"""Computational solid mechanics standard computations.
This module includes several functions to compute common tensorial quantities
arising in computational solid mechanics.
Classes
-------
MaterialQuantitiesComputer:
Computation of quantities based on material state variables.
Functions
---------
compute_rotation_tensor
Rotation tensor from polar decomposition of deformation gradient.
compute_material_log_strain
Material logarithmic strain from deformation gradient.
compute_spatial_log_strain
Spatial logarithmic strain from deformation gradient.
cauchy_from_kirchhoff
Cauchy stress tensor from Kirchhoff stress tensor.
first_piola_from_kirchhoff
First Piola-Kirchhoff stress tensor from Kirchhoff stress tensor.
cauchy_from_first_piola
Cauchy stress tensor from first Piola-Kirchhoff stress tensor.
cauchy_from_second_piola
Cauchy stress tensor from second Piola-Kirchhoff stress tensor.
first_piola_from_cauchy
First Piola-Kirchhoff stress tensor from Cauchy stress tensor.
first_piola_from_second_piola
First Piola-Kirchhoff from second Piola-Kirchhoff stress tensor.
kirchhoff_from_first_piola
Kirchhoff stress tensor from first Piola-Kirchhoff stress tensor.
material_from_spatial_tangent_modulus
Material consistent tangent modulus from spatial counterpart.
conjugate_material_log_strain
Stress conjugate of material logarithmic strain tensor.
conjugate_spatial_log_strain
Stress conjugate of spatial logarithmic strain tensor.
"""
#
# Modules
# =============================================================================
# Third-party
import numpy as np
# Local
import tensor.matrixoperations as mop
import tensor.tensoroperations as top
#
# Authorship & Credits
# =============================================================================
__author__ = 'Bernardo Ferreira (bernardo_ferreira@brown.edu)'
__credits__ = ['Bernardo Ferreira', ]
__status__ = 'Stable'
# =============================================================================
#
# Rotation tensor
# =============================================================================
[docs]def compute_rotation_tensor(def_gradient):
"""Rotation tensor from polar decomposition of deformation gradient.
.. math::
\\boldsymbol{R} = \\boldsymbol{F}(\\boldsymbol{F}^{T}
\\boldsymbol{F})^{\\frac{1}{2}}
where :math:`\\boldsymbol{R}` is the rotation tensor and
:math:`\\boldsymbol{F}` is the deformation gradient.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
Returns
-------
r : numpy.ndarray (2d)
Rotation tensor.
"""
# Compute right stretch tensor
right_stretch = mop.matrix_root(np.matmul(
np.transpose(def_gradient), def_gradient), p=0.5)
# Compute rotation tensor
r = np.matmul(def_gradient, np.linalg.inv(right_stretch))
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return r
#
# Strain tensors conversions
# =============================================================================
def compute_material_log_strain(def_gradient):
"""Material logarithmic strain from deformation gradient.
.. math::
\\boldsymbol{E} = \\frac{1}{2} \\ln (\\boldsymbol{F}^{T}
\\boldsymbol{F})
where :math:`\\boldsymbol{E}` is the material logarithmic strain tensor and
:math:`\\boldsymbol{F}` is the deformation gradient.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
Returns
-------
material_log_strain : numpy.ndarray (2d)
Material logarithmic strain.
"""
# Compute material logarithmic strain tensor
material_log_strain = 0.5*top.isotropic_tensor(
'log', np.matmul(np.transpose(def_gradient), def_gradient))
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return material_log_strain
# =============================================================================
[docs]def compute_spatial_log_strain(def_gradient):
"""Spatial logarithmic strain from deformation gradient.
.. math::
\\boldsymbol{\\varepsilon} = \\frac{1}{2} \\ln (\\boldsymbol{F}
\\boldsymbol{F}^{T})
where :math:`\\boldsymbol{\\varepsilon}` is the spatial logarithmic strain
tensor and :math:`\\boldsymbol{F}` is the deformation gradient.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
Returns
-------
spatial_log_strain : numpy.ndarray (2d)
Spatial logarithmic strain.
"""
# Compute spatial logarithmic strain tensor
spatial_log_strain = 0.5*top.isotropic_tensor(
'log', np.matmul(def_gradient, np.transpose(def_gradient)))
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return spatial_log_strain
#
# Stress tensors conversions
# =============================================================================
[docs]def cauchy_from_kirchhoff(def_gradient, kirchhoff_stress):
"""Cauchy stress tensor from Kirchhoff stress tensor.
.. math::
\\boldsymbol{\\sigma} = \\frac{1}{\\det (\\boldsymbol{F})} \\,
\\boldsymbol{\\tau}
where :math:`\\boldsymbol{\\sigma}` is the Cauchy stress tensor,
:math:`\\boldsymbol{F}` is the deformation gradient, and
:math:`\\boldsymbol{\\tau}` is the Kirchhoff stress tensor.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
kirchhoff_stress : numpy.ndarray (2d)
Kirchhoff stress tensor.
Returns
-------
cauchy_stress : numpy.ndarray (2d)
Cauchy stress tensor.
"""
# Compute Cauchy stress tensor
cauchy_stress = (1.0/np.linalg.det(def_gradient))*kirchhoff_stress
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return cauchy_stress
# =============================================================================
[docs]def first_piola_from_kirchhoff(def_gradient, kirchhoff_stress):
"""First Piola-Kirchhoff stress tensor from Kirchhoff stress tensor.
.. math::
\\boldsymbol{P} = \\boldsymbol{\\tau} \\boldsymbol{F}^{-T}
where :math:`\\boldsymbol{P}` is the first Piola-Kirchhoff stress tensor,
:math:`\\boldsymbol{\\tau}` is the Kirchhoff stress tensor, and
:math:`\\boldsymbol{F}` is the deformation gradient.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
kirchhoff_stress : numpy.ndarray (2d)
Kirchhoff stress tensor.
Returns
-------
first_piola_stress : numpy.ndarray (2d)
First Piola-Kirchhoff stress tensor.
"""
# Compute First Piola-Kirchhoff stress tensor
first_piola_stress = np.matmul(kirchhoff_stress,
np.transpose(np.linalg.inv(def_gradient)))
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return first_piola_stress
# =============================================================================
[docs]def cauchy_from_first_piola(def_gradient, first_piola_stress):
"""Cauchy stress tensor from first Piola-Kirchhoff stress tensor.
.. math::
\\boldsymbol{\\sigma} = \\frac{1}{\\det (\\boldsymbol{F})} \\,
\\boldsymbol{P} \\boldsymbol{F}^{T}
where :math:`\\boldsymbol{\\sigma}` is the Cauchy stress tensor,
:math:`\\boldsymbol{F}` is the deformation gradient, and
:math:`\\boldsymbol{P}` is the first Piola-Kirchhoff stress tensor.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
first_piola_stress : numpy.ndarray (2d)
First Piola-Kirchhoff stress tensor.
Returns
-------
cauchy_stress : numpy.ndarray (2d)
Cauchy stress tensor.
"""
# Compute Cauchy stress tensor
cauchy_stress = (1.0/np.linalg.det(def_gradient))*np.matmul(
first_piola_stress, np.transpose(def_gradient))
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return cauchy_stress
# =============================================================================
[docs]def cauchy_from_second_piola(def_gradient, second_piola_stress):
"""Cauchy stress tensor from second Piola-Kirchhoff stress tensor.
.. math::
\\boldsymbol{\\sigma} = \\frac{1}{\\det (\\boldsymbol{F})} \\,
\\boldsymbol{F} \\boldsymbol{S}
\\boldsymbol{F}^{T}
where :math:`\\boldsymbol{\\sigma}` is the Cauchy stress tensor,
:math:`\\boldsymbol{F}` is the deformation gradient, and
:math:`\\boldsymbol{S}` is the second Piola-Kirchhoff stress tensor.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
second_piola_stress : numpy.ndarray (2d)
Second Piola-Kirchhoff stress tensor.
Returns
-------
cauchy_stress : numpy.ndarray (2d)
Cauchy stress tensor.
"""
# Compute Cauchy stress tensor
cauchy_stress = (1.0/np.linalg.det(def_gradient))*np.matmul(
def_gradient, np.matmul(second_piola_stress,
np.transpose(def_gradient)))
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return cauchy_stress
# =============================================================================
def first_piola_from_cauchy(def_gradient, cauchy_stress):
"""First Piola-Kirchhoff stress tensor from Cauchy stress tensor.
.. math::
\\boldsymbol{P} = \\det (\\boldsymbol{F}) \\,
\\boldsymbol{\\sigma} \\boldsymbol{F}^{-T}
where :math:`\\boldsymbol{P}` is the first Piola-Kirchhoff stress tensor,
:math:`\\boldsymbol{F}` is the deformation gradient, and
:math:`\\boldsymbol{\\sigma}` is the Cauchy stress tensor.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
cauchy_stress : numpy.ndarray (2d)
Cauchy stress tensor.
Returns
-------
first_piola_stress : numpy.ndarray (2d)
First Piola-Kirchhoff stress tensor.
"""
# Compute first Piola_Kirchhoff stress tensor
first_piola_stress = np.linalg.det(def_gradient)*np.matmul(
cauchy_stress, np.transpose(np.linalg.inv(def_gradient)))
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return first_piola_stress
# =============================================================================
[docs]def first_piola_from_second_piola(def_gradient, second_piola_stress):
"""First Piola-Kirchhoff from second Piola-Kirchhoff stress tensor.
.. math::
\\boldsymbol{P} = \\boldsymbol{F} \\boldsymbol{S}
where :math:`\\boldsymbol{P}` is the first Piola-Kirchhoff stress tensor
and :math:`\\boldsymbol{S}` is the second Piola-Kirchhoff stress tensor.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
second_piola_stress : numpy.ndarray (2d)
Second Piola-Kirchhoff stress tensor.
Returns
-------
first_piola_stress : numpy.ndarray (2d)
First Piola-Kirchhoff stress tensor.
"""
# Compute first Piola_Kirchhoff stress tensor
first_piola_stress = np.matmul(def_gradient, second_piola_stress)
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return first_piola_stress
# =============================================================================
def kirchhoff_from_first_piola(def_gradient, first_piola_stress):
"""Kirchhoff stress tensor from first Piola-Kirchhoff stress tensor.
.. math::
\\boldsymbol{\\tau} = \\boldsymbol{P} \\boldsymbol{F}^{T}
where :math:`\\boldsymbol{\\tau}` is the Kirchhoff stress tensor,
:math:`\\boldsymbol{P}` is the first Piola-Kirchhoff stress tensor,
and :math:`\\boldsymbol{F}` is the deformation gradient.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
first_piola_stress : numpy.ndarray (2d)
First Piola-Kirchhoff stress tensor.
Returns
-------
kirchhoff_stress : numpy.ndarray (2d)
Kirchhoff stress tensor.
"""
# Compute Kirchhoff stress tensor
kirchhoff_stress = np.matmul(first_piola_stress,
np.transpose(def_gradient))
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return kirchhoff_stress
# =============================================================================
[docs]def material_from_spatial_tangent_modulus(spatial_consistent_tangent,
def_gradient):
"""Material consistent tangent modulus from spatial counterpart.
.. math::
\\mathsf{A}_{ijkl} = \\det (\\boldsymbol{F}) \\, \\mathsf{a}_{ipkq} \\,
F_{lq}^{-1} F_{jp}^{-1}
where :math:`\\mathbf{\\mathsf{A}}` is the material consistent tangent
modulus, :math:`\\boldsymbol{F}` is the deformation gradient, and
:math:`\\mathbf{\\mathsf{a}}` is the spatial consistent tangent modulus.
----
Parameters
----------
spatial_consistent_tangent : numpy.ndarray (4d)
Spatial consistent tangent modulus.
def_gradient : numpy.ndarray (2d)
Deformation gradient.
Returns
-------
material_consistent_tangent : numpy.ndarray (4d)
Material consistent tangent modulus.
"""
# Compute inverse of deformation gradient
def_gradient_inv = np.linalg.inv(def_gradient)
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Compute material consistent tangent modulus
material_consistent_tangent = np.linalg.det(def_gradient)*top.dot42_2(
top.dot42_1(spatial_consistent_tangent, def_gradient_inv),
def_gradient_inv)
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return material_consistent_tangent
#
# Stress-strain conjugate pairs
# =============================================================================
def conjugate_material_log_strain(def_gradient, first_piola_stress):
"""Stress conjugate of material logarithmic strain tensor.
*Material logarithmic strain tensor*:
.. math::
\\boldsymbol{E} = \\frac{1}{2} \\ln (\\boldsymbol{F}^{T}
\\boldsymbol{F})
where :math:`\\boldsymbol{E}` is the material logarithmic strain tensor and
:math:`\\boldsymbol{F}` is the deformation gradient.
----
*Stress conjugate of material logarithmic strain tensor*:
.. math::
\\boldsymbol{T} = \\boldsymbol{R}^{T} \\boldsymbol{P}
\\boldsymbol{F}^{T} \\boldsymbol{R}
where :math:`\\boldsymbol{T}` is the stress conjugate of the material
logarithmic strain tensor, :math:`\\boldsymbol{R}` is the rotation tensor,
:math:`\\boldsymbol{P}` is the first Piola-Kirchhoff stress tensor,
and :math:`\\boldsymbol{F}` is the deformation gradient.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
first_piola_stress : numpy.ndarray (2d)
First Piola-Kirchhoff stress tensor.
Returns
-------
material_log_strain : numpy.ndarray (2d)
Material logarithmic strain.
stress_conjugate : numpy.ndarray (2d)
Stress conjugate to material logarithmic strain.
"""
# Compute material logarithmic strain tensor
material_log_strain = compute_material_log_strain(def_gradient)
# Compute rotation tensor (polar decomposition of deformation gradient)
rotation = compute_rotation_tensor(def_gradient)
# Compute stress conjugate to material logarithmic strain tensor
stress_conjugate = np.matmul(
np.transpose(rotation), np.matmul(first_piola_stress,
np.matmul(np.transpose(def_gradient),
rotation)))
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return material_log_strain, stress_conjugate
# =============================================================================
def conjugate_spatial_log_strain(def_gradient, first_piola_stress):
"""Stress conjugate of spatial logarithmic strain tensor.
*Spatial logarithmic strain tensor*:
.. math::
\\boldsymbol{\\varepsilon} = \\frac{1}{2} \\ln (\\boldsymbol{F}
\\boldsymbol{F}^{T})
where :math:`\\boldsymbol{\\varepsilon}` is the spatial logarithmic strain
tensor and :math:`\\boldsymbol{F}` is the deformation gradient.
----
*Stress conjugate of spatial logarithmic strain tensor*:
.. math::
\\boldsymbol{\\tau} = \\boldsymbol{P} \\boldsymbol{F}^{T}
where :math:`\\boldsymbol{\\tau}` is the Kirchhoff stress tensor,
:math:`\\boldsymbol{P}` is the first Piola-Kirchhoff stress tensor,
and :math:`\\boldsymbol{F}` is the deformation gradient.
----
Parameters
----------
def_gradient : numpy.ndarray (2d)
Deformation gradient.
first_piola_stress : numpy.ndarray (2d)
First Piola-Kirchhoff stress tensor.
Returns
-------
spatial_log_strain : numpy.ndarray (2d)
Spatial logarithmic strain.
stress_conjugate : numpy.ndarray (2d)
Stress conjugate to spatial logarithmic strain.
"""
# Compute spatial logarithmic strain tensor
spatial_log_strain = compute_spatial_log_strain(def_gradient)
# Compute stress conjugate to spatial logarithmic strain tensor
stress_conjugate = kirchhoff_from_first_piola(def_gradient,
first_piola_stress)
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Return
return spatial_log_strain, stress_conjugate
#
# Computation of material-related quantities
# =============================================================================
[docs]class MaterialQuantitiesComputer:
"""Computation of quantities based on material state variables.
Material-related quantities computations are always performed assuming the
three-dimensional strain and/or stress state.
Attributes
----------
_n_dim : int
Problem dimension.
_comp_order_sym : list[str]
Strain/Stress components symmetric order.
_comp_order_nsym : list[str]
Strain/Stress components nonsymmetric order.
_fodevprojsym_mf : numpy.ndarray (2d)
Fourth-order deviatoric projection tensor (second order symmetric
tensors) (matricial form).
Methods
-------
get_vm_stress(self, stress_mf)
Compute von Mises equivalent stress.
get_vm_strain(self, strain_mf)
Compute von Mises equivalent strain.
"""
# -------------------------------------------------------------------------
[docs] def __init__(self):
"""Constructor."""
# Get 3D problem parameters
n_dim, comp_order_sym, comp_order_nsym = \
mop.get_problem_type_parameters(problem_type=4)
# Set material-related quantities computer parameters
self._n_dim = n_dim
self._comp_order_sym = comp_order_sym
self._comp_order_nsym = comp_order_nsym
# Get fourth-order tensors
_, _, _, _, _, _, fodevprojsym = top.get_id_operators(self._n_dim)
# Get fourth-order tensors matricial form
fodevprojsym_mf = mop.get_tensor_mf(fodevprojsym, self._n_dim,
self._comp_order_sym)
self._fodevprojsym_mf = fodevprojsym_mf
# -------------------------------------------------------------------------
[docs] def get_vm_stress(self, stress_mf):
"""Compute von Mises equivalent stress.
.. math::
\\sigma_{\\text{VM}} = \\frac{3}{2}
|| \\boldsymbol{\\sigma_{d}} ||
where :math:`\\sigma_{\\text{VM}}` is the von Mises equivalent stress
and :math:`\\boldsymbol{\\sigma_{d}}` is the deviatoric Cauchy stress
tensor.
----
Parameters
----------
stress_mf : numpy.ndarray (1d)
Cauchy stress tensor (matricial form).
"""
# Compute deviatoric stress tensor (matricial form)
dev_stress_mf = np.matmul(self._fodevprojsym_mf, stress_mf)
# Compute von Mises equivalent stress
vm_stress = np.sqrt(3.0/2.0)*np.linalg.norm(dev_stress_mf)
# Return
return vm_stress
# -------------------------------------------------------------------------
[docs] def get_vm_strain(self, strain_mf):
"""Compute von Mises equivalent strain.
.. math::
\\varepsilon_{\\text{VM}} = \\frac{2}{3}
|| \\boldsymbol{\\varepsilon_{d}} ||
where :math:`\\varepsilon_{\\text{VM}}` is the von Mises equivalent
strain and :math:`\\boldsymbol{\\varepsilon_{d}}` is either the
deviatoric infinitesimal strain tensor (infinitesimal strains) or the
spatial logarithmic strain tensor (finite strains).
----
Parameters
----------
strain_mf : numpy.ndarray (1d)
Strain tensor (matricial form): infinitesimal strain tensor
(infinitesimal strains) or spatial logarithmic strain tensor
(finite strains).
"""
# Compute deviatoric strain tensor (matricial form)
dev_strain_mf = np.matmul(self._fodevprojsym_mf, strain_mf)
# Compute von Mises equivalent strain
vm_strain = np.sqrt(2.0/3.0)*np.linalg.norm(dev_strain_mf)
# Return
return vm_strain