Elasticipy.FourthOrderTensor module

class Elasticipy.FourthOrderTensor.ComplianceTensor(C, **kwargs)[source]

Bases: StiffnessTensor

Class for manipulating compliance tensors

C11_C12_factor = 2.0

Compute the (Voigt-Reuss-)Hill average ofthe stiffness tensor. If the tensor contains no orientation, we assume isotropic behaviour. Otherwise, the mean is computed over all orientations.

Parameters:: orientations (np.ndarray or None) – Set of m orientation matrices, defined as a [m, 3, 3] array. If None, uniform distribution is assumed, resulting in isotropic tensor
Returns:: Voigt-Reuss-Hill average of tensor
Return type:: StiffnessTensor

See also

Voigt_average: compute the Voigt average
Reuss_average: compute the Reuss average

Reuss_average()[source]

Compute the Reuss average of the stiffness tensor. If the tensor contains no orientation, we assume isotropic behaviour. Otherwise, the mean is computed over all orientations.

Returns:: Reuss average of stiffness tensor
Return type:: StiffnessTensor

See also

Voigt_average: compute the Voigt average
Hill_average: compute the Voigt-Reuss-Hill average

Voigt_average()[source]

Compute the Voigt average of the stiffness tensor. If the tensor contains no orientation, we assume isotropic behaviour. Otherwise, the mean is computed over all orientations.

Returns:: Voigt average of stiffness tensor
Return type:: StiffnessTensor

See also

Reuss_average: compute the Reuss average
Hill_average: compute the Voigt-Reuss-Hill average

component_prefix = 'S'

inv()[source]

Compute the reciprocal stiffness tensor

Returns:: Reciprocal tensor
Return type:: StiffnessTensor

classmethod isotropic(**kwargs)[source]

Create an isotropic compliance tensor for either E, nu, lame1 or lame2. Exactly two of these parameters must be provided.

Parameters:: kwargs (keyword arguments) – E, nu, lame1 or lame2, passed to the StiffnessTensor.isotropic constructor

See also

StressTensor.isotropic

tensor_name = 'Compliance'

voigt_map = array([[1., 1., 1., 2., 2., 2.], [1., 1., 1., 2., 2., 2.], [1., 1., 1., 2., 2., 2.], [2., 2., 2., 4., 4., 4.], [2., 2., 2., 4., 4., 4.], [2., 2., 2., 4., 4., 4.]])

class Elasticipy.FourthOrderTensor.StiffnessTensor(S, **kwargs)[source]

Bases: SymmetricTensor

Class for manipulating fourth-order stiffness tensors.

C11_C12_factor = 0.5

Hill_average()[source]

Compute the (Voigt-Reuss-)Hill average ofthe stiffness tensor. If the tensor contains no orientation, we assume isotropic behaviour. Otherwise, the mean is computed over all orientations.

Parameters:: orientations (np.ndarray or None) – Set of m orientation matrices, defined as a [m, 3, 3] array. If None, uniform distribution is assumed, resulting in isotropic tensor
Returns:: Voigt-Reuss-Hill average of tensor
Return type:: StiffnessTensor

See also

Voigt_average: compute the Voigt average
Reuss_average: compute the Reuss average

property Poisson_ratio

Directional Poisson’s ratio

Returns:: Poisson’s ratio
Return type:: HyperSphericalFunction

Reuss_average()[source]

Compute the Reuss average of the stiffness tensor. If the tensor contains no orientation, we assume isotropic behaviour. Otherwise, the mean is computed over all orientations.

Returns:: Reuss average of stiffness tensor
Return type:: StiffnessTensor

See also

Voigt_average: compute the Voigt average
Hill_average: compute the Voigt-Reuss-Hill average

Voigt_average()[source]

Compute the Voigt average of the stiffness tensor. If the tensor contains no orientation, we assume isotropic behaviour. Otherwise, the mean is computed over all orientations.

Returns:: Voigt average of stiffness tensor
Return type:: StiffnessTensor

See also

Reuss_average: compute the Reuss average
Hill_average: compute the Voigt-Reuss-Hill average

property Young_modulus

Directional Young’s modulus

Returns:: Young’s modulus
Return type:: SphericalFunction

inv()[source]

Compute the reciprocal compliance tensor

Returns:: Reciprocal tensor
Return type:: ComplianceTensor

classmethod isotropic(E=None, nu=None, lame1=None, lame2=None, phase_name=None)[source]

Create an isotropic stiffness tensor from two elasticity coefficients, namely: E, nu, lame1, or lame2. Exactly two of these coefficients must be provided.

Parameters:

E (float, None) – Young modulus
nu (float, None) – Poisson ratio
lame1 (float, None) – First Lamé coefficient
lame2 (float, None) – Second Lamé coefficient
phase_name (str, None) – Name to print

Return type:

Corresponding isotropic stiffness tensor

Examples

On can check that the shear modulus for steel is around 80 GPa: >>> C=StiffnessTensor.isotropic(E=210e3, nu=0.28)

… C.shear_modulus Hyperspherical function Min=82031.24999999991, Max=82031.24999999997

property shear_modulus

Directional shear modulus

Returns:: Shear modulus
Return type:: HyperSphericalFunction

tensor_name = 'Stiffness'

class Elasticipy.FourthOrderTensor.SymmetricTensor(M, phase_name='', symmetry='Triclinic', orientations=None)[source]

Bases: object

Template class for manipulating symmetric fourth-order tensors.

matrix

(6,6) matrix gathering all the components of the tensor, using the Voigt notation.

Type:: np.ndarray

symmetry

Symmetry of the tensor

Type:: str

C11_C12_factor = 0.5

component_prefix = 'C'

classmethod fromCrystalSymmetry(symmetry='Triclinic', point_group=None, diad='x', phase_name=None, **kwargs)[source]

Create a fourth-order tensor from limited number of components, taking advantage of crystallographic symmetries

Parameters:

symmetry (str, default Triclinic) – Name of the crystallographic symmetry
point_group (str) – Point group of the considered crystal. Only used (and mandatory) for tetragonal and trigonal symmetries.
diad (str {'x', 'y'}, default 'x') – Alignment convention. Sets whether x||a or y||b. Only used for monoclinic symmetry.
phase_name (str, default None) – Name to use when printing the tensor
kwargs – Keywords describing all the necessary components, depending on the crystal’s symmetry and the type of tensor. For Stiffness, they should be named as ‘Cij’ (e.g. C11=…, C12=…). For Comliance, they should be named as ‘Sij’ (e.g. S11=…, S12=…). See examples below.

Return type:

FourthOrderTensor

See also

StiffnessTensor.isotropic: creates an isotropic stiffness tensor from two paremeters (e.g. E and v).

Notes

The relationships between the tensor’s components depend on the crystallogrpahic symmetry [1].

References

Examples

>>> from Elasticipy.FourthOrderTensor import StiffnessTensor

>>> StiffnessTensor.fromCrystalSymmetry(symmetry='monoclinic', diad='y', phase_name='TiNi',
...                                     C11=231, C12=127, C13=104,
...                                     C22=240, C23=131, C33=175,
...                                     C44=81, C55=11, C66=85,
...                                     C15=-18, C25=1, C35=-3, C46=3)
Stiffness tensor (in Voigt notation) for TiNi:
[[231. 127. 104.   0. -18.   0.]
 [127. 240. 131.   0.   1.   0.]
 [104. 131. 175.   0.  -3.   0.]
 [  0.   0.   0.  81.   0.   3.]
 [-18.   1.  -3.   0.  11.   0.]
 [  0.   0.   0.   3.   0.  85.]]
Symmetry: monoclinic

>>> from Elasticipy.FourthOrderTensor import ComplianceTensor

>>> ComplianceTensor.fromCrystalSymmetry(symmetry='monoclinic', diad='y', phase_name='TiNi',
...                                      S11=8, S12=-3, S13=-2,
...                                      S22=8, S23=-5, S33=10,
...                                      S44=12, S55=116, S66=12,
...                                      S15=14, S25=-8, S35=0, S46=0)
Compliance tensor (in Voigt notation) for TiNi:
[[  8.  -3.  -2.   0.  14.   0.]
 [ -3.   8.  -5.   0.  -8.   0.]
 [ -2.  -5.  10.   0.   0.   0.]
 [  0.   0.   0.  12.   0.   0.]
 [ 14.  -8.   0.   0. 116.   0.]
 [  0.   0.   0.   0.   0.  12.]]
Symmetry: monoclinic

full_tensor()[source]

Returns the full (unvoigted) tensor, as a [3, 3, 3, 3] array

Returns:: Full tensor (4-index notation)
Return type:: np.ndarray

rotate(rotation)[source]

Apply a single rotation to a tensor, and return its component into the rotated frame.

Parameters:: rotation (Rotation) – Rotation to apply
Returns:: Rotated tensor
Return type:: SymmetricTensor

tensor_name = 'Symmetric'

voigt_map = array([[1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1.], [1., 1., 1., 1., 1., 1.]])

Elasticipy.FourthOrderTensor.unvoigt_index(i)[source]

Translate the one-index notation to two-index notation

Parameters:: i (int or np.ndarray) – Index to translate

Elasticipy.FourthOrderTensor.voigt_indices(i, j)[source]

Translate the two-index notation to one-index notation

Parameters:

i (int or np.ndarray) – First index
j (int or np.ndarray) – Second index

Return type:

Index in the vector of length 6