nicoguaro / continuum_mechanics

Utilities for doing calculations in continuum mechanics.

Nicolas Guarín-Zapata / Homepage / Twitter / GitHub / Gist / Computational Science Exchange / Speakerdeck

PhD. in Computational Engineering with interest and experience in simulation-based engineering, numerical methods, biomimetics, mechanics of solids, design of materials, and wave propagation.

Courses & tutorials

• Advanced Mathematics for Engineers This is a repository with material for the course Advanced Mathematics for Engineers.

• Computational Modeling This is a repository with material for the course Computational Modeling (in Spanish).

• Continuum Mechanics This is a repository with material for the course Continuum Mechanics (in Spanish).

• Introduction to the Finite Element Method This is a repository with material for the course Introduction to the Finite Element Method.

• Numerical Methods for PDEs This is a repository with material for the course Numerical Methods for PDE.

• Tutorial on Scientific Visualization Given in PyCon Colombia 2019

Development

• continuum_mechanics. Utilities for doing calculations in continuum mechanics.

• Pyrobo contains phrases from the Colombian film La Vendedora de Rosas by Víctor Gaviria

• SolidsPy. A simple finite element analysis code for 2D elasticity problems

Science

• Analysis of the directionality on periodic materials (2023).

• Variational principles and finite element Bloch analysis in couple stress elastodynamics (2021).

• Flipping a Computational Modeling Class: Strategies to Engage Students and Foster Active Learning (2020).

• A natural impact-resistant bicontinuous composite nanoparticle coating (2020).

• Finite element modeling of micropolar-based phononic crystals (2020).

• A General-Purpose Element-Based Approach to Compute Dispersion Relations in Periodic Materials with Existing Finite Element Codes (2019).

• Bandgap tuning in bioinspired helicoidal composites (2019).

continuum_mechanics's Issues

Add optional parameters to ``mohr2d``

Regarding Mohr circles in 2D it would be useful to have optional parameters to turn off some of the details in the graphics, such as the center or the line connecting the current state.

Error in gradient of a vector

The gradient of a vector returns wrong results for coordinates different from Cartesian.

Add conversion between isotropic elasticity paremeters

Depending on the context people use different combinations of the material constants in elasticity. We need a way to get any combination of two of them from any other two, as shown in:

https://en.wikipedia.org/wiki/Lam%C3%A9_parameters

Tensor divergence seems to be wrong

• Continuum Mechanics version: 0.2.2
• Python version: 3.7.10

Description

The divergence of the stress tensor is not 0 while the Navier-Cauchy operator is zero for the displacement vector.

What I Did

Paste the command(s) you ran and the output.
If there was a crash, please include the traceback here.

from sympy import *
from continuum_mechanics.vector import div_tensor
from continuum_mechanics.solids import navier_cauchy

r, theta, phi = symbols("r theta phi")
T, mu = symbols("T mu")

h = (1, r, r*sin(theta))
coords = (r, theta, phi)
disp = Matrix([[0], [0], [T*sin(theta)/(8*pi*mu*r**2)]])
stress = Matrix([[0, 0, -3*T*sin(theta)/(8*pi*r**3)], [0, 0, 0], [-3*T*sin(theta)/(8*pi*r**3), 0, 0]])

And

navier_cauchy(disp, [1, mu], coords, h)

gives Matrix([[0], [0], [0]]), while

div_tensor(stress, coords, h)

gives Matrix([[0], [0], [3*T*sin(theta)/(8*pi*r**4)]]).

Add coordinate transformations

Add coordinate transformations for the supported coordinate systems.

Add Airy stress functions

We can add Airy stress functions to the package since the vector operators are implemented. A good starting point is given in the following notebook:

https://github.com/nicoguaro/notebooks_examples/blob/master/Airy%20stress%20functions.ipynb

It could be possible to add Michelle general solution for 2D elasticity in polar coordinates.

• J. H. Michell, 1899, On the direct determination of stress in an elastic solid,
  with application to the theory of plates, Proceedings of the London Mathematical
  Society, vol. 31, pages 100-124.

Rotation matrices don't match reference solution?

• Continuum Mechanics version: 0.3.0

Description

The rotation matrix given in http://solidmechanics.org/Text/Chapter3_2/Chapter3_2.php under section 3.2.11 is not the same as what I get from the package. It might be an issue with clockwise and counterclockwise rotation, as the numbers are the same, the signs are not always.

As far as I understand it, a counterclockwise rotation should be positive about the rotation axis according to the right-hand rule. This is also what's shown in his book and the figure under 3.2.11 on the webpage.

I know this is a version under development, but I checked your code and did not find any error.

What I Did

Computed the two rotation matrices as below. They don't match.

theta = sym.pi/6
c = sym.cos(theta)
s = sym.sin(theta)

# counterclockwise, z-axis, 30°
# Taken from https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations
Q = sym.Matrix([[c, -s, 0], [s, c, 0], [0, 0, 1]])     

M = sym.zeros(6)
for i in range(3):
    for j in range(3):
        M[i, j] = Q[i, j]**2
        M[i, j + 3] = 2 * Q[i, (j + 1) % 3] * Q[i, (j + 2) % 3]
        M[i + 3, j] = Q[(i + 1) % 3, j] * Q[(i + 2) % 3, j]
        M[i + 3, j + 3] = Q[(i + 1) % 3, (j + 1) % 3] * \
            Q[(i + 2) % 3, (j + 2) % 3] + \
            Q[(i + 1) % 3, (j + 2) % 3] * \
            Q[(i + 2) % 3, (j + 1) % 3]

# Counterclockwise, z-axis
M_ref = np.array([[c**2, s**2, 0, 0, 0, 2*c*s],
                  [s**2, c**2, 0, 0, 0, -2*c*s],
                  [0, 0, 1, 0, 0, 0],
                  [0, 0, 0, c, s, 0],
                  [0, 0, 0, -s, c, 0],
                  [-c*s, c*s, 0, 0, 0, c**2-s**2]])

print(M_ref == np.array(M))

Output:

[[ True  True  True  True  True False]
 [ True  True  True  True  True False]
 [ True  True  True  True  True  True]
 [ True  True  True  True  True  True]
 [ True  True  True  True  True  True]
 [False False  True  True  True  True]]