def postprocess_cauchy_stress(self, displacement):
"""
Compute Cauchy stress from given numerical solution.
Parameters
----------
displacement: Function
Computed numerical displacement
"""
assert hasattr(self, "_elastic_ratio")
assert hasattr(self, "_I")
assert isinstance(displacement, dlfn.function.function.Function)
# displacement gradient
H = grad(displacement)
# deformation gradient
F = self._I + H
# right Cauchy-Green tensor
C = F.T * F
# volume ratio
J = dlfn.det(F)
# dimensionless 2. Piola-Kirchhoff stress tensor (symbolic)
S = self._I - J**(-self._elastic_ratio) * inv(C)
# dimensionless Cauchy stress tensor (symbolic)
sigma = (F * S * F.T) / J
return sigma
def dw_int(self, u, v):
"""
Construct internal energy.
Parameters
----------
u: Function
v: TestFunction
"""
# u Function (in the nonlinear case)
assert isinstance(u, dlfn.function.function.Function)
# v TestFunction
assert isinstance(v, dlfn.function.argument.Argument)
assert hasattr(self, "_elastic_ratio")
assert hasattr(self, "_I")
# deformation gradient
F = self._I + grad(u)
# right Cauchy-Green tensor
C = F.T * F
# volume ratio
J = dlfn.det(F)
# 2. Piola-Kirchhoff stress
S = self._I - J**(-self._elastic_ratio) * inv(C)
dE = dlfn.Constant(0.5) * (F.T * grad(v) + grad(v).T * F)
return inner(S, dE)
def evolEqG(C, Cv, Cv_step):
Iv1 = tr(Cv)
Ce = C * inv(Cv)
CvinvC = inv(Cv) * C
Ie1 = tr(Ce)
# define etaK
A2 = m1 * (Ie1 / 3)**(a1 - 1) + m2 * (Ie1 / 3)**(a2 - 1)
G = (
A2 /
(etaInf + (eta0 - etaInf + K1 * (Iv1**bta1 - 3**bta1)) /
(1 +
(K2 *
((-(Ie1**2) / 6.0 + 1.0 / 2 * inner(CvinvC, Ce)) * A2**2))**bta2)) *
(C - Ie1 / 3.0 * Cv))
dl_interp(G, Cv_step)
def freeEnergy(C, Cv):
J = sqrt(det(C))
I1 = tr(C)
Ce = C * inv(Cv)
Ie1 = tr(Ce)
Je = J / sqrt(det(Cv))
psiEq = (3**(1 - alph1) / (2.0 * alph1) * mu1 * (I1**alph1 - 3**alph1) +
3**(1 - alph2) / (2.0 * alph2) * mu2 * (I1**alph2 - 3**alph2) -
(mu1 + mu2) * ln(J) + mu_pr / 2 * (J - 1)**2)
psiNeq = (3**(1 - a1) / (2.0 * a1) * m1 * (Ie1**a1 - 3**a1) + 3**(1 - a2) /
(2.0 * a2) * m2 * (Ie1**a2 - 3**a2) - (m1 + m2) * ln(Je))
return psiEq + psiNeq
def dS_dFg(self, u_, p_, ivar, theta_old_, dt):
theta_ = ivar["theta"]
Cmat = self.S(u_, p_, ivar, tang=True)
i, j, k, l, m, n = indices(6)
# elastic material tangent (living in intermediate growth configuration)
Cmat_e = dot(
self.F_g(theta_),
dot(self.F_g(theta_),
dot(Cmat, dot(self.F_g(theta_).T,
self.F_g(theta_).T))))
Fginv_outertop_S = as_tensor(
inv(self.F_g(theta_))[i, k] * self.S(u_, p_, ivar=ivar)[j, l],
(i, j, k, l))
S_outerbot_Fginv = as_tensor(
self.S(u_, p_, ivar=ivar)[i, l] * inv(self.F_g(theta_))[j, k],
(i, j, k, l))
Fginv_outertop_Fginv = as_tensor(
inv(self.F_g(theta_))[i, k] * inv(self.F_g(theta_))[j, l],
(i, j, k, l))
FginvT_outertop_Ce = as_tensor(
inv(self.F_g(theta_)).T[i, k] *
self.C_e(self.kin.C(u_), theta_)[j, l], (i, j, k, l))
Ce_outerbot_FginvT = as_tensor(
self.C_e(self.kin.C(u_), theta_)[i, l] *
inv(self.F_g(theta_)).T[j, k], (i, j, k, l))
Cmat_e_with_C_e = 0.5 * as_tensor(
Cmat_e[i, j, m, n] *
(FginvT_outertop_Ce[m, n, k, l] + Ce_outerbot_FginvT[m, n, k, l]),
(i, j, k, l))
Fginv_outertop_Fginv_with_Cmat_e_with_C_e = as_tensor(
Fginv_outertop_Fginv[i, j, m, n] * Cmat_e_with_C_e[m, n, k, l],
(i, j, k, l))
return -(Fginv_outertop_S +
S_outerbot_Fginv) - Fginv_outertop_Fginv_with_Cmat_e_with_C_e
def A(self):
return self.F() * inv(self.G)
def e(self, u_):
return 0.5*(self.I - inv(self.b(u_)))
def L(self, theta_, theta_old, dt):
return dot((self.F_g(theta_) - self.F_g(theta_old))/dt, inv(self.F_g(theta_)))
def J_e(self, u_, theta_):
return det(self.kin.F(u_)*inv(self.F_g(theta_)))
def C_e(self, C_, theta_):
return inv(self.F_g(theta_)) * C_ * inv(self.F_g(theta_)).T
# External forces T = Coefficient(u_element) p0 = Coefficient(p_element) # Material parameters FIXME rho = Constant(cell) K = Constant(cell) c00 = Constant(cell) c11 = Constant(cell) c22 = Constant(cell) # Deformation gradient I = Identity(d) F = I + grad(u) F = variable(F) Finv = inv(F) J = det(F) # Left Cauchy-Green deformation tensor B = F * F.T I1_B = tr(B) I2_B = (I1_B**2 - tr(B * B)) / 2 I3_B = J**2 # Right Cauchy-Green deformation tensor C = F.T * F I1_C = tr(C) I2_C = (I1_C**2 - tr(C * C)) / 2 I3_C = J**2 # Green strain tensor