Continuum mechanics
This static webpage shows the tangible meaning of kinematics quantities usually used in continuum mechanics.
Gradient of the transformation
Let \(\mathcal{F}\) be the transformation such that \(\forall \underline{X} \in \Omega_0\) we can define its counter part \(\underline{x} = \mathcal{F}\left\{\underline{X}\right\} \in \Omega_t\)
Most kinematic quantities that describe the transformation of an elementary volume in the bulk are defined based on \[\underline{\underline{F}} = \underline{\underline{\nabla}}\mathcal{F},\] the gradient of the transformation.
Let’s try to grasp the physical meaning of the component of \(\underline{\underline{F}}\) in a Cartesian framework.
import marimo as mo
from wigglystuff import Matrix
import numpy as np
F = mo.ui.anywidget(Matrix(np.eye(3), step=0.01))
mo.hstack([mo.md(r"$\underline{\underline{F}}=$"), F])You can play with its components (hover and then click + drag directly on the matrix’s components) and observe the impact it has on the deformation of a unit cube. In the following figure we can observe a unit cube initially undeformed (black) that gets deformed by the action of \(\underline{\underline{F}}\) (red).
import plotly.graph_objects as go
nodes_0 = np.array([
[0, 1, 1, 0, 0, 1, 1, 0], # X
[0, 0, 1, 1, 0, 0, 1, 1], # Y
[0, 0, 0, 0, 1, 1, 1, 1] # Z
])
nodes_t = F.matrix@nodes_0
edges = np.array([
[0, 1], [1, 2], [2, 3], [3, 0], # bottom face
[4, 5], [5, 6], [6, 7], [7, 4], # top face
[0, 4], [1, 5], [2, 6], [3, 7] # vertical edges
])
# Create figure
fig = go.Figure()
# Add cube edges
for edge in edges:
x = nodes_0[0, edge]
y = nodes_0[1, edge]
z = nodes_0[2, edge]
fig.add_trace(go.Scatter3d(
x=x, y=y, z=z,
mode='lines',
line=dict(color='black', width=3)
))
for edge in edges:
x = nodes_t[0, edge]
y = nodes_t[1, edge]
z = nodes_t[2, edge]
fig.add_trace(go.Scatter3d(
x=x, y=y, z=z,
mode='lines',
line=dict(color='red', width=6)
))
# Configure layout with turntable mode
fig.update_layout(
scene=dict(
aspectmode='cube',
xaxis=dict(range=x_range.value),
yaxis=dict(range=y_range.value),
zaxis=dict(range=z_range.value),
),
# scene_camera=dict(
# eye=dict(x=1.5, y=1.5, z=1),
# projection=dict(type='perspective')
# ),
dragmode='turntable',
title="Initial and deformed unit cubes",
showlegend = False
)
mo.ui.plotly(fig)From this second order tensor, we can also compute the associated right Cauchy–Green deformation tensor
F_mat = np.array(F.matrix)
C = mo.ui.anywidget(Matrix(F_mat.T@F_mat, step=0.0))
mo.hstack([mo.md(r"$\underline{\underline{C}}=$"), C])that is not impacted by rigid body motions (translations or rotations).
You can change the limits of the boundary box to zoom in and out on the cubes with the following sliders.
x_range = mo.ui.range_slider(
start=-5, stop=5, value=[-1,2],
label="X range", step=0.1,full_width=True
)
y_range = mo.ui.range_slider(
start=-5, stop=5, value=[-1,2],
label="Y range", step=0.1,full_width=True
)
z_range = mo.ui.range_slider(
start=-5, stop=5, value=[-1,2],
label="Z range", step=0.1,full_width=True
)
mo.vstack([x_range, y_range, z_range])The variation of the volume is given by the jacobian \[J=\text{det}\left(\underline{\underline{F}}\right).\] The current volume \(V_t\) is computed as \[V_t = J~V_0\] with \(V_0\) being the volume of the initial cube.
jacobian = np.linalg.det(F.matrix)
mo.md(rf"See the new volume $V_t$ ={jacobian:.1f}$\times V_0$")Stress tensor
The Cauchy stress tensor \(\underline{\underline{\sigma}}\) describes internal traction forces. To get an idea of the physical interpretation of each of its component let’s manipulate this tensor and look at the impact it has on the corresponding traction forces on an elementary volume element.
normal_axis_dropdown = mo.ui.dropdown(options=["e_1", "e_2", "e_3","-e_1", "-e_2", "-e_3"], label="Select the normal vector you want here: ", value = "e_1")
normal_axis_dropdownAnd see how the components of
sigma = mo.ui.anywidget(Matrix(np.eye(3), step=0.01,mirror=True))
mo.hstack([mo.md(r"$\underline{\underline{\sigma}}=$"), sigma])affect the traction vector \(\underline{T} = \underline{\underline{\sigma}} \underline{n}\)
normal_axis = normal_axis_dropdown.value
match normal_axis:
case "e_1":
n_axis = np.array([[1,0,0]]).T
T_0 = np.array([[1,0.5,0.5]]).T
case "-e_1":
n_axis = np.array([[-1,0,0]]).T
T_0 = np.array([[0,0.5,0.5]]).T
case "e_2":
n_axis = np.array([[0,1,0]]).T
T_0 = np.array([[0.5,1,0.5]]).T
case "-e_2":
n_axis = np.array([[0,-1,0]]).T
T_0 = np.array([[0.5,0,0.5]]).T
case "e_3":
n_axis = np.array([[0,0,1]]).T
T_0 = np.array([[0.5,0.5,1]]).T
case "-e_3":
n_axis = np.array([[0,0,-1]]).T
T_0 = np.array([[0.5,0.5,0]]).T
T_sigma = sigma.matrix@n_axis/2
x_T_sigma = np.hstack([T_0,T_0+T_sigma])
fig_sigma = go.Figure()
match normal_axis:
case "-e_3":
fig_sigma.add_trace(go.Surface(
# x=x_n,
# y=x_n,
# z=z_n, # z = 0 plane
x=[[0,1],[0,1]],
y=[[0,0],[1,1]],
z=[[0,0],[0,0]], # z = 0 plane
showscale=False,
opacity=0.5,
colorscale=[[0, 'blue'], [1, 'blue']]
))
case "e_3":
fig_sigma.add_trace(go.Surface(
# x=x_n,
# y=x_n,
# z=z_n, # z = 0 plane
x=[[0,1],[0,1]],
y=[[0,0],[1,1]],
z=[[1,1],[1,1]], # z = 0 plane
showscale=False,
opacity=0.5,
colorscale=[[0, 'blue'], [1, 'blue']]
))
case "e_1":
fig_sigma.add_trace(go.Surface(
# x=x_n,
# y=x_n,
# z=z_n, # z = 0 plane
x=[[1,1],[1,1]],
y=[[0,1],[0,1]],
z=[[0,0],[1,1]], # z = 0 plane
showscale=False,
opacity=0.5,
colorscale=[[0, 'blue'], [1, 'blue']]
))
case "-e_1":
fig_sigma.add_trace(go.Surface(
# x=x_n,
# y=x_n,
# z=z_n, # z = 0 plane
x=[[0,0],[0,0]],
y=[[0,1],[0,1]],
z=[[0,0],[1,1]], # z = 0 plane
showscale=False,
opacity=0.5,
colorscale=[[0, 'blue'], [1, 'blue']]
))
case "-e_2":
fig_sigma.add_trace(go.Surface(
# x=x_n,
# y=x_n,
# z=z_n, # z = 0 plane
x=[[0,0],[1,1]],
y=[[0,0],[0,0]],
z=[[0,1],[0,1]], # z = 0 plane
showscale=False,
opacity=0.5,
colorscale=[[0, 'blue'], [1, 'blue']]
))
case "e_2":
fig_sigma.add_trace(go.Surface(
# x=x_n,
# y=x_n,
# z=z_n, # z = 0 plane
x=[[0,0],[1,1]],
y=[[1,1],[1,1]],
z=[[0,1],[0,1]], # z = 0 plane
showscale=False,
opacity=0.5,
colorscale=[[0, 'blue'], [1, 'blue']]
))
fig_sigma.add_trace(go.Scatter3d(
x=x_T_sigma[0,:], y=x_T_sigma[1,:], z=x_T_sigma[2,:],
mode='lines',
name='Traction vector',
line=dict(color='red', width=5)
))
for edge_sigma in edges:
x_sigma = nodes_0[0, edge_sigma]
y_sigma = nodes_0[1, edge_sigma]
z_sigma = nodes_0[2, edge_sigma]
fig_sigma.add_trace(go.Scatter3d(
x=x_sigma, y=y_sigma, z=z_sigma,
mode='lines',
showlegend=False,
line=dict(color='black', width=3)
))
fig_sigma.update_layout(
scene=dict(
aspectmode='cube',
# xaxis=dict(range=[-1, 2]),
# yaxis=dict(range=[-1, 2]),
# zaxis=dict(range=[-1, 2]),
),
# scene_camera=dict(
# eye=dict(x=1.5, y=1.5, z=1),
# projection=dict(type='perspective')
# ),
dragmode='turntable',
title="Infinitesimal volume and traction vector",
showlegend = True
)
mo.ui.plotly(fig_sigma)