Source code for hyperelastic.frameworks._generalized
from ..models.stretches import GeneralizedInvariantsModel
from ._stretches import Stretches
[docs]
class GeneralizedInvariants(Stretches):
r"""Generalized-invariants isotropic hyperelastic material formulation based on the
principal stretches.
.. math::
\psi = \psi \left(
I_1\left( E_1, E_2, E_3 \right),
I_2\left( E_1, E_2, E_3 \right),
I_3\left( E_1, E_2, E_3 \right) \right)
The three principal invariants
.. math::
J_1 &= E_1 + E_2 + E_3
J_2 &= E_1 E_2 + E_2 E_3 + E_1 E_3
J_3 &= E_1 E_2 E_3
are formulated on a one-dimensional strain-stretch relation.
.. math::
E_\alpha &= f(\lambda_\alpha)
E'_\alpha &= f'(\lambda_\alpha) = \frac{\partial f(\lambda_\alpha)}
{\partial \lambda_\alpha}
E''_\alpha &= f''(\lambda_\alpha) = \frac{\partial^2 f(\lambda_\alpha)}
{\partial \lambda_\alpha~\partial \lambda_\alpha}
Depending on the strain-stretch relation, the invariants contain deformation-
independent values.
.. math::
J_{1,0} &= J_1(E_\alpha(\lambda_\alpha=1))
J_{2,0} &= J_2(E_\alpha(\lambda_\alpha=1))
J_{3,0} &= J_3(E_\alpha(\lambda_\alpha=1))
The deformation-dependent parts of the invariants are scaled by deformation-
independent coefficients of normalization. The deformation-independent parts are
re-added after the scaling.
.. math::
I_1 &= c_1 (J_1 - J_{1,0}) + J_{1,0}
I_2 &= c_2 (J_2 - J_{2,0}) + J_{2,0}
I_3 &= J_3
Note that the scaling is only applied to the first and second invariant, as the
third invariant does not contribute to the strain energy function at the undeformed
state.
.. math::
E_0 &= E(\lambda=1)
E'_0 &= E'(\lambda=1)
E''_0 &= E''(\lambda=1)
The second partial derivative of the strain w.r.t. the stretch must be
provided for a reference strain, e.g. the Green-Lagrange strain measure (at the
undeformed state).
.. math::
J''_{1,0} &= \frac{3}{2} \left( E''_0 + E'_0 \right)
J''_{2,0} &= \frac{3}{2} \left( (2 E_0 (E''_0 + E'_0)) - E'^2_0 \right)
.. math::
c_1 &= \frac{J''_{1,0,ref}}{J''_{1,0}}
c_2 &= \frac{J''_{2,0,ref}}{J''_{2,0}}
The first partial derivatives of the strain energy function w.r.t. the invariants
.. math::
\psi_{,1} &= \frac{\partial \psi}{\partial I_1}
\psi_{,2} &= \frac{\partial \psi}{\partial I_2}
\psi_{,3} &= \frac{\partial \psi}{\partial I_3}
and the partial derivatives of the invariants w.r.t. the principal stretches are
defined. From here on, this is consistent with any invariant-based hyperelastic
material formulation, except for the factors of normalization.
.. math::
\frac{\partial I_1}{\partial E_\alpha} &= c_1
\frac{\partial I_2}{\partial E_\alpha} &= c_2 \left( E_\beta + E_\gamma \right)
\frac{\partial I_3}{\partial E_\alpha} &= E_\beta E_\gamma
The first partial derivatives of the strain energy density w.r.t. the
principal stretches are required for the principal values of the stress.
.. math::
\frac{\partial \psi}{\partial \lambda_\alpha} =
\frac{\partial \psi}{\partial I_1} \frac{\partial I_1}{\partial E_\alpha}
\frac{\partial E_\alpha}{\partial \lambda_\alpha}
+ \frac{\partial \psi}{\partial I_2} \frac{\partial I_2}{\partial E_\alpha}
\frac{\partial E_\alpha}{\partial \lambda_\alpha}
+ \frac{\partial \psi}{\partial I_3} \frac{\partial I_3}{\partial E_\alpha}
\frac{\partial E_\alpha}{\partial \lambda_\alpha}
Furthermore, the second partial derivatives of the strain energy density w.r.t. the
principal stretches, necessary for the principal components of the elastic tangent
moduli, are carried out. This is done in two steps: first, the second partial
derivatives w.r.t. the principal strain components are carried out, followed by the
projection to the derivatives w.r.t. the principal stretches.
.. math::
\frac{\partial^2 \psi}{\partial E_\alpha~\partial E_\beta} &=
\frac{\partial^2 \psi}{\partial I_1~\partial I_1}
\frac{\partial I_1}{\partial E_\alpha}
\frac{\partial I_1}{\partial E_\beta}
+
\frac{\partial^2 \psi}{\partial I_2~\partial I_2}
\frac{\partial I_2}{\partial E_\alpha}
\frac{\partial I_2}{\partial E_\beta}
+
\frac{\partial^2 \psi}{\partial I_3~\partial I_3}
\frac{\partial I_3}{\partial E_\alpha}
\frac{\partial I_3}{\partial E_\beta}
&+
\frac{\partial^2 \psi}{\partial I_1~\partial I_2}
\left(
\frac{\partial I_1}{\partial E_\alpha}
\frac{\partial I_2}{\partial E_\beta}
+
\frac{\partial I_2}{\partial E_\alpha}
\frac{\partial I_1}{\partial E_\beta}
\right)
&+
\frac{\partial^2 \psi}{\partial I_2~\partial I_3}
\left(
\frac{\partial I_2}{\partial E_\alpha}
\frac{\partial I_3}{\partial E_\beta}
+
\frac{\partial I_3}{\partial E_\alpha}
\frac{\partial I_2}{\partial E_\beta}
\right)
&+
\frac{\partial^2 \psi}{\partial I_1~\partial I_3}
\left(
\frac{\partial I_1}{\partial E_\alpha}
\frac{\partial I_3}{\partial E_\beta}
+
\frac{\partial I_3}{\partial E_\alpha}
\frac{\partial I_1}{\partial E_\beta}
\right)
&+
\frac{\partial \psi}{\partial I_1}
\frac{\partial^2 I_1}{\partial E_\alpha~\partial E_\beta}
+
\frac{\partial \psi}{\partial I_2}
\frac{\partial^2 I_1}{\partial E_\alpha~\partial E_\beta}
+
\frac{\partial \psi}{\partial I_3}
\frac{\partial^2 I_1}{\partial E_\alpha~\partial E_\beta}
.. math::
\frac{\partial^2 \psi}{\partial \lambda_\alpha~\partial \lambda_\beta} =
\frac{\partial E_\alpha}{\partial \lambda_\alpha}
\frac{\partial^2 \psi}{\partial E_\alpha~\partial E_\beta}
\frac{\partial E_\beta} {\partial \lambda_\beta}
+
\left(
\frac{\partial \psi}{\partial I_1} \frac{\partial I_1}{\partial E_\alpha}
+
\frac{\partial \psi}{\partial I_2} \frac{\partial I_2}{\partial E_\alpha}
+
\frac{\partial \psi}{\partial I_3} \frac{\partial I_3}{\partial E_\alpha}
\right)
\frac{\partial^2 E_\alpha}{\partial \lambda_\alpha \partial \lambda_\alpha}
"""
def __init__(self, material, fun, nstatevars=0, parallel=False, **kwargs):
"""Initialize the generalized invariant-based isotropic hyperelastic material
formulation."""
model = GeneralizedInvariantsModel(material, fun, **kwargs)
super().__init__(material=model, nstatevars=nstatevars, parallel=parallel)
