Elasticipy.tensors.fourth_order module
- class Elasticipy.tensors.fourth_order.FourthOrderTensor(M, mapping='Kelvin', check_minor_symmetry=True, force_minor_symmetry=False)[source]
Bases:
objectTemplate class for manipulating symmetric fourth-order tensors.
- matrix[source]
(6,6) matrix gathering all the components of the tensor, using the Voigt notation.
- Type:
np.ndarray
Construct of Fourth-order tensor with minor symmetry.
- Parameters:
M (np.ndarray) – (6,6) matrix corresponding to the stiffness tensor, written using the Voigt notation, or array of shape (3,3,3,3).
mapping (str or list of list, or numpy/ndarray, optional) – Mapping convention to translate the (3,3,3,3) array to (6,6) matrix
check_minor_symmetry (bool, optional) – If true (default), check that the input array have minor symmetries (see Notes). Only used if an array of shape (…,3,3,3,3) is passed.
force_minor_symmetry – Ensure that the tensor displays minor symmetry.
Notes
The minor symmetry is defined so that:
\[M_{ijkl}=M_{jikl}=M_{jilk}=M_{ijlk}\]- ddot(other, mode='pair')[source]
Perform tensor product contracted twice (“:”) between two fourth-order tensors
- Parameters:
other (FourthOrderTensor or SecondOrderTensor) – Right-hand side of “:” symbol
mode (str, optional) – If mode==”pair”, the tensors must be broadcastable, and the tensor product are performed on the last axes. If mode==”cross”, all cross-combinations are considered.
- Returns:
If both the tensors are 0D (no orientation), the return value will be of type SymmetricTensor Otherwise, the return value will be the full tensor, of shape (…,3,3,3,3).
- Return type:
FourthOrderTensor or numpy.ndarray
- flatten()[source]
Flatten the tensor
If the tensor array is of shape (m,n,o…,r), the flattened array will be of shape (m*n*o*…*r,).
- Returns:
Flattened tensor
- Return type:
- 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 identity(shape=(), return_full_tensor=False, mapping='Kelvin')[source]
Create a 4th-order identity tensor
- Parameters:
shape (int or tuple, optional) – Shape of the tensor to create
return_full_tensor (bool, optional) – If True, return the full tensor as a (3,3,3,3) or a (…,3,3,3,3) array. Otherwise, the tensor is returned as a SymmetricTensor object.
- Returns:
Identity tensor
- Return type:
numpy.ndarray or SymmetricTensor
- inv()[source]
Invert the tensor. The inverted tensors inherits the properties (if any)
- Returns:
Inverse tensor
- Return type:
- mean(axis=None)[source]
Compute the mean value of the tensor T
- Parameters:
axis (int or list of int or tuple of int, optional) – axis along which to compute the mean. If None, the mean is computed on the flattened tensor
- Returns:
If no axis is given, the result will be of shape (3,3,3,3). Otherwise, if T.ndim=m, and len(axis)=n, the returned value will be of shape (…,3,3,3,3), with ndim=m-n+4
- Return type:
numpy.ndarray
- property ndim[source]
Returns the dimensionality of the tensor (number of dimensions in the orientation array)
- Returns:
Number of dimensions
- Return type:
int
- rotate(rotation)[source]
Apply a single rotation to a tensor, and return its component into the rotated frame.
- Parameters:
rotation (Rotation or orix.quaternion.rotation.Rotation) – Rotation to apply
- Returns:
Rotated tensor
- Return type:
- property shape[source]
Return the shape of the tensor array :returns: Shape of the tensor array :rtype: tuple
- transpose_array()[source]
Transpose the orientations of the tensor array
- Returns:
The same tensor, but with transposed axes
- Return type:
- class Elasticipy.tensors.fourth_order.SymmetricFourthOrderTensor(M, check_symmetries=True, force_symmetries=False, **kwargs)[source]
Bases:
FourthOrderTensorConstruct a fully symmetric fourth-order tensor from a (…,6,6) or a (3,3,3,3) array.
The input matrix must be symmetric, otherwise an error is thrown (except if check_symmetry==False, see below)
- Parameters:
M (np.ndarray) – (6,6) matrix corresponding to the stiffness tensor, written using the Voigt notation, or array of shape (3,3,3,3).
check_symmetries (bool, optional) – Whether to check or not that the tensor to built displays both major and minor symmetries (see Notes).
force_symmetries (bool, optional) – If true, ensure that the tensor displays both minor and major symmetries.
Notes
The major symmetry is defined so that:
\[M_{ijkl}=M_{klij}\]whereas the minor symmetry is:
\[M_{ijkl}=M_{jikl}=M_{jilk}=M_{ijlk}\]- invariants(order='all')[source]
Compute the invariants of the tensor.
Compute the linear or/and quadratic invariant of the fourth-order tensor (see notes)
- Parameters:
order (str, optional) – If ‘linear’, only A1 and A2 are returned If ‘quadratic’, A1², A2², B1, B2, B3, B4 and B5 are returned If ‘all’ (default), A1, A2, A1², A2², B1, B2, B3, B4 and B5 are returned
- Returns:
invariants of the given order (see above)
- Return type:
tuple
Notes
The nomenclature of the invariants follows that of [4]. The linear invariants are:
\[ \begin{align}\begin{aligned}A_1=C_{ijij}\\A_2=C_{iijj}\end{aligned}\end{align} \]whereas the quadratic invariants are:
\[ \begin{align}\begin{aligned}B_1 = C_{ijkl}C_{ijkl}\\B_2 = C_{iikl}C_{jjkl}\\B_3 = C_{iikl}C_{jkjl}\\B_4 = C_{kiil}C_{kjjl}\\B_5 = C_{ijkl}C_{ikjl}\end{aligned}\end{align} \]References
- Elasticipy.tensors.fourth_order.rotate_tensor(full_tensor, r)[source]
Rotate a (full) fourth-order tensor.
- Parameters:
full_tensor (numpy.ndarray) – array of shape (3,3,3,3) or (…,3,3,3,3) containing all the components
r (scipy.spatial.Rotation or orix.quaternion.Rotation) – Rotation, or set of rotations, to apply
- Returns:
Rotated tensor. If r is an array, the corresponding axes will be added as first axes in the result array.
- Return type:
numpy.ndarray