tensors module¶

class apsg.tensors.DefGrad¶

Bases: numpy.ndarray

DefGrad store deformation gradient tensor derived from numpy.ndarray.

Parameters:	a (3x3 array_like) – Input data, that can be converted to 3x3 2D array. This includes lists, tuples and ndarrays.
Returns:	`DefGrad` object

Example

>>> F = DefGrad(np.diag([2, 1, 0.5]))

classmethod from_axis(vector, theta)¶

Return DefGrad representing rotation around axis.

Parameters:	vector – Rotation axis as `Vec3` like object theta – Angle of rotation in degrees

Example

>>> F = DefGrad.from_axis(Lin(120, 30), 45)

classmethod from_comp(xx=1, xy=0, xz=0, yx=0, yy=1, yz=0, zx=0, zy=0, zz=1)¶

Return DefGrad tensor defined by individual components. Default is identity tensor.

Keyword Arguments:
	xy, xz, yx, yy, yz, zx, zy, zz (xx,) – tensor components

Example

>>> F = DefGrad.from_comp(xy=1, zy=-0.5)
>>> F
DefGrad:
[[ 1.   1.   0. ]
 [ 0.   1.   0. ]
 [ 0.  -0.5  1. ]]

classmethod from_pair(p)¶

classmethod from_ratios(Rxy=1, Ryz=1)¶

Return isochoric DefGrad tensor with axial stretches defined by strain ratios. Default is identity tensor.

Keyword Arguments:
	Ryz (Rxy,) – strain ratios

Example

>>> F = DefGrad.from_ratios(Rxy=2, Ryz=3)
>>> F
DefGrad:
[[2.28942849 0.         0.        ]
 [0.         1.14471424 0.        ]
 [0.         0.         0.38157141]]

rotate(vector, theta=0)¶

Rotate tensor around axis by angle theta.

Using rotation matrix it returns F = R * F * R . T.

velgrad(time=1)¶: Return VelGrad for given time

E1¶: Max eigenvalue

E2¶: Middle eigenvalue

E3¶: Min eigenvalue

I¶: Returns the inverse tensor.

R¶: Return rotation part of DefGrad from polar decomposition.

U¶: Return stretching part of DefGrad from right polar decomposition.

V¶: Return stretching part of DefGrad from left polar decomposition.

axisangle¶: Return rotation part of DefGrad axis, angle tuple.

eigenfols¶: Return `Group` of principal eigenvectors as Fol objects.

eigenlins¶: Return `Group` of principal eigenvectors as Lin objects.

eigenvals¶: Return tuple of sorted eigenvalues.

eigenvects¶: Return `Group` of principal eigenvectors as Vec3 objects.

class apsg.tensors.VelGrad¶

Bases: numpy.ndarray

VelGrad store velocity gradient tensor derived from numpy.ndarray.

Parameters:	a (3x3 array_like) – Input data, that can be converted to 3x3 2D array. This includes lists, tuples and ndarrays.
Returns:	`VelGrad` object

Example

>>> L = VelGrad(np.diag([0.1, 0, -0.1]))

defgrad(time=1)¶

Return DefGrad tensor accumulated after given time.

When time is iterable, return list of DefGrad tensors for each time

classmethod from_comp(xx=0, xy=0, xz=0, yx=0, yy=0, yz=0, zx=0, zy=0, zz=0)¶

Return VelGrad tensor. Default is zero tensor.

Keyword Arguments:
	xy, xz, yx, yy, yz, zx, zy, zz (xx,) – tensor components

Example

>>> L = VelGrad.from_comp(xx=0.1, zz=-0.1)
>>> L
VelGrad:
[[ 0.1  0.   0. ]
 [ 0.   0.   0. ]
 [ 0.   0.  -0.1]]

rotate(vector, theta=0)¶

Rotate tensor around axis by angle theta.

Using rotation matrix it returns F = R * F * R . T.

rate¶: Return rate of deformation tensor

spin¶: Return spin tensor

class apsg.tensors.Stress¶

Bases: numpy.ndarray

Stress store stress tensor derived from numpy.ndarray.

Parameters:	a (3x3 array_like) – Input data, that can be converted to 3x3 2D array. This includes lists, tuples and ndarrays.
Returns:	`Stress` object

Example

>>> S = Stress([[-8, 0, 0],[0, -5, 0],[0, 0, -1]])

cauchy(n)¶

Return stress vector associated with plane given by normal vector.

Parameters:	n – normal given as `Vec3` or `Fol` object

Example

>>> S = Stress.from_comp(xx=-5, yy=-2, zz=10, xy=1)
>>> S.cauchy(Fol(160, 30))
V(-2.520, 0.812, 8.660)

fault(n)¶

Return Fault object derived from given by normal vector.

Parameters:	n – normal given as `Vec3` or `Fol` object

Example

>>> S = Stress.from_comp(xx=-5, yy=-2, zz=10, xy=8)
>>> S.fault(Fol(160, 30))
F:160/30-141/29 +

classmethod from_comp(xx=0, xy=0, xz=0, yy=0, yz=0, zz=0)¶

Return Stress tensor. Default is zero tensor.

Note that stress tensor must be symmetrical.

Keyword Arguments:
	xy, xz, yy, yz, zz (xx,) – tensor components

Example

>>> S = Stress.from_comp(xx=-5, yy=-2, zz=10, xy=1)
>>> S
Stress:
[[-5  1  0]
 [ 1 -2  0]
 [ 0  0 10]]

normal_stress(n)¶: Return normal stress component on plane given by normal vector.

rotate(vector, theta=0)¶

Rotate tensor around axis by angle theta.

Using rotation matrix it returns S = R * S * R . T

shear_stress(n)¶: Return magnitude of shear stress component on plane given by normal vector.

stress_comp(n)¶: Return normal and shear stress Vec3 components on plane given by normal vector.

E1¶: Max eigenvalue

E2¶: Middle eigenvalue

E3¶: Min eigenvalue

I1¶: First invariant

I2¶: Second invariant

I3¶: Third invariant

deviatoric¶: A stress deviator tensor component

diagonalized¶: Returns Stress tensor in a coordinate system with axes oriented to the principal directions and orthogonal matrix R, which brings actual coordinate system to principal one.

eigenfols¶: Returns Group of three eigenvectors represented as Fol

eigenlins¶: Returns Group of three eigenvectors represented as Lin

eigenvects¶: Returns Group of three eigenvectors represented as Vec3

hydrostatic¶: Mean hydrostatic stress tensor component

mean_stress¶: Mean stress

class apsg.tensors.Ortensor(matrix, **kwargs)¶

Bases: apsg.tensors.Tensor

Represents an orientation tensor, which characterize data distribution using eigenvalue method. See (Watson 1966, Scheidegger 1965).

See following methods and properties for additional operations.

Keyword Arguments:
Parameters:	matrix (3x3 array_like) – Input data, that can be converted to 3x3 2D matrix. This includes lists, tuples and ndarrays. Array could be also `Group` (for backward compatibility)
	name (str) – name od tensor scaled (bool) – When True eigenvectors are scaled by eigenvalues. Otherwise unit length. Default False
Returns:	`Ortensor` object

Example

>>> ot = Ortensor([[8, 0, 0], [0, 2, 0], [0, 0, 1]])
>>> ot
Ortensor:  Kind: LLS
(E1:8,E2:2,E3:1)
[[8 0 0]
 [0 2 0]
 [0 0 1]]

classmethod from_comp(xx=0, xy=0, xz=0, yy=0, yz=0, zz=0, **kwargs)¶

Return Ortensor tensor. Default is identity tensor.

Note that Ortensor tensor must be symmetrical.

Example

>>> ot = Ortensor.from_comp(xx=5, yy=2, zz=10, xy=1)
>>> ot
Ortensor:  Kind: SSL
(E1:10,E2:5.303,E3:1.697)
[[ 5  1  0]
 [ 1  2  0]
 [ 0  0 10]]

classmethod from_group(g, **kwargs)¶

Return Ortensor of data in Group

Parameters:	g – `Group` of `Vec3`, `Lin` or `Fol`

Example

>>> g = Group.examples('B2')
>>> ot = Ortensor.from_group(g)
>>> ot
Ortensor: B2 Kind: L
(E1:0.9825,E2:0.01039,E3:0.007101)
[[ 0.19780807 -0.13566589 -0.35878837]
 [-0.13566589  0.10492993  0.25970594]
 [-0.35878837  0.25970594  0.697262  ]]
>>> ot.eigenlins.data
[L:144/57, L:360/28, L:261/16]

B¶: Cylindricity index - Vollmer, 1990

E1¶: Max eigenvalue

E2¶: Middle eigenvalue

E3¶: Min eigenvalue

G¶: Girdle index - Vollmer, 1990

Intensity¶: Intensity index - Lisle, 1985

MAD¶: Return approximate deviation according to shape

MADo¶: Return approximate deviation from the plane normal to E3

MADp¶: Return approximate angular deviation from the major axis along E1

P¶: Point index - Vollmer, 1990

R¶: Random index - Vollmer, 1990

class apsg.tensors.Ellipsoid(matrix, **kwargs)¶

Bases: apsg.tensors.Tensor

Ellipsoid class

See following methods and properties for additional operations.

Parameters:	matrix (3x3 array_like) – Input data, that can be converted to 3x3 2D matrix. This includes lists, tuples and ndarrays.
Returns:	`Ellipsoid` object

Example

>>> E = Ellipsoid([[8, 0, 0], [0, 2, 0], [0, 0, 1]])
>>> E
Ellipsoid:  Kind: LLS
(E1:2.828,E2:1.414,E3:1)
[[8 0 0]
 [0 2 0]
 [0 0 1]]

classmethod from_axes(x=1, y=1, z=1, **kwargs)¶

Return Ellipsoid tensor defined by principal axes.

Example

>>> E = Ellipsoid.from_axes(x=4, y=0.5, z=0.5)
>>> E
Ellipsoid:  Kind: L
(E1:4,E2:0.5,E3:0.5)
[[16.    0.    0.  ]
 [ 0.    0.25  0.  ]
 [ 0.    0.    0.25]]

classmethod from_defgrad(F, **kwargs)¶

Return Finger (Left Cauchy-Green) deformation tensor resulting from deformation F

Parameters:	F – `DefGrad` or any 3x3 array_like object

Example

>>> F = DefGrad.from_comp(xx=2, xz=1, zz=0.5)
>>> E = Ellipsoid.from_defgrad(F)
>>> E
Ortensor: D Kind: LS
(E1:0.9825,E2:0.01039,E3:0.007101)
[[ 0.19780807 -0.13566589 -0.35878837]
 [-0.13566589  0.10492993  0.25970594]
 [-0.35878837  0.25970594  0.697262  ]]

transform(F)¶: Return Ellipsoid representing result of deformation F

E1¶: Max eigenvalue

E2¶: Middle eigenvalue

E3¶: Min eigenvalue