def update_state(self, dt, d, stress, xtra):
"""Compute updated stress given strain increment
Parameters
----------
dt : float
Time step
d : array_like
Deformation rate
stress : array_like
Stress at beginning of step
xtra : array_like
Extra variables
Returns
-------
S : array_like
Updated stress
xtra : array_like
Updated extra variables
"""
dstrain = d * dt
dstress = 3. * self.K * iso(dstrain) + 2 * self.G * dev(dstrain)
return stress + dstress, xtra
def update_state(self, dt, d, stress, xtra):
"""Compute updated stress given strain increment
Parameters
----------
dt : float
Time step
d : array_like
Deformation rate
stress : array_like
Stress at beginning of step
xtra : array_like
Extra variables
Returns
-------
S : array_like
Updated stress
xtra : array_like
Updated extra variables
"""
dstrain = d * dt
dstress = 3. * self.K * iso(dstrain) + 2 * self.G * dev(dstrain)
return stress + dstress, xtra
def update_state(self, dt, d, stress, xtra):
"""Compute updated stress given strain increment
Parameters
----------
dt : float
Time step
d : array_like
Deformation rate
stress : array_like
Stress at beginning of step
xtra : array_like
Extra variables at beginning of step
Returns
-------
S : array_like
Updated stress
dep : float
Updated equivalent plastic strain
"""
dstrain = d * dt
dep = self.deplas(dt, stress, xtra, dstrain)
xtra[0] += dep
# Bulk modulus, Youngs modulus and Poissons ratio
E = self._params[self.Ei]
nu = self._params[self.Nui]
K = E / (3. * (1. - 2. * nu))
devol = np.sum(dstrain[:3])
p = trace(stress)
# S is the deviatoric stress predictor
S = stress - p / 3. * I6 + E / (1 + nu) * dev(dstrain)
se = np.sqrt(1.5 * np.sum(S * S))
if se > 0:
beta = 1. - 1.5 * E * dep / ((1 + nu) * se)
else:
beta = 1.;
# S now stores the full stress tensor
S = beta * S + (p / 3. + K * devol) * I6
return S, xtra
def update_state(self, dt, d, stress, xtra):
"""Compute updated stress given strain increment
Parameters
----------
dt : float
Time step
d : array_like
Deformation rate
stress : array_like
Stress at beginning of step
xtra : array_like
Extra variables at beginning of step
Returns
-------
S : array_like
Updated stress
dep : float
Updated equivalent plastic strain
"""
dstrain = d * dt
dep = self.deplas(dt, stress, xtra, dstrain)
xtra[0] += dep
# Bulk modulus, Youngs modulus and Poissons ratio
E = self._params[self.Ei]
nu = self._params[self.Nui]
K = E / (3. * (1. - 2. * nu))
devol = np.sum(dstrain[:3])
p = trace(stress)
# S is the deviatoric stress predictor
S = stress - p / 3. * I6 + E / (1 + nu) * dev(dstrain)
se = np.sqrt(1.5 * np.sum(S * S))
if se > 0:
beta = 1. - 1.5 * E * dep / ((1 + nu) * se)
else:
beta = 1.
# S now stores the full stress tensor
S = beta * S + (p / 3. + K * devol) * I6
return S, xtra
def stiffness(self, dt, d, stress, xtra):
"""Compute the material stiffness tensor
Parameters
----------
dt : float
time step
d : array_like
Deformation rate
stress : array_like
Stress at beginning of step
xtra : float
Extra variables
Returns
-------
C : array_like
The material stiffness
Notes
-----
Currently coded either for plane strain or general 3D. Note that in
this procedure `stress' is the current estimate for stress at the end
of the increment S_n+1
"""
dstrain = d * dt
eplas = xtra[0]
E = self._params[self.Ei]
nu = self._params[self.Nui]
K = E / (3. * (1. - 2. * nu))
G = 3. * K * E / (9. * K - E)
e0 = self._params[self.E0i]
n = self._params[self.Ni]
m = self._params[self.Mi]
devol = trace(dstrain)
p = trace(stress)
S = dev(stress)
se = np.sqrt(1.5 * np.sum(S * S))
# tjfulle: fix model
dep = 0.
if se * dep > 0:
beta = 1. / (1. + 1.5 * E * dep / ((1. + nu) * se))
gamma = beta * (1.5 * E / ((1 + nu) * se)
+ (1 / (n * (e0 + eplas + dep)) + 1. / (m * dep)))
factor = 1.5 * 1.5 * E * (dep - 1. / gamma) / ((1. + nu) * se ** 3)
else:
beta = 1.
factor = 0.
C = np.zeros((6, 6))
for i in range(3):
for j in range(3):
for k in range(3):
for l in range(3):
ik = reduce_map((i, k), 3)
jl = reduce_map((j, l), 3)
jk = reduce_map((j, k), 3)
il = reduce_map((i, l), 3)
ij = reduce_map((i, j), 3)
kl = reduce_map((k, l), 3)
w = ((I6[ik] * I6[jl] + I6[jk] * I6[il]) / 2.,
-I6[ij] * I6[kl] / 3.,
factor * S[ij] * S[kl])
c = (beta * E / (1 + nu) * (w[0] + w[1] + w[2]),
K * I6[ij] * I6[kl])
C[ij, kl] = c[0] + c[1]
continue
continue
continue
continue
return C
def deplas(self, dt, stress, xtra, dstrain):
"""Compute plastic strain increment given strain increment
Parameters
----------
dt : float
Time step
stress : array_like
Stress at beginning of step
xtra : array_like
Extra variables
dstrain : array_like
Strain increment
Returns
-------
e : float
Plastic strain
"""
# Material properties
E = self._params[self.Ei]
nu = self._params[self.Nui]
Y = self._params[self.Yi]
e0 = self._params[self.E0i]
n = self._params[self.Ni]
edot0 = self._params[self.Edot0i]
m = self._params[self.Mi]
devol = np.sum(dstrain[:3])
p = np.sum(stress[:3])
# deviatoric stress predictor
S = -I6 * p / 3 + E / (1 + nu) * dev(dstrain)
sequiv = np.sqrt(1.5 * np.sum(S * S))
e = 10e-15
err = Y
tol = 10e-06 * Y
if sequiv * edot0 == 0:
e = 0.
else:
while err > tol:
c = (1 + (xtra + e) / e0) ** (1 / n)
c *= (e / (dt * edot0)) ** (1 / m)
f = sequiv / Y - 1.5 * e * E / (Y * (1 + nu)) - c
dfde = (-1.5 * E / (Y * (1 + nu))
- c * (1 / (n * (xtra +e + e0)) + 1 / (m * e)))
enew = e - f / dfde
if enew < 0.:
# e must be > 0, so if new approx to e < 0 the solution
# must lie between current approx to e and zero.
e = e / 10.
else:
e = enew
err = np.abs(f)
continue
return e
def stiffness(self, dt, d, stress, xtra):
"""Compute the material stiffness tensor
Parameters
----------
dt : float
time step
d : array_like
Deformation rate
stress : array_like
Stress at beginning of step
xtra : float
Extra variables
Returns
-------
C : array_like
The material stiffness
Notes
-----
Currently coded either for plane strain or general 3D. Note that in
this procedure `stress' is the current estimate for stress at the end
of the increment S_n+1
"""
dstrain = d * dt
eplas = xtra[0]
E = self._params[self.Ei]
nu = self._params[self.Nui]
K = E / (3. * (1. - 2. * nu))
G = 3. * K * E / (9. * K - E)
e0 = self._params[self.E0i]
n = self._params[self.Ni]
m = self._params[self.Mi]
devol = trace(dstrain)
p = trace(stress)
S = dev(stress)
se = np.sqrt(1.5 * np.sum(S * S))
# tjfulle: fix model
dep = 0.
if se * dep > 0:
beta = 1. / (1. + 1.5 * E * dep / ((1. + nu) * se))
gamma = beta * (1.5 * E / ((1 + nu) * se) +
(1 / (n * (e0 + eplas + dep)) + 1. / (m * dep)))
factor = 1.5 * 1.5 * E * (dep - 1. / gamma) / ((1. + nu) * se**3)
else:
beta = 1.
factor = 0.
C = np.zeros((6, 6))
for i in range(3):
for j in range(3):
for k in range(3):
for l in range(3):
ik = reduce_map((i, k), 3)
jl = reduce_map((j, l), 3)
jk = reduce_map((j, k), 3)
il = reduce_map((i, l), 3)
ij = reduce_map((i, j), 3)
kl = reduce_map((k, l), 3)
w = ((I6[ik] * I6[jl] + I6[jk] * I6[il]) / 2.,
-I6[ij] * I6[kl] / 3., factor * S[ij] * S[kl])
c = (beta * E / (1 + nu) * (w[0] + w[1] + w[2]),
K * I6[ij] * I6[kl])
C[ij, kl] = c[0] + c[1]
continue
continue
continue
continue
return C
def deplas(self, dt, stress, xtra, dstrain):
"""Compute plastic strain increment given strain increment
Parameters
----------
dt : float
Time step
stress : array_like
Stress at beginning of step
xtra : array_like
Extra variables
dstrain : array_like
Strain increment
Returns
-------
e : float
Plastic strain
"""
# Material properties
E = self._params[self.Ei]
nu = self._params[self.Nui]
Y = self._params[self.Yi]
e0 = self._params[self.E0i]
n = self._params[self.Ni]
edot0 = self._params[self.Edot0i]
m = self._params[self.Mi]
devol = np.sum(dstrain[:3])
p = np.sum(stress[:3])
# deviatoric stress predictor
S = -I6 * p / 3 + E / (1 + nu) * dev(dstrain)
sequiv = np.sqrt(1.5 * np.sum(S * S))
e = 10e-15
err = Y
tol = 10e-06 * Y
if sequiv * edot0 == 0:
e = 0.
else:
while err > tol:
c = (1 + (xtra + e) / e0)**(1 / n)
c *= (e / (dt * edot0))**(1 / m)
f = sequiv / Y - 1.5 * e * E / (Y * (1 + nu)) - c
dfde = (-1.5 * E / (Y * (1 + nu)) - c *
(1 / (n * (xtra + e + e0)) + 1 / (m * e)))
enew = e - f / dfde
if enew < 0.:
# e must be > 0, so if new approx to e < 0 the solution
# must lie between current approx to e and zero.
e = e / 10.
else:
e = enew
err = np.abs(f)
continue
return e