def inplaneTension(fYLD, iopt=1, **kwargs):
"""
inplane tesion tests provides
R-value and yield stress variations along difference
direction from the rolling direction of sheet metals
fYLD: yield function (a function of stress state only)
Returns
-------
psis (rotation angles)
rvs (r-values)
phis (yield stresses)
"""
psis = np.linspace(0, +np.pi / 2., 100)
## stress state for inplane tension in the lab axes
## is of course uniaxial stress state:
sLab = np.zeros(6)
sLab[0] = 1.
phis = []
rvs = []
## rotate this stress state to 'material' axes for
## each psi angles and collect results
for i in xrange(len(psis)):
sMat = rot_6d(sLab, psis[i])
ysMat, Phi, dPhiMat, d2PhiMat = fYLD(s=sMat, **kwargs)
ysLab = rot_6d(ysMat, -psis[i])
deLab = rot_6d(dPhiMat, -psis[i])
if iopt == 0:
phis.append(Phi)
elif iopt == 1:
phis.append(ysLab[0])
e1, e2, e3 = deLab[0], deLab[1], -deLab[0] - deLab[1]
rvs.append(e2 / e3)
return psis, rvs, phis
def inplaneTension(fYLD,iopt=1,**kwargs):
"""
inplane tesion tests provides
R-value and yield stress variations along difference
direction from the rolling direction of sheet metals
fYLD: yield function (a function of stress state only)
Returns
-------
psis (rotation angles)
rvs (r-values)
phis (yield stresses)
"""
psis = np.linspace(0,+np.pi/2.,100)
## stress state for inplane tension in the lab axes
## is of course uniaxial stress state:
sLab=np.zeros(6)
sLab[0]=1.
phis=[];rvs=[]
## rotate this stress state to 'material' axes for
## each psi angles and collect results
for i in xrange(len(psis)):
sMat = rot_6d(sLab, psis[i])
ysMat, Phi, dPhiMat, d2PhiMat = fYLD(s=sMat,**kwargs)
ysLab = rot_6d(ysMat, -psis[i])
deLab = rot_6d(dPhiMat,-psis[i])
if iopt==0:
phis.append(Phi)
elif iopt==1:
phis.append(ysLab[0])
e1,e2,e3 = deLab[0],deLab[1],- deLab[0]-deLab[1]
rvs.append(e2/e3)
return psis, rvs, phis
def func_fld2(
ndim,
T,
# s,
b,
x,
yold,
matA,
matB,
verbose):
"""
Arguments
---------
ndim
T : axis along integration occurs
s : stress of region A
b : variables?
b[0] : unknown
b[1] : f0
b[2] : unknown
b[3] : unknown
b[4] : unknown
b[5] : unknown
b[6] : unknown
b[7] : unknown
b[8] : delta t: delta equivalent strain increment: \dot{lambda}d(time)=dleta lambda^B for region B
b[9] : psi_old
b[10]: psi_new
x :
x[0]: d\lambda^A, i.e., the increment of equivalent strain pertaining to region A
x[1]: s11 component of stress
x[2]: s22 component of stress
x[3]: s12 component of stress
yold: yold defined in pasapas [deps,]
y[1]: accumulative strain RD
y[2]: accumulative strain TD
matA : A material
matB : B material
verbose
** yold
Initially, it is determined:
[0,0,0,tzero*fb[0],tzero*fb[1]]
** bn
b[1] = f0
b[8] = deltt
b[9] = psi_old
b[10] = delta_psi
Returns
-------
F,J,fa,fb,b,siga,s (region A strss)
"""
import os, time
from mk.library.lib import rot_6d
t0 = time.time()
t0_grand = time.time()
f0 = b[1]
deltat = b[8] # delta t for region B
## stress in region B
# xb = x[1]
# yb = x[2]
# zb = x[3]
xb, yb, zb = x[1:4]
## (guessed) stress referred in band
sb_dump = np.array([xb, yb, 0., 0., 0., zb])
## Parameters in region A
matA.update_yld(matA.stress)
s, phia, fa, f2a = matA.o_yld
t1 = time.time() - t0_grand
# ------------------------------------------------------------#
t0 = time.time()
## psi0 * ()
delta_psi = x[0] * (fa[0] - fa[1]) * tan(b[9]) / (1 + tan(b[9])**2)
b[10] = delta_psi * 1.
psi_new = b[9] + delta_psi
sa_rot = rot_6d(s, -psi_new) ## A stress referred in the band axes
xa, ya, za = sa_rot[0], sa_rot[1], sa_rot[5]
da_rot = rot_6d(fa, -psi_new) ## A's dphi/dsig referred in the band axes
matA.update_hrd(yold[0] + x[0])
# na,ma,siga,dsiga,dma,qqa = matA.o_hrd
ma, siga, dsiga, dma, qqa = matA.o_hrd[:5]
## parameters in region B
sb = rot_6d(sb_dump, psi_new) ## use updated (tilde) psi
matB.update_yld(sb)
sb, phib, fb, f2b = matB.o_yld
db_rot = rot_6d(fb, -psi_new)
matB.update_hrd(T + deltat) ## delta t (incremental effective strain)
mb, sigb, dsigb, dmb, qqb = matB.o_hrd[:5]
E = -yold[3]-yold[4]-deltat*(fb[0]+fb[1])\
+yold[1]+yold[2]+ x[0]*(fa[0]+fa[1])
t2 = time.time() - t0
# ------------------------------------------------------------#
t0 = time.time()
cp = cos(psi_new)
sp = sin(psi_new)
c2 = cp * cp
s2 = sp * sp
sc = sp * cp
t_bench_0 = time.time()
d2fb = calcD2FB(psi=psi_new, f2b=f2b)
dt_bench = time.time() - t_bench_0
dxp = np.zeros(6)
dxp[0] = 2 * sc * (yb - xb) - 2 * (c2 - s2) * zb
dxp[1] = -dxp[0]
dxp[5] = (c2 - s2) * (xb - yb) - 4 * sc * zb
dpe = (fa[0] - fa[1]) * tan(psi_new) / (1 + tan(psi_new)**2)
Q = x[0] / deltat # ratio between A / B equivalent strain rate increment?
## x[0] = \delta \Lambda^A?
## x[1] = xb
## x[2] = yb ???
## x[3] = zb
t3 = time.time() - t0
# print 'dt_bench/t3',(dt_bench/t3)*100
# ------------------------------------------------------------#
t0 = time.time()
F = np.zeros(4)
## conditions to satisfy
eE = np.exp(E)
F[0] = f0 * eE * sigb * xb - xa * siga * (Q**mb) * qqb**(ma - mb)
F[1] = xb * za - zb * xa
F[2] = phib - 1.0
F[3] = deltat * db_rot[1] - x[0] * da_rot[1]
J = np.zeros((4, 4))
J[0,0] = (fa[0]+fa[1])*f0*eE*sigb*xb\
-xa*dsiga*(Q**mb)*qqb**(ma-mb)\
-xa*siga*(
(mb/deltat)*(Q**(mb-1.))*qqb**(ma-mb)\
+(dma/x[0])*np.log(qqb)*qqb*(ma-mb)
)
J[0,1] =-deltat*(d2fb[0,0]+d2fb[1,0])*f0*eE*sigb*xb\
+f0*eE*sigb
J[0, 2] = -deltat * (d2fb[0, 1] + d2fb[1, 1]) * f0 * eE * sigb * xb
J[0, 3] = -deltat * (d2fb[0, 5] + d2fb[1, 5]) * f0 * eE * sigb * xb
J[1, 1] = za
J[1, 3] = -xa
J[2, 0] = (fb[0] * dxp[0] + fb[1] * dxp[1] + 2 * fb[5] * dxp[5]) * dpe
J[2, 1] = db_rot[0]
J[2, 2] = db_rot[1]
J[2, 3] = 2 * db_rot[5]
J[3, 0] = (f2b[0, 0] * dxp[0] + f2b[0, 1] * dxp[1] + f2b[0, 5] * dxp[5] /
2.) * s2 + (f2b[1, 0] * dxp[0] + f2b[1, 1] * dxp[1] +
f2b[1, 5] * dxp[5] / 2) * c2
J[3, 0] = (J[3, 0] - 2 * (f2b[5, 0] * dxp[0] + f2b[5, 1] * dxp[1] +
f2b[5, 5] * dxp[5] / 2) * sc) * deltat * dpe
J[3, 0] = J[3, 0] - da_rot[1] + x[0] * (
2 * sc * (fa[0] - fa[1]) - 2 * fa[5] *
(c2 - s2)) * dpe + deltat * (2 * sc * (fb[0] - fb[1]) - 2 * fb[5] *
(c2 - s2)) * dpe
J[3,
1] = deltat * (d2fb[0, 0] * s2 + d2fb[1, 0] * c2 - 2 * d2fb[5, 0] * sc)
J[3,
2] = deltat * (d2fb[0, 1] * s2 + d2fb[1, 1] * c2 - 2 * d2fb[5, 1] * sc)
J[3,
3] = deltat * (d2fb[0, 5] * s2 + d2fb[1, 5] * c2 - 2 * d2fb[5, 5] * sc)
t4 = time.time() - t0
# ------------------------------------------------------------#
dt_grand = time.time() - t0_grand
t1 = t1 / dt_grand * 100
t2 = t2 / dt_grand * 100
t3 = t3 / dt_grand * 100
t4 = t4 / dt_grand * 100
print '---------------------------------------'
print('%.1f ' * 5) % (t1, t2, t3, t4, (t1 + t2 + t3 + t4))
print '---------------------------------------'
# raise IOError
return F, J, fa, fb, b # ,s
def func_fld1(
ndim,
b,
x,
# f_hard,f_yld,
matA,
matB,
verbose):
"""
Arguments
=========
ndim
----
b
-
an array determined in the onepath
that is initially set as:
[ psi0, f0, matA.sig, matA.m, matA.qq, sx[5]/sx[0] ]
sx is initially the stress of A that is rotated by psi
b is *NOT* iteratively changing within the Newton Raphson
as func_fld1 is to find the initial state of region B
that corresponding to the initial state of region A
b[6], b[7] are changing though - they are functions of fb[0], fb[1]
x
-
initially given as [1,1,0,0] but should be
iteratively determined
x[0] = s11; x[1] == s22 and x[2] == s12 for the region B
x[3] is unknown yet, but seems related with some kind of
incremental step size (?)
- potentially x[3] is d\lambda, that is the plastic multiplier.
- Usually, d\lambda determines the incremental
size of plastic strain rate
F_{i} = J_{ij}: x_{j}
matA
----
Material description of region A
(including the evolved state varaiables)
matB
----
Material description of region B
(including the evolved state varaiables)
verbose
Returns
-------
F : objective functions
J : jacobian
fb : first derivative of the yield function
pertaining to the region B)
"""
import os
from mk.library.lib import rot_6d
psi0, f0, siga, ma, qq, b6 = b[:6]
### currently (guessed) stress states of region B in band axes
s11, s22, s12 = x[:3]
s = np.array([s11, s22, 0., 0., 0., s12])
## rotate it back to 'RD/TD/ND' axes
## This is primarily due to the fact that yield functions
## are usually written in the frame of rolled sheet metals.
## The stress (as the argument) to the yield function should
## be referred in the proper material axes.
sb = rot_6d(s, psi0)
## stress of region B in the rd-td-nd axes
sb, phib, fb, f2b = matB.f_yld(sb)
"""
For reminding the defintion of b-array:
b[0] = psi0
b[1] = f0
b[2] = matA.sig
b[3] = matA.m
b[4] = matA.qq
## stress state ratio within the band
## from region A stress state
b[5] = sx[5]/sx[0]
if x[3] is d\lambda(
that means, in the context of func_fld1,
the first incremental strain.
b[6] = $\dot{e}_{RD}^B$
b[7] = $\dot{e}_{TD}^B$
"""
b[6] = x[3] * fb[0]
## equivalent accumulative strain x
## (or the first incremental equivalent strain)
b[7] = x[3] * fb[1]
db = rot_6d(fb, -psi0) ## Region B's strain rate in the band axes
if verbose: print 'db:', db
## Seems to rotate the second derivative of yield function
## into the band axes
## May not be required when psi0 = 0,
## which is often the case in forming limit simulations
f2xb = calcF2XB(psi0, f2b)
mb, sigb, dsigb, dmb, qq = matB.f_hrd(x[3])[:5]
if verbose: print 'x(4):', x[3]
f = np.zeros(4)
f[0] = db[
1] ## ettb (transverse strain rate or region B, should it be ettb - etta???)
## force equilibrium condition
f[1] = x[2] - x[0] * b[5] ## s12b - s11b * s12a/s11a ( all in band axes)
f[2] = phib - 1. ## determine if the stress is on the yield locus (consistency condition?)
f[3] = -log(b[1] * sigb / b[2]) + (b[3] - mb) * log(b[4]) - x[3] * db[0]
if not (np.isfinite(f[3])):
print '-' * 20
print 'f[3] is not finite:', f[3]
print 'b[:]:', b[:]
print 'mb:', mb
print 'sigb:', sigb
print 'dsigb:', dsigb
print 'dmb:', dmb
print 'qq:', qq
print 'x[:]', x[:]
print '-' * 20
cp = cos(psi0)
sp = sin(psi0)
c2 = cp * cp
s2 = sp * sp
sc = sp * cp
J = np.zeros((4, 4))
J = np.zeros((4, 4))
J[0, 0] = s2 * f2xb[0, 0] + c2 * f2xb[1, 0] - 2 * sc * f2xb[5, 0]
J[0, 1] = s2 * f2xb[0, 1] + c2 * f2xb[1, 1] - 2 * sc * f2xb[5, 1]
J[0, 2] = s2 * f2xb[0, 5] + c2 * f2xb[1, 5] - 2 * sc * f2xb[5, 5]
J[1, 0] = -b[5] ## sx[5]/sx[0]
J[1, 2] = 1.
J[2, 0] = db[0] ## e_nn^B
J[2, 1] = db[1] ## e_tt^B
J[2, 2] = 2 * db[5] ## 2e_tn^B
J[3, 3] = -dsigb / sigb - dmb * log(
b[4]) - db[0] ## (dH^B/H^B) - dm^B (qq^A) - enn^B
return f, J, fb
def onepath(matA,matB,psi0,f0,T,snapshot):
"""
Run under the given condition that is
characterized by the passed arguments
Arguments
---------
matA
matB
psi0
f0
T
snapshot
"""
import os
from mk.library.lib import rot_6d
from mk.materials.func_hard_for import return_swift
## A stress state referred in band axes
sx = rot_6d(matA.stress,-psi0)
# ## strain hardening can be passed
# independently as the was f_yld is passed.
matA.update_hrd(0.) ## initialize hardening parameters
## initial_conditions
ndim = 4
b = np.zeros(20)
b[0] = psi0
b[1] = f0
b[2] = matA.sig
b[3] = matA.m
b[4] = matA.qq
## stress state ratio within the band from
## region A stress state
b[5] = sx[5]/sx[0]
xzero = np.array([1,1,0,0])
## Determine the initial states
## x[:3]: stress of region b referred in the band axes
## x[:3] = [s11, s22, s12] of the region B referred in the band axes
xfinal, fb = new_raph_fld(
ndim=ndim,ncase=1,
xzero=xzero,b=b,
matA=matA,
matB=matB,
verbose=False)
## fb is the first derivative of region B yield function
## that gives the 'directions' of strain rate according to the AFR
## Initial values
tzero = xfinal[3] ## d\labmda
yzero = np.zeros(5)
## fb: first derivative in region B
yzero[3] = tzero*fb[0] ## B strain 'increment' along RD
yzero[4] = tzero*fb[1] ## B strain 'increment' along TD
ndds = 5 ## dimension differential system
dydx = np.zeros(ndds)
dydx[0] = 1.
## xbb = [psi0,s1,s2,s3,s4,s5,s6]
xbb = np.zeros(7)
xbb[0] = psi0
## caution: xbb[1:] corresponds to the stress components
## sx: A stress state referred in band axes
xbb[1] = sx[0]
xbb[2] = sx[1]
xbb[6] = sx[5]
t0=time.time()
## integrate through monotonic loading
ynew,absciss,xbb\
= integrateMono(
f0,
tzero,
yzero,
ndds,
dydx,
xbb,
matA,
matB,
snapshot,
verbose=False)
psif = xbb[0]
print ('%8.3f'*5)%(ynew[0],ynew[1],ynew[2],ynew[3],ynew[4])
uet(time.time()-t0,'Elapsed time in step by step integration')
print '\nAfter integrateMono'
print 'absciss:',absciss
## check the hardening curve?
return ynew,absciss,xbb
def func_fld2(
ndim,
T,
# s,
b,
x,
yold,
matA,
matB,
verbose):
"""
Arguments
---------
ndim
T : axis along integration occurs
s : stress of region A
b : variables?
b[0] : unknown
b[1] : f0
b[2] : unknown
b[3] : unknown
b[4] : unknown
b[5] : unknown
b[6] : unknown
b[7] : unknown
b[8] : delta t: delta equivalent strain increment: \dot{lambda}d(time)=dleta lambda^B for region B
b[9] : psi_old
b[10]: psi_new
x :
x[0]: d\lambda^A, i.e., the increment of equivalent strain pertaining to region A
x[1]: s11 component of stress
x[2]: s22 component of stress
x[3]: s12 component of stress
yold: yold defined in pasapas [deps,]
y[1]: accumulative strain RD
y[2]: accumulative strain TD
matA : A material
matB : B material
verbose
** yold
Initially, it is determined:
[0,0,0,tzero*fb[0],tzero*fb[1]]
** bn
b[1] = f0
b[8] = deltt
b[9] = psi_old
b[10] = delta_psi
Returns
-------
F,J,fa,fb,b,siga,s (region A strss)
"""
import os, time
from mk.library.lib import rot_6d
t0 = time.time()
t0_grand = time.time()
f0 = b[1]
deltat = b[8] # delta t for region B
## stress in region B
# xb = x[1]
# yb = x[2]
# zb = x[3]
xb,yb,zb = x[1:4]
## (guessed) stress referred in band
sb_dump = np.array([xb,yb,0.,0.,0.,zb])
## Parameters in region A
matA.update_yld(matA.stress)
s,phia,fa,f2a = matA.o_yld
t1 = time.time()-t0_grand
# ------------------------------------------------------------#
t0 = time.time()
## psi0 * ()
delta_psi= x[0] * (fa[0]-fa[1])*tan(b[9])/(1+tan(b[9])**2)
b[10] = delta_psi*1.
psi_new = b[9] + delta_psi
sa_rot = rot_6d(s,-psi_new) ## A stress referred in the band axes
xa,ya,za = sa_rot[0], sa_rot[1],sa_rot[5]
da_rot = rot_6d(fa,-psi_new) ## A's dphi/dsig referred in the band axes
matA.update_hrd(yold[0]+x[0])
# na,ma,siga,dsiga,dma,qqa = matA.o_hrd
ma, siga, dsiga, dma, qqa = matA.o_hrd[:5]
## parameters in region B
sb = rot_6d(sb_dump,psi_new) ## use updated (tilde) psi
matB.update_yld(sb)
sb,phib,fb,f2b = matB.o_yld
db_rot = rot_6d(fb,-psi_new)
matB.update_hrd(T+deltat) ## delta t (incremental effective strain)
mb,sigb,dsigb,dmb,qqb = matB.o_hrd[:5]
E = -yold[3]-yold[4]-deltat*(fb[0]+fb[1])\
+yold[1]+yold[2]+ x[0]*(fa[0]+fa[1])
t2 = time.time()-t0
# ------------------------------------------------------------#
t0 = time.time()
cp = cos(psi_new); sp = sin(psi_new)
c2 = cp*cp; s2 = sp*sp; sc = sp*cp
t_bench_0 = time.time()
d2fb = calcD2FB(psi=psi_new,f2b=f2b)
dt_bench = time.time() - t_bench_0
dxp = np.zeros(6)
dxp[0] = 2*sc*(yb-xb)-2*(c2-s2)*zb
dxp[1] = -dxp[0]
dxp[5] = (c2-s2)*(xb-yb)-4*sc*zb
dpe = (fa[0]-fa[1])*tan(psi_new)/(1+tan(psi_new)**2)
Q = x[0]/deltat # ratio between A / B equivalent strain rate increment?
## x[0] = \delta \Lambda^A?
## x[1] = xb
## x[2] = yb ???
## x[3] = zb
t3 = time.time()-t0
# print 'dt_bench/t3',(dt_bench/t3)*100
# ------------------------------------------------------------#
t0 = time.time()
F = np.zeros(4)
## conditions to satisfy
eE = np.exp(E)
F[0] = f0*eE*sigb*xb - xa*siga*(Q**mb)*qqb**(ma-mb)
F[1] = xb*za - zb*xa
F[2] = phib - 1.0
F[3] = deltat*db_rot[1] - x[0]*da_rot[1]
J=np.zeros((4,4))
J[0,0] = (fa[0]+fa[1])*f0*eE*sigb*xb\
-xa*dsiga*(Q**mb)*qqb**(ma-mb)\
-xa*siga*(
(mb/deltat)*(Q**(mb-1.))*qqb**(ma-mb)\
+(dma/x[0])*np.log(qqb)*qqb*(ma-mb)
)
J[0,1] =-deltat*(d2fb[0,0]+d2fb[1,0])*f0*eE*sigb*xb\
+f0*eE*sigb
J[0,2] =-deltat*(d2fb[0,1]+d2fb[1,1])*f0*eE*sigb*xb
J[0,3] =-deltat*(d2fb[0,5]+d2fb[1,5])*f0*eE*sigb*xb
J[1,1] = za
J[1,3] =-xa
J[2,0] = (fb[0]*dxp[0]+fb[1]*dxp[1]+2*fb[5]*dxp[5])*dpe
J[2,1] = db_rot[0]
J[2,2] = db_rot[1]
J[2,3] = 2*db_rot[5]
J[3,0] = (f2b[0,0]*dxp[0]+f2b[0,1]*dxp[1]+f2b[0,5]*dxp[5]/2.)*s2+(f2b[1,0]*dxp[0]+f2b[1,1]*dxp[1]+f2b[1,5]*dxp[5]/2)*c2
J[3,0] = (J[3,0]-2*(f2b[5,0]*dxp[0]+f2b[5,1]*dxp[1]+f2b[5,5]*dxp[5]/2)*sc)*deltat*dpe
J[3,0] = J[3,0]-da_rot[1]+x[0]*(2*sc*(fa[0]-fa[1])-2*fa[5]*(c2-s2))*dpe + deltat*(2*sc*(fb[0]-fb[1])-2*fb[5]*(c2-s2))*dpe
J[3,1] = deltat*(d2fb[0,0]*s2+d2fb[1,0]*c2-2*d2fb[5,0]*sc)
J[3,2] = deltat*(d2fb[0,1]*s2+d2fb[1,1]*c2-2*d2fb[5,1]*sc)
J[3,3] = deltat*(d2fb[0,5]*s2+d2fb[1,5]*c2-2*d2fb[5,5]*sc)
t4 = time.time()-t0
# ------------------------------------------------------------#
dt_grand = time.time() - t0_grand
t1=t1/dt_grand * 100
t2=t2/dt_grand * 100
t3=t3/dt_grand * 100
t4=t4/dt_grand * 100
print '---------------------------------------'
print ('%.1f '*5)%(t1,t2,t3,t4, (t1+t2+t3+t4))
print '---------------------------------------'
# raise IOError
return F,J,fa,fb,b# ,s
def func_fld1(
ndim,b,x,
# f_hard,f_yld,
matA, matB,
verbose):
"""
Arguments
=========
ndim
----
b
-
an array determined in the onepath
that is initially set as:
[ psi0, f0, matA.sig, matA.m, matA.qq, sx[5]/sx[0] ]
sx is initially the stress of A that is rotated by psi
b is *NOT* iteratively changing within the Newton Raphson
as func_fld1 is to find the initial state of region B
that corresponding to the initial state of region A
b[6], b[7] are changing though - they are functions of fb[0], fb[1]
x
-
initially given as [1,1,0,0] but should be
iteratively determined
x[0] = s11; x[1] == s22 and x[2] == s12 for the region B
x[3] is unknown yet, but seems related with some kind of
incremental step size (?)
- potentially x[3] is d\lambda, that is the plastic multiplier.
- Usually, d\lambda determines the incremental
size of plastic strain rate
F_{i} = J_{ij}: x_{j}
matA
----
Material description of region A
(including the evolved state varaiables)
matB
----
Material description of region B
(including the evolved state varaiables)
verbose
Returns
-------
F : objective functions
J : jacobian
fb : first derivative of the yield function
pertaining to the region B)
"""
import os
from mk.library.lib import rot_6d
psi0, f0, siga, ma, qq, b6 = b[:6]
### currently (guessed) stress states of region B in band axes
s11,s22,s12 = x[:3]
s = np.array([s11,s22,0.,0.,0.,s12])
## rotate it back to 'RD/TD/ND' axes
## This is primarily due to the fact that yield functions
## are usually written in the frame of rolled sheet metals.
## The stress (as the argument) to the yield function should
## be referred in the proper material axes.
sb = rot_6d(s, psi0)
## stress of region B in the rd-td-nd axes
sb, phib, fb, f2b = matB.f_yld(sb)
"""
For reminding the defintion of b-array:
b[0] = psi0
b[1] = f0
b[2] = matA.sig
b[3] = matA.m
b[4] = matA.qq
## stress state ratio within the band
## from region A stress state
b[5] = sx[5]/sx[0]
if x[3] is d\lambda(
that means, in the context of func_fld1,
the first incremental strain.
b[6] = $\dot{e}_{RD}^B$
b[7] = $\dot{e}_{TD}^B$
"""
b[6]=x[3]*fb[0]
## equivalent accumulative strain x
## (or the first incremental equivalent strain)
b[7]=x[3]*fb[1]
db = rot_6d(fb,-psi0) ## Region B's strain rate in the band axes
if verbose: print 'db:',db
## Seems to rotate the second derivative of yield function
## into the band axes
## May not be required when psi0 = 0,
## which is often the case in forming limit simulations
f2xb = calcF2XB(psi0,f2b)
mb,sigb,dsigb,dmb,qq=matB.f_hrd(x[3])[:5]
if verbose:print 'x(4):',x[3]
f=np.zeros(4)
f[0] = db[1] ## ettb (transverse strain rate or region B, should it be ettb - etta???)
## force equilibrium condition
f[1] = x[2] - x[0]*b[5] ## s12b - s11b * s12a/s11a ( all in band axes)
f[2] = phib - 1. ## determine if the stress is on the yield locus (consistency condition?)
f[3] =-log(b[1]*sigb/b[2])+(b[3]-mb)*log(b[4])-x[3]*db[0]
if not(np.isfinite(f[3])):
print '-'*20
print 'f[3] is not finite:',f[3]
print 'b[:]:',b[:]
print 'mb:',mb
print 'sigb:',sigb
print 'dsigb:',dsigb
print 'dmb:',dmb
print 'qq:',qq
print 'x[:]',x[:]
print '-'*20
cp = cos(psi0)
sp = sin(psi0)
c2 = cp*cp
s2 = sp*sp
sc = sp*cp
J=np.zeros((4,4))
J = np.zeros((4,4))
J[0,0] = s2 * f2xb[0,0] + c2 * f2xb[1,0] - 2*sc*f2xb[5,0]
J[0,1] = s2 * f2xb[0,1] + c2 * f2xb[1,1] - 2*sc*f2xb[5,1]
J[0,2] = s2 * f2xb[0,5] + c2 * f2xb[1,5] - 2*sc*f2xb[5,5]
J[1,0] = -b[5] ## sx[5]/sx[0]
J[1,2] = 1.
J[2,0] = db[0] ## e_nn^B
J[2,1] = db[1] ## e_tt^B
J[2,2] = 2*db[5] ## 2e_tn^B
J[3,3] = -dsigb / sigb - dmb*log(b[4]) - db[0] ## (dH^B/H^B) - dm^B (qq^A) - enn^B
return f, J, fb