stress-tensor.jl

### A Pluto.jl notebook ###
# v0.19.46

#> [frontmatter]
#> order = "4"
#> title = "Stress Tensor"
#> tags = ["introduction"]
#> layout = "layout.jlhtml"

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ 13a3429e-12f6-11ed-326f-c154f5debceb
begin
	using LinearAlgebra
	using PlutoUI
	using LaTeXStrings
	using PlutoTeachingTools
	using PlutoUI
	using Symbolics
	using SymbolicUtils
	using Einsum
	using PlutoPlotly
	using TikzPictures
end

# ╔═╡ 261ec03b-6d67-4cf9-8567-df00d124fb3d
ChooseDisplayMode()

# ╔═╡ 3f1bc2d1-2327-48cc-984a-df09c936da87
TableOfContents()

# ╔═╡ fd14256d-8c69-4d8a-91b5-924a32479866
md"""
# Stress Tensor


##### [Interactive Seismology Notebooks](https://pawbz.github.io/Interactive-Seismology.jl/)


Instructor: *Pawan Bharadwaj*,
Indian Institute of Science, Bengaluru, India
"""

# ╔═╡ ff0744cf-c4a7-4430-b3ee-b04c96a64dcf
md"""
The stress tensor three dimensions is given by:

```math
\sigma = \begin{bmatrix}
\sigma_{11} & \sigma_{12} & \sigma_{13} \\
\sigma_{12} & \sigma_{22} & \sigma_{23} \\
\sigma_{13} & \sigma_{23} & \sigma_{33} \\

\end{bmatrix}
```

|  | | |
|:--------:|:-------:|:-------:|
|$σ_{11}$ = $(@bind σ11 Scrubbable(-1:0.01:1, default=1)) | $σ_{22}$ = $(@bind σ22 Scrubbable(-1:0.1:1, default=0))| $σ_{33}$ = $(@bind σ33 Scrubbable(-1:0.1:1, default=0.5))|
|$σ_{12}$ = $(@bind σ12 Scrubbable(-1:0.1:1, default=0.5)) | $σ_{23}$ = $(@bind σ23 Scrubbable(-1:0.1:1, default=0))| $σ_{13}$ = $(@bind σ13 Scrubbable(-1:0.1:1, default=0))|



which is symmetric.
 
"""

# ╔═╡ 47a0a800-4384-43e8-a2b8-b6a219c964ff
@bind resample_normal CounterButton("Resample Imaginary Plane")

# ╔═╡ ca079784-f5ac-4e9b-9d10-8501564c47f0
md"## UI and Plotting"

# ╔═╡ 07845efc-0ac1-49a5-9714-8c77c4d93089
md"""
Let us define the functions that we will be using in this notebook.
"""

# ╔═╡ ca99052e-ff24-4e8a-9d06-74bc24974873
σinput = let 
	[σ11 σ12 σ13; σ12 σ22 σ23; σ13 σ23 σ33]
end

# ╔═╡ 987d9253-2f7d-4ca8-8eef-aae8e8d53bc6
eigsigma = eigen(σinput)

# ╔═╡ c807659b-6fc2-43e2-86ca-9a1953b4d75a
nsample = let
	resample_normal
	n = randn(3)
	n = n ./ sqrt(sum(abs2, n))
end

# ╔═╡ c69f6bf9-1aa4-4ff2-a0c4-1aab285bcb9b
Tinput = σinput * nsample;

# ╔═╡ 2f8aa0b9-37a7-4d77-8cd7-57ed53834544
function plot3lines_1(v, color, name)
	p1 = scatter3d(x=[0, v[1]], y=[0, 0], z=[0, 0], mode="lines", name=name, line=attr(color=color), )
	p2 = scatter3d(x=[0, 0], y=[0, v[2]], z=[0, 0], mode="lines", name=name, line=attr(color=color), )
	p3 = scatter3d(x=[0, 0], y=[0, 0], z=[0, v[3]], mode="lines", name=name, line=attr(color=color), )
	return [p1, p2, p3]
end

# ╔═╡ dfcfae3f-f8df-4e30-8997-f126bbc564b3
function plot3lines_2(v1,v2,v3, color, name)
	p1 = scatter3d(x=[0, v1[1]], y=[0, v1[2]], z=[0, v1[3]], mode="lines", name=name, line=attr(color=color), )
		p2 = scatter3d(x=[0, v2[1]], y=[0, v2[2]], z=[0, v2[3]], mode="lines", name=name, line=attr(color=color), )
		p3 = scatter3d(x=[0, v3[1]], y=[0, v3[2]], z=[0, v3[3]], mode="lines", name=name, line=attr(color=color), )
	return [p1, p2, p3]
end

# ╔═╡ 4ee07ab7-b089-43a5-9928-cb5eca293fad
function stress_plot()
	
	p1= cone(x=[0], y=[0], z=[0], u=[nsample[1]], v=[nsample[2]], w=[nsample[3]], name="Surface Normal", hoverinfo="name", showscale=false, opacity=0.5, anchor="tail", color="black", colorscale=colors.Blues_8, sizeref=0.5)
	p2 = scatter3d(x=[0, nsample[1]], y=[0, nsample[2]], z=[0, nsample[3]], mode="lines", name="Normal Vector", line=attr(color="gray"), )
	p3 = plot3lines_1(Tinput, "red", "Traction")
	p4 = plot3lines_2((collect(eachcol(eigsigma.vectors)) .* eigsigma.values)..., "blue", "Principal")
	layout = Layout(scene_camera_eye=attr(x=0, y=0, z=0), 
		width=600,
		height=500,
		uirevision=1,
		 scene=attr(
        xaxis=attr(
            nticks=10,
            range=[-1,1]
        ),
        yaxis=attr(
            nticks=10,
            range=[-1,1]
        ),
        zaxis=attr(
            nticks=10,
            range=[-1,1]
        ),),
)
	return plot([p1, p2, p3..., p4...], layout)
end

# ╔═╡ 491a8065-e82d-474a-b0e3-c3875c338c67
stress_plot()

# ╔═╡ 50eb4916-b3d7-4f2a-90c6-8661cbbd8e7a
@variables σ[1:3, 1:3]

# ╔═╡ 383136ab-f7b1-4a4b-b61f-03a35d1108c8
σ[3]

# ╔═╡ 7b88bfc2-fd25-4c9b-9cfe-b6ca65a70e94
Symbolics.scalarize(σ * [0,0,1])

# ╔═╡ 88b33835-d993-4396-8605-bb3456200eb1
@variables n[1:3]

# ╔═╡ 1be2182b-c05b-4b1b-b48e-2ba8c749cf56
@einsum aa[i] :=  σ[i, j] * f[j]

# ╔═╡ 6f33c7ed-e8a8-440d-a771-84789ee1f397
md"""
#### Appendix
"""

# ╔═╡ 3a1e536b-064a-44ef-948d-66b96adc2b0c
md"### Tikz"

# ╔═╡ 8546b3ea-bc7b-46c2-8b36-31176e4da001
tikz_default_options = raw"""
  background rectangle/.style={fill=white}, show background rectangle,
  """

# ╔═╡ 74f7ce13-a920-4504-91c4-a864001d86cc
tikz_preamble = raw"""
  \usepackage{tikz}
  \usepackage{tikz-3dplot}
  \usetikzlibrary{fit, matrix, shapes.geometric}
  \tikzset{% use tikzset, not tikzstyle
      cell/.style={
          rectangle, rounded corners=5pt, draw,
      }
  }
  \tikzset{% use tikzset, not tikzstyle
      cellv/.style={
          rectangle, rounded corners=5pt, draw, rotate=90,
      }
  }

  \usepackage{xifthen}
  \usetikzlibrary{hobby}
  \usepackage{pgfplots}
  \usepackage{fontawesome}
  \usepackage{bm,amsfonts,amsmath}
  \usetikzlibrary{backgrounds,pgfplots.groupplots,snakes}
  \usepgfplotslibrary{patchplots}
  \pgfplotsset{try min ticks=2}
  \usepackage{pgfplotstable} 
  \usetikzlibrary{plotmarks,positioning,spy}
  \usetikzlibrary{shapes.geometric, arrows, fadings}
  \usepgfplotslibrary{groupplots, polar}
  \usepackage[space]{grffile}

  \usetikzlibrary{%
              decorations.pathreplacing,%
                  decorations.pathmorphing%
                  }
                  \usetikzlibrary{positioning,fit,backgrounds}



  \usetikzlibrary{shapes,arrows}
  \usetikzlibrary{decorations.markings}
  \usetikzlibrary{patterns}
  \usetikzlibrary{plotmarks}
  \usetikzlibrary{fit}
  \usetikzlibrary{intersections}
  \usepgfplotslibrary{fillbetween}

    \pgfplotsset{
                axis line style={black!10},
                    every axis label/.append style ={black!10},
                    every axis title/.append style ={black!10},
                        every tick label/.append style={black!10}  
                          }

  % need for pgfplots
  \newcommand{\axisz}{0cm}
  \newcommand{\axisx}{0cm}

  \usetikzlibrary{positioning}
  \usetikzlibrary{shapes.geometric}
  \usetikzlibrary{backgrounds}
  """

# ╔═╡ cbec50f6-1da8-4a1e-9a88-660771c732a1
# t1, t2, t3, tv are texts
# s1 and s2 are labels with sizes
plot_plane_normal() = TikzPicture(L"""
\tdplotsetmaincoords{105}{-30}
\tdplotsetrotatedcoords{00}{30}{0}
 \begin{scope}[tdplot_rotated_coords]
  \begin{scope}[canvas is xy plane at z=0]
   \fill[gray,fill opacity=0.3] (-2,-3) rectangle (2,3); 
   \draw[very thick] (-2,0) -- (2,0);
   \path (-150:2) coordinate (H) (-1.5,0) coordinate(X);
   \pgflowlevelsynccm
   \draw[very thick,-stealth,gray] (0,0) -- (-30:1.5);
  \end{scope} 
  \draw[stealth-] (H) -- ++ (-1,0,0.2) node[pos=1.3]{$H$};
  \draw[stealth-] (X) -- ++ (0,1,0.2) node[pos=1.3]{$X$};
  \draw[very thick,-stealth] (0,0,0) coordinate (O) -- (0,0,3) node[right]{$p$};
 \end{scope}
 \pgfmathsetmacro{\Radius}{1.5}
 \draw[-stealth]  (O)-- (2.5*\Radius,0,0) node[pos=1.15] {spot $0$};
 \draw[-stealth] (O) -- (0,3.5*\Radius,0) node[pos=1.15] {spot $2$};
 \draw[-stealth] (O) -- (0,0,2.5*\Radius) node[pos=1.05] {spot $1$};
  """, options=tikz_default_options, preamble=tikz_preamble, width="12cm")

# ╔═╡ 4d566fea-0823-44fd-ac31-a571a182506f
plot_plane_normal()

# ╔═╡ d5417cd4-30e2-4e23-837b-df3a98e59f40
TikzPicture(L"""
\tdplotsetmaincoords{60}{125}
\tikzset{
		cube/.style={very thick,black},
			grid/.style={very thin,gray},
			axis/.style={->,blue,thick},}

	%draw a grid in the x-y plane
	\foreach \x in {-0.5,0,...,2.5}
		\foreach \y in {-0.5,0,...,2.5}
		{
			\draw[grid] (\x,-0.5) -- (\x,2.5);
			\draw[grid] (-0.5,\y) -- (2.5,\y);
		}
			
	%draw the axes
	\draw[axis] (0,0,0) -- (3,0,0) node[anchor=west]{$x_1$};
	\draw[axis] (0,0,0) -- (0,3,0) node[anchor=west]{$x_2$};
	\draw[axis] (0,0,0) -- (0,0,3) node[anchor=west]{$x_3$};

	%draw the top and bottom of the cube
	\draw[cube] (0,0,0) -- (0,2,0) -- (2,2,0) -- (2,0,0) -- cycle;
	\draw[cube] (0,0,2) -- (0,2,2) -- (2,2,2) -- (2,0,2) -- cycle;
	
	%draw the edges of the cube
	\draw[cube] (0,0,0) -- (0,0,2);
	\draw[cube] (0,2,0) -- (0,2,2);
	\draw[cube] (2,0,0) -- (2,0,2);
	\draw[cube] (2,2,0) -- (2,2,2);
	
""", options=tikz_default_options, preamble=tikz_preamble, width="12cm")

# ╔═╡ 857ff10b-385e-417d-83c1-ebb2252174e4
begin
	
# Define the normal vector
normal_vector = [0.0, 0.577, 0.07]

# Calculate two vectors perpendicular to the normal vector using Gram-Schmidt process
u = [normal_vector[2], -normal_vector[1], 0]
v = cross(normal_vector, u)

# Normalize the perpendicular vectors
u = u / norm(u)
v = v / norm(v)

# Define the plane points
a = 1.0  # Half the side length of the plane
plane_points = [
    a * u + a * v,
    a * u - a * v,
    -a * u - a * v,
    -a * u + a * v
]

# Create a grid of points for the surface
x_range = -a:0.1:a
y_range = -a:0.1:a
z_plane = (x, y) -> (-normal_vector[1]*x - normal_vector[2]*y) / normal_vector[3]

x_grid, y_grid = [x for x in x_range, y in y_range], [y for x in x_range, y in y_range]
z_grid = [z_plane(x, y) for x in x_range, y in y_range]
end

# ╔═╡ 5bb5af41-e820-496d-8b00-ebe7cb40b6e5
plot(
    [scatter3d(x=x, y=y, z=z, mode="lines", name="Plane"),
    scatter3d(x=[0, normal_vector[1]], y=[0, normal_vector[2]], z=[0, normal_vector[3]], mode="lines+markers", name="Normal Vector", line=attr(color="red"))]
)

# ╔═╡ 957b8f08-65ac-4868-aead-23dbc2b4c5f5
plot(scatter3d(x=x, y=y, z=z, mode="lines", name="Plane"))

# ╔═╡ f7826dd2-8584-4f96-a23a-575fb3f565c5
TikzPicture(L"""
    % Define the unit normal vector
    \def\nx{0.5} % x-component of the normal vector
    \def\ny{0.5} % y-component of the normal vector
    \def\nz{0.5} % z-component of the normal vector

    % Define the plane equation ax + by + cz = d
    \def\d{1} % Distance from the origin

    % Calculate the vertices of the small plane
    \pgfmathsetmacro{\a}{0.5} % Half the side length of the small plane

    % Calculate two vectors perpendicular to the normal vector using Gram-Schmidt process
    \pgfmathsetmacro{\ux}{\ny}
    \pgfmathsetmacro{\uy}{-\nx}
    \pgfmathsetmacro{\uz}{0}
    \pgfmathsetmacro{\vx}{\nx*\nz}
    \pgfmathsetmacro{\vy}{\ny*\nz}
    \pgfmathsetmacro{\vz}{-(\nx^2 + \ny^2)}

    % Normalize the perpendicular vectors
    \pgfmathsetmacro{\ulen}{sqrt(\ux*\ux + \uy*\uy + \uz*\uz)}
    \pgfmathsetmacro{\vlen}{sqrt(\vx*\vx + \vy*\vy + \vz*\vz)}
    \pgfmathsetmacro{\ux}{\ux/\ulen}
    \pgfmathsetmacro{\uy}{\uy/\ulen}
    \pgfmathsetmacro{\uz}{\uz/\ulen}
    \pgfmathsetmacro{\vx}{\vx/\vlen}
    \pgfmathsetmacro{\vy}{\vy/\vlen}
    \pgfmathsetmacro{\vz}{\vz/\vlen}

    % Draw the coordinate axes
    \draw[->] (0,0,0) -- (3,0,0) node[anchor=north east] {$x$};
    \draw[->] (0,0,0) -- (0,3,0) node[anchor=north west] {$y$};
    \draw[->] (0,0,0) -- (0,0,3) node[anchor=south] {$z$};

    % Draw the small plane
    \filldraw[opacity=0.5] 
        ({\a*\ux}, {\a*\uy}, {\a*\uz}) -- 
        ({\a*\vx}, {\a*\vy}, {\a*\vz}) -- 
        ({-\a*\ux}, {-\a*\uy}, {-\a*\uz}) -- 
        ({-\a*\vx}, {-\a*\vy}, {-\a*\vz}) -- 
        cycle;

    % Draw the normal vector
    \draw[->, thick, color=red] (0,0,0) -- (\nx,\ny,\nz);
    \node at (\nx,\ny,\nz) [anchor=south west] {$\vec{n}$};
""", options=tikz_default_options, preamble=tikz_preamble, width="12cm")

# ╔═╡ 703da67d-b63f-40d0-88bc-8cf8e0510658


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git-tree-sha1 = "afead5aba5aa507ad5a3bf01f58f82c8d1403495"
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git-tree-sha1 = "bcd466676fef0878338c61e655629fa7bbc69d8e"
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[[deps.libpng_jll]]
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version = "0.13.1+0"
"""

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