-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathfvk.html
154 lines (154 loc) · 7.61 KB
/
fvk.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>Buckling of a sheet</title>
<style>
html {
line-height: 1.7;
font-family: Open Sans, Arial, sans-serif;
font-size: 20px;
color: #1a1a1a;
background-color: #fdfdfd;
}
body {
margin: 0 auto;
max-width: 40em;
padding-left: 50px;
padding-right: 50px;
padding-top: 50px;
padding-bottom: 50px;
hyphens: auto;
word-wrap: break-word;
text-rendering: optimizeLegibility;
font-kerning: normal;
}
@media (max-width: 600px) {
body {
font-size: 0.9em;
padding: 1em;
}
}
@media print {
body {
background-color: transparent;
color: black;
}
p, h2, h3 {
orphans: 3;
widows: 3;
}
h2, h3, h4 {
page-break-after: avoid;
}
}
p {
margin-top: 1.7em;
}
a {
color: #1a1a1a;
}
a:visited {
color: #1a1a1a;
}
img {
max-width: 100%;
}
h1, h2, h3, h4, h5, h6 {
margin-top: 1.7em;
}
ol, ul {
padding-left: 1.7em;
margin-top: 1.7em;
}
li > ol, li > ul {
margin-top: 0;
}
blockquote {
margin: 1.7em 0 1.7em 1.7em;
padding-left: 1em;
border-left: 2px solid #e6e6e6;
font-style: italic;
}
code {
font-family: Menlo, Monaco, 'Lucida Console', Consolas, monospace;
background-color: #f0f0f0;
font-size: 85%;
margin: 0;
padding: .2em .4em;
}
pre {
line-height: 1.5em;
padding: 1em;
background-color: #f0f0f0;
overflow: auto;
}
pre code {
padding: 0;
overflow: visible;
}
hr {
background-color: #1a1a1a;
border: none;
height: 1px;
margin-top: 1.7em;
}
table {
border-collapse: collapse;
width: 100%;
overflow-x: auto;
display: block;
}
th, td {
border-bottom: 1px solid lightgray;
padding: 1em 3em 1em 0;
}
header {
margin-bottom: 6em;
text-align: center;
}
nav a:not(:hover) {
text-decoration: none;
}
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
span.underline{text-decoration: underline;}
div.column{display: inline-block; vertical-align: top; width: 50%;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
</style>
<script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.js"></script>
<script>document.addEventListener("DOMContentLoaded", function () {
var mathElements = document.getElementsByClassName("math");
var macros = [];
for (var i = 0; i < mathElements.length; i++) {
var texText = mathElements[i].firstChild;
if (mathElements[i].tagName == "SPAN") {
katex.render(texText.data, mathElements[i], {
displayMode: mathElements[i].classList.contains('display'),
throwOnError: false,
macros: macros,
fleqn: false
});
}}});
</script>
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/katex.min.css" />
<!--[if lt IE 9]>
<script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script>
<![endif]-->
</head>
<body>
<h1 id="buckling-of-a-sheet">Buckling of a sheet</h1>
<p>Föppl-von Karman equations in 2D captures the buckling of a sheet/strip/beam. The morphology of the sheet is captured using the deformation field along the in-plane direction is <span class="math inline">u(x)</span> and the displacement along the out-of-plane direction is <span class="math inline">w(x)</span>. The governing equations for this deformation field is <span class="math display">\begin{aligned}
S (u_{xx} + w_{x} w_{xx}) + f =& \ 0, \\
-B w_{xxxx} + S [ w_{x} u_{xx} + \frac{1}{2} w_{xx} (2 u_{x} + 3 w_{x}^2 ) ] + p =& \ 0.\end{aligned}</span> <span class="math inline">S</span> is the stretching modulus, given by <span class="math inline">S = E \xi</span> and <span class="math inline">B</span> the bending modulus is <span class="math inline">E I</span> where <span class="math inline">I</span> is the second moment of inertial of the organism, where <span class="math inline">\xi</span> is the thickness of the sheet, <span class="math inline">E</span> the Young’s modulus, <span class="math inline">f</span> and <span class="math inline">p</span> are the external forces per unit length.</p>
<p><img src="./figs/fig3.jpeg" alt="image" /></p>
<p>We can non-dimensionalize the force balance equations using the body-length of the organism <span class="math inline">L</span> as the length-scale and <span class="math inline">E</span> the Young’s modulus of the material as the stress-scale. Further we know that <span class="math inline">S \sim E\xi, B \sim E \xi^3</span> and using this we can then write the non-dimensional equations as <span class="math display">\begin{aligned}
(d_{ss} + h_{s} h_{ss}) + \mathsf{f}=& \ 0, \\
- K h_{ssss} + h_{s} d_{ss} + \frac{1}{2} h_{ss} (2 d_{s} + 3 h_{s}^2) + \mathsf{p} =& \ 0,\end{aligned}</span> where <span class="math inline">x= sL, u = dL, w = hL, K = (\xi/L)^2</span> is the geometric quantity highlighting slenderness, also known as the von Kármán number, <span class="math inline">\mathsf{f}= (fL/E\xi), \mathsf{p} = (p L/E\xi)</span> are the non-dimensional forces. We can solve this system with the following boundary conditions: fixed ends, <span class="math inline">d|_0 = h|_0^1 = 0</span>; zero applied moments, <span class="math inline">h_{ss}|_0^1 = 0</span>; applied tangential strain, <span class="math inline">d_s|_1 = -\alpha</span>. We can again perform a similar analysis to that of the elastica for a fixed value of <span class="math inline">K</span> and changing the applied strain and is shown in the figure below. The code corresponding to these plots is in the <code>1parameter</code> folder inside <a href="https://github.com/sgangaprasath/autoTutorial/tree/main/Tutorial2_FvK"><code>Tutorial2_FvK</code></a>.</p>
<p>The leverage we have using the <code>.auto</code> script is that we can now steer the solution branch by varying different parameters in the equation. Once you have shifted from the unbuckled configuration to a bucked one using <code>ISW=-1</code> (just as in the elastica case), we can change the continuation parameter to a different one, say <span class="math inline">K</span> just by manipulating the <code>.auto</code> script. We can go further using <code>auto-07p</code> and continue the solution by varying both <span class="math inline">(K, \alpha)</span> by using the variable <code>ICP</code> and setting limits using the <code>UZSTOP</code> command. For example we continue along two variables here by adding an integral constraint. This is because we need to satisfy the constraint <code>NCONT = NBC+NINT-NDIM+1</code> where <code>NCONT</code>-number of continuation parameters, <code>NINT</code>-number of integral constraints, <code>NDIM</code>-number of dimensions. In the current problem we have <code>NBC=6, NDIM=6</code> and thus we need an integral constraint to be able to perform continuation along the two-parameter phase space. After performing this, we get a isocontours in <span class="math inline">(K, \alpha)</span>-plane representing morphologies of the sheet (a sample is shown in the figure below).</p>
<p><img src="./figs/2param.png" style="width:50.0%" alt="image" /></p>
</body>
</html>