Elasticipy.SecondOrderTensor module

class Elasticipy.SecondOrderTensor.SecondOrderTensor(matrix)[source]

Bases: object

Template class for manipulation of second order tensors or arrays of second order tensors

matrix

(…,3,3) matrix storing all the components of the tensor

Type:: np.ndarray

property C

Return tensor components

For instance T.C[i,j] returns all the (i,j)-th components of each tensor in the array.

Returns:: Tensor components
Return type:: np.ndarray

property I1

First invariant of the tensor (trace)

Returns:: First invariant(s) of the tensor(s)
Return type:: np.ndarray or float

See also

I2: Second invariant of the tensors
I3: Third invariant of the tensors (det)

property I2

Second invariant of the tensor

For a matrix M, it is defined as:

I_2 = 0.5 * ( np.trace(M)**2 + np.trace(np.matmul(M, M.T)) )

Returns:: Second invariant(s) of the tensor(s)
Return type:: np.array or float

See also

I1: First invariant of the tensors (trace)
I3: Third invariant of the tensors (det)

property I3

Third invariant of the tensor (determinant)

Returns:: Third invariant(s) of the tensor(s)
Return type:: np.array or float

See also

I1: First invariant of the tensors (trace)
I2: Second invariant of the tensors

property T

Transpose the array of tensors.

It is actually an alias for transposeArray()

Returns:: Transposed array
Return type:: SecondOrderTensor

ddot(other)[source]

Double dot product (contraction of tensor product, usually denoted “:”) of two tensors.

For two tensors whose matrices are M1 and M2:

M1.ddot(M2) == np.trace(np.matmul(M1, M2))

Parameters:: other (SecondOrderTensor or np.ndarray) – Tensor or tensor array to multiply by before contraction.
Returns:: Result of double dot product
Return type:: float or np.ndarray

See also

matmul: matrix-like product between two tensor arrays.

deviatoricPart()[source]

Deviatoric part of the tensor

Return type:: SecondOrderTensor

See also

sphericalPart: spherical part of the tensor

eig()[source]

Eigenvalues of the tensor

Returns:

lambda (np.ndarray) – Eigenvalues of each tensor.
v (np.ndarray) – Eigenvectors of teach tensor.

See also

principalDirections: return only the principal directions (without eigenvalues)

classmethod eye(shape=())[source]

Create an array of tensors populated with identity matrices

Parameters:: shape (tuple or int, default ()) – If not provided, it just creates a single identity tensor. Otherwise, the tensor array will be of the specified shape.
Returns:: Array of identity tensors
Return type:: SecondOrderTensor

See also

ones: creates an array of tensors full of ones
zeros: creates an array full of zero tensors

flatten()[source]

Flatten the array of tensors.

If T is of shape [s1, s2, …, sn], the shape for T.flatten() is [s1*s2*…*sn].

Returns:: Flattened array (vector) of tensor
Return type:: SecondOrderTensor

See also

ndim: number of dimensions of the tensor array
shape: shape of the tensor array

matmul(other)[source]

Perform matrix-like product between tensor arrays. Each “product” is a matrix product between the tensor components.

If A.shape=(a1, a2, …, an) and B.shape=(b1, b2, …, bn), with C=A.matmul(B), we have:

C.shape = (a1, a2, ..., an, b1, b2, ..., bn)

and:

C[i,j,k,...,p,q,r...] = np.matmul(A[i,j,k,...], B[p,q,r,...])

Parameters:: other (SecondOrderTensor or np.ndarray or Rotation) – Tensor array or rotation to right-multiply by. If Rotation is provided, the rotations are applied on each tensor.
Returns:: Tensor array
Return type:: SecondOrderTensor

See also

__mul__: Element-wise matrix product

max(axis=None)[source]

Maximum value

Parameters:: axis (int or None, default None) – Axis to compute maximum along with. If None, returns the overall maximum (max of flattened array)
Returns:: Maximum value of tensors
Return type:: SecondOrderTensor

See also

min: Minimum value
mean: Mean value
std: Standard deviation

mean(axis=None)[source]

Arithmetic mean value

Parameters:: axis (int or None, default None) – Axis to compute the mean along with. If None, returns the overall mean (mean of flattened array)
Returns:: Mean tensor
Return type:: SecondOrderTensor

See also

std: Standard deviation
min: Minimum value
max: Maximum value

min(axis=None)[source]

Minimum value

Parameters:: axis (int or None, default None) – Axis to compute minimum along with. If None, returns the overall minimum (min of flattened array)
Returns:: Minimum value of tensors
Return type:: SecondOrderTensor

See also

max: Maximum value
mean: Mean value
std: Standard deviation

name = 'Second-order tensor': Name to use when printing the tensor

property ndim

Return the number of dimensions of the tensor array

Returns:: number of dimensions
Return type:: float

See also

shape: shape of tensor array

classmethod ones(shape=())[source]

Create an array of tensors populated with matrices of full of ones.

Parameters:: shape (tuple or int, default ()) – If not provided, it just creates a single tensor of ones. Otherwise, the tensor array will be of the specified shape.
Returns:: Array of ones tensors
Return type:: SecondOrderTensor

See also

eye: creates an array of identity tensors
zeros: creates an array full of zero tensors

principalDirections()[source]

Principal directions of the tensors

Returns:: Principal directions of each tensor of the tensor array
Return type:: np.nparray

See also

eig: Return both eigenvalues and corresponding principal directions

property shape

Return the shape of the tensor array

Returns:: Shape of array
Return type:: tuple

See also

ndim: number of dimensions

classmethod shear(u, v, magnitude)[source]

Create an array of tensors corresponding to shear state along two orthogonal directions.

Parameters:

u (np.ndarray or list) – First direction. Must be a 3D vector.
v (np.ndarray or list) – Second direction. Must be a 3D vector.
magnitude (float or np.ndarray or list) – Magnitude of the shear state to consider. If a list or an array is provided, the shape of the tensor array will be of the same shape as magnitude.

Returns:

tensor or tensor array

Return type:

SecondOrderTensor

skewPart()[source]

Skew-symmetric part of the tensor

Returns:: Skew-symmetric tensor
Return type:: SecondOrderTensor

sphericalPart()[source]

Spherical (hydrostatic) part of the tensor

Returns:: Spherical part
Return type:: SecondOrderTensor

See also

I1: compute the first invariant of the tensor
deviatoricPart: deviatoric the part of the tensor

std(axis=None)[source]

Standard deviation

Parameters:: axis (int or None, default None) – Axis to compute standard deviation along with. If None, returns the overall standard deviation (std of flattened array)
Returns:: Tensor of standard deviation
Return type:: SecondOrderTensor

See also

mean: Mean value
min: Minimum value
max: Maximum value

symmetricPart()[source]

Symmetric part of the tensor

Returns:: Symmetric tensor
Return type:: SecondOrderTensor

See also

skewPart: Skew-symmetric part of the tensor

classmethod tensile(u, magnitude)[source]

Create an array of tensors corresponding to tensile state along a given direction.

Parameters:

u (np.ndarray or list) – Tensile direction. Must be a 3D vector.
magnitude (float or np.ndarray or list) – Magnitude of the tensile state to consider. If a list or an array is provided, the shape of the tensor array will be of the same shape as magnitude.

Returns:

tensor or tensor array

Return type:

SecondOrderTensor

trace()[source]

Return the traces of the tensor array

Returns:: traces of each tensor of the tensor array
Return type:: np.ndarray or float

See also

I1: First invariant of the tensors (trace)
I2: Second invariant of the tensors
I3: Third invariant of the tensors (det)

transposeArray()[source]

Transpose the array of tensors

If A is a tensor array of shape [s1, s2, …, sn], A.T is of shape [sn, …, s2, s1].

Returns:: Transposed array
Return type:: SecondOrderTensor

See also

T: transpose the tensor array (not the components)

transposeTensor()[source]

Transpose of tensors of the tensor array

Returns:: Array of transposed tensors of the tensor array
Return type:: SecondOrderTensor

See also

Transpose: transpose the array (not the components)

classmethod zeros(shape=())[source]

Create an array of tensors populated with matrices full of zeros.

Parameters:: shape (tuple or int, default ()) – If not provided, it just creates a single tensor of ones. Otherwise, the tensor array will be of the specified shape.
Returns:: Array of ones tensors
Return type:: SecondOrderTensor

See also

eye: creates an array of identity tensors
ones: creates an array of tensors full of ones

class Elasticipy.SecondOrderTensor.SymmetricSecondOrderTensor(mat)[source]

Bases: SecondOrderTensor

classmethod from_Voigt(array)[source]

Construct a SymmetricSecondOrderTensor from a Voigt vector, or slices of Voigt vectors.

If the array is of shape (6,), a single tensor is returned. If the array is of shape (m,n,o,…,6), the tensor will be of shape (m,n,o,…).

Parameters:: array (np.ndarray or list) – array to build the SymmetricSecondOrderTensor from. We must have array.ndim>0 and array.shape[-1]==6.
Return type:: SymmetricSecondOrderTensor

Examples

>>> SymmetricSecondOrderTensor.from_Voigt([11, 22, 33, 23, 13, 12])
Second-order tensor
[[11. 12. 13.]
 [12. 22. 23.]
 [13. 23. 33.]]

classmethod shear(u, v, magnitude)[source]

Create an array of tensors corresponding to shear state along two orthogonal directions.

Parameters:

u (np.ndarray or list) – First direction. Must be a 3D vector.
v (np.ndarray or list) – Second direction. Must be a 3D vector.
magnitude (float or np.ndarray or list) – Magnitude of the shear state to consider. If a list or an array is provided, the shape of the tensor array will be of the same shape as magnitude.

Returns:

tensor or tensor array

Return type:

SecondOrderTensor

voigt_map = [1, 1, 1, 1, 1, 1]: List of factors to use for building a tensor from Voigt vector(s)