def dEdρ(ρ, p, A, MP):
""" Returns the partial derivative of E by ρ (holding p,A constant)
"""
G = gram(A)
dC0dρ = dC_0dρ(ρ, MP)
ret = mg.dedρ(ρ, p, MP) + dC0dρ / 4 * L2_2D(dev(G))
return ret
def temperature_hyp(S, A, HYP):
G = gram(A)
I3 = I_3(G)
T0 = HYP.T0
γ = HYP.γ
cv = HYP.cv
return T0 * I3**(γ / 2) * exp(S / cv)
def total_energy_hyp(A, S, v, HYP):
""" Returns the total energy, given the distortion tensor and entropy
"""
G = gram(A)
I1 = I_1(G)
I2 = I_2(G)
I3 = I_3(G)
return U_1(I3, HYP) + U_2(I3, S, HYP) + W(I1, I2, I3, HYP) + dot(v, v) / 2
def Sigma_hyp(ρ, A, S, HYP):
""" Returns the total stress tensor
"""
G = gram(A)
I1 = I_1(G)
I2 = I_2(G)
I3 = I_3(G)
GdedG = GdU_1dG(I3, HYP) + GdU_2dG(I3, S, HYP) + GdWdG(G, I1, I2, I3, HYP)
return -2 * ρ * GdedG
def dsigmadAdd(ρ, A, d, MP):
""" Returns dσ_dj / dA_md, holding ρ constant.
NOTE: Only valid for EOS with E_2A = cs^2/4 * |devG|^2
"""
cs2 = c_s2(ρ, MP)
G = gram(A)
ret = AdevG(A, G)
ret[:, d] *= 2
ret += 1 / 3 * outer(A[:, d], G[d])
ret += G[d, d] * A
return -ρ * cs2 * ret.T
def entropy_hyp(E, A, v, HYP):
cv = HYP.cv
T0 = HYP.T0
γ = HYP.γ
G = gram(A)
I1 = I_1(G)
I2 = I_2(G)
I3 = I_3(G)
U2 = E - (U_1(I3, HYP) + W(I1, I2, I3, HYP) + dot(v, v) / 2)
return cv * log(1 + U2 / (cv * T0 * I3**(γ / 2)))
def dsigmadA(ρ, A, MP):
""" Returns T_ijmn = dσ_ij / dA_mn, holding ρ constant.
NOTE: Only valid for EOS with E_2A = cs^2/4 * |devG|^2
"""
cs2 = c_s2(ρ, MP)
G = gram(A)
AdevGT = AdevG(A, G).T
GA = dot(G[:, :, None], A[:, None])
ret = GA.swapaxes(0, 3) + GA.swapaxes(1, 3) - 2 / 3 * GA
for i in range(3):
ret[i, :, :, i] += AdevGT
ret[:, i, :, i] += AdevGT
return -ρ * cs2 * ret
def pressure_hyp(ρ, A, S, HYP):
G = gram(A)
I1 = I_1(G)
I2 = I_2(G)
I3 = I_3(G)
K0 = HYP.K0
α = HYP.α
cv = HYP.cv
T0 = HYP.T0
γ = HYP.γ
b02 = HYP.b02
β = HYP.β
const = b02 / 2 * I3**(β / 2)
ret = K0 / (2 * α) * (I3**α - I3**(α / 2))
ret += cv * T0 * γ / 2 * (exp(S / cv) - 1) * I3**(γ / 2)
ret += const * ((β / 2) * (I1**2 / 3 - I2))
return 2 * ρ * ret
def dEdA_s(ρ, A, MP):
""" Returns the partial derivative of E by A (holding ρ,s constant)
"""
cs2 = c_s2(ρ, MP)
G = gram(A)
return cs2 * AdevG(A, G)
def dEdA(ρ, A, MP):
""" Returns the partial derivative of E by A (holding ρ,s constant)
"""
C0 = C_0(ρ, MP)
G = gram(A)
return C0 * AdevG(A, G)
def E_2A(ρ, A, MP):
""" Returns the mesoscale energy dependent on the distortion
"""
C0 = C_0(ρ, MP)
G = gram(A)
return C0 / 4 * L2_2D(dev(G))
def G(self):
return gram(self.A)