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polycrystal.py
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__author__ = ['titrian', 'aydin', 'Martin Saip']
# Conversion to Python >= 3.10, corrections, improvements & refactoring done by Martin Saip in 2020 & 2021.
import copy, sys
from sympy import * #Symbol, nsolve, var, subs
from mpmath import *
from sympy.matrices import *
from numpy import linalg
#import crystalStack as st
EPS = 1.e-8
import numpy as np
import tools.utilities as u
#global crystallist, varlist
#crystallist = []
#varlist = []
def __call__(self, *args, **kwargs):
return self.subs(kwargs)
class polycrystal:
def __init__(self):
self.f1 = None
self.f2 = None
self.conc = 1.
self.K0 = None
self.mue0 = None
# def addCrystal(self, name, params):
# crystallist.append(str(name) + str(params))
# def removeCrystal(self, name, params):
# for k in crystallist:
# if k == str(name) + str(params):
# crystallist.remove(k)
def setYoungsMod(self):
# """ This module calculates the youngs modulus E.
# K0 and mue0 (bulk modulus and shear modulus) is
# assumed to be known
# """
if self.mue0 is None or self.K0 is None:
print("bulk modulus or shear modulus unknown")
exit()
self.E = 2 * self.mue0 * (1 + (3 * self.K0 - 2 * self.mue0)/(6 * self.K0 + 2 * self.mue0))
def setpoissonratio(self):
# """ This module calculates the poisson's ratio v.
# K0 and mue0 (bulk modulus and shear modulus) is
# assumed to be known
# """
if self.mue0 is None or self.K0 is None:
print("poission's ratio unknown")
exit()
self.v = (3 * self.K0 - 2 * self.mue0) / (2 * (3 * self.K0 + self.mue0 ))
def areamoduli(self):
if self.EE1 is None:
print("area moduli unknown")
exit()
if self.EE2 is None:
print("area moduli unknown")
exit()
if self.EE3 is None:
print("area moduli unknown")
exit()
if self.EE1 == 0:
self.A1 = "not calculated"
if self.EE2 == 0:
self.A2 = "not calculated"
if self.EE3 == 0:
self.A3 = "not calculated"
else:
self.A1 = 1 / (self.EE1 + (3*self.K0)**-1)
self.A2 = 1 / (self.EE2 + (3*self.K0)**-1)
self.A3 = 1 / (self.EE3 + (3*self.K0)**-1)
def __add__(self, other):
newFunction = copy.copy(self)
newFunction.f1 = self.f1 + other.f1
newFunction.f2 = self.f2 + other.f2
return newFunction
def setK0andMue0(self):
f = [lambda a, b: self.f1.subs(dict(K0=a, mue0=b)),
lambda a, b: self.f2.subs(dict(K0=a, mue0=b))]
k = 3; a = -1.; b = -1.;
while a <= EPS or b <= EPS:
a = 0.5; b = 0.5
try:
a, b = findroot(f, (k *0.1 , k *0.1 ))# TODO: we need an accurate initial guess
#print (a, b)
except:
a, b = -1.,-1.
k += 1
self.K0 = a#a*100.
self.mue0 = b#b*100.
def setConc(self, conc):
if conc > 1 or conc < 0:
print("Concentration not in the range <0, 1> => exiting!")
exit()
self.conc = conc
self.f1 = self.conc * self.local_1
self.f2 = self.conc * self.local_2
class cubic(polycrystal):
def __init__(self, C11 = None, C12 = None, C44 = None):
"""
This is the cubic polycrystal class. Inherits from polycrystal class... Containing the mathematical functions for
"""
if (C11 is None or C12 is None or C44 is None):
#u.inputError("C parameters not set")
exit()
self.C11 = C11/100 # Thanks to Python 3, we do not need to explicitly state "float" in these 'C's
self.C12 = C12/100
self.C44 = C44/100
correctInput, msg = self.checkCond()
if correctInput == True:
polycrystal.__init__(self)
self.crystalname = "cubic"
self.Cparamlist = (C11, C12, C44)
# self.addCrystal(self.crystalname, self.Cparamlist)
K0 = Symbol('K0')
mue0 = Symbol('mue0')
m_cubic = Matrix (([self.C11,self.C12,self.C12, 0, 0, 0],
[self.C12,self.C11,self.C12, 0, 0, 0],
[self.C12,self.C12,self.C11, 0, 0, 0],
[ 0, 0, 0,self.C44, 0, 0],
[ 0, 0, 0, 0,self.C44, 0],
[ 0, 0, 0, 0, 0,self.C44]))
S_cubic = linalg.inv(np.asarray(m_cubic,dtype=np.float64))
L11 = 0
L12 = 0
L13 = 1.
L21 = 1 / sqrt(2) # Thanks to Python 3, we do not need to explicitly state "float"
L22 = L21
L23 = 0
L31 = 1 / sqrt(3) # Thanks to Python 3, we do not need to explicitly state "float"
L32 = L31
L33 = L31
self.EE1 = ( L11**4 + 2 * (L11**2) * (L12**2) * S_cubic [0,1] + 2 * (L11**2) * (L13**2) * S_cubic [0,2] + L12**4 * S_cubic [1,1]\
+ 2 * (L12**2) * (L13**2) * S_cubic [1,2] + L13**4 * S_cubic [2,2] + (L12**2) * (L13**2) * S_cubic [3,3]\
+ (L11**2) * (L13**2) * S_cubic [4,4] + (L11**2) * (L12**2) * S_cubic [5,5] )
self.EE2 = ( L21**4 + 2 * (L21**2) * (L22**2) * S_cubic [0,1] + 2 * (L21**2) * (L23**2) * S_cubic [0,2] + L22**4 * S_cubic [1,1]\
+ 2 * (L22**2) * (L23**2) * S_cubic [1,2] + L23**4 * S_cubic [2,2] + (L22**2) * (L23**2) * S_cubic [3,3]\
+ (L21**2) * (L23**2) * S_cubic [4,4] + (L21**2) * (L22**2) * S_cubic [5,5] )
self.EE3 = ( L31**4 + 2 * (L31**2) * (L32**2) * S_cubic [0,1] + 2 * (L31**2) * (L33**2) * S_cubic [0,2] + L32**4 * S_cubic [1,1]\
+ 2 * (L32**2) * (L33**2) * S_cubic [1,2] + L33**4 * S_cubic [2,2] + (L32**2) * (L33**2) * S_cubic [3,3]\
+ (L31**2) * (L33**2) * S_cubic [4,4] + (L31**2) * (L32**2) * S_cubic [5,5] )
mue = self.C44
nue = 0.5 * (self.C11 - self.C12)
# K = K0
beta = -3 * (K0 + 2 * mue0)/(5 * mue0 * (3 * K0 + 4 * mue0))
coeff = 1/5 # Thanks to Python 3, we do not need to explicitly state "float"
denom = (3 - ( self.C11 + 2 * self.C12 - 3 * K0))
# these are the main functions
self.local_1 = coeff*((2*nue - 2*mue0)/(1 - 2*beta*(nue - mue0)) + (3*mue - 3*mue0)/(1 - 2*beta*(mue - mue0)))
#coeff * (1/(self.C11 - self.C12 - 2* mue0) - beta)**(-1)\
# + 3 * (1/(self.C44 - mue0) - 2*beta)**(-1)
self.local_2 = (3 * (self.C11 + 2 * self.C12) - 9. * K0)/denom
self.f1 = self.local_1
self.f2 = self.local_2
# ------------------------------- Upper & Lower Bound ------------------------------- #
#[NOT IN GPa yet]
# BULK MODULUS
# 1/ Voigt
Kvoigt = (1/3. * (self.C11 + 2 * self.C12)) #Bulk Modulus
# 2/ Reuss = Voigt
Kreuss = Kvoigt
# 3/ HS-
KHSn = Kvoigt
# 4/ HS+
KHSp = Kvoigt
# SHEAR MODULUS
# S11 = (self.C11 + self.C12) / ((self.C11 - self.C12) * (self.C11 + 2 * self.C12))
# S12 = -(self.C12)/ ((self.C11 - self.C12) * (self.C11 + 2 * self.C12))
# S44 = 1/self.C44
G1 = 0.5*(self.C11 - self.C12)
G2 = self.C44
K = 1/3 *(self.C11 + 2*self.C12)
# 1/ Voigt
# Gvoigt = (1/5.*(self.C11 - self.C12 + 3 * self.C44))*100.
Gvoigt = 1/5.*( 2 * G1 + 3 * G2)
# 2/ Reuss
# Greuss = 5 / (4 * (S11 - S12) + 3 * S44)
# Greuss = (5 * (self.C11 - self.C12) * self.C44 / 4 * self.C44 + 3 * (self.C11 - self.C12))*100 # Thanks to Python 3, we do not need to explicitly state "float"
Greuss = (5 * G1 * G2) / (2 * G2 + 3 * G1)
# 3/ HS-
beta1 = - (3 * (K + 2 * G1)) / (5 * G1 * (3 * K + 4 * G1))
GHSn = G1 + 3*((5/(G2-G1))- 4*beta1)**(-1)
# 4/ HS+
beta2 = - (3 * (K + 2 * G2)) / (5 * G2 * (3 * K + 4 * G2))
GHSp = G2 + 2*((5/(G1-G2))- 6*beta2)**(-1)
self.voigt_bulk = Kvoigt
self.reuss_bulk = Kreuss
self.HS_min_bulk = KHSn
self.HS_post_bulk = KHSp
self.voigt_shear = Gvoigt
self.reuss_shear = Greuss
self.HS_min_shear = GHSn
self.HS_post_shear = GHSp
else:
print('warning' + '<br>')
print(msg)
exit()
def getElasticDict(self):
mydict = {'C11' : self.C11*100, # Thanks to Python 3, we do not need to explicitly state "float" in this dict
'C12' : self.C12*100,
'C44' : self.C44*100}
return mydict
def checkCond(self):
# Check condition
m_cubic = Matrix (([self.C11,self.C12,self.C12, 0, 0, 0],
[self.C12,self.C11,self.C12, 0, 0, 0],
[self.C12,self.C12,self.C11, 0, 0, 0],
[ 0, 0, 0,self.C44, 0, 0],
[ 0, 0, 0, 0,self.C44, 0],
[ 0, 0, 0, 0, 0,self.C44]))
cubic_det = det(m_cubic)
cubic_condition_1 = self.C11 + 2 * self.C12
cubic_condition_2 = self.C44
cubic_condition_3 = self.C11 - self.C12
if cubic_condition_1 > 0:
pass
else:
#u.inputError('Please enter right value: C11 + 2 C12 > 0')
return False, 'Please enter right value: C11 + 2 C12 > 0 (cubic phase).'
if cubic_condition_2 > 0:
pass
else:
#u.inputError('Please enter right value: C44 > 0')
return False, 'Please enter right value: C44 > 0 (cubic phase).'
if cubic_condition_3 > 0:
pass
else:
#u.inputError('Please enter right value: C11 - C12 > 0')
return False, 'Please enter right value: C11 - C12 > 0 (cubic phase).'
return True, ''
class tetragonal(polycrystal):
def __init__(self, C11 = None, C12 = None, C13 = None, C33 = None, C44 = None, C66 = None):
if (C11 is None or C12 is None or C13 is None or C33 is None or C44 is None or C66 is None):
u.inputError("C parameters not set!")
exit()
self.C11 = C11/100 # Thanks to Python 3, we do not need to explicitly state "float" in these 'C's
self.C12 = C12/100
self.C13 = C13/100
self.C33 = C33/100
self.C44 = C44/100
self.C66 = C66/100
correctInput, msg = self.checkCond()
if correctInput == True:
polycrystal.__init__(self)
# self.conc = 1
self.crystalname = "tetragonal"
self.Cparamlist = (C11, C12, C13, C33, C44, C66)
# self.addCrystal(self.crystalname, self.Cparamlist)
K0 = Symbol('K0')
mue0 = Symbol('mue0')
m_tetragonal = Matrix (([self.C11,self.C12,self.C13, 0, 0, 0],
[self.C12,self.C11,self.C13, 0, 0, 0],
[self.C13,self.C13,self.C33, 0, 0, 0],
[ 0, 0, 0,self.C44, 0, 0],
[ 0, 0, 0, 0,self.C44, 0],
[ 0, 0, 0, 0, 0,self.C66]))
S_tetragonal = linalg.inv(np.asarray(m_tetragonal,dtype=np.float64))
# print m_tetragonal
# print S_tetragonal
# print S_tetragonal [0,5]
L11 = 0
L12 = 0
L13 = 1.
L21 = 1 / sqrt(2) # Thanks to Python 3, we do not need to explicitly state "float"
L22 = L21
L23 = 0
L31 = 1 / sqrt(3) # Thanks to Python 3, we do not need to explicitly state "float"
L32 = L31
L33 = L31
self.EE1 = ( L11**4 + 2 * (L11**2) * (L12**2) * S_tetragonal [0,1] + 2 * (L11**2) * (L13**2) * S_tetragonal [0,2] + L12**4 * S_tetragonal [1,1]\
+ 2 * (L12**2) * (L13**2) * S_tetragonal [1,2] + L13**4 * S_tetragonal [2,2] + (L12**2) * (L13**2) * S_tetragonal [3,3]\
+ (L11**2) * (L13**2) * S_tetragonal [4,4] + (L11**2) * (L12**2) * S_tetragonal [5,5] )
self.EE2 = ( L21**4 + 2 * (L21**2) * (L22**2) * S_tetragonal [0,1] + 2 * (L21**2) * (L23**2) * S_tetragonal [0,2] + L22**4 * S_tetragonal [1,1]\
+ 2 * (L22**2) * (L23**2) * S_tetragonal [1,2] + L23**4 * S_tetragonal [2,2] + (L22**2) * (L23**2) * S_tetragonal [3,3]\
+ (L21**2) * (L23**2) * S_tetragonal [4,4] + (L21**2) * (L12**2) * S_tetragonal [5,5] )
self.EE3 = ( L31**4 + 2 * (L31**2) * (L32**2) * S_tetragonal [0,1] + 2 * (L31**2) * (L33**2) * S_tetragonal [0,2] + L32**4 * S_tetragonal [1,1]\
+ 2 * (L32**2) * (L33**2) * S_tetragonal [1,2] + L33**4 * S_tetragonal [2,2] + (L32**2) * (L33**2) * S_tetragonal [3,3]\
+ (L31**2) * (L33**2) * S_tetragonal [4,4] + (L31**2) * (L32**2) * S_tetragonal [5,5] )
Knue = 1./9 * ( self.C33 + 2 * (self.C11 + self.C12) + 4 * self.C13)
M = self.C11 + self.C12 + 2 * self.C33 - 4 * self.C13
C2 = self.C33 * (self.C11 + self.C12) - 2 * self.C13**2
psi = self.C11 + self.C12 + self.C33 - 3 * K0 - 2 * mue0
deltakk = C2 - K0 * ( M - 6 * mue0 ) - 6 * mue0 * Knue
sigma = 9 * Knue- 9./2 * K0 - 6 * mue0 + 3./2 * self.C33 - 6 * self.C13
beta = -3 * (K0 + 2 * mue0)/(5 * mue0 * (3 * K0 + 4 * mue0))
eta = -3./(3 * K0 + 4 * mue0)
gamma = 1./9 * (eta - 3 * beta)
denom1 = (1 - beta * psi - 9 * gamma *(Knue - K0) + 1./3 * eta * beta * deltakk)
denom2 = 2 *(1 - beta *(self.C11 - self.C12 -2 * mue0))
coeff = 1 / 15 # Thanks to Python 3, we do not need to explicitly state "float"
fone = M - 6 * mue0
ftwo = (self.C11 - self.C12 - 2 * mue0)*(3 - 2 * beta * sigma - 27 * gamma * (Knue - K0) - 2 * eta * beta * deltakk) - eta * deltakk
fa = (fone + ftwo) / (denom2 * denom1)
fb = 6 * (self.C44 - mue0) / (1 - 2 * beta * (self.C44 - mue0))
fc = 3 * (self.C66 - mue0) / (1 - 2 * beta * (self.C66 - mue0))
self.local_1 = coeff *( fa + fb + fc )
self.local_2 = (3 * (Knue - K0) - beta * deltakk)/denom1 #/(1 - beta * psi - 9 * gamma *(Knue -K0)+ 1./3 * eta * beta * deltakk)
self.f1 = self.local_1
self.f2 = self.local_2
# ------------------------------- Upper & Lower Bound ------------------------------- #
# Bulk Modulus
G3 = 0.5 * (self.C11 - self.C12)
G_low = [self.C44, self.C66, G3, C2 / (6* Knue)]
G1 = max(G_low)
G_up = [self.C44, self.C66, G3, (1/6. * M)]
G2 = min(G_up)
if abs(M - 6 * G2) < EPS: # Avoid division by zero. Take the next larger number
G_up.sort()
G2 = G_up[1]
K1 = (C2 - 6 * G1 * Knue) / (M * 6 * G1)
K2 = (C2 - 6 * G2 * Knue) / (M * 6 * G2)
beta1 = -3 * (K1 + 2 * G1) / (5 * G1 * (3 * K1 + 4 * G1))
beta2 = -3 * (K2 + 2 * G2) / (5 * G2 * (3 * K2 + 4 * G2))
delta1 = C2 - K1*(M - 6*G1) - 6*G1*Knue
delta2 = C2 - K2*(M - 6*G2) - 6*G2*Knue
# 1/ Voigt
Kvoigt = 1/9. * (2 * (self.C11 + self.C12) + self.C33 + 4 * self.C13)
# 2/ Reuss
Kreuss = C2 / M
# 3/ HS-
KHSn = K1 + ((Knue - K1) -1/3. * beta1 * delta1 ) / ( 1 - 1/3.*beta1 * (M - 6*G1))
# 4/ HS+
KHSp = K2 + ((Knue - K2) -1/3. * beta2 * delta2 ) / ( 1 - 1/3.*beta2 * (M - 6*G2))
# Shear Modulus
eta1 = -3/(3 * K1 + 4 * G1) # Thanks to Python 3, we do not need to explicitly state "float" in these
eta2 = -3/(3 * K2 + 4 * G2)
gamma1 = 1/9 * (eta1 - 3 * beta1)
gamma2 = 1/9 * (eta2 - 3 * beta2)
sigma1 = 9 * Knue- 9./2 * K1 - 6 * G1 + 3./2 * self.C33 - 6 * self.C13
sigma2 = 9 * Knue- 9./2 * K2 - 6 * G2 + 3./2 * self.C33 - 6 * self.C13
psi1 = self.C11 + self.C12 + self.C33 - 3 * K1 - 2 * G1
psi2 = self.C11 + self.C12 + self.C33 - 3 * K2 - 2 * G2
a1 = (M - 6 * G1 + (self.C11 - self.C12 - 2*G1) * (3 - 2 * beta1 * sigma1 - 27 * gamma1 * (Knue - K1) - 2 * eta1 * beta1 * delta1) - eta1 * delta1) /\
(2 * (1 - beta1 * (self.C11 - self.C12 - 2 * G1)) * (1 - beta1 * psi1 - 9 * gamma1 * (Knue - K1) + 1/3. * eta1 * beta1 * delta1))
a2 = (M - 6 * G2 + (self.C11 - self.C12 - 2*G2) * (3 - 2 * beta2 * sigma2 - 27 * gamma2 * (Knue - K2) - 2 * eta2 * beta2 * delta2) - eta2 * delta2) /\
(2 * (1 - beta2 * (self.C11 - self.C12 - 2 * G2)) * (1 - beta2 * psi2 - 9 * gamma2 * (Knue - K2) + 1/3. * eta2 * beta2 * delta2))
b1 = 6 * (C44 - G1) / (1 - 2 * beta1 * (C44 - G1))
b2 = 6 * (C44 - G2) / (1 - 2 * beta2 * (C44 - G2))
c1 = 3 * (C66 - G1) / (1 - 2 * beta1 * (C66 - G1))
c2 = 3 * (C66 - G2) / (1 - 2 * beta2 * (C66 - G2))
B21 = 1/15 * (a1 + b1 + c1) # Thanks to Python 3, we do not need to explicitly state "float" in these 'B's
B22 = 1/15 * (a2 + b2 + c2)
# 1/ Voigt
Gvoigt = 1/30 * (M + 3 * self.C11 - 3 * self.C12 + 12 * self.C44 + 6 * self.C66) # Thanks to Python 3, we do not need to explicitly state "float"
# 2/ Reuss
Greuss = 15 * (18 * Knue /C2 + 6 / (self.C11 - self.C12) + 6 / self.C44 + 3 / self.C66)**-1
# 3/ HS-
GHSp = G1 + B21 / (1 + 2 * beta1 * B21)
# 4/ HS+
GHSn = G2 + B22 / (1 + 2 * beta2 * B22)
self.voigt_bulk = Kvoigt
self.reuss_bulk = Kvoigt
self.HS_min_bulk = KHSn
self.HS_post_bulk = KHSp
self.voigt_shear = Gvoigt
self.reuss_shear = Greuss
self.HS_min_shear = KHSn
self.HS_post_shear = KHSp
else:
print('warning' + '<br>')
print(msg)
exit()
def getElasticDict(self):
mydict = {'C11' : self.C11*100, # Thanks to Python 3, we do not need to explicitly state "float" in these 'C's
'C12' : self.C12*100,
'C13' : self.C13*100,
'C33' : self.C33*100,
'C44' : self.C44*100,
'C66' : self.C66*100}
return mydict
def checkCond(self):
# Check condition
m_tetragonal = Matrix (([self.C11,self.C12,self.C13, 0, 0, 0],
[self.C12,self.C11,self.C13, 0, 0, 0],
[self.C13,self.C13,self.C33, 0, 0, 0],
[ 0, 0, 0,self.C44, 0, 0],
[ 0, 0, 0, 0,self.C44, 0],
[ 0, 0, 0, 0, 0,self.C66]))
tetragonal_det = det(m_tetragonal)
tetragonal_condition_1 = self.C11
tetragonal_condition_2 = det(Matrix(([self.C11,self.C12],
[self.C12,self.C11])))
tetragonal_condition_3 = det(Matrix(([self.C11,self.C12,self.C13],
[self.C12,self.C11,self.C13],
[self.C13,self.C13,self.C33])))
tetragonal_condition_4 = self.C44
tetragonal_condition_5 = self.C66
if tetragonal_condition_1 > 0:
pass
else:
# u.inputError('Please enter right value: C11 > 0')
return False, 'Please enter such a value that C11 > 0 (tetragonal phase).'
if tetragonal_condition_2 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12), (C12, C11) )')
return False, 'Please enter such a value that det (matrix A) > 0!A = [(C11, C12),(C12, C11)] (tetragonal phase).'
if tetragonal_condition_3 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12, C13), (C12, C11, C13), (C13, C13, C33) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12, C13), (C12, C11, C13), (C13, C13, C33)] (tetragonal phase).'
if tetragonal_condition_4 > 0:
pass
else:
# u.inputError('Please enter right value: C44 > 0')
return False, 'Please enter such a value that C44 > 0 (tetragonal phase).'
if tetragonal_condition_5 > 0:
pass
else:
# u.inputError('Please enter right value: C66 > 0')
return False, 'Please enter such a value that C66 > 0 (tetragonal phase).'
return True, ''
class trigonal(polycrystal):
def __init__(self, C11 = None, C12 = None, C13 = None, C14 = None, C33 = None, C44 = None):
if (C11 is None or C12 is None or C13 is None or C14 is None or C33 is None or C44 is None):
# u.inputError("C parameters not set")
exit()
self.C11 = C11/100 # Thanks to Python 3, we do not need to explicitly state "float" in these 'C's
self.C12 = C12/100
self.C13 = C13/100
self.C14 = C14/100
self.C33 = C33/100
self.C44 = C44/100
correctInput, msg = self.checkCond()
if correctInput == True:
polycrystal.__init__(self)
self.crystalname = "trigonal"
self.Cparamlist = (C11, C12, C13, C14, C33, C44)
# self.addCrystal(self.crystalname, self.Cparamlist)
self.EE1 = 0
self.EE2 = 0
self.EE3 = 0
K0 = Symbol('K0')
mue0 = Symbol('mue0')
Knue = 1./9 * ( self.C33 + 2 * (self.C11 + self.C12) + 4 * self.C13)
M = self.C11 + self.C12 + 2 * self.C33 - 4 * self.C13
C2 = self.C33 * (self.C11 + self.C12) - 2 * self.C13**2
psi = self.C11 + self.C12 + self.C33 - 3 * K0 - 2 * mue0
deltakk = C2 - K0 * ( M - 6 * mue0 ) - 6 * mue0 * Knue
C66 = 0.5 * (self.C11 - self.C12)
chi = 4 * (self.C14**2 - (self.C44 - mue0) - (C66 - mue0))
beta = -3 * (K0 + 2 * mue0) / (5 * mue0 * (3 * K0 + 4 * mue0))
eta = -3./(3 * K0 + 4 * mue0)
gamma = 1./9 * (eta - 3 * beta)
denom1 = (1 - beta * psi - 9 * gamma *(Knue -K0) + 1./3 * eta * beta * deltakk)
denom2 = (1 - 2 * beta * (self.C44 + C66 - 2 * mue0) - beta**2 * chi)
coeff = 1 / 30.
fa = (M - 6 * mue0 - eta * deltakk)/denom1
fb = 12 * (self.C44 + C66 - 2 * mue0 + beta * chi)/denom2
self.local_1 = coeff *( fa + fb )
self.local_2 = (3 * (Knue - K0) - beta * deltakk)/denom1
self.f1 = self.local_1
self.f2 = self.local_2
# -------------------- Upper and Lower Bound ----------------------------- #
# Bulk Modulus
i_G1 = 0.5 * (self.C44 + C66) - (0.25 * (self.C44 - C66)**2 + self.C14**2)**0.5
i_G2 = 0.5 * (self.C44 + C66) + (0.25 * (self.C44 - C66)**2 + self.C14**2)**0.5
G_low = [i_G1, C2 / (6* Knue)]
G1 = max(G_low)
G_up = [i_G2,(1/6. * M)]
G2 = min(G_up)
if abs(M - 6 * G2) < EPS: # Avoid division by zero. Take the next larger number
G_up.sort()
G2 = G_up[1]
K1 = (C2 - 6 * G1 * Knue) / (M * 6 * G1)
K2 = (C2 - 6 * G2 * Knue) / (M * 6 * G2)
beta1 = -3 * (K1 + 2 * G1) / (5 * G1 * (3 * K1 + 4 * G1))
beta2 = -3 * (K2 + 2 * G2) / (5 * G2 * (3 * K2 + 4 * G2))
delta1 = C2 - K1 * (M - 6 * G1) - 6 * G1 * Knue
delta2 = C2 - K2 * (M - 6 * G2) - 6 * G2 * Knue
# 1/ Voigt
Kvoigt = 1/9. * (2 * (self.C11 + self.C12) + self.C33 + 4 * self.C13)
# 2/ Reuss
Kreuss = C2 / M
# 3/ HS-
KHSn = K1 + ((Knue - K1) - 1/3. * beta1 * delta1 ) / ( 1 - 1/3. * beta1 * (M - 6 * G1))
# 4/ HS+
KHSp = K2 + ((Knue - K2) - 1/3. * beta2 * delta2 ) / ( 1 - 1/3. * beta2 * (M - 6 * G2))
# Shear Modulus
eta1 = -3/(3 * K1 + 4 * G1) # Thanks to Python 3, we do not need to explicitly state "float" in these
eta2 = -3/(3 * K2 + 4 * G2)
gamma1 = 1/9 * (eta1 - 3 * beta1)
gamma2 = 1/9 * (eta2 - 3 * beta2)
omega1 = 4 * (self.C14**2 - (self.C44 - G1) * (C66 - G1))
omega2 = 4 * (self.C14**2 - (self.C44 - G2) * (C66 - G2))
a1 = (M - 6 * G1 - eta1 * delta1) / (1 - beta1 * (self.C11 + self.C12 + self.C33 - 3*K1 - 2*G1) - 9 * gamma1 * (Knue - K1) + 1/3. * eta1 * beta1 * delta1)
a2 = (M - 6 * G2 - eta2 * delta2) / (1 - beta2 * (self.C11 + self.C12 + self.C33 - 3*K2 - 2*G2) - 9 * gamma2 * (Knue - K2) + 1/3. * eta2 * beta2 * delta2)
b1 = 12 * (self.C44 + C66 - 2 * G1 + beta1 * omega1) / (1- 2 * beta1 * (self.C44 + C66 - 2 * G1) - (beta1**2 * omega1))
b2 = 12 * (self.C44 + C66 - 2 * G2 + beta2 * omega2) / (1- 2 * beta2 * (self.C44 + C66 - 2 * G2) - (beta2**2 * omega2))
B21 = 1/30 * (a1 + b1) # Thanks to Python 3, we do not need to explicitly state "float" in these
B22 = 1/30 * (a2 + b2)
# 1/ Voigt
Gvoigt = 1/30* (M + 12 * self.C44 + 12 * C66)
# 2/ Reuss
Greuss = 5/2 * (C2 * (self.C44 * C66 - self.C14**2) / (3* Knue * (self.C44 * C66 - self.C14**2) + C2 * (self.C44 + C66)))
# 3/ HS-
GHSp = G1 + B21 / (1 + 2 * beta1 * B21)
# 4/ HS+
GHSn = G2 + B22 / (1 + 2 * beta2 * B22)
D = [self.C44 , C66]
D1 = min (D)
D2 = max (D)
if G1 < D2:
# print 'D1 accepted'
pass
else:
print('D1 is False --> Please enter a value corresponding to a mechanically stable system!')
if G2 > D1:
# print 'D2 accepted'
pass
else:
print('D2 is False --> Please enter a value corresponding to a mechanically stable system!')
self.voigt_bulk = Kvoigt
self.reuss_bulk = Kvoigt
self.HS_min_bulk = KHSn
self.HS_pos_bulk = KHSp
self.voigt_shear = Gvoigt
self.reuss_shear = Greuss
self.HS_min_shear = KHSn
self.HS_pos_shear = KHSp
else:
print('warning' + '<br>')
print(msg)
exit()
def getElasticDict(self):
mydict = {'C11' : self.C11*100, # Thanks to Python 3, we do not need to explicitly state "float" in these 'C's
'C12' : self.C12*100,
'C13' : self.C13*100,
'C14' : self.C14*100,
'C33' : self.C33*100,
'C44' : self.C44*100}
return mydict
def checkCond(self):
# Check condition
C66 = 0.5 * (self.C11 - self.C12)
m_trigonal = Matrix (([self.C11, self.C12,self.C13, self.C14, 0, 0],
[self.C12, self.C11,self.C13,-self.C14, 0, 0],
[self.C13, self.C13,self.C33, 0, 0, 0],
[self.C14,-self.C14, 0, self.C44, 0, 0],
[ 0, 0, 0, 0,self.C44, self.C14/2.],
[ 0, 0, 0, self.C14/2., 0,C66]))
trigonal_det = det(m_trigonal)
trigonal_condition_1 = self.C11
trigonal_condition_2 = det(Matrix(([self.C11, self.C12],
[self.C12, self.C11])))
trigonal_condition_3 = det(Matrix(([self.C11, self.C12,self.C13],
[self.C12, self.C11,self.C13],
[self.C13, self.C13,self.C33])))
trigonal_condition_4 = det(Matrix(([self.C11, self.C12,self.C13, self.C14],
[self.C12, self.C11,self.C13,-self.C14],
[self.C13, self.C13,self.C33, 0],
[self.C14,-self.C14, 0, self.C44])))
trigonal_condition_5 = det(Matrix(([self.C11, self.C12,self.C13, self.C14, 0],
[self.C12, self.C11,self.C13,-self.C14, 0],
[self.C13, self.C13,self.C33, 0, 0],
[self.C14,-self.C14, 0, self.C44, 0],
[ 0, 0, 0, 0,self.C44])))
trigonal_condition_6 = det(m_trigonal)
if trigonal_condition_1 > 0:
pass
else:
# u.inputError('Please enter right value: C11 > 0')
return False, 'Please enter such a value that C11 > 0 (trigonal phase).'
if trigonal_condition_2 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12), (C12, C11) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12), (C12, C11)] (trigonal phase).'
if trigonal_condition_3 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12, C13), (C12, C11, C13), (C13, C13, C33) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12, C13), (C12, C11, C13), (C13, C13, C33)] (trigonal phase).'
if trigonal_condition_4 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12, C13, C14), (C12, C11, C13, -C14), (C13, C13, C33, 0), (C14, -C14, 0, C44) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12, C13, C14), (C12, C11, C13, -C14), (C13, C13, C33, 0), (C14, -C14, 0, C44)] (trigonal phase).'
if trigonal_condition_5 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12, C13, C14, 0), (C12, C11, C13, -C14, 0), (C13, C13, C33, 0, 0), (C14, -C14, 0, C44, 0), (0, 0, 0, 0, C44) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12, C13, C14, 0), (C12, C11, C13, -C14, 0), (C13, C13, C33, 0, 0), (C14, -C14, 0, C44, 0), (0, 0, 0, 0, C44)] (trigonal phase).'
if trigonal_condition_6 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12, C13, C14, 0, 0), (C12, C11, C13, -C14, 0, 0), (C13, C13, C33, 0, 0, 0), (C14, -C14, 0, C44, 0, 0), (0, 0, 0, 0, C44, 0), (0, 0, 0, 0, 0, C66) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12, C13, C14, 0, 0), (C12, C11, C13, -C14, 0, 0), (C13, C13, C33, 0, 0, 0), (C14, -C14, 0, C44, 0, 0), (0, 0, 0, 0, C44, 0), (0, 0, 0, 0, 0, C66)] (trigonal phase).'
return True, ''
class hexagonal(polycrystal):
def __init__(self, C11 = None, C12 = None, C13 = None, C33 = None, C55 = None):
if (C11 is None or C12 is None or C13 is None or C33 is None or C55 is None):
# u.inputError("C parameters not set")
exit()
self.C11 = C11/100 # Thanks to Python 3, we do not need to explicitly state "float" in these 'C's
self.C12 = C12/100
self.C13 = C13/100
self.C33 = C33/100
self.C55 = C55/100
correctInput, msg = self.checkCond()
if correctInput == True:
polycrystal.__init__(self)
self.crystalname = "hexagonal"
self.Cparamlist = (C11, C12, C13, C33, C55)
# self.addCrystal(self.crystalname, self.Cparamlist)
K0 = Symbol('K0')
mue0 = Symbol('mue0')
self.EE1 = 0
self.EE2 = 0
self.EE3 = 0
Knue = 1./9 * ( self.C33 + 2 * (self.C11 + self.C12) + 4 * self.C13)
M = self.C11 + self.C12 + 2 * self.C33 - 4 * self.C13
C2 = self.C33 * (self.C11 + self.C12) - 2 * self.C13**2
psi = self.C11 + self.C12 + self.C33 - 3 * K0 - 2 * mue0
deltakk = C2 - K0 * ( M - 6 * mue0 ) - 6 * mue0 * Knue
C66 = 0.5 * (self.C11 - self.C12)
C44 = self.C55
beta = -3 * (K0 + 2 * mue0) / (5 * mue0 * (3 * K0 + 4 * mue0))
eta = -3./(3 * K0 + 4 * mue0)
gamma = 1./9 * (eta - 3 * beta)
denom = (1 - beta * psi - 9 * gamma *(Knue -K0) + 1./3 * eta * beta * deltakk)
coeff = 1 / 30 # Thanks to Python 3, we do not need to explicitly state "float"
fa = (M - 6 * mue0 - eta * deltakk)/(1 - beta * psi - 9 * gamma * (Knue - K0) + 1./3 * eta * beta * deltakk)
fb = 12 * ((C44 - mue0) / (1 - 2 * beta * (C44 - mue0))
+ (C66 - mue0) / (1 - 2 * beta * (C66 - mue0)))
self.local_1 = self.conc * coeff *( fa + fb )
self.local_2 = self.conc * (3 * (Knue - K0) - beta * deltakk)/denom
self.f1 = self.local_1
self.f2 = self.local_2
# ------------------------------- Upper & Lower Bound ------------------------------- #
# Bulk Modulus
G_low = [C44, C66, C2 / (6* Knue)]
G1 = max(G_low)
G_up = [C44, C66, (1/6 * M)] # Thanks to Python 3, we do not need to explicitly state "float"
G2 = min(G_up)
if abs(M - 6 * G2) < EPS: # Avoid division by zero. Take the next larger number
G_up.sort()
G2 = G_up[1]
K1 = (C2 - 6 * G1 * Knue) / (M - 6 * G1)
K2 = (C2 - 6 * G2 * Knue) / (M - 6 * G2)
beta1 = -3 * (K1 + 2 * G1) / (5 * G1 * (3 * K1 + 4 * G1))
beta2 = -3 * (K2 + 2 * G2) / (5 * G2 * (3 * K2 + 4 * G2))
delta1 = C2 - K1 * (M - 6 * G1) - 6 * G1 * Knue
delta2 = C2 - K2 * (M - 6 * G2) - 6 * G2 * Knue
# 1/ Voigt
Kvoigt = (1/9. * (2 * (self.C11 + self.C12) + self.C33 + 4 * self.C13))
# 2/ Reuss
Kreuss = (C2 / M)
# 3/ HS-
KHSn = (K1 + ((Knue - K1) - 1/3. * beta1 * delta1 ) / ( 1 - 1/3.*beta1 * (M - 6*G1)))
# 4/ HS+
KHSp = (K2 + ((Knue - K2) - 1/3. * beta2 * delta2 ) / ( 1 - 1/3.*beta2 * (M - 6*G2)))
# Shear Modulus
eta1 = -3/(3 * K1 + 4 * G1) # Thanks to Python 3, we do not need to explicitly state "float" in these
eta2 = -3/(3 * K2 + 4 * G2)
gamma1 = 1/9 * (eta1 - 3 * beta1)
gamma2 = 1/9 * (eta2 - 3 * beta2)
a1 = M - 6 * G1 - eta1 * delta1 / (1 - (beta1 * (self.C11 + self.C12 + self.C33 - 3*K1 - 2*G1)) - (9*gamma1*(Knue - K1)) + (1/3. * eta1 * beta1 * delta1))
a2 = M - 6 * G2 - eta2 * delta2 / (1 - (beta2 * (self.C11 + self.C12 + self.C33 - 3*K2 - 2*G2)) - (9*gamma2*(Knue - K2)) + (1/3. * eta2 * beta2 * delta2))
b1 = 12 * (C44 - G1) / (1 - 2 * beta1 * (C44 - G1))
b2 = 12 * (C44 - G2) / (1 - 2 * beta2 * (C44 - G2))
c1 = 12 * (C66 - G1) / (1 - 2 * beta1 * (C66 - G1))
c2 = 12 * (C66 - G2) / (1 - 2 * beta2 * (C66 - G2))
B21 = 1/30 * (a1 + b1 + c1) # Thanks to Python 3, we do not need to explicitly state "float" in these 'B's
B22 = 1/30 * (a2 + b2 + c2)
# 1/ Voigt
Gvoigt = (1/30.* (M + 12 * C44 + 12 * C66))
# 2/ Reuss
Greuss = (5/2. * (C2 * C44 * C66) / (3* Knue * C44 * C66 + C2 * (C44 + C66)))
# 3/ HS-
GHSn = (G1 + B21 / (1 + 2 * beta1 * B21))
# 4/ HS+
GHSp = (G2 + B22 / (1 + 2 * beta2 * B22))
self.voigt_bulk = Kvoigt
self.reuss_bulk = Kvoigt
self.HS_min_bulk = KHSn
self.HS_pos_bulk = KHSp
self.voigt_shear = Gvoigt
self.reuss_shear = Greuss
self.HS_min_shear = KHSn
self.HS_pos_shear = KHSp
else:
print('warning' + '<br>')
print(msg)
exit()
def getElasticDict(self):
mydict = {'C11' : self.C11*100, # Thanks to Python 3, we do not need to explicitly state "float" in these 'C's
'C12' : self.C12*100,
'C13' : self.C13*100,
'C33' : self.C33*100,
'C55' : self.C55*100}
return mydict
def checkCond(self):
# Check condition
C66 = 0.5 * (self.C11 - self.C12)
m_hexagonal = Matrix (([self.C11,self.C12,self.C13, 0, 0, 0],
[self.C12,self.C11,self.C13, 0, 0, 0],
[self.C13,self.C13,self.C33, 0, 0, 0],
[ 0, 0, 0,self.C55, 0, 0],
[ 0, 0, 0, 0,self.C55, 0],
[ 0, 0, 0, 0, 0,C66]))
hexagonal_det = det(m_hexagonal)
hexagonal_condition_1 = self.C11
hexagonal_condition_2 = det(Matrix(([self.C11,self.C12],
[self.C12,self.C11])))
hexagonal_condition_3 = det(Matrix(([self.C11,self.C12,self.C13],
[self.C12,self.C11,self.C13],
[self.C13,self.C13,self.C33])))
hexagonal_condition_4 = det(Matrix(([self.C11,self.C12,self.C13, 0],
[self.C12,self.C11,self.C13, 0],
[self.C13,self.C13,self.C33, 0],
[ 0, 0, 0,self.C55])))
hexagonal_condition_5= det(Matrix(([self.C11,self.C12,self.C13, 0, 0],
[self.C12,self.C11,self.C13, 0, 0],
[self.C13,self.C13,self.C33, 0, 0],
[ 0, 0, 0,self.C55, 0],
[ 0, 0, 0, 0,self.C55])))
hexagonal_condition_6 = det(m_hexagonal)
if hexagonal_condition_1 > 0:
pass
else:
# u.inputError('Please enter right value: C11 > 0')
return False, 'Please enter such a value that C11 > 0 (hexagonal phase).'
if hexagonal_condition_2 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12), (C12, C11) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12), (C12, C11)] (hexagonal phase).'
if hexagonal_condition_3 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12, C13), (C12, C11, C13), (C13, C13, C33) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12, C13), (C12, C11, C13), (C13, C13, C33)] (hexagonal phase).'
if hexagonal_condition_4 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12, C13, 0), (C12, C11, C13, 0), (C13, C13, C33, 0), (0, 0, 0, C55) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12, C13, 0), (C12, C11, C13, 0), (C13, C13, C33, 0), (0, 0, 0, C55)] (hexagonal phase).'
if hexagonal_condition_5 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12, C13, 0, 0), (C12, C11, C13, 0, 0), (C13, C13, C33, 0, 0), (0, 0, 0, C55, 0), (0, 0, 0, 0, C55) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12, C13, 0, 0), (C12, C11, C13, 0, 0), (C13, C13, C33, 0, 0), (0, 0, 0, C55, 0), (0, 0, 0, 0, C55)] (hexagonal phase).'
if hexagonal_condition_6 > 0:
pass
else:
# u.inputError('Please enter right value: det (matrix A) > 0!\n A = ( (C11, C12, C13, 0, 0, 0), (C12, C11, C13, 0, 0, 0), (C13, C13, C33, 0, 0, 0), (0, 0, 0, C55, 0, 0), (0, 0, 0, 0, C55, 0), (0, 0, 0, 0, 0, C66) )')
return False, 'Please enter such a value that det (matrix A) > 0! A = [(C11, C12, C13, 0, 0, 0), (C12, C11, C13, 0, 0, 0), (C13, C13, C33, 0, 0, 0), (0, 0, 0, C55, 0, 0), (0, 0, 0, 0, C55, 0), (0, 0, 0, 0, 0, C66)] (hexagonal phase).'
return True, ''
class orthorombic(polycrystal):
def __init__(self, C11 = None, C12 = None, C13 = None, C22 = None,
C23 = None, C33 = None, C44 = None, C55 = None, C66 = None):
if (C11 is None or C12 is None or C13 is None or C22 is None or C23 is None or C33 is None or C44 is None or C55 is None or C66 is None):
u.inputError("C parameters not set!")
exit()
self.C11 = C11/100 # Thanks to Python 3, we do not need to explicitly state "float" in these 'C's
self.C12 = C12/100
self.C13 = C13/100
self.C22 = C22/100
self.C23 = C23/100
self.C33 = C33/100
self.C44 = C44/100
self.C55 = C55/100
self.C66 = C66/100
correctInput, msg = self.checkCond()
if correctInput == True:
polycrystal.__init__(self)
self.crystalname = "orthorombic"
self.Cparamlist = (C11, C12, C13, C22, C23, C33, C44, C55, C66)
# self.addCrystal(self.crystalname, self.Cparamlist)