scikit-elasticity¶

A library for working with anisotropic elasticity in Python.

Homogenization¶

Calculation of elastic constants from elastic stiffness and a list of orientations.

>>> import skel
>>> import damask
>>> import numpy as np

>>> N = 1000
>>> C_Al = skel.stiffness('cubic',C_11=106.9e9,C_12=60.5e9,C_44=28.4e9)
>>> O = damask.Rotation.from_random(N)
>>> p = skel.Polycrystal(C_Al,O)
>>> with np.printoptions(precision=2,suppress=True):
>>>    print(p.C_Reuss/1e9)
>>>    print(p.C_Voigt/1e9)
>>>    print(p.C_Hill_C/1e9)
>>>    print(p.C_Hill_S/1e9)

class skel.VolumeElement(C, O)[source]¶

property C_Hill¶

Homogenized stiffness according to Hill.

Average of C and S average from Voigt and Reuss.

property C_Hill_C¶

Homogenized stiffness according to Hill.

Average of C from Voigt and Reuss.

property C_Hill_S¶

Homogenized stiffness according to Hill.

Average of S from Voigt and Reuss.

property C_Hill_geometric¶: Homogenized stiffness according to Hill.

property C_Reuss¶: Homogenized compliance according to Reuss.

property C_Voigt¶: Homogenized stiffness according to Voigt.

property S_Hill¶

Homogenized compliance according to Hill.

Average of S and C average from Voigt and Reuss.

property S_Hill_C¶

Homogenized compliance according to Hill.

Average of C from Voigt and Reuss.

property S_Hill_S¶

Homogenized compliance according to Hill.

Average of S from Voigt and Reuss.

property S_Hill_geometric¶: Homogenized compliance according to Hill.

property S_Reuss¶: Homogenized stiffness according to Reuss.

property S_Voigt¶: Homogenized compliance according to Voigt.

Conversions and Representations¶

skel.Voigt.rotate_compliance(O, S_tilde)[source]¶

Rotate symmetric 4th order compliance tensor in Voigt notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
S_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order compliance tensor in Voigt notation.

Returns¶

S_tilde’numpy.ndarray of shape (…,6,6): Rotated 4th order compliance tensor in Voigt notation.

skel.Voigt.rotate_stiffness(O, C_tilde)[source]¶

Rotate symmetric 4th order stiffness tensor in Voigt notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
C_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order stiffness tensor in Voigt notation.

Returns¶

C_tilde’numpy.ndarray of shape (…,6,6): Rotated 4th order stiffness tensor in Voigt notation.

skel.Voigt.rotate_strain(O, epsilon_tilde)[source]¶

Rotate symmetric 2nd order strain tensor in Voigt notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
epsilon_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order strain tensor in Voigt notation.

Returns¶

epsilon_tilde’numpy.ndarray of shape (…,6): Rotated 2nd order strain tensor in Voigt notation.

skel.Voigt.rotate_stress(O, sigma_tilde)[source]¶

Rotate symmetric 2nd order stress tensor in Voigt notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
sigma_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order stress tensor in Voigt notation.

Returns¶

sigma_tilde’numpy.ndarray of shape (…,6): Rotated 2nd order stress tensor in Voigt notation.

skel.Voigt.to_3x3_strain(epsilon_tilde)[source]¶

Convert symmetric 2nd order strain tensor in Voigt notation to matrix notation.

Parameters¶

epsilon_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order strain tensor in Voigt notation.

Returns¶

epsilonnumpy.ndarray of shape (…,3,3): Symmetric 2nd order strain tensor in matrix notation.

skel.Voigt.to_3x3_stress(sigma_tilde)[source]¶

Convert symmetric 2nd order stress tensor in Voigt notation to matrix notation.

Parameters¶

sigma_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order stress tensor in Voigt notation.

Returns¶

sigmanumpy.ndarray of shape (…,3,3): Symmetric 2nd order stress tensor in matrix notation.

skel.Voigt.to_3x3x3x3_compliance(S_tilde)[source]¶

Convert symmetric 4th order compliance tensor in Voigt notation to matrix notation.

Parameters¶

S_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order tensor in Voigt notation.

Returns¶

Snumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order compliance tensor in matrix notation.

skel.Voigt.to_3x3x3x3_stiffness(C_tilde)[source]¶

Convert symmetric 4th order stiffness tensor in Voigt notation to matrix notation.

Parameters¶

C_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order stiffness tensor in Voigt notation.

Returns¶

Cnumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order stiffness tensor in matrix notation.

skel.Voigt.to_6_strain(epsilon)[source]¶

Convert symmetric 2nd order strain tensor in matrix notation to Voigt notation.

Parameters¶

epsilonnumpy.ndarray of shape (…,3,3): Symmetric 2nd order strain tensor in matrix notation.

Returns¶

epsilon_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order strain tensor in Voigt notation.

skel.Voigt.to_6_stress(sigma)[source]¶

Convert symmetric 2nd order stress tensor in matrix notation to Voigt notation.

Parameters¶

sigmanumpy.ndarray of shape (…,3,3): Symmetric 2nd order stress tensor in matrix notation.

Returns¶

sigma_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order stress tensor in Voigt notation.

skel.Voigt.to_6x6_compliance(S)[source]¶

Convert symmetric 4th order compliance tensor in matrix notation to Voigt notation.

Parameters¶

Snumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order compliance tensor in matrix notation.

Returns¶

S_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order compliance tensor in Voigt notation.

skel.Voigt.to_6x6_stiffness(C)[source]¶

Convert symmetric 4th order stiffness tensor in matrix notation to Voigt notation.

Parameters¶

Cnumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order stiffness tensor in matrix notation.

Returns¶

C_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order stiffness tensor in Voigt notation.

skel.Mandel.rotate_2nd(O, t_tilde)[source]¶

Rotate symmetric 2nd order tensor in Mandel notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
t_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order tensor in Mandel notation.

Returns¶

t_tilde’numpy.ndarray of shape (…,6): Rotated 2nd order tensor in Mandel notation.

Notes¶

Due to the definition of the Mandel notation, this function can be used to rotate strain and stress tensors.

skel.Mandel.rotate_4th(O, T_tilde)[source]¶

Rotate symmetric 4th order tensor in Mandel notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
T_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order tensor in Mandel notation.

Returns¶

T_tilde’numpy.ndarray of shape (…,6,6): Rotated 4th order tensor in Mandel notation.

Notes¶

Due to the definition of the Mandel notation, this function can be used to rotate compliance and stiffness tensors.

skel.Mandel.rotate_compliance(O, S_tilde)[source]¶

Rotate symmetric 4th order compliance tensor in Mandel notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
S_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order compliance tensor in Mandel notation.

Returns¶

S_tilde’numpy.ndarray of shape (…,6,6): Rotated 4th order compliance tensor in Mandel notation.

skel.Mandel.rotate_stiffness(O, C_tilde)[source]¶

Rotate symmetric 4th order stiffness tensor in Mandel notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
C_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order stiffness tensor in Mandel notation.

Returns¶

C_tilde’numpy.ndarray of shape (…,6,6): Rotated 4th order stiffness tensor in Mandel notation.

skel.Mandel.rotate_strain(O, epsilon_tilde)[source]¶

Rotate symmetric 2nd order strain tensor in Mandel notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
epsilon_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order strain tensor in Mandel notation.

Returns¶

epsilon_tilde’numpy.ndarray of shape (…,6): Rotated 2nd order strain tensor in Mandel notation.

skel.Mandel.rotate_stress(O, sigma_tilde)[source]¶

Rotate symmetric 2nd order stress tensor in Mandel notation.

Parameters¶

O: damask.Orientation of shape (…): Orientations.
sigma_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order stress tensor in Mandel notation.

Returns¶

sigma_tilde’numpy.ndarray of shape (…,6): Rotated 2nd order stress tensor in Mandel notation.

skel.Mandel.to_3x3(t_tilde)[source]¶

Convert symmetric 2nd order tensor in Mandel notation to matrix notation.

Parameters¶

t_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order tensor in Mandel notation.

Returns¶

tnumpy.ndarray of shape (…,3,3): Symmetric 2nd order tensor in matrix notation.

Notes¶

Due to the definition of the Mandel notation, this function can be used to convert strain and stress tensors.

skel.Mandel.to_3x3_strain(epsilon_tilde)[source]¶

Convert symmetric 2nd order strain tensor in Mandel notation to matrix notation.

Parameters¶

epsilon_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order strain tensor in Mandel notation.

Returns¶

epsilonnumpy.ndarray of shape (…,3,3): Symmetric 2nd order strain tensor in matrix notation.

skel.Mandel.to_3x3_stress(sigma_tilde)[source]¶

Convert symmetric 2nd order stress tensor in Mandel notation to matrix notation.

Parameters¶

sigma_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order stress tensor in Mandel notation.

Returns¶

sigmanumpy.ndarray of shape (…,3,3): Symmetric 2nd order stress tensor in matrix notation.

skel.Mandel.to_3x3x3x3(T_tilde)[source]¶

Convert symmetric 4th order tensor in Mandel notation to matrix notation.

Parameters¶

T_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order tensor in Mandel notation.

Returns¶

Tnumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order tensor in matrix notation.

Notes¶

Due to the definition of the Mandel notation, this function can be used to convert compliance and stiffness tensors.

skel.Mandel.to_3x3x3x3_compliance(S_tilde)[source]¶

Convert symmetric 4th order compliance tensor in Mandel notation to matrix notation.

Parameters¶

S_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order tensor in Mandel notation.

Returns¶

Snumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order compliance tensor in matrix notation.

skel.Mandel.to_3x3x3x3_stiffness(C_tilde)[source]¶

Convert symmetric 4th order stiffness tensor in Mandel notation to matrix notation.

Parameters¶

C_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order stiffness tensor in Mandel notation.

Returns¶

Cnumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order stiffness tensor in matrix notation.

skel.Mandel.to_6(t)[source]¶

Convert symmetric 2nd order tensor in matrix notation to Mandel notation.

Parameters¶

tnumpy.ndarray of shape (…,3,3): Symmetric 2nd order tensor in matrix notation.

Returns¶

t_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order tensor in Mandel notation.

Notes¶

Due to the definition of the Mandel notation, this function can be used to convert strain and stress tensors.

skel.Mandel.to_6_strain(epsilon)[source]¶

Convert symmetric 2nd order strain tensor in matrix notation to Mandel notation.

Parameters¶

epsilonnumpy.ndarray of shape (…,3,3): Symmetric 2nd order strain tensor in matrix notation.

Returns¶

epsilon_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order strain tensor in Mandel notation.

skel.Mandel.to_6_stress(sigma)[source]¶

Convert symmetric 2nd order stress tensor in matrix notation to Mandel notation.

Parameters¶

sigmanumpy.ndarray of shape (…,3,3): Symmetric 2nd order stress tensor in matrix notation.

Returns¶

sigma_tilde: numpy.ndarray of shape (…,6): Symmetric 2nd order stress tensor in Mandel notation.

skel.Mandel.to_6x6(T)[source]¶

Convert symmetric 4th order tensor in matrix notation to Mandel notation.

Parameters¶

Tnumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order tensor in matrix notation.

Returns¶

T_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order tensor in Mandel notation.

Notes¶

Due to the definition of the Mandel notation, this function can be used to convert compliance and stiffness tensors.

skel.Mandel.to_6x6_compliance(S)[source]¶

Convert symmetric 4th order compliance tensor in matrix notation to Mandel notation.

Parameters¶

Snumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order compliance tensor in matrix notation.

Returns¶

S_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order compliance tensor in Mandel notation.

skel.Mandel.to_6x6_stiffness(C)[source]¶

Convert symmetric 4th order stiffness tensor in matrix notation to Mandel notation.

Parameters¶

Cnumpy.ndarray of shape (…,3,3,3,3): Symmetric 4th order stiffness tensor in matrix notation.

Returns¶

C_tilde: numpy.ndarray of shape (…,6,6): Symmetric 4th order stiffness tensor in Mandel notation.