left(\frac{V}{V_{0}}\right)^{n+1}\left[\ln \left(\frac{V}{V_{0}}\right)-\frac{1}{n+1}\right]+E_{\infty}.\]

Evaluate pressure

The $P(V)$ relation of equations of state are listed as below:

1. Murnaghan:

   \[P(V) = \frac{B_{0}}{B_{0}^{\prime}}\left[\left(\frac{V_{0}}{V}\right)^{B_{0}^{\prime}}-1\right].\]

2. BirchMurnaghan2nd:

   \[P(V) = \frac{3}{2} B_{0}\left(x^{-7 / 3}-x^{-5 / 3}\right).\]

3. BirchMurnaghan3rd:

   \[P(V) = \frac{3}{8} B_{0} \frac{x^{2 / 3}-1}{x^{10 / 3}}\left\{3 B_{0}^{\prime} x-16 x-3 x^{1 / 3}\left(B_{0}^{\prime}-4\right)\right\}.\]

4. BirchMurnaghan4th:

   \[P(V) = \frac{1}{2} B_{0}(2 f+1)^{5 / 2}\left\{\left(9 H-63 B_{0}^{\prime}+143\right) f^{2}\right.\left.+9\left(B_{0}^{\prime}-4\right) f+6\right\}.\]

5. PoirierTarantola2nd:

   \[P(V) = -\frac{B_{0}}{x} \ln x.\]

6. PoirierTarantola3rd:

   \[P(V) = -\frac{B_{0} \ln x}{2 x}\left[\left(B_{0}^{\prime}+2\right) \ln x+2\right].\]

7. PoirierTarantola4th:

   \[P(V) = -\frac{B_{0} \ln x}{6 x}\left\{\left(H+3 B_{0}^{\prime}+3\right) \ln ^{2} x+3\left(B_{0}^{\prime}+6\right) \ln x+6\right\}.\]

8. Vinet:

   \[P(V) = 3 B_{0} \frac{1-\eta}{\eta^{2}} \exp \left\{-\frac{3}{2}\left(B_{0}^{\prime}-1\right)(\eta-1)\right\}.\]

9. AntonSchmidt:

   \[P(V) = -\beta\left(\frac{V}{V_{0}}\right)^{n} \ln \left(\frac{V}{V_{0}}\right).\]

Evaluate bulk modulus

The $B(V)$ relation of equations of state are listed as below:

1. BirchMurnaghan2nd:

   \[B(V) = B_{0}(7 f+1)(2 f+1)^{5 / 2}.\]

2. BirchMurnaghan3rd:

   \[B(V) = B_{0}(2 f+1)^{5 / 2} \left\{ 1 + (3B_{0}^{\prime} - 5) f + \frac{ 27 }{ 2 }(B_{0}^{\prime} - 4) f^2 \right\}\]

3. BirchMurnaghan4th:

   \[B(V) = \frac{1}{6} B_{0}(2 f+1)^{5 / 2}\left\{\left(99 H-693 B_{0}^{\prime}+1573\right) f^{3}\right.\left.+\left(27 H-108 B_{0}^{\prime}+105\right) f^{2}+6\left(3 B_{0}^{\prime}-5\right) f+6\right\}.\]

4. PoirierTarantola2nd:

   \[B(V) = \frac{B_{0}}{x}(1-\ln x).\]

5. PoirierTarantola3rd:

   \[B(V) = -\frac{B_{0}}{2 x}\left[\left(B_{0}^{\prime}+2\right) \ln x(\ln x-1)-2\right].\]

6. PoirierTarantola4th:

   \[B(V) = -\frac{B_{0}}{6 x}\left\{\left(H+3 B_{0}^{\prime}+3\right) \ln ^{3} x-3\left(H+2 B_{0}^{\prime}+1\right) \ln ^{2} x\right.\left.-6\left(B_{0}^{\prime}+1\right) \ln x-6\right\}.\]

7. Vinet:

   \[B(V) = -\frac{B_{0}}{2 \eta^{2}}\left[3 \eta(\eta-1)\left(B_{0}^{\prime}-1\right)+2(\eta-2)\right]\times \exp \left\{-\frac{3}{2}\left(B_{0}^{\prime}-1\right)(\eta-1)\right\}.\]

8. AntonSchmidt:

   \[B(V) = \beta\left(\frac{V}{V_{0}}\right)^{n}\left[1+n \ln \frac{V}{V_{0}}\right].\]

Public interfaces

Murnaghan1st(v0, b0, b′0, e0=zero(v0 * b0))

BirchMurnaghan(args...)

BirchMurnaghan2nd(v0, b0, e0=zero(v0 * b0))

BirchMurnaghan3rd(v0, b0, b′0, e0=zero(v0 * b0))

BirchMurnaghan4th(v0, b0, b′0, b″0, e0=zero(v0 * b0))

PoirierTarantola(args...)

PoirierTarantola2nd(v0, b0, e0=zero(v0 * b0))

PoirierTarantola3rd(v0, b0, b′0, e0=zero(v0 * b0))

Vinet(v0, b0, b′0, e0=zero(v0 * b0))

EnergyEquation{T} <: EquationOfStateOfSolids{T}
EnergyEquation(parameters::Parameters)

PressureEquation{T} <: EquationOfStateOfSolids{T}
PressureEquation(parameters::Parameters)

BulkModulusEquation{T} <: EquationOfStateOfSolids{T}
BulkModulusEquation(parameters::Parameters)

getparam(eos::EquationOfStateOfSolids)

orderof(x::FiniteStrainParameters)

julia> orderof(BirchMurnaghan(40, 0.5, 4, 0)) == 3
true