import numpy as np
[docs]
class GeneralizedInvariantsModel:
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, **kwargs):
"""Initialize the generalized invariant-based isotropic hyperelastic material
formulation."""
self.material = material
self.strain = fun
self.kwargs = kwargs
E, dEdλ, d2Edλdλ, E0, dEdλ0, d2Edλdλ0 = self.strain(1, **self.kwargs)
# normalize invariants
c1_upper = 3 / 2 * (d2Edλdλ0 + dEdλ0)
c1_lower = 3 / 2 * (d2Edλdλ + dEdλ)
c2_upper = 3 * (d2Edλdλ0 + dEdλ0) * E0 - 3 / 2 * dEdλ0**2
c2_lower = 3 * (d2Edλdλ + dEdλ) * E - 3 / 2 * dEdλ**2
self.c1 = c1_upper / c1_lower
self.c2 = c2_upper / c2_lower
self.I10 = 3 * E
self.I20 = 3 * E**2
self.I30 = E**3
[docs]
def gradient(self, stretches, statevars):
"""The gradient as the partial derivative of the strain energy function w.r.t.
the principal stretches."""
# principal strains
self.E, self.dEdλ, self.d2Edλdλ = self.strain(stretches, **self.kwargs)[:3]
E = self.E
# strain invariants
I1 = E[0] + E[1] + E[2]
I2 = E[0] * E[1] + E[1] * E[2] + E[2] * E[0]
I3 = E[0] * E[1] * E[2]
self.I1 = self.c1 * I1 - self.I10 * (self.c1 - 1)
self.I2 = self.c2 * I2 - self.I20 * (self.c2 - 1)
self.I3 = I3
self.dWdI1, self.dWdI2, self.dWdI3, statevars_new = self.material.gradient(
self.I1, self.I2, self.I3, statevars
)
Eβ = E[[1, 0, 0]]
Eγ = E[[2, 2, 1]]
self.dI1dE = self.c1 * np.ones_like(E)
self.dI2dE = self.c2 * (Eβ + Eγ)
self.dI3dE = Eβ * Eγ
dWdλ = np.zeros_like(stretches)
if self.dWdI1 is not None:
dWdλ += self.dWdI1 * self.dI1dE * self.dEdλ
if self.dWdI2 is not None:
dWdλ += self.dWdI2 * self.dI2dE * self.dEdλ
if self.dWdI3 is not None:
dWdλ += self.dWdI3 * self.dI3dE * self.dEdλ
return dWdλ, statevars_new
[docs]
def hessian(self, stretches, statevars):
"""The hessian as the second partial derivatives of the strain energy function
w.r.t. the principal stretches."""
dWdλα, statevars_new = self.gradient(stretches, statevars)
(
d2WdI1dI1,
d2WdI2dI2,
d2WdI3dI3,
d2WdI1dI2,
d2WdI2dI3,
d2WdI1dI3,
) = self.material.hessian(self.I1, self.I2, self.I3, statevars)
dI1dE = self.dI1dE
dI2dE = self.dI2dE
dI3dE = self.dI3dE
dEdλ = self.dEdλ
d2Edλdλ = self.d2Edλdλ
Eγ = [2, 0, 1]
d2WdEαdEβ = np.zeros((6, *dWdλα.shape[1:]))
α = [0, 1, 2, 0, 1, 0]
β = [0, 1, 2, 1, 2, 2]
if d2WdI1dI1 is not None:
d2WdEαdEβ += d2WdI1dI1 * dI1dE[α] * dI1dE[β]
if d2WdI2dI2 is not None:
d2WdEαdEβ += d2WdI2dI2 * dI2dE[α] * dI2dE[β]
if d2WdI3dI3 is not None:
d2WdEαdEβ += d2WdI3dI3 * dI3dE[α] * dI3dE[β]
if d2WdI1dI2 is not None:
d2WdEαdEβ += d2WdI1dI2 * (dI1dE[α] * dI2dE[β] + dI2dE[α] * dI1dE[β])
if d2WdI2dI3 is not None:
d2WdEαdEβ += d2WdI2dI3 * (dI2dE[α] * dI3dE[β] + dI3dE[α] * dI2dE[β])
if d2WdI1dI3 is not None:
d2WdEαdEβ += d2WdI1dI3 * (dI1dE[α] * dI3dE[β] + dI3dE[α] * dI1dE[β])
# if self.dWdI1 is not None:
# d2I1dEαdEβ = 0
# d2WdEαdEβ[3:] += self.dWdI2 * d2I1dEαdEβ
if self.dWdI2 is not None:
d2I2dEαdEβ = self.c2
d2WdEαdEβ[3:] += self.dWdI2 * d2I2dEαdEβ
if self.dWdI3 is not None:
d2I3dEαdEβ = Eγ
d2WdEαdEβ[3:] += self.dWdI3 * d2I3dEαdEβ
# in-place modification (avoid the creation of a new array)
d2Wdλαdλβ = d2WdEαdEβ
d2Wdλαdλβ *= dEdλ[α] * dEdλ[β]
if self.dWdI1 is not None:
d2Wdλαdλβ[:3] += self.dWdI1 * dI1dE * d2Edλdλ
if self.dWdI2 is not None:
d2Wdλαdλβ[:3] += self.dWdI2 * dI2dE * d2Edλdλ
if self.dWdI3 is not None:
d2Wdλαdλβ[:3] += self.dWdI3 * dI3dE * d2Edλdλ
return d2Wdλαdλβ
