import numpy as np
from ..math import astensor, asvoigt, cdya, cdya_ik, ddot, det, dya, eye, inv, transpose
from ._space import Space
[docs]
class Distortional(Space):
r"""The distortional (part of the deformation) space is a partial deformation with
constant volume. For a given deformation map :math:`\boldsymbol{x}(\boldsymbol{X})`
and its deformation gradient :math:`\boldsymbol{F}`, the distortional part of the
deformation gradient :math:`\hat{\boldsymbol{F}}` is obtained by a multiplicative
(consecutive) split into a volume-changing (dilatational) and a constant-volume
(distortional) part of the deformation gradient. Due to the fact that the
dilatational part is proportional to the unit tensor, the order of these partial
deformations is not unique.
.. math::
\boldsymbol{F} = \overset{\circ}{\boldsymbol{F}} \hat{\boldsymbol{F}} =
\hat{\boldsymbol{F}} \overset{\circ}{\boldsymbol{F}}
This class takes a Total-Lagrange material formulation and applies it only on the
distortional space.
.. math::
\hat{\psi} = \psi(\hat{\boldsymbol{C}}(\boldsymbol{F}))
The distortional (unimodular) part of the right Cauchy-Green deformation tensor is
evaluated by the help of its third invariant (the determinant). The determinant of
a distortional (an unimodular) tensor equals to one.
.. math::
\hat{\boldsymbol{C}} = I_3^{-1/3} \boldsymbol{C}
The gradient of the strain energy function is carried out w.r.t. the Green Lagrange
strain tensor. Hence, the work-conjugate stress tensor used in this space
projection refers to the second Piola-Kirchhoff stress tensor.
.. math::
\boldsymbol{S}' = \frac{\partial \hat{\psi}}{\partial \frac{1}{2}\boldsymbol{C}}
The distortional space projection leads to a **physically** deviatoric second
Piola-Kirchhoff stress tensor, evaluated by the application of the chain rule.
.. math::
\hat{\boldsymbol{S}} = \frac{\partial \hat{\psi}}
{\partial \frac{1}{2}\hat{\boldsymbol{C}}}
The (**phyiscally**) deviatoric projection is obtained by the partial derivative of
the distortional part of the right Cauchy-Green deformation tensor w.r.t. the right
Cauchy-Green deformation tensor.
.. math::
\frac{\partial \hat{\boldsymbol{C}}}{\partial \boldsymbol{C}} =
\frac{\partial I_3^{-1/3} \boldsymbol{C}}{\partial \boldsymbol{C}} =
I_3^{-1/3} \left( \boldsymbol{1} \odot \boldsymbol{1}
- \frac{1}{3} \boldsymbol{C} \otimes \boldsymbol{C}^{-1} \right)
This partial derivative is used to perform the distortional space projection of the
second Piola-Kirchhoff stress tensor. Instead of asserting the determinant-scaling
to the fourth-order projection tensor, this factor is combined with the second
Piola-Kirchhoff stress tensor in the distortional space. Hence, the stress tensor in
the distortional space, scaled by :math:`I_3^{-1/3}`, is introduced as a new
(frequently re-used) variable, denoted by an overset bar.
.. math::
\boldsymbol{S}' &= \mathbb{P} : \bar{\boldsymbol{S}}
\bar{\boldsymbol{S}} &= I_3^{-1/3} \hat{\boldsymbol{S}}
\mathbb{P} &= \boldsymbol{1} \odot \boldsymbol{1}
- \frac{1}{3} \boldsymbol{C}^{-1} \otimes \boldsymbol{C}
The evaluation of the double-dot product for the distortional space projection leads
to the mathematical deviator of the product between the scaled distortional space
stress tensor and the right Cauchy-Green deformation tensor, right multiplied by the
inverse of the right Cauchy-Green deformation tensor.
.. math::
\boldsymbol{S}' = \bar{\boldsymbol{S}}
- \frac{\bar{\boldsymbol{S}}:\boldsymbol{C}}{3} \boldsymbol{C}^{-1}
= \text{dev}(\bar{\boldsymbol{S}} \boldsymbol{C}) \boldsymbol{C}^{-1}
The hessian of the strain energy function is carried out w.r.t. the Green-Lagrange
strain tensor. Hence, the work-conjugate elasticity tensor used in this space
projection refers to the fourth-order Total-Lagrangian elasticity tensor.
.. math::
\mathbb{C}' = \frac{\partial^2 \hat{\psi}}{\partial \frac{1}{2}\boldsymbol{C}~
\frac{1}{2}\boldsymbol{C}}
The evaluation of this second partial derivative leads to the elasticity tensor of
the distortional space projection. The remaining determinant scaling terms of the
projection tensor are included in the determinant-modified fourth-order elasticity
tensor, denoted with an overset bar.
.. math::
\mathbb{C}' = \mathbb{P} : \bar{\mathbb{C}} : \mathbb{P}^T + \frac{2}{3} \left(
\left( \bar{\boldsymbol{S}}:\boldsymbol{C} \right)
\boldsymbol{C}^{-1} \odot \boldsymbol{C}^{-1}
- \bar{\boldsymbol{S}} \otimes \boldsymbol{C}^{-1}
- \boldsymbol{C}^{-1} \otimes \bar{\boldsymbol{S}}
+ \frac{1}{3} \left( \bar{\boldsymbol{S}}:\boldsymbol{C} \right)
\boldsymbol{C}^{-1} \otimes \boldsymbol{C}^{-1}
\right)
.. math::
\bar{\mathbb{C}} = I_3^{-2/3} \hat{\mathbb{C}}
For the variation and linearization of the virtual work of internal forces, the
output quantities have to be transformed: The second Piola-Kirchhoff stress tensor
is converted into the deformation gradient work-conjugate first Piola-Kirchhoff
stress tensor, along with its fourth-order elasticity tensor. Also, the so-called
geometric tangent stiffness component (initial stress matrix) is added to the
fourth-order elasticity tensor.
.. math::
\delta W_{int} &=
- \int_V \boldsymbol{P} : \delta \boldsymbol{F} ~ dV
\Delta \delta W_{int} &=
- \int_V \delta \boldsymbol{F} : \mathbb{A} : \Delta \boldsymbol{F} ~ dV
where
.. math::
\boldsymbol{P} &= \boldsymbol{F} \boldsymbol{S}
\mathbb{A}_{iJkL} &= F_{iI} F_{kK} \mathbb{C}_{IJKL} + \delta_{ik} S_{JL}
"""
def __init__(self, material, parallel=False, finalize=True, force=None, area=0):
self.material = material
# initial variables for calling
# ``self.gradient(self.x)`` and ``self.hessian(self.x)``
self.x = [material.x[0], material.x[-1]]
super().__init__(parallel=parallel, finalize=finalize, force=force, area=area)
[docs]
def gradient(self, x):
"""The gradient as the partial derivative of the strain energy function w.r.t.
the deformation gradient."""
F, statevars = x[0], x[-1]
self.C = C = asvoigt(self.einsum("kI...,kJ...->IJ...", F, F))
self.I3 = I3 = det(C)
self.Cu = I3 ** (-1 / 3) * C
dWudCu, statevars_new = self.material.gradient(self.Cu, statevars)
self.Sb = Sb = 2 * I3 ** (-1 / 3) * dWudCu
self.SbC = SbC = ddot(Sb, C)
self.invC = invC = inv(C, determinant=I3)
self.S = Sb - SbC / 3 * invC
return [self.piola(F=F, S=self.S, detF=np.sqrt(self.I3)), statevars_new]
[docs]
def hessian(self, x):
"""The hessian as the second partial derivative of the strain energy function
w.r.t. the deformation gradient."""
F, statevars = x[0], x[-1]
dWudF, statevars_new = self.gradient(x)
I = eye(self.C)
d2WdCdC = self.material.hessian(self.Cu, statevars)
C4b = 4 * self.I3 ** (-2 / 3) * d2WdCdC
P4 = cdya(I, I) - dya(self.invC, self.C) / 3
I4 = cdya(self.invC, self.invC)
SbinvC = dya(self.Sb, self.invC)
invCSb = transpose(SbinvC)
invCinvC = dya(self.invC, self.invC)
C4 = 2 / 3 * (self.SbC * I4 - SbinvC - invCSb + self.SbC / 3 * invCinvC)
if not np.allclose(C4b, 0):
C4 += ddot(ddot(P4, C4b, mode=(4, 4)), transpose(P4), mode=(4, 4))
return [self.piola(F=F, S=self.S, detF=np.sqrt(self.I3), C4=C4, invC=self.invC)]
