Elasticipy.FourthOrderTensor module

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

Bases: StiffnessTensor

Class for manipulating compliance tensors

C11_C12_factor = 2.0

C46_C56_factor = 2.0

Compute the (Voigt-Reuss-)Hill 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-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

fromCrystalSymmetry: Define a stiffness tensor, taking advantage of crystal symmetry

classmethod orthotropic(**kwargs)[source]

Create a stiffness tensor corresponding to orthotropic symmetry, given the engineering constants.

Parameters:

Ex (float) – Young modulus along the x axis
Ey (float) – Young modulus along the y axis
Ez (float) – Young modulus along the z axis
nu_xy (float) – Poisson ratio between x and y axes
nu_xz (float) – Poisson ratio between x and z axes
nu_yz (float) – Poisson ratio between y and z axes
Gxy (float) – Shear modulus in the x-y plane
Gxz (float) – Shear modulus in the x-z plane
Gyz (float) – Shear modulus in the y-z plane
kwargs (dict, optional) – Keyword arguments to pass to the StiffnessTensor constructor

Return type:

StiffnessTensor

See also

transverse_isotropic: create a stiffness tensor for transverse-isotropic symmetry

tensor_name = 'Compliance'

classmethod transverse_isotropic(**kwargs)[source]

Create a stiffness tensor corresponding to the transverse isotropic symmetry, given the engineering constants.

Parameters:

Ex (float) – Young modulus along the x axis
Ez (float) – Young modulus along the y axis
nu_xy (float) – Poisson ratio between x and y axes
nu_xz (float) – Poisson ratio between x and z axes
Gxz (float) – Shear modulus in the x-z plane
kwargs (dict) – Keyword arguments to pass to the StiffnessTensor constructor

Return type:

StiffnessTensor

See also

orthotropic: create a stiffness tensor for orthotropic symmetry

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

Christoffel_tensor(u)[source]

Create the Christoffel tensor along a given direction, or set or directions.

Parameters:: u (list or np.ndarray) – 3D direction(s) to compute the Christoffel tensor along with
Returns:: Gamma – Array of Christoffel tensor(s). if u is a list of directions, Gamma[i] is the Christoffel tensor for direction u[i].
Return type:: np.ndarray

See also

wave_velocity: computes the p- and s-wave velocities.

Notes

For a given stiffness tensor C and a given unit vector u, the Christoffel tensor is defined as [2] :

\[M_{ij} = C_{iklj}.u_k.u_l\]

Hill_average()[source]

Compute the (Voigt-Reuss-)Hill 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-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

See also

transverse_isotropic: create a transverse-isotropic tensor

Examples

On can check that the shear modulus for steel is around 82 GPa:

>>> from Elasticipy.FourthOrderTensor import StiffnessTensor
>>> C=StiffnessTensor.isotropic(E=210e3, nu=0.28)
>>> C.shear_modulus
Hyperspherical function
Min=82031.24999999991, Max=82031.24999999997

classmethod orthotropic(*, Ex, Ey, Ez, nu_xy, nu_xz, nu_yz, Gxy, Gxz, Gyz, **kwargs)[source]

Create a stiffness tensor corresponding to orthotropic symmetry, given the engineering constants.

Parameters:

Ex (float) – Young modulus along the x axis
Ey (float) – Young modulus along the y axis
Ez (float) – Young modulus along the z axis
nu_xy (float) – Poisson ratio between x and y axes
nu_xz (float) – Poisson ratio between x and z axes
nu_yz (float) – Poisson ratio between y and z axes
Gxy (float) – Shear modulus in the x-y plane
Gxz (float) – Shear modulus in the x-z plane
Gyz (float) – Shear modulus in the y-z plane
kwargs (dict, optional) – Keyword arguments to pass to the StiffnessTensor constructor

Return type:

StiffnessTensor

See also

transverse_isotropic: create a stiffness tensor for transverse-isotropic symmetry

property shear_modulus

Directional shear modulus

Returns:: Shear modulus
Return type:: HyperSphericalFunction

tensor_name = 'Stiffness'

classmethod transverse_isotropic(*, Ex, Ez, nu_xy, nu_xz, Gxz, **kwargs)[source]

Create a stiffness tensor corresponding to the transverse isotropic symmetry, given the engineering constants.

Parameters:

Ex (float) – Young modulus along the x axis
Ez (float) – Young modulus along the y axis
nu_xy (float) – Poisson ratio between x and y axes
nu_xz (float) – Poisson ratio between x and z axes
Gxz (float) – Shear modulus in the x-z plane
kwargs (dict) – Keyword arguments to pass to the StiffnessTensor constructor

Return type:

StiffnessTensor

See also

orthotropic: create a stiffness tensor for orthotropic symmetry

wave_velocity(rho)[source]

Compute the wave velocities, given the mass density.

Parameters:: rho (float) – mass density. Its unit must be consistent with that of the stiffness tensor. See notes for hints.

See also

ChristoffelTensor: Computes the Christoffel tensor along a given direction

Returns:

c_p (SphericalFunction) – Velocity of the primary (compressive) wave
c_s1 (SphericalFunction) – Velocity of the fast secondary (shear) wave
c_s2 (SphericalFunction) – Velocity of the slow secondary (shear) wave

Notes

The estimation of the wave velocities is made by finding the eigenvalues of the Christoffel tensor [2].

One should double-check the units. The table below provides hints about the unit you get, depending on the units you use for stiffness and the mass density:

Stiffness	Mass density	Velocities	Notes
Pa (N/m²)	kg/m³	m/s	SI units
GPa (10⁹ Pa)	kg/dm³	km/s	Conversion factor
GPa (10³ N/mm²)	kg/mm³	m/s	Consistent units
MPa (10⁶ Pa)	kg/m³	km/s	Conversion factor
MPa (10³ N/mm²)	g/mm³	m/s	Consistent units

References

class Elasticipy.FourthOrderTensor.SymmetricTensor(M, phase_name=None, 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

C46_C56_factor = 1.0

component_prefix = 'C'

classmethod cubic(*, C11=0.0, C12=0.0, C44=0.0, phase_name=None)[source]

Create a fourth-order tensor from cubic symmetry.

Parameters:

C11 (float)
C12 (float)
C44 (float)
phase_name (str, optional) – Phase name to display

Return type:

StiffnessTensor

See also

hexagonal: create a tensor from hexagonal symmetry
orthorhombic: create a tensor from orthorhombic symmetry

classmethod fromCrystalSymmetry(symmetry='Triclinic', point_group=None, diad='y', phase_name=None, prefix=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
prefix (str, default None) – Define the prefix to use when providing the components. By default, it is ‘C’ for stiffness tensors, ‘S’ for compliance.
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. The behaviour can be overriten with the prefix option (see above)

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

classmethod hexagonal(*, C11=0.0, C12=0.0, C13=0.0, C33=0.0, C44=0.0, phase_name=None)[source]

Create a fourth-order tensor from hexagonal symmetry.

Parameters:

C11 (float)
C12 (float)
C13 (float)
C33 (float)
C44 (float)
phase_name (str, optional) – Phase name to display

Return type:

FourthOrderTensor

See also

transverse_isotropic: creates a transverse-isotropic tensor from engineering parameters
cubic: create a tensor from cubic symmetry
tetragonal: create a tensor from tetragonal symmetry

classmethod monoclinic(*, C11=0.0, C12=0.0, C13=0.0, C22=0.0, C23=0.0, C33=0.0, C44=0.0, C55=0.0, C66=0.0, C15=None, C25=None, C35=None, C46=None, C16=None, C26=None, C36=None, C45=None, phase_name=None)[source]

Create a fourth-order tensor from monoclinic symmetry. It automatically detects whether the components are given according to the Y or Z diad, depending on the input arguments.

For Diad || y, C15, C25, C35 and C46 must be provided. For Diad || z, C16, C26, C36 and C45 must be provided.

Parameters:

C11 (float)
C12 (float)
C13 (float)
C22 (float)
C23 (float)
C33 (float)
C44 (float)
C55 (float)
C66 (float)
C15 (float, optional)
C25 (float, optional)
C35 (float, optional)
C46 (float, optional)
C16 (float, optional)
C26 (float, optional)
C36 (float, optional)
C45 (float, optional)
phase_name (str, optional) – Name to display

Return type:

FourthOrderTensor

See also

triclinic: create a tensor from triclinic symmetry
orthorhombic: create a tensor from orthorhombic symmetry

classmethod orthorhombic(*, C11=0.0, C12=0.0, C13=0.0, C22=0.0, C23=0.0, C33=0.0, C44=0.0, C55=0.0, C66=0.0, phase_name=None)[source]

Create a fourth-order tensor from orthorhombic symmetry.

Parameters:

C11 (float)
C12 (float)
C13 (float)
C22 (float)
C23 (float)
C33 (float)
C44 (float)
C55 (float)
C66 (float)
phase_name (str, optional) – Phase name to display

Return type:

FourthOrderTensor

See also

monoclinic: create a tensor from monoclinic symmetry
orthorhombic: create a tensor from orthorhombic symmetry

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'

classmethod tetragonal(*, C11=0.0, C12=0.0, C13=0.0, C33=0.0, C44=0.0, C16=0.0, C66=0.0, phase_name=None)[source]

Create a fourth-order tensor from tetragonal symmetry.

Parameters:

C11 (float)
C12 (float)
C13 (float)
C33 (float)
C44 (float)
C66 (float)
C16 (float, optional) – 16 component in Voigt notation (for point groups 4, -4 and 4/m only)
phase_name (str, optional) – Phase name to display

Return type:

FourthOrderTensor

See also

trigonal: create a tensor from trigonal symmetry
orthorhombic: create a tensor from orthorhombic symmetry

classmethod triclinic(C11=0.0, C12=0.0, C13=0.0, C14=0.0, C15=0.0, C16=0.0, C22=0.0, C23=0.0, C24=0.0, C25=0.0, C26=0.0, C33=0.0, C34=0.0, C35=0.0, C36=0.0, C44=0.0, C45=0.0, C46=0.0, C55=0.0, C56=0.0, C66=0.0, phase_name=None)[source]

Parameters:

C11 (float)
C12 (float)
C13 (float)
C14 (float)
C15 (float)
C16 (float)
C22 (float)
C23 (float)
C24 (float)
C25 (float)
C26 (float)
C33 (float)
C34 (float)
C35 (float)
C36 (float)
C44 (float)
C45 (float)
C46 (float)
C55 (float)
C56 (float)
C66 (float)
phase_name (str, optional) – Name to display

Return type:

FourthOrderTensor

See also

monoclinic: create a tensor from monoclinic symmetry
orthorhombic: create a tensor from orthorhombic symmetry

classmethod trigonal(*, C11=0.0, C12=0.0, C13=0.0, C14=0.0, C33=0.0, C44=0.0, C15=0.0, phase_name=None)[source]

Create a fourth-order tensor from trigonal symmetry.

Parameters:

C11 (float)
C12 (float)
C13 (float)
C14 (float)
C33 (float)
C44 (float)
C15 (float, optional)
phase_name (str, optional) – Phase name to display

Return type:

FourthOrderTensor

See also

tetragonal: create a tensor from tetragonal symmetry
orthorhombic: create a tensor from orthorhombic symmetry

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