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Ae 315 Homework 5 Solutions  2008/2/8
We have studied all the elements of a theory of elasticity. Here we put everything together with the simplest problems: We know
the displacements and we want to confirm the other equations.
Preliminaries
These are just some items that setup or describe the operating environment of this program.
In[1]:=
$Version
Out[1]=
6.0 for Mac OS X x86
H
32

bit
L H
April 20, 2007
L
In[2]:=
DateList
@D
Out[2]=
8
2008, 2, 7, 14, 54, 56.002228
<
In[3]:=
Off
@
General::
spell
D
;
Off
@
General::
spell1
D
;
Off
@
Solve::
svars
D
;
Off
@
ParametricPlot::
ppcom
D
;
Off
@
ParametricPlot3D::
ppcom
D
In[8]:=
Needs
@
"Units`"
D
Needs
@
"PhysicalConstants`"
D
Needs
@
"BarCharts`"
D
; Needs
@
"Histograms`"
D
; Needs
@
"PieCharts`"
D
Needs
@
"VectorAnalysis`"
D
Common functions for all problems
Here you are given a dispalcement field and you need to interperet what it means.
ü
reference region.
This is the undeformed configuration. Here I have to assume some numerical values for some of the parameters so that the
plotting can be accomplished. Here the radius "a" is 1.0, the length "c" is 5. I do not plot the inside radius. Also, note that I
make use of polar coordinates to parameterize the "surface".
In[12]:=
aValue
=
1.0;
cValue
=
5.0;
surface
=
8
aValue Cos
@
q
D
, aValue Sin
@
q
D
, z
<
;
Note that I plot the "z" direction first so my cylinder comes out nearly horizontal rather than vertical. I do this with the "Rotate
Right" command. NOTE: I am only plotting 1/2 of the section where
x
3
>= 0. The equations that are provided only work on this
subregion.
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View Full DocumentIn[15]:=
ParametricPlot3D
@
RotateRight
@
surface
D
,
8
q
,
p
,
p
<
,
8
z, 0, cValue
<D
Out[15]=
ü
displacement
First, lets define the displacement field: (Note I define the displacements here in such a manner so I can automate certain opera
tions.) Here I'm going to try andd make things a bit more clear in how the displacements are defined.
In[16]:=
u
@
1
D@8
x1
_
, x2
_
, x3
_
<
,
8
a_
,
b_
,
g_
,
d_
<
,
8
m
_
,
n_
, a
_
, c
_
<D
:
=
1
2
a
I
x3
2
 n
x2
2
+ n
x1
2
M
 n b
x1
 g
x3 x2
+ d
n
2
H
c

x3
L
I
x1
2

x2
2
M

x3
3
6
+
c x3
2
2
;
u
@
2
D@8
x1
_
, x2
_
, x3
_
<
,
8
a_
,
b_
,
g_
,
d_
<
,
8
m
_
,
n_
, a
_
, c
_
<D
:
=
n a
x1 x2
 n b
x2
+ g
x3 x1
+ d n
H
c

x3
L
x1 x2;
u
@
3
D@8
x1
_
, x2
_
, x3
_
<
,
8
a_
,
b_
,
g_
,
d_
<
,
8
m
_
,
n_
, a
_
, c
_
<D
:
=
a
x1 x3
+ b
x3
 d

3
4
+
n
2
a
2
x1
+
1
4
I
x1
3

3
x1 x2
2
M
+
x1 x2
2
+
x3 x1
c

x3
2
;
Here I have separated out each symbol so its easier to determine what the equation is doing. Here "x" is a vector of the coordi
nates and "p" is a column matrix representing the parameters "
g
", "
d
" and "
n
", "c", and "a" respectively. To check this out:
In[19]:=
u
@
1
D@8
x
1
, x
2
, x
3
<
,
8
a
,
b
,
g
,
d
<
,
8
m,
n
, a, c
<D
Out[19]=
b n
x
1
 g
x
2
x
3
+
1
2
a
I
n
x
1
2
 n
x
2
2
+
x
3
2
M
+ d
1
2
n
I
x
1
2

x
2
2
M H
c

x
3
L
+
c x
3
2
2

x
3
3
6
In[20]:=
u
@
2
D@8
x
1
, x
2
, x
3
<
,
8
a
,
b
,
g
,
d
<
,
8
m,
n
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 Winter '08
 washsba

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