Print available parameterizations of fourth-order FOT
[1]:
import vofotensors as vot
[2]:
print("Available parameterizations of fourth order fiber orientation tensors:\n")
vot.utils.print_nested(vot.fabric_tensors.N4s_parametric)
Available parameterizations of fourth order fiber orientation tensors:
####################
Group: cubic
d1
Matrix([
[1/5 - 2*d1, d1 + 1/15, d1 + 1/15, 0, 0, 0],
[ d1 + 1/15, 1/5 - 2*d1, d1 + 1/15, 0, 0, 0],
[ d1 + 1/15, d1 + 1/15, 1/5 - 2*d1, 0, 0, 0],
[ 0, 0, 0, 2*d1 + 2/15, 0, 0],
[ 0, 0, 0, 0, 2*d1 + 2/15, 0],
[ 0, 0, 0, 0, 0, 2*d1 + 2/15]])
####################
Group: planar
alpha1_d1_d8
Matrix([
[3*alpha1/4 - d1 + 27/70, d1 + 4/35, 0, 0, 0, sqrt(2)*d8],
[ d1 + 4/35, -3*alpha1/4 - d1 + 27/70, 0, 0, 0, -sqrt(2)*d8],
[ 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0],
[ sqrt(2)*d8, -sqrt(2)*d8, 0, 0, 0, 2*d1 + 8/35]])
la1_d1_d8
Matrix([
[-d1 + la1 - 4/35, d1 + 4/35, 0, 0, 0, sqrt(2)*d8],
[ d1 + 4/35, -d1 - la1 + 31/35, 0, 0, 0, -sqrt(2)*d8],
[ 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0],
[ sqrt(2)*d8, -sqrt(2)*d8, 0, 0, 0, 2*d1 + 8/35]])
####################
Group: transv_isotropic
alpha1_rho1
Matrix([
[6*alpha1/7 + 8*rho1 + 1/5, alpha1/14 - 4*rho1 + 1/15, alpha1/14 - 4*rho1 + 1/15, 0, 0, 0],
[alpha1/14 - 4*rho1 + 1/15, -3*alpha1/7 + 3*rho1 + 1/5, -alpha1/7 + rho1 + 1/15, 0, 0, 0],
[alpha1/14 - 4*rho1 + 1/15, -alpha1/7 + rho1 + 1/15, -3*alpha1/7 + 3*rho1 + 1/5, 0, 0, 0],
[ 0, 0, 0, -2*alpha1/7 + 2*rho1 + 2/15, 0, 0],
[ 0, 0, 0, 0, alpha1/7 - 8*rho1 + 2/15, 0],
[ 0, 0, 0, 0, 0, alpha1/7 - 8*rho1 + 2/15]])
alpha2_rho2
Matrix([
[-3*alpha2/7 + 3*rho2 + 1/5, alpha2/14 - 4*rho2 + 1/15, -alpha2/7 + rho2 + 1/15, 0, 0, 0],
[ alpha2/14 - 4*rho2 + 1/15, 6*alpha2/7 + 8*rho2 + 1/5, alpha2/14 - 4*rho2 + 1/15, 0, 0, 0],
[ -alpha2/7 + rho2 + 1/15, alpha2/14 - 4*rho2 + 1/15, -3*alpha2/7 + 3*rho2 + 1/5, 0, 0, 0],
[ 0, 0, 0, alpha2/7 - 8*rho2 + 2/15, 0, 0],
[ 0, 0, 0, 0, -2*alpha2/7 + 2*rho2 + 2/15, 0],
[ 0, 0, 0, 0, 0, alpha2/7 - 8*rho2 + 2/15]])
alpha3_rho3
Matrix([
[-3*alpha3/7 + 3*rho3 + 1/5, -alpha3/7 + rho3 + 1/15, alpha3/14 - 4*rho3 + 1/15, 0, 0, 0],
[ -alpha3/7 + rho3 + 1/15, -3*alpha3/7 + 3*rho3 + 1/5, alpha3/14 - 4*rho3 + 1/15, 0, 0, 0],
[ alpha3/14 - 4*rho3 + 1/15, alpha3/14 - 4*rho3 + 1/15, 6*alpha3/7 + 8*rho3 + 1/5, 0, 0, 0],
[ 0, 0, 0, alpha3/7 - 8*rho3 + 2/15, 0, 0],
[ 0, 0, 0, 0, alpha3/7 - 8*rho3 + 2/15, 0],
[ 0, 0, 0, 0, 0, -2*alpha3/7 + 2*rho3 + 2/15]])
####################
Group: tetragonal
alpha1_d1_d3
Matrix([
[6*alpha1/7 - 2*d1 + 1/5, alpha1/14 + d1 + 1/15, alpha1/14 + d1 + 1/15, 0, 0, 0],
[ alpha1/14 + d1 + 1/15, -3*alpha1/7 - d1 - d3 + 1/5, -alpha1/7 + d3 + 1/15, 0, 0, 0],
[ alpha1/14 + d1 + 1/15, -alpha1/7 + d3 + 1/15, -3*alpha1/7 - d1 - d3 + 1/5, 0, 0, 0],
[ 0, 0, 0, -2*alpha1/7 + 2*d3 + 2/15, 0, 0],
[ 0, 0, 0, 0, alpha1/7 + 2*d1 + 2/15, 0],
[ 0, 0, 0, 0, 0, alpha1/7 + 2*d1 + 2/15]])
####################
Group: trigonal
alpha1_d3_d9
Matrix([
[6*alpha1/7 + 8*d3 + 1/5, alpha1/14 - 4*d3 + 1/15, alpha1/14 - 4*d3 + 1/15, 0, 0, 0],
[alpha1/14 - 4*d3 + 1/15, -3*alpha1/7 + 3*d3 + 1/5, -alpha1/7 + d3 + 1/15, 0, 0, sqrt(2)*d9],
[alpha1/14 - 4*d3 + 1/15, -alpha1/7 + d3 + 1/15, -3*alpha1/7 + 3*d3 + 1/5, 0, 0, -sqrt(2)*d9],
[ 0, 0, 0, -2*alpha1/7 + 2*d3 + 2/15, -2*d9, 0],
[ 0, 0, 0, -2*d9, alpha1/7 - 8*d3 + 2/15, 0],
[ 0, sqrt(2)*d9, -sqrt(2)*d9, 0, 0, alpha1/7 - 8*d3 + 2/15]])
####################
Group: orthotropic
la1_la2_d1_d2_d3
Matrix([
[-d1 - d2 + 6*la1/7 - 3/35, d1 + la1/7 + la2/7 - 1/35, d2 - la2/7 + 4/35, 0, 0, 0],
[d1 + la1/7 + la2/7 - 1/35, -d1 - d3 + 6*la2/7 - 3/35, d3 - la1/7 + 4/35, 0, 0, 0],
[ d2 - la2/7 + 4/35, d3 - la1/7 + 4/35, -d2 - d3 - 6*la1/7 - 6*la2/7 + 27/35, 0, 0, 0],
[ 0, 0, 0, 2*d3 - 2*la1/7 + 8/35, 0, 0],
[ 0, 0, 0, 0, 2*d2 - 2*la2/7 + 8/35, 0],
[ 0, 0, 0, 0, 0, 2*d1 + 2*la1/7 + 2*la2/7 - 2/35]])
alpha1_alpha3_rho1_rho2_rho3
Matrix([
[6*alpha1/7 - 3*alpha3/7 + 8*rho1 + 3*rho2 + 3*rho3 + 1/5, alpha1/14 - alpha3/7 - 4*rho1 - 4*rho2 + rho3 + 1/15, alpha1/14 + alpha3/14 - 4*rho1 + rho2 - 4*rho3 + 1/15, 0, 0, 0],
[ alpha1/14 - alpha3/7 - 4*rho1 - 4*rho2 + rho3 + 1/15, -3*alpha1/7 - 3*alpha3/7 + 3*rho1 + 8*rho2 + 3*rho3 + 1/5, -alpha1/7 + alpha3/14 + rho1 - 4*rho2 - 4*rho3 + 1/15, 0, 0, 0],
[ alpha1/14 + alpha3/14 - 4*rho1 + rho2 - 4*rho3 + 1/15, -alpha1/7 + alpha3/14 + rho1 - 4*rho2 - 4*rho3 + 1/15, -3*alpha1/7 + 6*alpha3/7 + 3*rho1 + 3*rho2 + 8*rho3 + 1/5, 0, 0, 0],
[ 0, 0, 0, -2*alpha1/7 + alpha3/7 + 2*rho1 - 8*rho2 - 8*rho3 + 2/15, 0, 0],
[ 0, 0, 0, 0, alpha1/7 + alpha3/7 - 8*rho1 + 2*rho2 - 8*rho3 + 2/15, 0],
[ 0, 0, 0, 0, 0, alpha1/7 - 2*alpha3/7 - 8*rho1 - 8*rho2 + 2*rho3 + 2/15]])
alpha1_alpha3_d1_d2_d3
Matrix([
[6*alpha1/7 - 3*alpha3/7 - d1 - d2 + 1/5, alpha1/14 - alpha3/7 + d1 + 1/15, alpha1/14 + alpha3/14 + d2 + 1/15, 0, 0, 0],
[ alpha1/14 - alpha3/7 + d1 + 1/15, -3*alpha1/7 - 3*alpha3/7 - d1 - d3 + 1/5, -alpha1/7 + alpha3/14 + d3 + 1/15, 0, 0, 0],
[ alpha1/14 + alpha3/14 + d2 + 1/15, -alpha1/7 + alpha3/14 + d3 + 1/15, -3*alpha1/7 + 6*alpha3/7 - d2 - d3 + 1/5, 0, 0, 0],
[ 0, 0, 0, -2*alpha1/7 + alpha3/7 + 2*d3 + 2/15, 0, 0],
[ 0, 0, 0, 0, alpha1/7 + alpha3/7 + 2*d2 + 2/15, 0],
[ 0, 0, 0, 0, 0, alpha1/7 - 2*alpha3/7 + 2*d1 + 2/15]])
####################
Group: monoclinic
alpha1_alpha3_d1_d2_d3_d4_d5
Matrix([
[6*alpha1/7 - 3*alpha3/7 - d1 - d2 + 1/5, alpha1/14 - alpha3/7 + d1 + 1/15, alpha1/14 + alpha3/14 + d2 + 1/15, -d4 - d5, 0, 0],
[ alpha1/14 - alpha3/7 + d1 + 1/15, -3*alpha1/7 - 3*alpha3/7 - d1 - d3 + 1/5, -alpha1/7 + alpha3/14 + d3 + 1/15, d4, 0, 0],
[ alpha1/14 + alpha3/14 + d2 + 1/15, -alpha1/7 + alpha3/14 + d3 + 1/15, -3*alpha1/7 + 6*alpha3/7 - d2 - d3 + 1/5, d5, 0, 0],
[ -d4 - d5, d4, d5, -2*alpha1/7 + alpha3/7 + 2*d3 + 2/15, 0, 0],
[ 0, 0, 0, 0, alpha1/7 + alpha3/7 + 2*d2 + 2/15, -sqrt(2)*(d4 + d5)],
[ 0, 0, 0, 0, -sqrt(2)*(d4 + d5), alpha1/7 - 2*alpha3/7 + 2*d1 + 2/15]])
####################
Group: triclinic
la1_la2_d1_d2_d3_d4_d5_d6_d7_d8_d9
Matrix([
[-d1 - d2 + 6*la1/7 - 3/35, d1 + la1/7 + la2/7 - 1/35, d2 - la2/7 + 4/35, -sqrt(2)*(d4 + d5), sqrt(2)*d6, sqrt(2)*d8],
[d1 + la1/7 + la2/7 - 1/35, -d1 - d3 + 6*la2/7 - 3/35, d3 - la1/7 + 4/35, sqrt(2)*d4, -sqrt(2)*(d6 + d7), sqrt(2)*d9],
[ d2 - la2/7 + 4/35, d3 - la1/7 + 4/35, -d2 - d3 - 6*la1/7 - 6*la2/7 + 27/35, sqrt(2)*d5, sqrt(2)*d7, -sqrt(2)*(d8 + d9)],
[ -sqrt(2)*(d4 + d5), sqrt(2)*d4, sqrt(2)*d5, 2*d3 - 2*la1/7 + 8/35, -2*d8 - 2*d9, -2*d6 - 2*d7],
[ sqrt(2)*d6, -sqrt(2)*(d6 + d7), sqrt(2)*d7, -2*d8 - 2*d9, 2*d2 - 2*la2/7 + 8/35, -2*d4 - 2*d5],
[ sqrt(2)*d8, sqrt(2)*d9, -sqrt(2)*(d8 + d9), -2*d6 - 2*d7, -2*d4 - 2*d5, 2*d1 + 2*la1/7 + 2*la2/7 - 2/35]])