# -*- coding: utf-8 -*-
"""
This file is part of FElupe.
FElupe is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
FElupe is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with FElupe. If not, see <http://www.gnu.org/licenses/>.
"""
from functools import wraps
import numpy as np
from tensortrax.math import log, sqrt
from tensortrax.math import sum as sum1
from tensortrax.math import trace
from tensortrax.math._linalg import det, eigvalsh, inv
from tensortrax.math._special import from_triu_1d, triu_1d
def isochoric_volumetric_split(fun):
"""Apply the material formulation only on the isochoric part of the
multiplicative split of the deformation gradient."""
@wraps(fun)
def apply_iso(C, *args, **kwargs):
return fun(det(C) ** (-1 / 3) * C, *args, **kwargs)
return apply_iso
[docs]def saint_venant_kirchhoff(C, mu, lmbda):
"""Strain energy function of the Saint Venant-Kirchhoff material formulation.
Here, ``I1`` and ``I2`` are strain invariants of the Green-Lagrange strain tensor.
"""
I1 = trace(C) / 2 - 3 / 2
I2 = trace(C @ C) / 4 - trace(C) / 2 + 3 / 4
return mu * I2 + lmbda * I1**2 / 2
[docs]def neo_hooke(C, mu):
"Strain energy function of the Neo-Hookean material formulation."
return mu / 2 * (det(C) ** (-1 / 3) * trace(C) - 3)
[docs]def mooney_rivlin(C, C10, C01):
"Strain energy function of the Mooney-Rivlin material formulation."
J3 = det(C) ** (-1 / 3)
I1 = J3 * trace(C)
I2 = (I1**2 - J3**2 * trace(C @ C)) / 2
return C10 * (I1 - 3) + C01 * (I2 - 3)
[docs]def yeoh(C, C10, C20, C30):
I1 = det(C) ** (-1 / 3) * trace(C)
return C10 * (I1 - 3) + C20 * (I1 - 3) ** 2 + C30 * (I1 - 3) ** 3
[docs]def ogden(C, mu, alpha):
"Strain energy function of the Ogden material formulation."
wC = det(C) ** (-1 / 3) * eigvalsh(C)
return sum1([2 * m / a**2 * (sum1(wC ** (a / 2)) - 3) for m, a in zip(mu, alpha)])
[docs]def arruda_boyce(C, C1, limit):
"Strain energy function of the Arruda-Boyce material formulation."
I1 = det(C) ** (-1 / 3) * trace(C)
alpha = [1 / 2, 1 / 20, 11 / 1050, 19 / 7000, 519 / 673750]
beta = 1 / limit**2
out = []
for i, a in enumerate(alpha):
j = i + 1
out.append(a * beta ** (2 * j - 2) * (I1**j - 3**j))
return C1 * sum1(out)
[docs]def extended_tube(C, Gc, delta, Ge, beta):
"Strain energy function of the Extended-Tube material formulation."
J3 = det(C) ** (-1 / 3)
D = J3 * trace(C)
wC = J3 * eigvalsh(C)
g = (1 - delta**2) * (D - 3) / (1 - delta**2 * (D - 3))
Wc = Gc / 2 * (g + log(1 - delta**2 * (D - 3)))
We = 2 * Ge / beta**2 * sum1(wC ** (-beta / 2) - 1)
return Wc + We
[docs]def van_der_waals(C, mu, limit, a, beta):
"Strain energy function of the Van der Waals material formulation."
J3 = det(C) ** (-1 / 3)
I1 = J3 * trace(C)
I2 = (trace(C) ** 2 - J3**2 * trace(C @ C)) / 2
Im = (1 - beta) * I1 + beta * I2
Im.x[np.isclose(Im.x, 3)] += 1e-8
eta = sqrt((Im - 3) / (limit**2 - 3))
return mu * (
-(limit**2 - 3) * (log(1 - eta) + eta) - 2 / 3 * a * ((Im - 3) / 2) ** (3 / 2)
)
@isochoric_volumetric_split
def finite_strain_viscoelastic(C, Cin, mu, eta, dtime):
"Finite strain viscoelastic material formulation."
# update of state variables by evolution equation
Ci = from_triu_1d(Cin, like=C) + mu / eta * dtime * C
Ci = det(Ci) ** (-1 / 3) * Ci
# first invariant of elastic part of right Cauchy-Green deformation tensor
I1 = trace(C @ inv(Ci))
# strain energy function and state variable
return mu / 2 * (I1 - 3), triu_1d(Ci)