# -*- 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 numpy.linalg import inv
[docs]
def lame_converter(E, nu):
r"""Convert the pair of given material parameters Young's modulus :math:`E` and
Poisson ratio :math:`\nu` to first and second Lamé - constants :math:`\lambda` and
:math:`\mu`.
Parameters
----------
E : float
Young's modulus.
nu : float
Poisson ratio.
Returns
-------
lmbda : float
First Lamé - constant.
mu : float
Second Lamé - constant (shear modulus).
Notes
-----
.. math::
\lambda &= \frac{E \nu}{(1 + \nu) (1 - 2 \nu)}
\mu &= \frac{E}{2 (1 + \nu)}
"""
lmbda = E * nu / ((1 + nu) * (1 - 2 * nu))
mu = E / (2 * (1 + nu))
return lmbda, mu
[docs]
def lame_converter_orthotropic(E, nu, G):
r"""Convert elastic orthotropic material parameters to Lamé constants.
Parameters
----------
E : list of float
List of the three elastic moduli :math:`E_1, E_2, E_3`.
nu : list of float
List of three poisson ratios :math:`\nu_{12}, \nu_{23}, \nu_{31}`.
G : list of float
List of three shear moduli :math:`G_{12}, G_{23}, G_{31}`.
Returns
-------
lmbda : list of float
List of six (upper triangle) first Lamé parameters :math:`\lambda_{11},
\lambda_{12}, \lambda_{13}, \lambda_{22}, \lambda_{23}, \lambda_{33}`.
mu : list of float
List of the three second Lamé parameters :math:`\mu_1,\mu_2, \mu_3`.
Notes
-----
The orthotropic material parameters are converted to orthotropic Lamé
constants.
The compliance matrix as the inverse of the stiffness matrix with the
parameters :math:`E_i`, :math:`\nu_{ij}` and :math:`G_{ij}` is given in
Eq. :eq:`ortho-matrix-inv`.
.. math::
:label: ortho-matrix-inv
\boldsymbol{C}^{-1} = \begin{bmatrix}
\frac{1}{E_1} & -\frac{\nu_{21}}{E_2} & -\frac{\nu_{31}}{E_3}
& 0 & 0 & 0 \\
-\frac{\nu_{12}}{E_1} & \frac{1}{E_2} & -\frac{\nu_{32}}{E_3}
& 0 & 0 & 0 \\
-\frac{\nu_{13}}{E_1} & -\frac{\nu_{23}}{E_2} & \frac{1}{E_3}
& 0 & 0 & 0 \\
0 & 0 & 0 & \frac{1}{G_{12}} & 0 & 0 \\
0 & 0 & 0 & 0 & \frac{1}{G_{23}} & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{1}{G_{31}}
\end{bmatrix}
The stiffness matrix with the Lamé constants is denoted in
Eq. :eq:`ortho-matrix`.
.. math::
:label: ortho-matrix-inv
\boldsymbol{C} = \begin{bmatrix}
\lambda_{11} + 2 \mu_1 & \lambda_{12} & \lambda_{13} & 0 & 0 & 0 \\
\lambda_{11} & \lambda_{12} + 2 \mu_2 & \lambda_{13} & 0 & 0 & 0 \\
\lambda_{11} & \lambda_{12} & \lambda_{13} + 2 \mu_3 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{\mu_1 + \mu_2}{2} & 0 & 0 \\
0 & 0 & 0 & 0 & \frac{\mu_2 + \mu_3}{2} & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{\mu_3 + \mu_1}{2}
\end{bmatrix}
Eq. :eq:`ortho-matrix-inv` is evaluated and inverted numerically to extract
the Lamé constants.
See Also
--------
felupe.LinearElasticOrthotropic : Orthotropic linear-elastic
material formulation.
felupe.constitution.tensortrax.models.hyperelastic.saint_venant_kirchhoff_orthotropic :
Strain energy function of the orthotropic hyperelastic Saint-Venant
Kirchhoff material formulation.
"""
# unpack orthotropic elastic material parameters
E1, E2, E3 = E
ν12, ν23, ν31 = nu
G12, G23, G31 = G
# orthotropic symmetry
ν21 = ν12 * E2 / E1
ν32 = ν23 * E3 / E2
ν13 = ν31 * E1 / E3
C = inv(
[
[1 / E1, -ν21 / E2, -ν31 / E3, 0, 0, 0],
[-ν12 / E1, 1 / E2, -ν32 / E3, 0, 0, 0],
[-ν13 / E1, -ν23 / E2, 1 / E3, 0, 0, 0],
[0, 0, 0, 1 / G12, 0, 0],
[0, 0, 0, 0, 1 / G23, 0],
[0, 0, 0, 0, 0, 1 / G31],
]
)
# take the components from this matrix
#
# [λ11 + 2 * μ1, λ12, λ13, 0, 0, 0]
# [λ12, λ22 + 2 * μ2, λ23, 0, 0, 0]
# [λ13, λ23, λ33 + 2 * μ3, 0, 0, 0]
# [0, 0, 0, (μ1 + μ2) / 2, 0, 0]
# [0, 0, 0, 0, (μ2 + μ3) / 2, 0]
# [0, 0, 0, 0, 0, (μ1 + μ3) / 2]
λ12 = C[0, 1]
λ23 = C[1, 2]
λ13 = C[0, 2]
μ1 = C[3, 3] - C[4, 4] + C[5, 5]
μ2 = C[3, 3] + C[4, 4] - C[5, 5]
μ3 = -C[3, 3] + C[4, 4] + C[5, 5]
λ11 = C[0, 0] - 2 * μ1
λ22 = C[1, 1] - 2 * μ2
λ33 = C[2, 2] - 2 * μ3
lmbda = [λ11, λ12, λ13, λ22, λ23, λ33]
mu = [μ1, μ2, μ3]
return lmbda, mu
