Continuum Mechanics package

Vector calculus module

vector.antisym_grad(A, coords=(x, y, z), h_vec=(1, 1, 1))

Antisymmetric part of the gradient of a vector function A.

A : Matrix (3, 1), list

Vector function to compute the gradient from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

antisym_grad: Matrix (3, 3)

Matrix with the components of the antisymmetric part of the gradient. The position (i, j) has as components diff(A[i], coords[j].

vector.biharmonic(u, coords=(x, y, z), h_vec=(1, 1, 1))

Bilaplacian of the scalar function u.

u : SymPy expression

Scalar function to compute the bilaplacian from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameters, and it takes a cartesian (x, y, z), as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

bilaplacian: Sympy expression

Bilaplacian of u.

vector.curl(A, coords=(x, y, z), h_vec=(1, 1, 1))

Curl of a function vector A.

A : Matrix, List

Vector function to compute the curl from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

curl : Matrix (3, 1)

Column vector with the curl of A.

vector.div(A, coords=(x, y, z), h_vec=(1, 1, 1))

Divergence of the vector function A.

A : Matrix, list

Vector function to compute the divergence from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

divergence: SymPy expression

Divergence of A.

vector.div_tensor(tensor, coords=(x, y, z), h_vec=(1, 1, 1))

Divergence of a (second order) tensor

tensor : Matrix (3, 3)

Tensor function function to compute the divergence from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

divergence: Matrix

Divergence of tensor.

[RICHARDS]

Rowland Richards. Principles of Solids Mechanics. CRC Press, 2011.

vector.dual_tensor(vec)

Compute the dual tensor for an axial vector

In index notation, the dual is defined by

\[C_{ij} = \epsilon_{ijk} C_k\]

where \(\epsilon_{ijk}\) is the Levi-Civita symbol.

vec : Matrix (3)

Axial vector.

dual: Matrix (3, 3)

Second order matrix that is dual of vec.

[ARFKEN]

George Arfken, Hans J. Weber and Frank Harris. Mathematical methods for physicists, Elsevier, 2013.

vector.dual_vector(tensor)

Compute the dual (axial) vector for an anti-symmetric tensor

In index notation, the dual is defined by

\[C_{i} = \frac{1}{2}\epsilon_{ijk} C_{jk}\]

where \(\epsilon_{ijk}\) is the Levi-Civita symbol.

tensor : Matrix (3, 3)

Second order tensor.

dual: Matrix (3)

Axial vector that is the dual of tensor.

[ARFKEN]

George Arfken, Hans J. Weber and Frank Harris. Mathematical methods for physicists, Elsevier, 2013.

vector.grad(u, coords=(x, y, z), h_vec=(1, 1, 1))

Compute the gradient of a scalara function phi.

u : SymPy expression

Scalar function to compute the gradient from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameters, and it takes a cartesian (x, y, z), as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

gradient: Matrix (3, 1)

Column vector with the components of the gradient.

vector.grad_vec(A, coords=(x, y, z), h_vec=(1, 1, 1))

Gradient of a vector function A.

A : Matrix (3, 1), list

Vector function to compute the gradient from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

gradient: Matrix (3, 3)

Matrix with the components of the gradient. The position (i, j) has as components diff(A[i], coords[j].

vector.lap(u, coords=(x, y, z), h_vec=(1, 1, 1))

Laplacian of the scalar function u.

u : SymPy expression

Scalar function to compute the laplacian from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameters, and it takes a cartesian (x, y, z), as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

laplacian: Sympy expression

Laplacian of u.

vector.lap_vec(A, coords=(x, y, z), h_vec=(1, 1, 1))

Laplacian of a vector function A.

A : Matrix, List

Vector function to compute the laplacian from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

laplacian : Matrix (3, 1)

Column vector with the components of the Laplacian.

vector.levi_civita(i, j, k)

Levi-Civita symbol

vector.scale_coeff(r_vec, coords)

Compute scale coefficients for the vector tranform given by r_vec.

r_vec : Matrix (3, 1)

Transform vector (x, y, z) as a function of coordinates u1, u2, u3.

coords : Tuple (3)

Coordinates for the new reference system.

h_vec : Tuple (3)

Scale coefficients.

vector.scale_coeff_coords(coord_sys, coords, a=1, b=1, c=1)

Return scale factors for predefined coordinate system.

coord_sys : string

Coordinate system.

coords : Tuple (3)

Coordinates for the new reference system.

a : SymPy expression, optional

Additional parameter for some coordinate systems.

b : SymPy expression, optional

Additional parameter for some coordinate systems.

c : SymPy expression, optional

Additional parameter for some coordinate systems.

h_vec : Tuple (3)

Scale coefficients.

[ORTHO]

Wikipedia contributors, ‘Orthogonal coordinates’, Wikipedia, The Free Encyclopedia, 2019

vector.sym_grad(A, coords=(x, y, z), h_vec=(1, 1, 1))

Symmetric part of the gradient of a vector function A.

A : Matrix (3, 1), list

Vector function to compute the gradient from.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

sym_grad: Matrix (3, 3)

Matrix with the components of the symmetric part of the gradient. The position (i, j) has as components diff(A[i], coords[j].

vector.transform_coords(coord_sys, coords, a=1, b=1, c=1)

Return transformation for predefined coordinate systems.

coord_sys : string

Coordinate system.

coords : Tuple (3)

Coordinates for the new reference system.

a : SymPy expression, optional

Additional parameter for some coordinate systems.

b : SymPy expression, optional

Additional parameter for some coordinate systems.

c : SymPy expression, optional

Additional parameter for some coordinate systems.

h_vec : Tuple (3)

Scale coefficients.

[ORTHO]

Wikipedia contributors, ‘Orthogonal coordinates’, Wikipedia, The Free Encyclopedia, 2019

vector.unit_vec_deriv(vec_i, coord_j, coords=(x, y, z), h_vec=(1, 1, 1))

Compute the derivatives of unit vectors with respect to coordinates

The derivative is defined as

\[\begin{split}\frac{\partial{\hat{\mathbf{e}}_i}}{\partial u_j} = \begin{cases} \hat{\mathbf{e}_j} \frac{1}{h_i} \frac{\partial{h_j}}{\partial u_i} &\text{if } i\neq j\\ -\sum_{\substack{k=1\\ k\neq i}}^3 \hat{\mathbf{e}_k} \frac{1}{h_k} \frac{\partial{h_i}}{\partial u_k} &\text{if } i = j\\ \end{cases}\, ,\end{split}\]

as presented in [ARFKEN].

vec_i : int

Number of the unit vector (0, 1, 2).

coord_j : int

Number of the coordinate (0, 1, 2).

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

deriv: Matrix (3, 1)

Derivative of the i-th unit vector with respect to the j-th coordinate.

[ARFKEN]

George Arfken, Hans J. Weber and Frank Harris. Mathematical methods for physicists, Elsevier, 2013.

Solid mechanics module

solids.c_cst(u, parameters, coords=(x, y, z), h_vec=(1, 1, 1))

Corrected-Couple-Stress-Theory (C-CST) elasticity operator of a vector function u, as presented in [CST].

u : Matrix (3, 1), list

Vector function to apply the Navier-Cauchy operator.

parameters : tuple

Material parameters in the following order:

lamda : float

Lamé’s first parameter.

mu : float, > 0

Lamé’s second parameter.

eta : float, > 0

Couple stress modulus in C-CST.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

c_cst : Matrix (3, 1)

Components of the C-CST operator applied to the displacement vector

[CST]

(1, 2, 3) Ali R. Hadhesfandiari, Gary F. Dargush. Couple stress theory for solids. International Journal for Solids and Structures, 2011, 48, 2496-2510.

solids.disp_def_cst(u, coords=(x, y, z), h_vec=(1, 1, 1))

Compute strain measures for C-CST elasticity, as defined in [CST].

u : Matrix (3, 1), list

Displacement vector function to apply the micropolar operator.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

strain : Matrix (3, 3)

Strain tensor.

curvature : Matrix (3, 3)

Curvature tensor.

solids.disp_def_micropolar(u, phi, coords=(x, y, z), h_vec=(1, 1, 1))

Compute strain measures for micropolar elasticity, as defined in [NOW].

u : Matrix (3, 1), list

Displacement vector function to apply the micropolar operator.

phi : Matrix (3, 1), list

Microrrotation (pseudo)-vector function to apply the micropolar operator.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

strain : Matrix (3, 3)

Strain tensor.

curvature : Matrix (3, 3)

Curvature tensor.

solids.micropolar(u, phi, parameters, coords=(x, y, z), h_vec=(1, 1, 1))

Micropolar operator of a vector function u, as defined in [NOW].

u : Matrix (3, 1), list

Displacement vector function to apply the micropolar operator.

phi : Matrix (3, 1), list

Microrrotation (pseudo)-vector function to apply the micropolar operator.

parameters : tuple

Material parameters in the following order:

lamda : float

Lamé’s first parameter.

mu : float, > 0

Lamé’s second parameter.

alpha : float, > 0

Micropolar parameter.

beta : float

Micropolar parameter.

gamma : float, > 0

Micropolar parameter.

epsilon : float, > 0

Micropolar parameter.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

disp_op : Matrix (3, 1)

Displacement components.

rot_op : Matrix (3, 1)

Microrrotational components.

[NOW]

(1, 2) Witold Nowacki. Theory of micropolar elasticity. International centre for mechanical sciences, Courses and lectures, No. 25. Berlin: Springer, 1972.

solids.navier_cauchy(u, parameters, coords=(x, y, z), h_vec=(1, 1, 1))

Navier-Cauchy operator of a vector function u.

u : Matrix (3, 1), list

Vector function to apply the Navier-Cauchy operator.

parameters : tuple

Material parameters in the following order:

lamda : float

Lamé’s first parameter.

mu : float, > 0

Lamé’s second parameter.

coords : Tuple (3), optional

Coordinates for the new reference system. This is an optional parameter it takes (x, y, z) as default.

h_vec : Tuple (3), optional

Scale coefficients for the new coordinate system. It takes (1, 1, 1), as default.

navier_op : Matrix (3, 1)

Components of the Navier-Cauchy operator applied to the displacement vector.

solids.strain_stress(strain, parameters)

Return the stress tensor for a given strain tensor and material properties lambda and mu.

u : Matrix (3, 3)

Strain tensor.

parameters : tuple

Material parameters in the following order:

lamda : float

Lamé’s first parameter.

mu : float, > 0

Lamé’s second parameter.

disp_op : Matrix (3, 1)

Displacement components.

solids.strain_stress_cst(strain, curvature, parameters)

Return the force-stress and couple-stress tensor for given strain and curvature tensors and material parameters for the Corrected-Couple-Stress-Theory (C-CST), as presented in [CST].

strain : Matrix (3, 3)

Strain tensor.

curvature : Matrix (3, 3)

Curvature tensor.

parameters : tuple

Material parameters in the following order:

lamda : float

Lamé’s first parameter.

mu : float, > 0

Lamé’s second parameter.

eta : float, > 0

Couple stress modulus in C-CST.

solids.strain_stress_micropolar(strain, curvature, parameters)

Return the force-stress and couple-stress tensor for given strain and curvature tensors and material parameters.

strain : Matrix (3, 3)

Strain tensor.

curvature : Matrix (3, 3)

Curvature tensor.

parameters : tuple

Material parameters in the following order:

lamda : float

Lamé’s first parameter.

mu : float, > 0

Lamé’s second parameter.

alpha : float, > 0

Micropolar parameter.

beta : float

Micropolar parameter.

gamma : float, > 0

Micropolar parameter.

epsilon : float, > 0

Micropolar parameter.

Visualization module

Functions to aid the visualization of mathematical entities such as second rank tensors.

visualization.christofel_eig(C, nphi=30, nth=30)

Compute surfaces of eigenvalues for the Christoffel stiffness

For every direction

\[\mathbf{n} =(\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi),\cos(\theta))\]

computes the eigenvalues of the Christoffel stiffness tensor. These eigencalues are \(\rho v_p^2\), where \(\rho\) is the mass density and \(v_p\) is the phase speed.

C : (6,6) array

Stiffness tensor in Voigt notation.

nphi : (optional) int

Number of partitions in the azimut angle (phi).

nth : (optional) int

Number of partitions in the cenit angle (theta).

V1, V2, V3 : (nphi, nth) arrays

Eigenvalues for the desired discretization.

phi_vec : (nphi) array

Array with azimut angles.

theta_vec : (nth) array

Array with cenit angles.

visualization.mohr2d(stress, ax=None)

Plot Mohr circle for a 2D tensor

stress : ndarray

Stress tensor.

ax : Matplotlib axes, optional

Axes where the plot is going to be added.

[BRAN]

Brannon, R. (2003). Mohr’s Circle and more circles. Poslední revize, 29(10).

visualization.mohr3d(stress, ax=None)

Plot 3D Mohr circles

stress : ndarray

Stress tensor.

ax : Matplotlib axes, optional

Axes where the plot is going to be added.

visualization.plot_surf(R, phi, theta, title='Wave speed', **kwargs)

Plot the surface given by R(phi, theta).

R : (m,n) ndarray

Radius function.

phi_vec : (m,n) ndarray

Meshgrid for the azimut angle (phi).

theta_vec : (m,n) ndarray

Meshgrid for the cenit angle (theta).

**kwargs : keyword arguments (optional)

Keyword arguments for mlab.mesh.

surf : mayavi mesh

Mayavi mesh for the surface R(phi, theta).

visualization.traction_circle(stress, npts=48, ax=None)

Visualize a second order tensor as a collection of tractions vectors over a circle.

stress : ndarray

Stress tensor.

npts : int, optional

Number of vector to plot over the circle.

ax : Matplotlib axes, optional

Axes where the plot is going to be added.