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 (list 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 (list 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

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

axisint or None, default None
Axis to compute standard deviation along with. If None, returns the overall standard deviation (std of flattened array)

SecondOrderTensor
Tensor of standard deviation

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

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)

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

classmethod zeros(shape=())[source]

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

Parameters:: shape (list 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