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PROFESSOR: Ladies and gentlemen,
welcome to this

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00:00:23,650 --> 00:00:26,600
lecture on non-linear finite
element analysis.

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00:00:26,600 --> 00:00:29,320
In the previous lectures, I
introduced you to non-linear

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00:00:29,320 --> 00:00:32,810
finite element analysis, to
the solution methods that

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00:00:32,810 --> 00:00:36,980
we're using in non-linear finite
element analysis, using

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00:00:36,980 --> 00:00:39,250
primarily physical concepts.

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00:00:39,250 --> 00:00:43,290
We are now ready to discuss the
mathematical basis of the

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00:00:43,290 --> 00:00:47,790
procedures that we're using
to obtain solutions.

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00:00:47,790 --> 00:00:50,830
Of course, this material is very
complex, and there is a

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00:00:50,830 --> 00:00:52,580
lot we could discuss.

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00:00:52,580 --> 00:00:54,530
But let us have a go at it.

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00:00:54,530 --> 00:01:00,470
We have a body that undergoes
large displacements, large

20
00:01:00,470 --> 00:01:03,720
rotations, and large strains.

21
00:01:03,720 --> 00:01:05,940
So therefore, we consider
the body to

22
00:01:05,940 --> 00:01:08,210
undergo very large motions.

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00:01:08,210 --> 00:01:10,940
We will use a Lagrangian
description of the

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00:01:10,940 --> 00:01:13,190
motion of the body.

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00:01:13,190 --> 00:01:17,560
Here, on this viewgraph, I have
prepared schematically

26
00:01:17,560 --> 00:01:19,500
the body that we
are looking at.

27
00:01:19,500 --> 00:01:24,730
Here the body is in its
configuration at time zero.

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00:01:24,730 --> 00:01:28,710
Notice that the area is, the
surface area of the body is

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00:01:28,710 --> 00:01:35,980
described as 0 A, the volume
is described as 0 V, the 0

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00:01:35,980 --> 00:01:37,870
being a left superscript.

31
00:01:37,870 --> 00:01:41,390
I should point out, at this
point, that you will be

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00:01:41,390 --> 00:01:44,610
encountering now a lot of
superscripts and subscripts,

33
00:01:44,610 --> 00:01:47,740
and that actually makes the
discussion somewhat difficult.

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00:01:47,740 --> 00:01:51,210
And for that reason, I like to
take now, on this viewgraph,

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00:01:51,210 --> 00:01:55,190
just a minute to explain to you
why we have a particular

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00:01:55,190 --> 00:01:58,200
superscript, a particular
subscript, in that particular

37
00:01:58,200 --> 00:02:01,930
location, and what
it means to us.

38
00:02:01,930 --> 00:02:09,199
So, a left superscript here, as
shown on A and V, means at

39
00:02:09,199 --> 00:02:12,470
time indicated by that, at the
time indicated by that

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00:02:12,470 --> 00:02:13,210
superscript.

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00:02:13,210 --> 00:02:20,270
So here we have 0 A, 0 V once
again, area at time 0,

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00:02:20,270 --> 00:02:22,520
volume at time 0.

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00:02:22,520 --> 00:02:30,160
A particular point here, point
P, has the coordinates 0 X 1,

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00:02:30,160 --> 00:02:32,250
0 X 2, 0 X 3.

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00:02:32,250 --> 00:02:35,960
Now, notice that these upper
0s here correspond to the

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00:02:35,960 --> 00:02:40,250
time, as I just pointed out,
whereas 1, 2, and 3, of

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00:02:40,250 --> 00:02:43,420
course, correspond to
the coordinate axis.

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00:02:43,420 --> 00:02:46,940
This is a particular
material particle.

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00:02:46,940 --> 00:02:51,970
This material particle will move
with the body from time 0

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00:02:51,970 --> 00:02:56,680
to time t as shown on
this viewgraph.

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00:02:56,680 --> 00:02:59,820
Of course, at this particular
time now, the surface area of

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00:02:59,820 --> 00:03:03,780
the body has become t A,
t now meaning surface

53
00:03:03,780 --> 00:03:07,170
area at time t.

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00:03:07,170 --> 00:03:11,290
And the volume of the body
is t V. Notice that the

55
00:03:11,290 --> 00:03:15,142
coordinates of this material
particle now are t x

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00:03:15,142 --> 00:03:18,870
1, t x 2, t x 3.

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00:03:18,870 --> 00:03:22,960
We will assume, later on in our
actual solution, that the

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00:03:22,960 --> 00:03:26,970
solution up to time t is known,
in other words, that

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00:03:26,970 --> 00:03:31,080
the area, the volume, in fact,
the stresses, the strains, at

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00:03:31,080 --> 00:03:33,380
time t are known.

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00:03:33,380 --> 00:03:36,870
And that we will want to march
ahead one time increment, to

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00:03:36,870 --> 00:03:38,770
time, t plus delta t.

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00:03:38,770 --> 00:03:42,450
Notice the volume of the body,
now, is given by t plus delta

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00:03:42,450 --> 00:03:47,730
t V. The surface area is given
by t plus delta t A. The

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00:03:47,730 --> 00:03:51,590
coordinates of the material
particle have now become t

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00:03:51,590 --> 00:03:58,460
plus delta t x 1, t plus delta t
x 2, and t plus delta t x 3.

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00:03:58,460 --> 00:04:02,090
Well, there's one very
important point.

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00:04:02,090 --> 00:04:05,540
And that important point is that
we are considering that

69
00:04:05,540 --> 00:04:10,500
the Cartesian coordinate axis
in which the motion of the

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00:04:10,500 --> 00:04:13,650
body is measured remains
stationary.

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00:04:13,650 --> 00:04:15,260
You should always keep
that in mind.

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00:04:15,260 --> 00:04:20,940
In other words, here we have the
Cartesian axis, x 1, x 2,

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00:04:20,940 --> 00:04:25,250
x 3, and those remain
stationary, whereas the body

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00:04:25,250 --> 00:04:31,540
moves through those axes,
through the, well, system of

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00:04:31,540 --> 00:04:37,700
axes, And we want to, of course,
solve for that motion.

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00:04:37,700 --> 00:04:43,420
With the particle motion
described by the coordinates

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00:04:43,420 --> 00:04:46,510
that I just mentioned, we can
now define the following

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00:04:46,510 --> 00:04:48,370
quantities down here.

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00:04:48,370 --> 00:04:54,890
We can say that t x i, the
coordinates at time t of the

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00:04:54,890 --> 00:04:58,370
material particle, are
equal to 0 x i--

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00:04:58,370 --> 00:05:00,660
both of those quantities, of
course, we know already now

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00:05:00,660 --> 00:05:02,310
what they are--

83
00:05:02,310 --> 00:05:03,720
plus t u i.

84
00:05:03,720 --> 00:05:07,040
t u i are just the displacements
from

85
00:05:07,040 --> 00:05:09,800
time 0 to time t.

86
00:05:09,800 --> 00:05:13,660
Similarly for the situation
from time 0 to time

87
00:05:13,660 --> 00:05:14,670
t plus delta t.

88
00:05:14,670 --> 00:05:18,640
Notice that these are now the
displacements from time 0 to

89
00:05:18,640 --> 00:05:20,950
time t plus delta t.

90
00:05:20,950 --> 00:05:24,140
And we can, of course, talk
also about the incremental

91
00:05:24,140 --> 00:05:24,840
displacements.

92
00:05:24,840 --> 00:05:29,620
The incremental displacements
from time t plus delta t, or

93
00:05:29,620 --> 00:05:33,190
rather from time t, I should
say, to time t plus delta t.

94
00:05:33,190 --> 00:05:34,840
And those incremental
displacements

95
00:05:34,840 --> 00:05:36,450
are given by u i.

96
00:05:36,450 --> 00:05:39,730
Of course, the lower right
subscript always

97
00:05:39,730 --> 00:05:42,260
refers to the axis.

98
00:05:42,260 --> 00:05:46,410
In other words, i goes from
one to two to three.

99
00:05:46,410 --> 00:05:48,390
For the u's and for the x's.

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00:05:51,210 --> 00:05:56,670
Well, I mentioned already that
the Cartesian axes are

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00:05:56,670 --> 00:05:57,940
stationary.

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00:05:57,940 --> 00:06:01,380
And I have prepared this
viewgraph to point it out once

103
00:06:01,380 --> 00:06:03,680
more very strongly.

104
00:06:03,680 --> 00:06:08,236
The unit distances along the
xi-axis are the same for 0 x

105
00:06:08,236 --> 00:06:12,610
i, t x i, and t plus
delta t x i.

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00:06:12,610 --> 00:06:15,560
That is also a very
important point.

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00:06:15,560 --> 00:06:18,100
Let's look at this very
simple example.

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00:06:18,100 --> 00:06:20,732
Here we have an x 2- axis,
and we have the x

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00:06:20,732 --> 00:06:24,070
1-axis along here.

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00:06:24,070 --> 00:06:29,570
Notice the units along this axis
are 1, 2, 3, and so on.

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00:06:29,570 --> 00:06:32,670
The material particle
has initially the

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00:06:32,670 --> 00:06:34,640
coordinates, 2, say.

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00:06:34,640 --> 00:06:38,450
Of course, then, this being
the length of 0 x 1.

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00:06:38,450 --> 00:06:45,920
This particle, at time 0, moves
to time t by t u 1, the

115
00:06:45,920 --> 00:06:47,600
way we just said it earlier.

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00:06:47,600 --> 00:06:50,640
And of course, the sum of
these two quantities is

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00:06:50,640 --> 00:06:51,890
equal to t x 1.

118
00:06:54,510 --> 00:07:00,160
Notice that t x 1 is measured
along the same axis, with the

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00:07:00,160 --> 00:07:05,600
same unit distances, as 0 x 1.

120
00:07:05,600 --> 00:07:09,200
And therefore, for this reason,
we now use just x 1

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00:07:09,200 --> 00:07:12,410
here, without the left
superscript, to denote the

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00:07:12,410 --> 00:07:13,980
coordinate axis.

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00:07:13,980 --> 00:07:16,620
By the way, in my book, the
textbook for this video

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00:07:16,620 --> 00:07:20,620
course, I have frequently
put here, a t or 0.

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00:07:20,620 --> 00:07:23,200
I feel now, it is better,
actually, to drop that

126
00:07:23,200 --> 00:07:26,560
superscript, because this
description, the way I just

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00:07:26,560 --> 00:07:30,110
explained it to you, I think,
is even better.

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00:07:30,110 --> 00:07:32,960
Well, let us talk now,
then, about the

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00:07:32,960 --> 00:07:35,410
principle of virtual work.

130
00:07:35,410 --> 00:07:38,570
And now, we want to look at the
principle of virtual work

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00:07:38,570 --> 00:07:41,700
applied to time t
plus delta t.

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00:07:41,700 --> 00:07:45,510
The reason being, of course,
that we want to solve for the

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00:07:45,510 --> 00:07:49,370
static and kinematic variables
at time t plus delta t.

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00:07:49,370 --> 00:07:52,400
We assume that we know
the solution from

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00:07:52,400 --> 00:07:55,050
time 0 to time t.

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00:07:55,050 --> 00:07:56,830
We discussed in the previous

137
00:07:56,830 --> 00:08:00,780
lectures already the principle.

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00:08:00,780 --> 00:08:03,090
I'd like to now just review it
once more, because it's just

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00:08:03,090 --> 00:08:07,720
of utmost importance to very
well grasp, understand what

140
00:08:07,720 --> 00:08:10,930
the principle says, and
how we apply it then.

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00:08:10,930 --> 00:08:12,980
Here we have now the
Cauchy stresses.

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00:08:12,980 --> 00:08:15,830
I defined the Cauchy stresses
earlier as the

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00:08:15,830 --> 00:08:17,930
force per unit area.

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00:08:17,930 --> 00:08:22,580
The i j, of course, mean
coordinate axis, i and j.

145
00:08:22,580 --> 00:08:25,930
In three dimensional analysis,
going from 1 to 3.

146
00:08:25,930 --> 00:08:28,890
t plus delta t means the
Cauchy stress at time

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00:08:28,890 --> 00:08:30,350
t plus delta t.

148
00:08:30,350 --> 00:08:34,530
These are the real stresses
that we want to solve for.

149
00:08:34,530 --> 00:08:38,159
Here we have virtual strains.

150
00:08:38,159 --> 00:08:43,770
These are infinitesimally small
strains, referred to the

151
00:08:43,770 --> 00:08:47,050
configuration at time
t plus delta t.

152
00:08:47,050 --> 00:08:51,750
This t plus delta t means a
reference to the configuration

153
00:08:51,750 --> 00:08:54,480
at time t plus delta t.

154
00:08:54,480 --> 00:08:57,270
And here we have, of course,
the t plus delta t d V,

155
00:08:57,270 --> 00:09:01,190
because we're integrating over
that volume which, of course,

156
00:09:01,190 --> 00:09:04,160
is unknown in the incremental
solution.

157
00:09:04,160 --> 00:09:07,920
This is the internet virtual
work, which must be equal to

158
00:09:07,920 --> 00:09:10,480
the external virtual work.

159
00:09:10,480 --> 00:09:14,510
Here is again, the script R. And
this script R, at time t

160
00:09:14,510 --> 00:09:18,500
plus delta t, is defined down
here in the viewgraph.

161
00:09:18,500 --> 00:09:21,860
We're integrating over the
volume, at time t plus delta

162
00:09:21,860 --> 00:09:24,750
t, the body forces.

163
00:09:24,750 --> 00:09:27,910
And these are the real body
forces, of forces per unit

164
00:09:27,910 --> 00:09:32,640
volume at time t plus delta t,
multiplied by the virtual

165
00:09:32,640 --> 00:09:36,490
displacements, delta u i.

166
00:09:36,490 --> 00:09:39,830
And the integration, of course,
goes over the volume

167
00:09:39,830 --> 00:09:41,730
at time t plus delta t.

168
00:09:41,730 --> 00:09:48,600
We're adding to this part the
surface forces, forces per

169
00:09:48,600 --> 00:09:53,060
unit area, at time t plus delta
t, multiplied by the

170
00:09:53,060 --> 00:09:56,160
virtual displacements
on the surface.

171
00:09:56,160 --> 00:10:02,670
This capital S stands for "on
the surface." And this product

172
00:10:02,670 --> 00:10:07,050
is integrated over the total
surface of the body at time t

173
00:10:07,050 --> 00:10:09,370
plus delta t.

174
00:10:09,370 --> 00:10:14,520
We assume that we know this part
once we have imposed a

175
00:10:14,520 --> 00:10:17,410
virtual displacement.

176
00:10:17,410 --> 00:10:21,580
And therefore, we can calculate
the right-hand side.

177
00:10:21,580 --> 00:10:23,960
Corresponding to that virtual
displacement we

178
00:10:23,960 --> 00:10:27,410
obtain virtual strains.

179
00:10:27,410 --> 00:10:30,080
And therefore, we can
calculate also

180
00:10:30,080 --> 00:10:31,320
the left-hand side.

181
00:10:31,320 --> 00:10:34,890
And once again, the principle of
virtual work says, that the

182
00:10:34,890 --> 00:10:38,460
left-hand side, the internal
virtual work, must be equal to

183
00:10:38,460 --> 00:10:42,510
the right-hand side, the
external virtual work, for any

184
00:10:42,510 --> 00:10:45,880
arbitrary virtual displacements
that satisfy the

185
00:10:45,880 --> 00:10:48,050
displacement boundary
conditions.

186
00:10:48,050 --> 00:10:50,820
Of course, remember, as I
pointed out earlier, in an

187
00:10:50,820 --> 00:10:54,070
earlier lecture, that the
virtual strains must

188
00:10:54,070 --> 00:10:56,480
correspond to the virtual
displacements.

189
00:10:56,480 --> 00:11:01,040
And here we have the virtual
strains, once again.

190
00:11:01,040 --> 00:11:06,530
On the viewgraph we saw this
stress, and I call it the

191
00:11:06,530 --> 00:11:07,850
Cauchy stress.

192
00:11:07,850 --> 00:11:09,580
Here, you have it spelled
out once more.

193
00:11:09,580 --> 00:11:13,910
It's the force per unit area
at time t plus delta t.

194
00:11:13,910 --> 00:11:16,330
We also talked about the
virtual strains.

195
00:11:16,330 --> 00:11:20,580
And the virtual strains here,
this was the quantity R,

196
00:11:20,580 --> 00:11:22,790
defined as shown on the
right-hand side.

197
00:11:22,790 --> 00:11:26,170
Let me once again point strongly
out that we are

198
00:11:26,170 --> 00:11:29,320
differentiating here, with
respect to the coordinates, at

199
00:11:29,320 --> 00:11:31,490
time t plus delta t.

200
00:11:31,490 --> 00:11:34,710
Of course, the j and
the i coordinate.

201
00:11:34,710 --> 00:11:38,910
And also, please recognize that
this strain is really the

202
00:11:38,910 --> 00:11:43,000
infinitesimal strain tensor
that you are very well

203
00:11:43,000 --> 00:11:46,470
familiar with in infinitesimal
analysis, in infinitesimal

204
00:11:46,470 --> 00:11:51,140
displacement analysis, I should
say, except for one

205
00:11:51,140 --> 00:11:54,020
difference that we're using
here, the unknown coordinates

206
00:11:54,020 --> 00:11:55,650
at time t plus delta t.

207
00:11:59,180 --> 00:12:02,260
And that is spelled out down
here, that we are using a

208
00:12:02,260 --> 00:12:05,470
variation in the small strains
referred to the configuration,

209
00:12:05,470 --> 00:12:07,340
at time t plus delta t.

210
00:12:07,340 --> 00:12:11,000
So really, if you are familiar
with the principle of virtual

211
00:12:11,000 --> 00:12:14,820
work, as applied in
infinitesimal displacement

212
00:12:14,820 --> 00:12:20,080
analysis, then you recognize
that the principle of virtual

213
00:12:20,080 --> 00:12:23,980
work now applied to large
deformation analysis, large

214
00:12:23,980 --> 00:12:28,955
displacement analysis, is quite
the same, except that we

215
00:12:28,955 --> 00:12:32,020
are applying it to the current
geometry, the current

216
00:12:32,020 --> 00:12:36,060
configuration of the body.

217
00:12:36,060 --> 00:12:41,150
In my textbook, actually, I have
a little note, a footnote

218
00:12:41,150 --> 00:12:45,160
that you might also look at,
in the beginning of Chapter

219
00:12:45,160 --> 00:12:48,570
Six, where I discuss this
principle of virtual work,

220
00:12:48,570 --> 00:12:52,980
that one way to look at it is
that you see a body and, you

221
00:12:52,980 --> 00:12:55,900
see the body in front of
yourself moving through the

222
00:12:55,900 --> 00:12:57,820
stationary coordinate frame--

223
00:12:57,820 --> 00:13:00,130
once again, the Cartesian
coordinate frame is

224
00:13:00,130 --> 00:13:01,380
stationary.

225
00:13:01,380 --> 00:13:03,920
You see it move, you see the
body move through that

226
00:13:03,920 --> 00:13:05,180
coordinate frame.

227
00:13:05,180 --> 00:13:08,050
And there is somebody standing,
taking a picture.

228
00:13:08,050 --> 00:13:11,190
And that picture is taken
at time t plus delta t.

229
00:13:11,190 --> 00:13:12,990
And now you have a picture.

230
00:13:12,990 --> 00:13:16,320
And you apply the principle
of virtual work to that

231
00:13:16,320 --> 00:13:18,860
particular configuration
which has been

232
00:13:18,860 --> 00:13:20,930
captured in that picture.

233
00:13:20,930 --> 00:13:24,030
That's perhaps one way to look
at it, at least one way that I

234
00:13:24,030 --> 00:13:25,280
like to look at it.

235
00:13:27,970 --> 00:13:31,230
In order to solve, or to work
with the principal of virtual

236
00:13:31,230 --> 00:13:35,320
work, you need to rewrite the
principal, really, because we

237
00:13:35,320 --> 00:13:39,640
cannot integrate over an unknown
volume, and we cannot

238
00:13:39,640 --> 00:13:44,090
directly work with increments
in the Cauchy stresses.

239
00:13:44,090 --> 00:13:47,760
The volume t plus delta t,
remember, is unknown.

240
00:13:47,760 --> 00:13:52,150
And the reason why we cannot
directly work, or simply work

241
00:13:52,150 --> 00:13:54,740
with increments in Cauchy
stresses is because a Cauchy

242
00:13:54,740 --> 00:14:00,380
stress is always referred to the
current geometry, current

243
00:14:00,380 --> 00:14:01,490
configuration.

244
00:14:01,490 --> 00:14:05,005
So if we talk about the Cauchy
stress at time t, then it's

245
00:14:05,005 --> 00:14:08,200
the force per unit
area at time t.

246
00:14:08,200 --> 00:14:11,440
And the Cauchy stress at time
t plus delta t is the force

247
00:14:11,440 --> 00:14:14,260
per unit area at time
t plus delta t.

248
00:14:14,260 --> 00:14:19,290
Now you can't add a quantity
that is referred to time t to

249
00:14:19,290 --> 00:14:22,010
a quality that is referred to
time t plus delta t, because

250
00:14:22,010 --> 00:14:25,430
the areas, the reference
areas, have changed.

251
00:14:25,430 --> 00:14:28,590
And we have to somehow take
care of that in our

252
00:14:28,590 --> 00:14:29,790
formulation.

253
00:14:29,790 --> 00:14:33,090
And for that reason, we
introduce these two new

254
00:14:33,090 --> 00:14:34,210
quantities.

255
00:14:34,210 --> 00:14:39,720
The t 0 s is the 2nd
Piola-Kirchhoff stress tensor.

256
00:14:39,720 --> 00:14:43,680
And the t 0 epsilon is the
Green-Lagrange strain tensor.

257
00:14:43,680 --> 00:14:47,180
We introduce these quantities
because these are well known

258
00:14:47,180 --> 00:14:47,840
quantities.

259
00:14:47,840 --> 00:14:52,810
If you were to look into
continuum mechanics texts,

260
00:14:52,810 --> 00:14:55,790
they have been described
in many, many books.

261
00:14:55,790 --> 00:14:58,520
And they are well known
for some time.

262
00:14:58,520 --> 00:15:02,530
But we have extracted,
basically, this information

263
00:15:02,530 --> 00:15:05,660
from the continuum mechanics
literature, because it's a

264
00:15:05,660 --> 00:15:08,680
convenient, these are convenient
stress and strain

265
00:15:08,680 --> 00:15:12,170
measures to work with in
large deformation,

266
00:15:12,170 --> 00:15:14,200
finite element analysis.

267
00:15:14,200 --> 00:15:17,190
So, this is the stress
and that is the

268
00:15:17,190 --> 00:15:18,600
strain we will be using.

269
00:15:18,600 --> 00:15:20,750
And let me say the
following now.

270
00:15:20,750 --> 00:15:28,160
This t here, the upper t, means
stress at time t, in the

271
00:15:28,160 --> 00:15:30,000
configuration at time t.

272
00:15:30,000 --> 00:15:34,570
This lower 0 means that the
stress is referred to the

273
00:15:34,570 --> 00:15:37,520
configuration at time 0.

274
00:15:37,520 --> 00:15:39,980
Similarly for the Green-Lagrange
strain.

275
00:15:39,980 --> 00:15:44,890
Upper left, the configuration in
which the strain, or stress

276
00:15:44,890 --> 00:15:52,300
measure is measured, and the
lower subscript means, to

277
00:15:52,300 --> 00:15:55,470
which the configuration,
to which the

278
00:15:55,470 --> 00:15:58,020
measure is referred to.

279
00:15:58,020 --> 00:16:01,280
So this is something to keep
in mind, because we will be

280
00:16:01,280 --> 00:16:04,240
encountering this quite a bit.

281
00:16:04,240 --> 00:16:10,080
Here is the definition of the
stress tensor, t 0 s i j.

282
00:16:10,080 --> 00:16:14,250
Once again, the stress in
configuration t, referred to

283
00:16:14,250 --> 00:16:16,570
the configuration at time 0.

284
00:16:16,570 --> 00:16:19,320
i j, of course, being the
Cartesian coordinate

285
00:16:19,320 --> 00:16:20,740
components.

286
00:16:20,740 --> 00:16:23,870
Here we have the mass
density ratio.

287
00:16:23,870 --> 00:16:29,400
The mass density at time 0, in
configuration 0, if you like.

288
00:16:29,400 --> 00:16:34,630
The mass density at time t, in
configuration at time t.

289
00:16:34,630 --> 00:16:39,510
And here we have a new quantity,
which is actually

290
00:16:39,510 --> 00:16:41,700
the inverse of the deformation
gradient.

291
00:16:41,700 --> 00:16:43,785
This is a component
of the inverse of

292
00:16:43,785 --> 00:16:44,880
the deformation gradient.

293
00:16:44,880 --> 00:16:48,890
We will talk about this quantity
quite a bit just now.

294
00:16:48,890 --> 00:16:52,090
Here we have the
Cauchy stress.

295
00:16:52,090 --> 00:16:56,510
And here we have another
component such as that one.

296
00:16:56,510 --> 00:16:58,260
This is the 2nd Piola-Kirchhoff
stress.

297
00:16:58,260 --> 00:17:02,780
Once again, I will talk about
this piece of information

298
00:17:02,780 --> 00:17:04,790
quite a bit just now.

299
00:17:04,790 --> 00:17:07,720
The Green-Lagrange strain tensor
is defined as shown

300
00:17:07,720 --> 00:17:12,990
here, t 0 epsilon i j, i j, of
course, being the components

301
00:17:12,990 --> 00:17:20,690
into the Cartesian coordinate
axes, 1/2 t 0 u i comma j plus

302
00:17:20,690 --> 00:17:30,070
t zero u j comma i plus t 0 u k
comma i times t zero u k j.

303
00:17:30,070 --> 00:17:36,240
Notice that here we would select
a particular i and j,

304
00:17:36,240 --> 00:17:39,110
say i equals 1, j equals 2.

305
00:17:39,110 --> 00:17:43,200
You would substitute those i and
j numbers in here, so to

306
00:17:43,200 --> 00:17:45,790
say, and here as well.

307
00:17:45,790 --> 00:17:49,610
And notice that this k then,
would actually run over all

308
00:17:49,610 --> 00:17:50,750
possibilities.

309
00:17:50,750 --> 00:17:54,390
k going from 1, then
to 2, then to 3.

310
00:17:54,390 --> 00:17:57,920
In other words, there is a
summation involved here.

311
00:17:57,920 --> 00:18:00,660
By the way, similarly, we have
a summation involved here,

312
00:18:00,660 --> 00:18:07,370
because m and n would run over
all the possibilities.

313
00:18:07,370 --> 00:18:10,020
I will talk just now a little
bit more about it.

314
00:18:10,020 --> 00:18:13,950
But let's first now look at how
this particular piece of

315
00:18:13,950 --> 00:18:16,610
information here is defined.

316
00:18:16,610 --> 00:18:22,600
It's defined as a partial of 0
x i with respect to t x m.

317
00:18:22,600 --> 00:18:26,200
Notice that this coordinate,
of course, is given, is

318
00:18:26,200 --> 00:18:27,990
assumed to be given.

319
00:18:27,990 --> 00:18:30,590
And that one here is also
now assumed to be given.

320
00:18:30,590 --> 00:18:35,930
Notice, of course, that t x
m is equal to 0 x m plus a

321
00:18:35,930 --> 00:18:38,720
displacement that
has occurred.

322
00:18:38,720 --> 00:18:42,390
Notice that this differentiation
here is

323
00:18:42,390 --> 00:18:46,300
defined as given here, partial
t u i with respect to the

324
00:18:46,300 --> 00:18:48,170
original coordinates.

325
00:18:48,170 --> 00:18:53,300
So, this lower 0 here, this
lower 0 here refers to the

326
00:18:53,300 --> 00:18:55,690
fact that we're differentiating
with respect

327
00:18:55,690 --> 00:18:57,670
to the original coordinates.

328
00:18:57,670 --> 00:19:02,450
This upper t here means that
we are differentiating the

329
00:19:02,450 --> 00:19:04,610
displacement at time t.

330
00:19:04,610 --> 00:19:08,450
Notice, of course, that i and
j are dummy indices here.

331
00:19:08,450 --> 00:19:11,740
So, in fact, this one here
is what I'm considering.

332
00:19:11,740 --> 00:19:14,240
But at the same time, I'm
considering that one, if I

333
00:19:14,240 --> 00:19:16,100
just switch around i and j.

334
00:19:18,670 --> 00:19:22,370
So these are the definitions
of these two quantities.

335
00:19:22,370 --> 00:19:25,710
And I mentioned briefly already
that we are summing in

336
00:19:25,710 --> 00:19:26,980
these definitions.

337
00:19:26,980 --> 00:19:32,050
For example, t s 1 1 would be
calculated from the formulas

338
00:19:32,050 --> 00:19:34,650
that I gave you as 0
rho over t rho--

339
00:19:34,650 --> 00:19:37,480
this, of course, is a scalar--

340
00:19:37,480 --> 00:19:41,220
times this product here.

341
00:19:41,220 --> 00:19:42,270
What do we do here?

342
00:19:42,270 --> 00:19:47,140
We are selecting the
1 as that 1 here.

343
00:19:47,140 --> 00:19:50,980
That 1 here refers to that 1.

344
00:19:50,980 --> 00:19:54,880
And then we are letting these
components here, these

345
00:19:54,880 --> 00:19:58,180
components here, and that one
here, run over all the

346
00:19:58,180 --> 00:19:59,050
possibilities.

347
00:19:59,050 --> 00:20:01,900
Notice, here we have 1 2.

348
00:20:01,900 --> 00:20:05,950
That is this 1 here,
and that 1, 2.

349
00:20:05,950 --> 00:20:09,660
And here we have
3, 3 3, and 3.

350
00:20:09,660 --> 00:20:13,440
So in other words, these
first two are fixed.

351
00:20:13,440 --> 00:20:17,290
These first two here are fixed,
this 1 and that 1.

352
00:20:17,290 --> 00:20:20,200
They correspond to this
1, to these two 1s.

353
00:20:20,200 --> 00:20:23,570
But the other components vary
over all the possibilities,

354
00:20:23,570 --> 00:20:26,690
namely, 1, to 2, to 3.

355
00:20:26,690 --> 00:20:28,740
This is here the summation
convention, which is

356
00:20:28,740 --> 00:20:30,980
abundantly used in continuum
mechanics.

357
00:20:30,980 --> 00:20:35,290
And you might be very well
familiar with it.

358
00:20:35,290 --> 00:20:38,050
Using this 2nd Piola-Kirchhoff
stress and Green-Lagrange

359
00:20:38,050 --> 00:20:39,630
strain tensors--

360
00:20:39,630 --> 00:20:42,220
and that is really the important
point now-- we can

361
00:20:42,220 --> 00:20:49,820
rewrite the basic equation of
the principle of virtual work

362
00:20:49,820 --> 00:20:51,770
in the form given here.

363
00:20:51,770 --> 00:20:55,590
Here on the left-hand side, we
have the Cauchy stresses times

364
00:20:55,590 --> 00:21:01,080
the variations on the
infinitesimal strains, or the

365
00:21:01,080 --> 00:21:06,210
small, infinitesimally virtual
small strains.

366
00:21:06,210 --> 00:21:10,160
We integrate this product
over the current volume.

367
00:21:10,160 --> 00:21:12,970
And we mentioned already that
we cannot deal with this

368
00:21:12,970 --> 00:21:15,090
integration very well.

369
00:21:15,090 --> 00:21:19,250
Well, we now rewrite, we can
rewrite this integral as shown

370
00:21:19,250 --> 00:21:23,800
here, 2nd Piola-Kirchhoff
stresses times variations in

371
00:21:23,800 --> 00:21:26,760
the Green-Lagrange strains.

372
00:21:26,760 --> 00:21:30,070
And we integrate this, we are
integrating this product over

373
00:21:30,070 --> 00:21:32,370
the original volume
of the body.

374
00:21:32,370 --> 00:21:33,580
And that is the clue.

375
00:21:33,580 --> 00:21:35,540
We know this volume.

376
00:21:35,540 --> 00:21:38,120
We asked in general,
increment analysis.

377
00:21:38,120 --> 00:21:43,560
We would not know this volume
if, for example, we only have

378
00:21:43,560 --> 00:21:47,840
calculated the response
up to t minus delta t.

379
00:21:47,840 --> 00:21:55,070
So, if we know this volume we
can directly deal with this

380
00:21:55,070 --> 00:21:56,270
integration.

381
00:21:56,270 --> 00:21:59,030
And we will see also that since
these stress components

382
00:21:59,030 --> 00:22:03,410
are always referred to the same
volume, the same area, we

383
00:22:03,410 --> 00:22:06,170
can also incrementally decompose
these stresses.

384
00:22:06,170 --> 00:22:09,480
And the same holds
for the strains.

385
00:22:09,480 --> 00:22:12,615
Notice this relation, of course,
holds for all times,

386
00:22:12,615 --> 00:22:16,560
delta t, 2 delta t,
t, t plus delta t.

387
00:22:16,560 --> 00:22:18,960
The way we talked about the
principal of virtual work just

388
00:22:18,960 --> 00:22:23,160
now, we really had, on the
left-hand side here,

389
00:22:23,160 --> 00:22:24,810
t plus delta t.

390
00:22:24,810 --> 00:22:29,360
Well, you would just exchange
this t, that t, and that t,

391
00:22:29,360 --> 00:22:32,480
and that t here, to
a t plus delta t.

392
00:22:32,480 --> 00:22:36,950
And similarly, of course, on the
right-hand side as well.

393
00:22:36,950 --> 00:22:40,670
I chose at the preparation of
the viewgraph to simply call

394
00:22:40,670 --> 00:22:44,910
it t, because after all, this
equation holds anyways, for

395
00:22:44,910 --> 00:22:47,110
any of those particular times.

396
00:22:49,930 --> 00:22:52,350
To develop, then, the
incremental finite element

397
00:22:52,350 --> 00:22:55,680
equations, we use the
principle of virtual

398
00:22:55,680 --> 00:22:57,940
work in this form.

399
00:22:57,940 --> 00:23:02,130
If we assume that at, that the
response has been calculated

400
00:23:02,130 --> 00:23:07,630
from time 0 to time t, we apply
the principle of virtual

401
00:23:07,630 --> 00:23:12,040
work at time t plus delta t.

402
00:23:12,040 --> 00:23:14,200
We integrate, now, over
a known volume,

403
00:23:14,200 --> 00:23:15,380
as I pointed out.

404
00:23:15,380 --> 00:23:21,560
And, we can directly decompose
the stresses at time t plus

405
00:23:21,560 --> 00:23:26,380
delta t, and the Green-Lagrange
strains at time

406
00:23:26,380 --> 00:23:30,770
t plus delta t as shown
in these equations.

407
00:23:30,770 --> 00:23:33,950
Now, notice that here we have
the stress at time t plus

408
00:23:33,950 --> 00:23:39,820
delta t referred to the
configuration at time 0 being

409
00:23:39,820 --> 00:23:43,580
equal to the stress at time t
referred to the configuration

410
00:23:43,580 --> 00:23:46,380
at time 0 plus an incremental
that is also

411
00:23:46,380 --> 00:23:48,790
referred to time 0.

412
00:23:48,790 --> 00:23:52,250
And this is what we
would like to see.

413
00:23:52,250 --> 00:23:55,150
Similarly for the Green-Lagrange
strain.

414
00:23:55,150 --> 00:23:59,650
These two items make if for us
possible to work very well

415
00:23:59,650 --> 00:24:04,340
with the principle of virtual
work in this particular form.

416
00:24:04,340 --> 00:24:06,810
Before developing, however,
the incremental continuum

417
00:24:06,810 --> 00:24:10,910
mechanics equations, I'd like
to discuss with you now some

418
00:24:10,910 --> 00:24:15,840
important kinematic
relationships that we use in

419
00:24:15,840 --> 00:24:18,720
general non-linear analysis.

420
00:24:18,720 --> 00:24:22,510
The 2nd Piola-Kirchhoff stress
and Green-Lagrange strains

421
00:24:22,510 --> 00:24:28,610
have some very interesting,
properties that I think I'd

422
00:24:28,610 --> 00:24:32,210
like to share with you
in a discussion.

423
00:24:32,210 --> 00:24:37,540
To explain some of these
properties, we want to

424
00:24:37,540 --> 00:24:40,100
introduce the deformation
gradient tensor.

425
00:24:40,100 --> 00:24:44,910
The deformation gradient tensor
is actually something

426
00:24:44,910 --> 00:24:47,680
very, very basic in continuum
mechanics.

427
00:24:47,680 --> 00:24:52,040
And so, let us spend a little
bit of time on this tensor, to

428
00:24:52,040 --> 00:24:56,190
get also a bit of a physical
feel for what it means, what

429
00:24:56,190 --> 00:24:59,120
it stands for, and what
can we do with it.

430
00:24:59,120 --> 00:25:04,340
The tensor captures the
straining of the body and the

431
00:25:04,340 --> 00:25:06,100
rotation of the body.

432
00:25:06,100 --> 00:25:09,120
And, as I mentioned, it is
really a very fundamental

433
00:25:09,120 --> 00:25:11,155
quantity used in continuum
mechanics.

434
00:25:14,550 --> 00:25:17,470
The definition of
the deformation

435
00:25:17,470 --> 00:25:19,570
gradient is as follows.

436
00:25:19,570 --> 00:25:27,070
Here, we have t 0 x being equal
to a matrix, a large

437
00:25:27,070 --> 00:25:30,210
matrix, which, of course, is
written in a Cartesian

438
00:25:30,210 --> 00:25:32,320
coordinate system.

439
00:25:32,320 --> 00:25:35,770
And notice what we're doing in
the elements of that matrix.

440
00:25:35,770 --> 00:25:40,320
We take the partial of t x
1 with respect to 0 x 1.

441
00:25:40,320 --> 00:25:45,180
In other words, we're taking the
coordinates at time t and

442
00:25:45,180 --> 00:25:47,490
differentiate them with respect
to the original

443
00:25:47,490 --> 00:25:48,690
coordinates.

444
00:25:48,690 --> 00:25:52,020
Here, the 1 component with
respect to the 1 component.

445
00:25:52,020 --> 00:25:57,210
Here, the t x 1 with respect
to the 0 x 2.

446
00:25:57,210 --> 00:26:01,970
And, t x 1 with respect
to 0 x 3.

447
00:26:01,970 --> 00:26:04,900
Like that, we get a three
by three matrix.

448
00:26:04,900 --> 00:26:08,640
Using indicial notation, we can
simply say that an element

449
00:26:08,640 --> 00:26:13,440
of that matrix, t 0 X i j,
capital X, because the matrix

450
00:26:13,440 --> 00:26:19,030
is defined as the capital t 0 X,
being equal partial of t X

451
00:26:19,030 --> 00:26:21,700
i with respect to 0 x j.

452
00:26:21,700 --> 00:26:24,930
Of course, i j are
those i j's here.

453
00:26:24,930 --> 00:26:28,340
And, you will also find in the
textbook the following

454
00:26:28,340 --> 00:26:31,750
notation here, t 0 x i j.

455
00:26:31,750 --> 00:26:36,070
Notice, when I have a comma
here, we denote the

456
00:26:36,070 --> 00:26:40,850
differentiation by that comma,
and we use a little x.

457
00:26:40,850 --> 00:26:44,570
That is different from using
the capital X here, with no

458
00:26:44,570 --> 00:26:50,010
comma, because that capital X
simply denotes a component of,

459
00:26:50,010 --> 00:26:56,650
let's go up here once more,
the t 0 x of that matrix.

460
00:26:56,650 --> 00:27:00,090
So this is a mathematical
definition, really, of the

461
00:27:00,090 --> 00:27:01,670
deformation gradient.

462
00:27:01,670 --> 00:27:05,380
And notice that in order to
calculate the elements of this

463
00:27:05,380 --> 00:27:11,620
matrix, of course what you need
to be given is this t x

464
00:27:11,620 --> 00:27:21,310
1, t x 2, t x 3, as a function
of 0 x 1, 0 x 2, and 0 x 3.

465
00:27:21,310 --> 00:27:25,000
If you are given the deformation
field, t x i as a

466
00:27:25,000 --> 00:27:29,600
function of the 0 x j's, then
you can calculate directly the

467
00:27:29,600 --> 00:27:31,700
components of the three
by three matrix.

468
00:27:34,820 --> 00:27:38,770
Another way to write the
deformation gradient is via

469
00:27:38,770 --> 00:27:41,700
this equation.

470
00:27:41,700 --> 00:27:47,600
t 0 x can be written
as 0 del, del 0.

471
00:27:47,600 --> 00:27:49,790
This here is a gradient
operator.

472
00:27:49,790 --> 00:27:52,240
Let's look down here.

473
00:27:52,240 --> 00:27:53,550
We have defined it here.

474
00:27:56,560 --> 00:28:00,410
Times t x, or this gradient
operator operates, I should

475
00:28:00,410 --> 00:28:06,230
say, on t x capital T. This
capital T means transposed.

476
00:28:06,230 --> 00:28:12,640
And this t x capital T
is given right here.

477
00:28:12,640 --> 00:28:17,130
Notice what we're listing in
this vector here now, in this

478
00:28:17,130 --> 00:28:21,390
row vector, are simply the
coordinates at time t, t x 1,

479
00:28:21,390 --> 00:28:24,330
t x 2, t x 3.

480
00:28:24,330 --> 00:28:26,630
Notice that there is
a transpose here.

481
00:28:26,630 --> 00:28:29,930
That transpose is simply
introduced to make sure that

482
00:28:29,930 --> 00:28:34,050
this right-hand side is equal to
what we have defined on the

483
00:28:34,050 --> 00:28:37,220
previous viewgraph this to be.

484
00:28:37,220 --> 00:28:41,030
If you leave out that capital
T, you would not quite get

485
00:28:41,030 --> 00:28:44,020
what we actually have defined
on the previous viewgraph.

486
00:28:44,020 --> 00:28:47,010
So that's why we put that
capital T in there.

487
00:28:47,010 --> 00:28:51,560
It would be a nice exercise
for you to just actually

488
00:28:51,560 --> 00:28:55,600
substitute from here into there,
plug into there, and

489
00:28:55,600 --> 00:28:58,600
write out the three by three
matrix, to make sure that you

490
00:28:58,600 --> 00:29:02,580
actually get what's given here
on the left-hand side the way

491
00:29:02,580 --> 00:29:04,390
we defined it on the
previous viewgraph.

492
00:29:07,390 --> 00:29:10,230
The deformation gradient, as I
briefly mentioned, describes

493
00:29:10,230 --> 00:29:13,110
the deformations, the rotations,
and stretches of

494
00:29:13,110 --> 00:29:14,880
the material fibers.

495
00:29:14,880 --> 00:29:17,340
Let's look at some
examples now.

496
00:29:17,340 --> 00:29:24,010
For example, if we have d 0 x
here, that is the material

497
00:29:24,010 --> 00:29:26,780
fiber in its original
configuration.

498
00:29:26,780 --> 00:29:29,270
Of course, that original
configuration is going to be

499
00:29:29,270 --> 00:29:33,520
deformed, moved through the
stationary coordinate system.

500
00:29:33,520 --> 00:29:37,385
That fiber moves into
this situation here.

501
00:29:37,385 --> 00:29:39,560
It becomes d t x.

502
00:29:39,560 --> 00:29:44,745
The deformation gradient gives
us this element as a function

503
00:29:44,745 --> 00:29:48,740
of that element, as given
in this equation.

504
00:29:48,740 --> 00:29:54,980
In other words, d t x is equal
to t 0 x times d 0 x.

505
00:29:54,980 --> 00:29:59,940
If you know this piece of
information, and you have

506
00:29:59,940 --> 00:30:01,920
calculated the deformation
gradient, you

507
00:30:01,920 --> 00:30:04,510
directly get d t x.

508
00:30:04,510 --> 00:30:07,993
That's what the deformation
gradient does for us.

509
00:30:07,993 --> 00:30:10,090
Well, an example.

510
00:30:10,090 --> 00:30:13,700
It's always nice to look at some
examples to demonstrate

511
00:30:13,700 --> 00:30:16,780
what we mean by these
theoretical formulas.

512
00:30:16,780 --> 00:30:19,780
And here we have a very
simple example.

513
00:30:19,780 --> 00:30:22,070
It's a one dimensional
deformation.

514
00:30:22,070 --> 00:30:27,780
We start off with this piece
of material, a rectangular

515
00:30:27,780 --> 00:30:29,590
piece of material.

516
00:30:29,590 --> 00:30:35,750
And that piece of material is
stretched out from time 0 to

517
00:30:35,750 --> 00:30:39,680
time t into this red
configuration.

518
00:30:39,680 --> 00:30:43,660
Quite a bit of stretching, but
the motion is simple because

519
00:30:43,660 --> 00:30:46,810
it's a uni-axial motion.

520
00:30:46,810 --> 00:30:51,470
Notice, the original length of
this piece of material is 1,

521
00:30:51,470 --> 00:30:56,160
and this side here
moves out 0.5.

522
00:30:56,160 --> 00:30:58,890
Of course, the material
particles here will move.

523
00:30:58,890 --> 00:31:03,550
For example, a particle that was
originally here, the black

524
00:31:03,550 --> 00:31:08,480
little lines there, will move
over to the right, and it

525
00:31:08,480 --> 00:31:12,020
becomes this red little line.

526
00:31:12,020 --> 00:31:16,230
This black line moves over
to become that red line.

527
00:31:16,230 --> 00:31:21,320
Notice the displacement of this
black line is given by

528
00:31:21,320 --> 00:31:24,240
this blue arrow here.

529
00:31:24,240 --> 00:31:27,710
This black line moves over to
there, and the displacement is

530
00:31:27,710 --> 00:31:31,340
given by that blue arrow.

531
00:31:31,340 --> 00:31:35,350
If we know the displacements of
all these particles here,

532
00:31:35,350 --> 00:31:38,160
we know the deformation field.

533
00:31:38,160 --> 00:31:41,620
Vice versa, if the deformation
field is given, and now we

534
00:31:41,620 --> 00:31:46,060
look at this equation here, we
know the displacements of all

535
00:31:46,060 --> 00:31:47,320
of the particles along.

536
00:31:47,320 --> 00:31:51,040
And t x 1 is equal to 0 x
1 for this particular

537
00:31:51,040 --> 00:31:56,470
deformation field plus
0.5 0 x 1 squared.

538
00:31:56,470 --> 00:31:58,480
Let's look at what this means.

539
00:31:58,480 --> 00:32:02,710
It tells that if I plug in a
particular coordinate on the

540
00:32:02,710 --> 00:32:06,430
right-hand side of a material
particle in its original

541
00:32:06,430 --> 00:32:13,760
configuration, then I'm going to
get the final coordinate of

542
00:32:13,760 --> 00:32:18,860
that same material particle,
that same material particle.

543
00:32:18,860 --> 00:32:23,140
Well, the deformation gradient
is defined via this

544
00:32:23,140 --> 00:32:25,590
relationship, and it's
therefore given

545
00:32:25,590 --> 00:32:27,650
as 1 plus 0 x 1.

546
00:32:27,650 --> 00:32:31,590
Notice, we have only motion in
one direction, so we only

547
00:32:31,590 --> 00:32:32,870
calculate this one component.

548
00:32:35,850 --> 00:32:38,700
Let's consider, quite
physically, a material

549
00:32:38,700 --> 00:32:43,050
particle initially at
x 1 equals 0.8.

550
00:32:43,050 --> 00:32:46,610
In other words, x 1 is
this direction, 0 x

551
00:32:46,610 --> 00:32:49,790
1 is equal to 0.8.

552
00:32:49,790 --> 00:32:53,920
Remember, x 1 just gives us
the coordinate direction.

553
00:32:53,920 --> 00:32:57,850
We now put a 0 on it to signify
that 0.8 is the

554
00:32:57,850 --> 00:33:01,330
position of the particle
initially.

555
00:33:01,330 --> 00:33:05,010
That particle moves with the
deformation field that I've

556
00:33:05,010 --> 00:33:13,670
been given to this point here,
and t x 1 is then 1.120.

557
00:33:13,670 --> 00:33:18,920
The same can, of course, be
repeated for an adjacent

558
00:33:18,920 --> 00:33:20,480
material particle.

559
00:33:20,480 --> 00:33:23,210
In other words, let us look
now at a material particle

560
00:33:23,210 --> 00:33:26,620
that is just a little bit to the
right-hand side, namely 0

561
00:33:26,620 --> 00:33:32,430
x 1 being 0.850, slightly to
the right-hand side of the

562
00:33:32,430 --> 00:33:34,660
other material particle
that we looked at.

563
00:33:34,660 --> 00:33:37,690
This material particle, with the
deformation field that we

564
00:33:37,690 --> 00:33:43,410
have been prescribing, moves
to t x 1, this number here.

565
00:33:43,410 --> 00:33:48,870
We can now compute t 0 x 1 1,
the element of the deformation

566
00:33:48,870 --> 00:33:52,610
gradient as this difference
here.

567
00:33:52,610 --> 00:33:54,870
And the numbers are
plugged in here.

568
00:33:54,870 --> 00:33:57,140
And we obtain 1.82.

569
00:33:57,140 --> 00:34:01,260
Of course this is an estimate,
because we did not take a

570
00:34:01,260 --> 00:34:02,210
differential.

571
00:34:02,210 --> 00:34:05,855
If we actually calculate the
deformation gradient the way

572
00:34:05,855 --> 00:34:12,010
it's being defined, we get 1.80,
which is quite close to

573
00:34:12,010 --> 00:34:13,570
the estimate here.

574
00:34:13,570 --> 00:34:15,690
The reason for that difference,
of course, is that

575
00:34:15,690 --> 00:34:20,850
0 x 1, for this particle, is not
close enough, close enough

576
00:34:20,850 --> 00:34:23,620
to the earlier particle,
the other particle that

577
00:34:23,620 --> 00:34:24,510
we also looked at.

578
00:34:24,510 --> 00:34:30,719
In other words, 0.850 is not
close enough to 0.8, in order

579
00:34:30,719 --> 00:34:35,120
to get a very close relationship
here between

580
00:34:35,120 --> 00:34:36,900
these two numbers.

581
00:34:36,900 --> 00:34:39,750
Let's look at another
example, a two

582
00:34:39,750 --> 00:34:41,989
dimensional deformation example.

583
00:34:41,989 --> 00:34:46,190
Here now, we have as the
original configuration, this

584
00:34:46,190 --> 00:34:50,480
black body here, black outline
shown, in the coordinate

585
00:34:50,480 --> 00:34:51,730
system x 1 X 2.

586
00:34:51,730 --> 00:34:53,790
Once again, remember the
coordinate system is

587
00:34:53,790 --> 00:34:55,380
stationary.

588
00:34:55,380 --> 00:34:58,410
And let us look at two
material fibers, this

589
00:34:58,410 --> 00:34:59,460
one and that one.

590
00:34:59,460 --> 00:35:01,170
This one does not carry a hat.

591
00:35:01,170 --> 00:35:03,140
That one carries a hat.

592
00:35:03,140 --> 00:35:06,070
So these are two different
material fibers.

593
00:35:06,070 --> 00:35:12,740
These fibers move to here
from time 0 to time t.

594
00:35:12,740 --> 00:35:16,340
This fiber, d 0 x
becomes d t x.

595
00:35:16,340 --> 00:35:20,415
That fiber, d 0 x hat,
becomes d t x hat.

596
00:35:23,360 --> 00:35:27,910
In other words, of course, this
here in red shown, is the

597
00:35:27,910 --> 00:35:30,580
configuration at time t.

598
00:35:30,580 --> 00:35:34,650
Now, notice that these fibers
here have particular

599
00:35:34,650 --> 00:35:38,250
coordinates, 0 x 1, 0 x 2.

600
00:35:38,250 --> 00:35:40,610
You can think of these
coordinates to be at the

601
00:35:40,610 --> 00:35:42,920
beginning of that fiber.

602
00:35:42,920 --> 00:35:44,390
Of course, these are
differential fibers.

603
00:35:44,390 --> 00:35:47,790
They are very, very small.

604
00:35:47,790 --> 00:35:51,080
These coordinates move
over to t x 1, t x 2.

605
00:35:51,080 --> 00:35:54,580
In other words, the particle
from here moves to here and

606
00:35:54,580 --> 00:35:56,890
takes on these coordinates.

607
00:35:56,890 --> 00:36:00,310
We also, say, know,
that the whole--

608
00:36:00,310 --> 00:36:04,700
I mean we also know, the whole
deformation field.

609
00:36:04,700 --> 00:36:07,910
And therefore we can directly
calculate the deformation

610
00:36:07,910 --> 00:36:11,460
gradient, the way I explained
it, by taking the

611
00:36:11,460 --> 00:36:15,420
differentiation of the current
coordinates with respect to

612
00:36:15,420 --> 00:36:16,830
the original coordinates.

613
00:36:16,830 --> 00:36:21,970
And those differentiations
give us these elements.

614
00:36:21,970 --> 00:36:28,700
If we do so, we have the
relationships between the d 0

615
00:36:28,700 --> 00:36:32,560
x and d t x given
right down here.

616
00:36:32,560 --> 00:36:41,610
Notice, d 0 x has these
components in the x 1 x 2

617
00:36:41,610 --> 00:36:43,320
coordinate frame.

618
00:36:43,320 --> 00:36:49,710
And you can see, for example,
this 0.866, the component in

619
00:36:49,710 --> 00:36:53,240
the x 1 direction, this
0.5 is the component

620
00:36:53,240 --> 00:36:56,580
into the x 2 direction.

621
00:36:56,580 --> 00:37:00,710
If we multiply this
out, we get d t x.

622
00:37:00,710 --> 00:37:05,500
And d t x, shown here, has
no component into the x 2

623
00:37:05,500 --> 00:37:09,450
direction, but has a component
of 0.75 long

624
00:37:09,450 --> 00:37:11,360
into the x 1 direction.

625
00:37:11,360 --> 00:37:16,950
What this tells us, is that if
we know the motion of the body

626
00:37:16,950 --> 00:37:22,810
from time 0 to time t, and
particularly at time t, and if

627
00:37:22,810 --> 00:37:25,570
we have calculated the
deformation gradient the way

628
00:37:25,570 --> 00:37:31,880
we have defined it, then we can
relate how the original

629
00:37:31,880 --> 00:37:36,040
fibers, or the fibers in the
original configuration will

630
00:37:36,040 --> 00:37:39,350
rotate and stretch.

631
00:37:39,350 --> 00:37:43,550
And that relationship is given
via the deformation gradient,

632
00:37:43,550 --> 00:37:46,420
as surely shown here
quite physically.

633
00:37:46,420 --> 00:37:53,560
Because this fiber, originally
here, has rotated into a new

634
00:37:53,560 --> 00:37:56,500
position, and has, in this
particular case, been

635
00:37:56,500 --> 00:37:57,750
compressed.

636
00:37:59,490 --> 00:38:05,160
So, this relationship tells us
how the fiber has moved,

637
00:38:05,160 --> 00:38:07,610
rotated, and compressed.

638
00:38:07,610 --> 00:38:11,370
It does not tell us how
much the movement was.

639
00:38:11,370 --> 00:38:13,780
That is not given by
this relationship.

640
00:38:13,780 --> 00:38:17,410
That we have already, because
we have to have that

641
00:38:17,410 --> 00:38:20,610
information to calculate
t 0 x.

642
00:38:20,610 --> 00:38:24,040
But what this information, this
equation gives us, is how

643
00:38:24,040 --> 00:38:29,800
much the fiber has rotated and
compressed, or stretched.

644
00:38:29,800 --> 00:38:34,420
Well, let's look at another
fiber, the one that we had on

645
00:38:34,420 --> 00:38:36,330
the earlier viewgraph.

646
00:38:36,330 --> 00:38:38,700
Here, another fiber.

647
00:38:38,700 --> 00:38:42,910
And if we recognize, of course,
that for that material

648
00:38:42,910 --> 00:38:46,690
particle here, the same material
particle still, we

649
00:38:46,690 --> 00:38:49,470
still have the same deformation
gradient.

650
00:38:49,470 --> 00:38:54,840
We can take this fiber, or
rather the vectorial

651
00:38:54,840 --> 00:38:59,780
representation of that fiber,
as given here, multiply this

652
00:38:59,780 --> 00:39:03,090
out, and we directly find out
what has happened to that

653
00:39:03,090 --> 00:39:07,180
infinitesimal fiber in the
motion that we're considering.

654
00:39:07,180 --> 00:39:10,900
Notice, in this particular case,
the components of this

655
00:39:10,900 --> 00:39:15,570
fiber are 0 in this direction,
1.5 into that direction.

656
00:39:15,570 --> 00:39:22,310
And it has moved to here, and
has taken on components 1 and

657
00:39:22,310 --> 00:39:24,935
1 into each of the coordinate
directions.

658
00:39:27,670 --> 00:39:34,360
The deformation gradient is
very useful for various

659
00:39:34,360 --> 00:39:37,920
measures, computations
that we perform in

660
00:39:37,920 --> 00:39:39,570
finite element analysis.

661
00:39:39,570 --> 00:39:44,480
And one such computation
is to calculate the

662
00:39:44,480 --> 00:39:46,380
mass density ratios.

663
00:39:46,380 --> 00:39:51,120
In other words, we have to
assess how the mass density of

664
00:39:51,120 --> 00:39:56,850
the body, as it moves through
the space, changes.

665
00:39:56,850 --> 00:39:59,900
Let us look at how we do that.

666
00:39:59,900 --> 00:40:01,820
Here we have our stationary
coordinate

667
00:40:01,820 --> 00:40:04,760
system, x 1, x 2, x 3.

668
00:40:04,760 --> 00:40:07,380
And in that stationary
coordinate system, I'd like to

669
00:40:07,380 --> 00:40:11,050
focus our attention onto a
differential volume, a very

670
00:40:11,050 --> 00:40:12,490
small volume.

671
00:40:12,490 --> 00:40:16,340
I call that 0 d V curl.

672
00:40:16,340 --> 00:40:19,400
A curl is there because it is a
specific volume that we now

673
00:40:19,400 --> 00:40:20,740
want to look at.

674
00:40:20,740 --> 00:40:25,620
Notice that the volume that
I'm talking about here is

675
00:40:25,620 --> 00:40:30,200
spanned out by the
differential.

676
00:40:30,200 --> 00:40:36,390
Differential is d 0 x curl
1, 2 and 3 not shown.

677
00:40:36,390 --> 00:40:39,800
But this is the infinitesimal
volume at time 0.

678
00:40:39,800 --> 00:40:44,230
Now, in the motion from time
0 to time t, of course this

679
00:40:44,230 --> 00:40:46,280
volume will change.

680
00:40:46,280 --> 00:40:51,870
And at time t, it
looks like this.

681
00:40:51,870 --> 00:40:53,310
Notice that these

682
00:40:53,310 --> 00:40:55,530
differentials here have changed.

683
00:40:55,530 --> 00:41:01,540
We show just one, d 0 x curl
one, has gone over into d t x

684
00:41:01,540 --> 00:41:06,870
curl 1, the t now
denoting time t.

685
00:41:06,870 --> 00:41:11,870
And this is the volume obtained
from, that can be

686
00:41:11,870 --> 00:41:15,300
calculated from these
vectors shown here.

687
00:41:15,300 --> 00:41:18,580
In fact, this is how we
calculate this volume, and can

688
00:41:18,580 --> 00:41:21,970
therefore relate, also,
the mass densities.

689
00:41:21,970 --> 00:41:27,090
Let us look at some of
the basic equations.

690
00:41:27,090 --> 00:41:30,810
First of all, we can say that
the mass in this volume, this

691
00:41:30,810 --> 00:41:34,130
differential volume that we have
focused our attention on,

692
00:41:34,130 --> 00:41:37,430
and that I have given a curl
for that reason, that that

693
00:41:37,430 --> 00:41:40,460
mass must be preserved.

694
00:41:40,460 --> 00:41:43,710
And this means that this
relationship must hold.

695
00:41:43,710 --> 00:41:48,120
Volume times mass density at
time t must be equal to volume

696
00:41:48,120 --> 00:41:51,300
times mass density at time 0.

697
00:41:51,300 --> 00:41:54,880
We can show, then, that this
relationship holds.

698
00:41:54,880 --> 00:41:57,570
And I will talk about that just
now it a little bit more.

699
00:41:57,570 --> 00:42:01,660
And these two relationships,
then, clearly give us that

700
00:42:01,660 --> 00:42:04,790
this is how the mass
density changes.

701
00:42:04,790 --> 00:42:09,100
In other words, the mass density
at time t can directly

702
00:42:09,100 --> 00:42:12,440
be calculated from the mass
density at time 0 and the

703
00:42:12,440 --> 00:42:16,590
determinant of the deformation
gradient tensor.

704
00:42:16,590 --> 00:42:19,620
Notice, in three dimensional
analysis, this is a three by

705
00:42:19,620 --> 00:42:20,620
three matrix.

706
00:42:20,620 --> 00:42:23,930
And you would have to calculate
this matrix, at the

707
00:42:23,930 --> 00:42:27,330
particular point that you're
focusing your attention on,

708
00:42:27,330 --> 00:42:30,750
and take the determinant
of that matrix.

709
00:42:30,750 --> 00:42:33,340
Let us look at this relationship
here, because

710
00:42:33,340 --> 00:42:35,470
that is a very basic
relationship.

711
00:42:35,470 --> 00:42:40,290
It tells how the volume, the
differential volumes change as

712
00:42:40,290 --> 00:42:43,250
a function of time.

713
00:42:43,250 --> 00:42:48,440
And to show you how we arrive
at this relationship, I've

714
00:42:48,440 --> 00:42:50,950
just compiled here a
few viewgraphs to

715
00:42:50,950 --> 00:42:53,290
give you the proof.

716
00:42:53,290 --> 00:42:57,660
Initially, we can say that d 0
x curl 1 is given via this

717
00:42:57,660 --> 00:42:58,520
relationship.

718
00:42:58,520 --> 00:43:02,630
Notice, this is the length
of this vector, simply.

719
00:43:02,630 --> 00:43:06,710
And, notice that I had aligned
this vector with the x 1

720
00:43:06,710 --> 00:43:10,170
coordinate axis, therefore
these 0s.

721
00:43:10,170 --> 00:43:13,920
Similar, d 0 x curl 2 was
aligned with the x 2

722
00:43:13,920 --> 00:43:17,280
coordinate axis, and therefore
these two 0s.

723
00:43:17,280 --> 00:43:21,450
And the length of the
vector is d s 2.

724
00:43:21,450 --> 00:43:26,510
d x curl 3 is given similarly,
but aligned, this vector is

725
00:43:26,510 --> 00:43:30,590
aligned in the 3 direction, and
the length was given as,

726
00:43:30,590 --> 00:43:32,900
is given as d s 3.

727
00:43:32,900 --> 00:43:37,330
Hence, by simple calculation,
of course, we find directly

728
00:43:37,330 --> 00:43:40,750
that the volume, the
differential volume that in

729
00:43:40,750 --> 00:43:46,200
the original configuration, is
given via this equation.

730
00:43:46,200 --> 00:43:53,240
But, if we now recognize that
we can express d t x curl i

731
00:43:53,240 --> 00:43:57,510
via this relationship here, and
that is nothing else than

732
00:43:57,510 --> 00:44:01,380
applying the deformation
gradient tensor the way we

733
00:44:01,380 --> 00:44:04,340
already talked about.

734
00:44:04,340 --> 00:44:09,070
We looked already at examples
where we put in a particular

735
00:44:09,070 --> 00:44:15,220
fiber, original configuration,
and calculated how this fiber

736
00:44:15,220 --> 00:44:19,160
stretched, deformed from
time 0 to time t.

737
00:44:19,160 --> 00:44:21,180
I just gave you some examples.

738
00:44:21,180 --> 00:44:26,840
And we use that information now,
here, to express the d t

739
00:44:26,840 --> 00:44:31,580
x curl i in terms
of d 0 x curl i.

740
00:44:31,580 --> 00:44:34,530
Of course, for i
equals 1, 2, 3.

741
00:44:34,530 --> 00:44:39,420
And now we use here a formula
that you must have encountered

742
00:44:39,420 --> 00:44:43,680
some time ago, in your studies
of mathematics.

743
00:44:43,680 --> 00:44:47,380
If you look at your old math
books, I'm sure you can

744
00:44:47,380 --> 00:44:51,560
extract that formula, which
simply says that the volume

745
00:44:51,560 --> 00:44:56,720
given, spanned by these vectors,
is given via this

746
00:44:56,720 --> 00:44:59,300
product here, as a cross product
of these two vectors,

747
00:44:59,300 --> 00:45:02,610
and the dot product of the
resulting two vectors.

748
00:45:02,610 --> 00:45:05,750
This, of course, gives you a
vector here, and you dot this

749
00:45:05,750 --> 00:45:09,760
vector with this vector, and you
get, directly, a number,

750
00:45:09,760 --> 00:45:13,450
which is the volume
of interest.

751
00:45:13,450 --> 00:45:18,450
However, if we now multiply
this out here, you find,

752
00:45:18,450 --> 00:45:21,710
directly, that this
is the answer.

753
00:45:21,710 --> 00:45:24,050
In other words, the deformation
gradient comes in

754
00:45:24,050 --> 00:45:24,890
right there.

755
00:45:24,890 --> 00:45:26,140
The determinant of
the deformation

756
00:45:26,140 --> 00:45:27,500
gradient comes in here.

757
00:45:27,500 --> 00:45:29,850
Of course, it comes in here
because you would substitute

758
00:45:29,850 --> 00:45:32,080
from here into there.

759
00:45:32,080 --> 00:45:35,800
And once again, since this
product here is nothing else

760
00:45:35,800 --> 00:45:39,090
than the original volume,
we're home.

761
00:45:39,090 --> 00:45:44,730
We have proven that d t V curl
is equal to the determinant of

762
00:45:44,730 --> 00:45:50,640
t 0 x times 0 d V curl, the
determinate of the deformation

763
00:45:50,640 --> 00:45:53,660
gradient in this relationship
between the

764
00:45:53,660 --> 00:45:55,350
differential volumes.

765
00:45:55,350 --> 00:45:59,860
It's really an interesting
proof, and maybe you want to

766
00:45:59,860 --> 00:46:03,570
think a little bit more about
it, and even go through this

767
00:46:03,570 --> 00:46:07,860
arithmetic here, to really
reinforce the understanding of

768
00:46:07,860 --> 00:46:10,240
what we are doing here.

769
00:46:10,240 --> 00:46:17,910
Let's look at a simple example
to just exemplify once again

770
00:46:17,910 --> 00:46:19,640
what we just talked about.

771
00:46:19,640 --> 00:46:23,020
Here we have a little example.

772
00:46:23,020 --> 00:46:28,150
A piece of material 1 long,
that in its original

773
00:46:28,150 --> 00:46:30,200
configuration is 1 long,
I should say.

774
00:46:30,200 --> 00:46:33,230
We are stretching that
piece of material up

775
00:46:33,230 --> 00:46:36,290
to time t, as shown.

776
00:46:36,290 --> 00:46:38,330
We assume uniform stretching.

777
00:46:38,330 --> 00:46:40,300
We assume plain strain
conditions.

778
00:46:40,300 --> 00:46:44,810
In other words, the strain
through the thickness is equal

779
00:46:44,810 --> 00:46:48,520
to 0 here, in this
particular case.

780
00:46:48,520 --> 00:46:53,290
And the deformation field is
assumed to be as shown here.

781
00:46:53,290 --> 00:46:57,500
Then, with the deformation field
given, clearly we can

782
00:46:57,500 --> 00:46:59,640
calculate this deformation
gradient.

783
00:46:59,640 --> 00:47:04,150
Here, this component is simply
obtained by differentiating t

784
00:47:04,150 --> 00:47:06,890
x 1 with respect to 0 x 1.

785
00:47:06,890 --> 00:47:14,590
If you do so, you get a 1 here
plus 0.25, giving you a 1.25.

786
00:47:14,590 --> 00:47:17,630
Because there is no shearing
in the material, and no

787
00:47:17,630 --> 00:47:21,370
rotation, these off diagonal
terms are 0.

788
00:47:21,370 --> 00:47:24,900
And, because there is no
stretching through this

789
00:47:24,900 --> 00:47:28,310
direction, and through the
thickness direction, we have

790
00:47:28,310 --> 00:47:29,960
1s right there.

791
00:47:29,960 --> 00:47:32,700
To take the determinant of this
matrix is rather simple.

792
00:47:32,700 --> 00:47:36,430
You just take the product of
the diagonal terms, and you

793
00:47:36,430 --> 00:47:40,590
directly obtain 1.25 for
its determinant.

794
00:47:40,590 --> 00:47:43,730
Now, let us plug this
into our formula.

795
00:47:43,730 --> 00:47:45,930
And here we have the result
of that formula.

796
00:47:45,930 --> 00:47:53,200
It tells us 0 rho is equal to
1.25 t rho for any particle,

797
00:47:53,200 --> 00:47:57,580
right here, because this
relationship is independent of

798
00:47:57,580 --> 00:47:59,410
the 0 x 1 component.

799
00:47:59,410 --> 00:48:02,410
Well, of course, this is also
a result of the uniform

800
00:48:02,410 --> 00:48:04,050
stretching.

801
00:48:04,050 --> 00:48:07,370
Notice that this makes
physical sense.

802
00:48:07,370 --> 00:48:11,930
If we have taken a piece
of material up here--

803
00:48:11,930 --> 00:48:13,580
once more let us look at it--

804
00:48:13,580 --> 00:48:16,810
that does not shrink here,
that does not stretch or

805
00:48:16,810 --> 00:48:20,840
shrink in this normal direction
as well, then surely

806
00:48:20,840 --> 00:48:24,920
the mass density has to decrease
as we stretch it.

807
00:48:24,920 --> 00:48:29,550
And in fact, this is being shown
right down here, t rho

808
00:48:29,550 --> 00:48:33,330
is equal to 0 rho
divided by 1.25.

809
00:48:33,330 --> 00:48:35,340
So, the result makes sense.

810
00:48:38,070 --> 00:48:41,930
We use also the inverse
deformation gradient.

811
00:48:41,930 --> 00:48:47,660
And here, on this viewgraph,
I've just summarized a few

812
00:48:47,660 --> 00:48:51,150
equations regarding this inverse
deformation gradient.

813
00:48:51,150 --> 00:48:53,400
Here, we now look at the
equation, where on the

814
00:48:53,400 --> 00:48:58,810
left-hand side we have the
material fiber at time 0, and

815
00:48:58,810 --> 00:49:01,980
on the right-hand side is the
material fiber at time t.

816
00:49:01,980 --> 00:49:06,270
In other words, d 0 x on the
left-hand side, and d t x on

817
00:49:06,270 --> 00:49:07,530
the right-hand side.

818
00:49:07,530 --> 00:49:11,340
The inverse deformation gradient
stands right here.

819
00:49:11,340 --> 00:49:15,050
Mathematically, we can show that
the inverse deformation

820
00:49:15,050 --> 00:49:19,040
gradient is nothing else
than t 0 x inverse.

821
00:49:19,040 --> 00:49:24,340
Of course, notice that 0 t
x would be calculated by

822
00:49:24,340 --> 00:49:28,950
differentiating the original
coordinates with respect to

823
00:49:28,950 --> 00:49:31,820
the coordinates at time t.

824
00:49:31,820 --> 00:49:36,350
However, we can also calculate
this matrix directly by taking

825
00:49:36,350 --> 00:49:41,690
this matrix here, where we
differentiate the coordinates

826
00:49:41,690 --> 00:49:45,080
at time t with respect to the
original coordinates.

827
00:49:45,080 --> 00:49:47,490
And then, having obtained this
matrix, calculated this

828
00:49:47,490 --> 00:49:49,980
matrix, we simply invert it.

829
00:49:49,980 --> 00:49:52,710
Well the proof that indeed
holds is given

830
00:49:52,710 --> 00:49:54,080
down here in green.

831
00:49:54,080 --> 00:49:59,755
We simply take the top equation
and substitute for d

832
00:49:59,755 --> 00:50:04,690
t x, as shown here, our
relationship that we had

833
00:50:04,690 --> 00:50:05,900
already earlier.

834
00:50:05,900 --> 00:50:10,590
And, if we then, well, look at
this equation, we can put

835
00:50:10,590 --> 00:50:12,130
brackets around here.

836
00:50:12,130 --> 00:50:18,580
And since d 0 x is equal to d
0 x for any material fiber,

837
00:50:18,580 --> 00:50:20,770
this must be the identity
matrix, which already

838
00:50:20,770 --> 00:50:22,790
completes the proof.

839
00:50:22,790 --> 00:50:25,610
An important theorem in
continuum mechanics is the

840
00:50:25,610 --> 00:50:30,080
polar decomposition theorem,
which tells that the

841
00:50:30,080 --> 00:50:35,040
deformation gradient, t 0 x, can
always be decomposed into

842
00:50:35,040 --> 00:50:39,150
a rotation matrix and
a stretch matrix.

843
00:50:39,150 --> 00:50:43,210
The rotation matrix is an
orthogonal matrix, of course,

844
00:50:43,210 --> 00:50:47,290
meaning that R, transposed R,
is the identity matrix.

845
00:50:47,290 --> 00:50:51,130
And the stretch matrix is
a symmetric matrix.

846
00:50:51,130 --> 00:50:54,890
This can always be
done for any x.

847
00:50:54,890 --> 00:50:58,580
And the proof, actually, is
given in the textbook.

848
00:50:58,580 --> 00:51:02,140
Please look at your study guide,
where I indicate where

849
00:51:02,140 --> 00:51:04,120
you can find the proof
in the textbook.

850
00:51:04,120 --> 00:51:07,180
And it might be quite good to
actually go through that proof

851
00:51:07,180 --> 00:51:09,890
in some detail, so as to
reinforce your understanding

852
00:51:09,890 --> 00:51:13,570
of what this theorem
really tells.

853
00:51:13,570 --> 00:51:16,370
The important point, once again,
is that we can always

854
00:51:16,370 --> 00:51:21,930
write the x matrix into this
form as R times u.

855
00:51:21,930 --> 00:51:27,810
And to give you an example of
what this means, let's look at

856
00:51:27,810 --> 00:51:29,280
this one here.

857
00:51:29,280 --> 00:51:34,560
Here we have, in the coordinate
frame x 1 x 2, a

858
00:51:34,560 --> 00:51:39,670
piece of material
3 long 2 wide.

859
00:51:39,670 --> 00:51:42,840
This is the original
configuration of the material.

860
00:51:42,840 --> 00:51:46,660
And that material moves
from time 0 to

861
00:51:46,660 --> 00:51:48,750
time t, as shown here.

862
00:51:48,750 --> 00:51:53,140
First it moves, it is being
stretched into the x 1

863
00:51:53,140 --> 00:51:56,230
direction and into the x 2
direction to take on the

864
00:51:56,230 --> 00:51:58,290
configuration shown in red.

865
00:51:58,290 --> 00:52:02,270
And then it rotates to take on
the configuration in green.

866
00:52:02,270 --> 00:52:06,160
And that, indeed, is the
configuration at time t.

867
00:52:06,160 --> 00:52:10,130
Let us calculate the x matrix
corresponding to that

868
00:52:10,130 --> 00:52:13,570
configuration, and see how
it would be decomposed

869
00:52:13,570 --> 00:52:15,530
into R times u.

870
00:52:15,530 --> 00:52:17,750
Well, we have done
that down here.

871
00:52:17,750 --> 00:52:21,200
Notice, this is the
x matrix, t 0 x.

872
00:52:21,200 --> 00:52:24,040
These are the components
of that matrix.

873
00:52:24,040 --> 00:52:28,970
And here we have the R matrix,
t 0 R. And here we

874
00:52:28,970 --> 00:52:31,240
have the t 0 u.

875
00:52:31,240 --> 00:52:34,440
In other words, this is the
rotation matrix, and this is

876
00:52:34,440 --> 00:52:35,890
the stretch matrix.

877
00:52:35,890 --> 00:52:39,330
Notice this stretch matrix
is indeed symmetric.

878
00:52:39,330 --> 00:52:41,910
Let's look at the components in
the stretch matrix a little

879
00:52:41,910 --> 00:52:43,330
bit closer.

880
00:52:43,330 --> 00:52:50,215
1.33 is nothing else than 4
divided by 3, in other words,

881
00:52:50,215 --> 00:52:53,610
the stretching into
the x 1 direction.

882
00:52:53,610 --> 00:52:58,880
1.5, down here, is nothing else
than the stretching into

883
00:52:58,880 --> 00:53:03,810
x 2 direction, namely
2 goes over into 3.

884
00:53:03,810 --> 00:53:06,850
3 divided by 2, 1.5.

885
00:53:06,850 --> 00:53:10,370
Notice that these off diagonal
elements are 0, the reason

886
00:53:10,370 --> 00:53:15,150
being that there has been no
shearing from the original

887
00:53:15,150 --> 00:53:19,760
configuration to the
red configuration.

888
00:53:19,760 --> 00:53:23,170
The rotation matrix really
expresses mathematically the

889
00:53:23,170 --> 00:53:27,500
movement of this red piece of
material into the green

890
00:53:27,500 --> 00:53:29,030
configuration.

891
00:53:29,030 --> 00:53:32,170
Notice that the entries in
this rotation matrix are

892
00:53:32,170 --> 00:53:34,790
nothing else than cosines
and sines of

893
00:53:34,790 --> 00:53:37,290
this 30-degree rotation.

894
00:53:37,290 --> 00:53:41,670
For example, here you have
the cosine of 30 degrees.

895
00:53:41,670 --> 00:53:43,230
This is the sine
of 30 degrees.

896
00:53:43,230 --> 00:53:46,010
In fact, it's the rotation of
this material fiber right

897
00:53:46,010 --> 00:53:48,590
there, from here into there.

898
00:53:48,590 --> 00:53:51,830
Well, this is the rotation
matrix and stretch matrix for

899
00:53:51,830 --> 00:53:53,090
this example.

900
00:53:53,090 --> 00:53:56,510
And we could easily construct
other examples.

901
00:53:56,510 --> 00:53:59,420
In fact, you find in the
textbook some more examples

902
00:53:59,420 --> 00:54:04,090
that you might want to study
regarding this particular

903
00:54:04,090 --> 00:54:06,400
phenomenon.

904
00:54:06,400 --> 00:54:09,950
Using the deformation gradient,
we can describe, we

905
00:54:09,950 --> 00:54:13,700
can define a Cauchy-Green
deformation tensor.

906
00:54:13,700 --> 00:54:15,730
There is a right Cauchy-Green
deformation tensor.

907
00:54:15,730 --> 00:54:18,060
There is also a left
Cauchy-Green deformation

908
00:54:18,060 --> 00:54:20,480
tensor which, however,
we will not be using.

909
00:54:20,480 --> 00:54:22,910
So, when I talk about the
Cauchy-Green deformation

910
00:54:22,910 --> 00:54:24,950
tensor, I really mean the right

911
00:54:24,950 --> 00:54:26,760
Cauchy-Green deformation tensor.

912
00:54:26,760 --> 00:54:28,670
And it's defined
as shown here.

913
00:54:28,670 --> 00:54:31,300
We take the deformation
gradient, transposed, times

914
00:54:31,300 --> 00:54:32,940
the deformation gradient.

915
00:54:32,940 --> 00:54:39,040
Now, notice the deformation
gradient is always a three by

916
00:54:39,040 --> 00:54:43,120
three matrix, in a three
dimensional coordinate space.

917
00:54:43,120 --> 00:54:45,230
However, it can be
a non-symmetric

918
00:54:45,230 --> 00:54:46,600
or symmetric matrix.

919
00:54:46,600 --> 00:54:48,660
Generally, it is a non-symmetric
matrix.

920
00:54:48,660 --> 00:54:51,470
It would be a symmetric matrix
if there has been no rotation

921
00:54:51,470 --> 00:54:52,740
points on it.

922
00:54:52,740 --> 00:54:54,960
But, in general, it's
non-symmetric.

923
00:54:54,960 --> 00:54:59,290
Notice that this non-symmetric
matrix, multiplied here, as

924
00:54:59,290 --> 00:55:03,320
shown by the non-symmetric
matrix transposed, makes c

925
00:55:03,320 --> 00:55:06,570
symmetric, a very important
information.

926
00:55:06,570 --> 00:55:12,980
Well, indeed, if we substitute
for x here, we find that c is

927
00:55:12,980 --> 00:55:17,600
nothing else than
t 0 u squared.

928
00:55:17,600 --> 00:55:21,540
In other words, the stretch
matrix squared, since R is

929
00:55:21,540 --> 00:55:21,940
orthogonal.

930
00:55:21,940 --> 00:55:22,980
R goes in here.

931
00:55:22,980 --> 00:55:26,410
But R transposed R, of course,
is the identity matrix.

932
00:55:26,410 --> 00:55:30,780
And therefore c is
simply u squared.

933
00:55:30,780 --> 00:55:35,660
Notice that t 0 c is independent
of the rotation,

934
00:55:35,660 --> 00:55:38,590
via this equation here, and
therefore we say that c is

935
00:55:38,590 --> 00:55:41,570
invariant under a rigid
body rotation.

936
00:55:41,570 --> 00:55:44,790
In other words, a rigid body
rotation of a material fiber

937
00:55:44,790 --> 00:55:49,590
does not enter into
the elements of c.

938
00:55:49,590 --> 00:55:52,280
Let's look at an example.

939
00:55:52,280 --> 00:55:55,500
Two dimensional motion, the
original configuration for the

940
00:55:55,500 --> 00:55:58,170
piece of material that I like
to consider is shown here.

941
00:55:58,170 --> 00:56:03,000
It moves from time 0 to time t,
to the red configuration.

942
00:56:03,000 --> 00:56:06,330
And then, in the increment of
time delta t to that green

943
00:56:06,330 --> 00:56:07,510
configuration.

944
00:56:07,510 --> 00:56:12,040
But this movement over time
delta t is simply a rigid body

945
00:56:12,040 --> 00:56:14,620
motion of 90 degrees.

946
00:56:14,620 --> 00:56:18,430
Well, we can calculate t 0
x corresponding to this

947
00:56:18,430 --> 00:56:19,500
configuration.

948
00:56:19,500 --> 00:56:20,800
Here it's done.

949
00:56:20,800 --> 00:56:23,930
And, by the formula that I gave
you, you get directly t 0

950
00:56:23,930 --> 00:56:27,480
c by taking x transposed
times x.

951
00:56:27,480 --> 00:56:29,480
You get this matrix here.

952
00:56:29,480 --> 00:56:33,550
Well, now we can also
calculate t 0 x.

953
00:56:33,550 --> 00:56:35,800
How do we obtain, by the
way, this matrix?

954
00:56:35,800 --> 00:56:37,140
There is a subtle point.

955
00:56:37,140 --> 00:56:41,570
You obtain this matrix simply by
pre-multiplying this matrix

956
00:56:41,570 --> 00:56:44,920
by a rotation matrix
corresponding to this

957
00:56:44,920 --> 00:56:47,150
90-degree rotation.

958
00:56:47,150 --> 00:56:49,040
This is the answer.

959
00:56:49,040 --> 00:56:54,180
And if we now take this matrix,
transposed by the

960
00:56:54,180 --> 00:56:58,030
matrix itself, in other words
this product x transpose x, we

961
00:56:58,030 --> 00:56:59,430
get the c matrix.

962
00:56:59,430 --> 00:57:04,670
And we notice that the c
matrices are the same for both

963
00:57:04,670 --> 00:57:06,500
of these configurations.

964
00:57:06,500 --> 00:57:11,750
In other words, this simply
exemplifies that the rigid

965
00:57:11,750 --> 00:57:17,370
body rotation from red to green
here did not enter into

966
00:57:17,370 --> 00:57:19,840
the elements of the c.

967
00:57:19,840 --> 00:57:23,405
They did not change from time
t to time t plus delta t.

968
00:57:25,940 --> 00:57:32,380
We use this c matrix to define
a strain measure, the

969
00:57:32,380 --> 00:57:34,750
Green-Lagrange strain measure.

970
00:57:34,750 --> 00:57:36,720
It's, of course, a strain
measure that is very well

971
00:57:36,720 --> 00:57:40,470
known, described in many,
many textbooks.

972
00:57:40,470 --> 00:57:43,600
And it is, however, very useful
for finite element

973
00:57:43,600 --> 00:57:47,170
analysis, as I briefly pointed
out already earlier.

974
00:57:47,170 --> 00:57:50,930
This is the definition of the
Green-Lagrange strain tensor.

975
00:57:50,930 --> 00:57:54,260
Notice, this here is a three
by three matrix.

976
00:57:54,260 --> 00:57:57,160
And we are taking this three
by three matrix.

977
00:57:57,160 --> 00:58:00,870
We subtract the identity
matrix.

978
00:58:00,870 --> 00:58:03,000
The identity matrix, of course,
being just 1s on the

979
00:58:03,000 --> 00:58:06,740
diagonal, and 0s on all the
off diagonal elements.

980
00:58:06,740 --> 00:58:10,150
And there is a 1/2 here that
gives us the definition of the

981
00:58:10,150 --> 00:58:12,670
Green-Lagrange strain tensor.

982
00:58:12,670 --> 00:58:16,140
There are a number of very
interesting properties that

983
00:58:16,140 --> 00:58:18,640
this strain tensor displays.

984
00:58:18,640 --> 00:58:20,580
And I've listed them here.

985
00:58:20,580 --> 00:58:24,170
We can see that this strain
tensor is symmetric.

986
00:58:24,170 --> 00:58:27,960
We can see that this strain
tensor does not change from

987
00:58:27,960 --> 00:58:33,340
time t to time t plus delta t if
the only motion from time t

988
00:58:33,340 --> 00:58:37,850
to time t plus delta t was
a rigid body motion.

989
00:58:37,850 --> 00:58:41,550
And we can see directly that
the Green-Lagrange strain

990
00:58:41,550 --> 00:58:49,570
tensor is 0 if, from time 0 to
time t, the only motion was a

991
00:58:49,570 --> 00:58:52,950
rigid body motion.

992
00:58:52,950 --> 00:58:57,690
One can prove these particular
items, these statements, and I

993
00:58:57,690 --> 00:59:00,040
would like to encourage
you to do so.

994
00:59:00,040 --> 00:59:02,770
While you are doing it, I
will do the same here.

995
00:59:02,770 --> 00:59:05,900
And then we share our
experiences regarding these

996
00:59:05,900 --> 00:59:07,150
proofs just afterwards.

997
00:59:12,160 --> 00:59:15,830
Well, here I am giving now to
you some information regarding

998
00:59:15,830 --> 00:59:17,350
these proofs.

999
00:59:17,350 --> 00:59:22,240
t 0 epsilon is symmetric because
t 0 c is symmetric.

1000
00:59:22,240 --> 00:59:25,480
And remember, we take here a
symmetric matrix from which we

1001
00:59:25,480 --> 00:59:28,910
subtract a symmetric matrix, and
clearly, the result must

1002
00:59:28,910 --> 00:59:30,460
be a symmetric matrix.

1003
00:59:30,460 --> 00:59:33,620
A very simple proof really.

1004
00:59:33,620 --> 00:59:36,780
Regarding the second item that I
mentioned earlier, for rigid

1005
00:59:36,780 --> 00:59:41,280
body motion, we know that from
time t to time t plus delta t,

1006
00:59:41,280 --> 00:59:47,270
we have that x at time t plus
delta t can be written as R

1007
00:59:47,270 --> 00:59:49,420
times t 0 x.

1008
00:59:49,420 --> 00:59:51,860
In other words, here I'm talking
about the deformation

1009
00:59:51,860 --> 00:59:55,420
gradient at time t plus delta
t, and here about the

1010
00:59:55,420 --> 00:59:57,020
deformation gradient
at time t.

1011
00:59:57,020 --> 01:00:01,040
And they are related merely
by a rotation matrix.

1012
01:00:01,040 --> 01:00:07,120
Now, if you substitute from here
into this equation, but

1013
01:00:07,120 --> 01:00:10,620
apply the time t plus delta t,
you obtain, of course, as we

1014
01:00:10,620 --> 01:00:14,520
saw actually on the earlier
example, that this c matrix

1015
01:00:14,520 --> 01:00:16,250
does not change.

1016
01:00:16,250 --> 01:00:19,700
And therefore, also the
Green-Lagrange stain tensor

1017
01:00:19,700 --> 01:00:21,160
does not change.

1018
01:00:21,160 --> 01:00:23,870
In fact, all of this information
was really

1019
01:00:23,870 --> 01:00:27,740
addressed in the earlier
examples that we discussed.

1020
01:00:27,740 --> 01:00:31,420
Also, for rigid body motion, we
have that the c matrix is

1021
01:00:31,420 --> 01:00:34,540
simply the identity matrix,
because remember, the c

1022
01:00:34,540 --> 01:00:38,190
matrix, Cauchy-Green deformation
tensor, does not

1023
01:00:38,190 --> 01:00:41,090
change with rigid
body rotation.

1024
01:00:41,090 --> 01:00:43,260
It is equal to the
identity matrix.

1025
01:00:43,260 --> 01:00:49,350
And if you then takes this c
matrix and substitute right up

1026
01:00:49,350 --> 01:00:53,640
there, on the top, you get that
the Green-Lagrange strain

1027
01:00:53,640 --> 01:00:55,640
tensor is 0.

1028
01:00:55,640 --> 01:01:01,080
So, 3 properties, really, that
we want to keep in mind

1029
01:01:01,080 --> 01:01:04,120
regarding the Green-Lagrange
strain tensor when we

1030
01:01:04,120 --> 01:01:07,040
later on use it.

1031
01:01:07,040 --> 01:01:09,710
Another definition of the
Green-Lagrange strain tensor

1032
01:01:09,710 --> 01:01:12,170
is given on this viewgraph.

1033
01:01:12,170 --> 01:01:15,180
In fact, this right-hand side
that I'm showing here is

1034
01:01:15,180 --> 01:01:20,130
simply obtained by substituting,
for the

1035
01:01:20,130 --> 01:01:26,080
deformation gradient, the
identity matrix plus the

1036
01:01:26,080 --> 01:01:29,380
deformation while the
displacement derivatives, with

1037
01:01:29,380 --> 01:01:31,190
respect to the coordinates.

1038
01:01:31,190 --> 01:01:34,520
In other words, what you want
to do, basically, is to

1039
01:01:34,520 --> 01:01:40,600
substitute for t 0 x, in
terms of displacements.

1040
01:01:40,600 --> 01:01:45,070
And if you go through and write
all the information out,

1041
01:01:45,070 --> 01:01:49,510
you end up that the t 0 epsilon
i j component of the

1042
01:01:49,510 --> 01:01:52,290
Green-Lagrange strain
tensor is given by

1043
01:01:52,290 --> 01:01:54,610
this equation here.

1044
01:01:54,610 --> 01:01:58,370
Notice that we have here i, j,
the same subscript that you

1045
01:01:58,370 --> 01:01:59,370
see here, too.

1046
01:01:59,370 --> 01:02:01,250
Of course, we're differentiating
with respect

1047
01:02:01,250 --> 01:02:02,960
to the j component here.

1048
01:02:02,960 --> 01:02:07,240
Here we have j, i, and
here k, i k,j.

1049
01:02:07,240 --> 01:02:09,860
We're summing over k here.

1050
01:02:09,860 --> 01:02:13,280
Notice also, that this part
here is non-linear in the

1051
01:02:13,280 --> 01:02:14,540
displacements.

1052
01:02:14,540 --> 01:02:17,510
This part is non-linear in the
displacements because these

1053
01:02:17,510 --> 01:02:20,120
are, so to say, quadratic
terms.

1054
01:02:20,120 --> 01:02:24,990
This here, this part, is linear
in displacements.

1055
01:02:24,990 --> 01:02:28,540
From time to time, students
approach me and say to me,

1056
01:02:28,540 --> 01:02:32,530
well, you must have neglected
something in the definition,

1057
01:02:32,530 --> 01:02:35,460
or in this Green-Lagrange
strain tensor.

1058
01:02:35,460 --> 01:02:38,150
Because you're only going
up to quadratic terms.

1059
01:02:38,150 --> 01:02:42,170
you must have neglected cubic
terms, and high order terms.

1060
01:02:42,170 --> 01:02:45,300
Well, the answer there is, we
have really neglected nothing.

1061
01:02:45,300 --> 01:02:50,520
The point is, that we have
defined this tensor to be

1062
01:02:50,520 --> 01:02:52,710
given by this equation.

1063
01:02:52,710 --> 01:02:56,300
Of course, my earlier definition
was in terms of

1064
01:02:56,300 --> 01:02:59,200
deformation gradients,
in terms of

1065
01:02:59,200 --> 01:03:00,990
the deformation gradient.

1066
01:03:00,990 --> 01:03:04,160
Now we actually have here
substituted displacements.

1067
01:03:04,160 --> 01:03:06,100
But it's still a definition.

1068
01:03:06,100 --> 01:03:09,660
This is, in fact,
the same tensor.

1069
01:03:09,660 --> 01:03:12,745
The same components here would
be calculated using the

1070
01:03:12,745 --> 01:03:15,830
earlier definition than
this one here.

1071
01:03:15,830 --> 01:03:17,560
Same thing.

1072
01:03:17,560 --> 01:03:21,140
And, as I say, it holds for any
deformation, any amount of

1073
01:03:21,140 --> 01:03:24,010
straining and stretching.

1074
01:03:24,010 --> 01:03:27,470
Let's look at one example, a
very simple example, where we

1075
01:03:27,470 --> 01:03:29,940
have a simple four-node
element that

1076
01:03:29,940 --> 01:03:31,150
we are pulling out.

1077
01:03:31,150 --> 01:03:33,420
Originally, it it is in the
black configuration.

1078
01:03:33,420 --> 01:03:36,300
It goes over into the
red configuration.

1079
01:03:36,300 --> 01:03:41,190
If we calculate t 0 epsilon 1 1
from the formula that I just

1080
01:03:41,190 --> 01:03:45,090
showed you, you get
these two terms.

1081
01:03:45,090 --> 01:03:48,110
This here is the engineering
strain term that we are very

1082
01:03:48,110 --> 01:03:49,500
well familiar with.

1083
01:03:49,500 --> 01:03:52,240
And this is here the non-linear
term, because it's

1084
01:03:52,240 --> 01:03:53,210
a quadratic term.

1085
01:03:53,210 --> 01:03:56,040
Non-linear in the displacement,
of course.

1086
01:03:56,040 --> 01:04:00,950
If we plot this information,
as shown here on the graph,

1087
01:04:00,950 --> 01:04:05,900
notice we're showing here t
delta over L, the original

1088
01:04:05,900 --> 01:04:10,190
length, where t delta
is the displacement.

1089
01:04:10,190 --> 01:04:14,670
And we have plotting here the
epsilon 1 1 component.

1090
01:04:14,670 --> 01:04:17,000
Actually, we should
have said here--

1091
01:04:17,000 --> 01:04:19,430
let me make this correction
right now here--

1092
01:04:19,430 --> 01:04:21,450
we should have had, really
have had here a

1093
01:04:21,450 --> 01:04:25,240
t 0 epsilon 1 1.

1094
01:04:25,240 --> 01:04:26,720
The t 0 was missing there.

1095
01:04:26,720 --> 01:04:30,380
Well, if we plot, in other
words, this component here as

1096
01:04:30,380 --> 01:04:35,310
a function of these elements
here, of the t delta over 0 L,

1097
01:04:35,310 --> 01:04:39,300
we get this red line,
the red line.

1098
01:04:39,300 --> 01:04:41,560
Notice the non-linearality
in that strain.

1099
01:04:41,560 --> 01:04:45,340
The engineering strain, which
is only the first term here,

1100
01:04:45,340 --> 01:04:48,300
gives us a straight line.

1101
01:04:48,300 --> 01:04:53,690
Notice that this strain here
increases quite rapidly as you

1102
01:04:53,690 --> 01:04:58,080
pull out, and it decreases in
magnitude here, with this

1103
01:04:58,080 --> 01:05:00,600
curvature, as you push in.

1104
01:05:00,600 --> 01:05:05,190
So there's no symmetry in the
strain when you extend or

1105
01:05:05,190 --> 01:05:07,600
compress a piece of material.

1106
01:05:07,600 --> 01:05:09,810
In the engineering strain
component, of course, we see a

1107
01:05:09,810 --> 01:05:11,790
symmetry, because we have
a straight line.

1108
01:05:15,430 --> 01:05:21,720
Let us look at another example
to calculate once the

1109
01:05:21,720 --> 01:05:23,200
Green-Lagrange strain.

1110
01:05:23,200 --> 01:05:27,180
Here, we have the original
configuration

1111
01:05:27,180 --> 01:05:28,710
of a four-node element.

1112
01:05:28,710 --> 01:05:31,810
And that four-node element,
shown black here, goes over

1113
01:05:31,810 --> 01:05:36,350
into this configuration at time
t, stretched into two

1114
01:05:36,350 --> 01:05:40,440
directions, and then, at time t
plus delta t, it has rotated

1115
01:05:40,440 --> 01:05:42,720
by 45 degrees.

1116
01:05:42,720 --> 01:05:46,340
The deformation gradient
corresponding to time t is

1117
01:05:46,340 --> 01:05:48,870
shown here.

1118
01:05:48,870 --> 01:05:51,620
The Cauchy-Green deformation
tensor shown here.

1119
01:05:51,620 --> 01:05:54,120
Notice no off diagonal
elements.

1120
01:05:54,120 --> 01:05:59,460
Because the piece of material,
the four-node element-- you

1121
01:05:59,460 --> 01:06:02,550
may think of this as a four-node
element, of course--

1122
01:06:02,550 --> 01:06:04,740
has simply been stretched
and compressed.

1123
01:06:04,740 --> 01:06:05,810
There's no shearing.

1124
01:06:05,810 --> 01:06:08,630
Therefore, we have
0 terms here.

1125
01:06:08,630 --> 01:06:12,030
And, if we calculate the
Green-Lagrange strain tensor,

1126
01:06:12,030 --> 01:06:13,780
we get this answer.

1127
01:06:13,780 --> 01:06:17,930
Once again, no shearing
components in

1128
01:06:17,930 --> 01:06:19,730
this tensor as well.

1129
01:06:19,730 --> 01:06:22,880
If we now do the same for the
configuration at time t plus

1130
01:06:22,880 --> 01:06:27,060
delta t, the green
configuration, we obtain this

1131
01:06:27,060 --> 01:06:29,690
deformation gradient, this
Cauchy-Green deformation

1132
01:06:29,690 --> 01:06:32,800
tensor, and this is a
Green-Lagrange strain tensor.

1133
01:06:32,800 --> 01:06:35,710
Where we notice that the
Cauchy-Green deformation

1134
01:06:35,710 --> 01:06:41,020
tensor has not changed, and
the Green-Lagrange strain

1135
01:06:41,020 --> 01:06:43,310
tensor has also not changed.

1136
01:06:43,310 --> 01:06:45,870
And this is what, of course,
we have proven already

1137
01:06:45,870 --> 01:06:50,580
earlier, that from time t to
time t plus delta t, the

1138
01:06:50,580 --> 01:06:53,260
Green-Lagrange strain tensor
and the Cauchy-Green

1139
01:06:53,260 --> 01:06:56,560
deformation tensor, both
of those do not change.

1140
01:06:56,560 --> 01:07:01,760
However for rigid body motion,
the x tensor, the deformation

1141
01:07:01,760 --> 01:07:06,360
gradient, does change, look at
here, because this deformation

1142
01:07:06,360 --> 01:07:10,160
gradient is affected
by the rotation.

1143
01:07:10,160 --> 01:07:14,160
Let's look at another example,
because all these examples

1144
01:07:14,160 --> 01:07:18,110
really enrich our understanding
of what these

1145
01:07:18,110 --> 01:07:21,240
kinematic quantities stand for,
and what they give us.

1146
01:07:21,240 --> 01:07:26,260
Here, we look at the example of
a simple shear deformation.

1147
01:07:26,260 --> 01:07:30,130
Originally, the element that we
are looking at was in this

1148
01:07:30,130 --> 01:07:32,420
configuration.

1149
01:07:32,420 --> 01:07:35,040
And, it is sheared over.

1150
01:07:35,040 --> 01:07:38,620
You can think of this top line
being simply moved over

1151
01:07:38,620 --> 01:07:42,360
horizontally into the
red configuration.

1152
01:07:42,360 --> 01:07:45,970
Notice that the original
length here is 1.

1153
01:07:45,970 --> 01:07:47,890
This length is also 1.

1154
01:07:47,890 --> 01:07:52,470
Notice that the movement
over is t delta.

1155
01:07:52,470 --> 01:07:54,990
And we calculate
the deformation

1156
01:07:54,990 --> 01:07:55,960
gradient once again.

1157
01:07:55,960 --> 01:07:58,670
The deformation gradient now
being given by via these

1158
01:07:58,670 --> 01:07:59,570
components.

1159
01:07:59,570 --> 01:08:02,280
Let's try to explain these
components a bit.

1160
01:08:02,280 --> 01:08:07,050
Well, this here is really the
differentiation of t x 1 with

1161
01:08:07,050 --> 01:08:09,140
respect to 0 x 1.

1162
01:08:09,140 --> 01:08:12,670
Now, t x 1 has not changed,
and therefore, this is 1.

1163
01:08:12,670 --> 01:08:15,410
t x 1 is equal to 0 x 1
in other words, and

1164
01:08:15,410 --> 01:08:17,270
therefore, this is 1.

1165
01:08:17,270 --> 01:08:21,760
Notice that the same holds also
for partial t x 2 with

1166
01:08:21,760 --> 01:08:24,120
respect to 0 x 2.

1167
01:08:24,120 --> 01:08:27,934
Notice that this element here
stands for partial t x 1 with

1168
01:08:27,934 --> 01:08:29,850
respect to 0 x 2.

1169
01:08:29,850 --> 01:08:34,149
And there, we look at partial
t x 1, a change into this

1170
01:08:34,149 --> 01:08:38,270
direction as we walk that
direction, and that gives us

1171
01:08:38,270 --> 01:08:43,800
this t delta, t delta over
this length here.

1172
01:08:43,800 --> 01:08:46,899
In other words, this element
is nothing else than this

1173
01:08:46,899 --> 01:08:49,609
length divided by that length.

1174
01:08:49,609 --> 01:08:53,630
And it's, of course, the same
for any material particle over

1175
01:08:53,630 --> 01:08:54,800
this domain.

1176
01:08:54,800 --> 01:08:57,810
Because all of these material
particles have

1177
01:08:57,810 --> 01:08:59,420
been sheared over.

1178
01:08:59,420 --> 01:09:02,990
Well, if you now take x
transposed times x, you get

1179
01:09:02,990 --> 01:09:04,779
this matrix here.

1180
01:09:04,779 --> 01:09:08,529
And you identify directly
these terms, if we then

1181
01:09:08,529 --> 01:09:11,040
calculate by the formula given,
the Green-Lagrange

1182
01:09:11,040 --> 01:09:13,450
strain, this is the result.

1183
01:09:13,450 --> 01:09:17,609
Notice that here we have now
shearing components.

1184
01:09:17,609 --> 01:09:20,880
And notice also one very
interesting phenomena, a

1185
01:09:20,880 --> 01:09:24,319
component appearing here.

1186
01:09:24,319 --> 01:09:26,939
This is an interesting
component, because it is

1187
01:09:26,939 --> 01:09:30,590
really an enormous strain
component in this direction.

1188
01:09:30,590 --> 01:09:33,500
And now there's an epsilon 1, 1,
epsilon 2, 2, I should have

1189
01:09:33,500 --> 01:09:38,130
said, component, an epsilon 2,
2 component that is non-zero

1190
01:09:38,130 --> 01:09:39,950
in this particular case,

1191
01:09:39,950 --> 01:09:43,840
In a material non-linear-only
formulation, where we only

1192
01:09:43,840 --> 01:09:47,740
include infinitesimal
displacements, in other words,

1193
01:09:47,740 --> 01:09:49,700
this part, of course,
would be 0.

1194
01:09:49,700 --> 01:09:52,220
And that is also signified
by the square term.

1195
01:09:52,220 --> 01:09:55,870
In other words, if indeed the
motion is small in this

1196
01:09:55,870 --> 01:10:01,610
direction, then this would
become 0, at least

1197
01:10:01,610 --> 01:10:05,370
approximately 0 when compared
to these terms, and it means

1198
01:10:05,370 --> 01:10:09,440
that the Green-Lagrange strain
tensor reduces to the

1199
01:10:09,440 --> 01:10:12,210
infinitesimal strain tensor that
we are so familiar with.

1200
01:10:15,660 --> 01:10:18,750
The 2nd Piola-Kirchhoff stress
tensor and the Green-Lagrange

1201
01:10:18,750 --> 01:10:22,120
strain tensors are energetically
conjugate.

1202
01:10:22,120 --> 01:10:25,060
This is a very important
statement.

1203
01:10:25,060 --> 01:10:29,670
And this is the reason why we
are working with these two

1204
01:10:29,670 --> 01:10:32,150
tensors, the 2nd Piola-Kirchhoff
stress tensor

1205
01:10:32,150 --> 01:10:33,980
and the Green-Lagrange
strain tensor.

1206
01:10:33,980 --> 01:10:40,590
In fact, what is happening here,
is that this product

1207
01:10:40,590 --> 01:10:45,340
here, which is the virtual work
at time t per current

1208
01:10:45,340 --> 01:10:51,990
volume, per unit current volume,
that this here is

1209
01:10:51,990 --> 01:10:57,210
given, of course, by the Cauchy
stress times the small,

1210
01:10:57,210 --> 01:11:03,410
or the virtual, infinitesimal
strain, is defined this way.

1211
01:11:03,410 --> 01:11:07,310
And here, we have the virtual
work at time t per unit

1212
01:11:07,310 --> 01:11:08,970
original volume.

1213
01:11:08,970 --> 01:11:13,650
And that is given by the 2nd
Piola-Kirchhoff stress times

1214
01:11:13,650 --> 01:11:17,520
the variation in the
Green-Lagrange strain.

1215
01:11:17,520 --> 01:11:22,170
We already had, earlier, that
the integration of this over

1216
01:11:22,170 --> 01:11:25,900
the current volume is equal
to the integration of this

1217
01:11:25,900 --> 01:11:29,070
quantity over the
original volume.

1218
01:11:29,070 --> 01:11:32,580
Well, this is here once more,
something to be kept in mind.

1219
01:11:32,580 --> 01:11:36,090
But the important point is that
the energy, that these

1220
01:11:36,090 --> 01:11:39,970
two quantities are energetically
conjugate, and

1221
01:11:39,970 --> 01:11:43,800
this is expressed via
this product here.

1222
01:11:43,800 --> 01:11:46,180
Let's look at the 2nd
Piola-Kirchhoff

1223
01:11:46,180 --> 01:11:47,660
stress tensor now.

1224
01:11:47,660 --> 01:11:51,180
The 2nd Piola-Kirchhoff stress
tensor in indicial notation is

1225
01:11:51,180 --> 01:11:52,980
defined as shown here.

1226
01:11:52,980 --> 01:11:54,080
And we had already that

1227
01:11:54,080 --> 01:11:56,000
definition early on a viewgraph.

1228
01:11:56,000 --> 01:12:01,710
In matrix notation, we can
directly say this is the way

1229
01:12:01,710 --> 01:12:03,330
you can also write it.

1230
01:12:03,330 --> 01:12:06,880
Notice that the entries in this
three by three matrix--

1231
01:12:06,880 --> 01:12:11,350
I'm always thinking about three
components, or rather,

1232
01:12:11,350 --> 01:12:13,430
three dimensional coordinate
space--

1233
01:12:13,430 --> 01:12:17,990
the entries here are nothing
else than those elements.

1234
01:12:17,990 --> 01:12:19,990
The entries of this matrix,
of course, are

1235
01:12:19,990 --> 01:12:21,550
those elements here.

1236
01:12:21,550 --> 01:12:24,840
And similarly here, this is
the scalar that we talked

1237
01:12:24,840 --> 01:12:26,810
about earlier already.

1238
01:12:26,810 --> 01:12:30,180
If we solve from here for Cauchy
stresses, we directly

1239
01:12:30,180 --> 01:12:32,250
obtain these two equations.

1240
01:12:32,250 --> 01:12:35,610
Notice that in this matrix
notation now, we have the

1241
01:12:35,610 --> 01:12:37,150
deformation gradient.

1242
01:12:37,150 --> 01:12:40,660
Whereas in the earlier equation
that we looked at

1243
01:12:40,660 --> 01:12:44,220
here, we had the inverse
deformation gradient.

1244
01:12:44,220 --> 01:12:47,930
Well, these are relations that
we want to keep in mind, and

1245
01:12:47,930 --> 01:12:51,750
that we will be referring to
quiet abundantly later on.

1246
01:12:51,750 --> 01:12:54,070
Let us look at some properties
of the 2nd Piola-Kirchhoff

1247
01:12:54,070 --> 01:12:55,510
stress tensor.

1248
01:12:55,510 --> 01:12:59,400
First of all, t 0
s is symmetric.

1249
01:12:59,400 --> 01:13:02,710
While that can be seen directly
by looking at the

1250
01:13:02,710 --> 01:13:07,540
definition, you might want to
prove it to yourself with a

1251
01:13:07,540 --> 01:13:09,340
small example as well.

1252
01:13:09,340 --> 01:13:12,760
t 0 s is invariant under
a rigid body motion,

1253
01:13:12,760 --> 01:13:15,720
translation, and/or rotation.

1254
01:13:15,720 --> 01:13:18,880
Hence, t 0 s only
changes when the

1255
01:13:18,880 --> 01:13:20,740
material is actually deformed.

1256
01:13:20,740 --> 01:13:25,900
This is a very important
statement, important fact, I

1257
01:13:25,900 --> 01:13:29,560
should say, and we want to look
at an example just now.

1258
01:13:29,560 --> 01:13:33,200
t 0 s has no direct physical
interpretation.

1259
01:13:33,200 --> 01:13:35,740
Well, if you look at some
books, you'll see some

1260
01:13:35,740 --> 01:13:40,090
pictures regarding t 0 s,
sorry, regarding the 2nd

1261
01:13:40,090 --> 01:13:41,400
Piola-Kirchhoff stress tensor.

1262
01:13:41,400 --> 01:13:43,830
In fact, I have some pictures
in my book,

1263
01:13:43,830 --> 01:13:45,160
in an example there.

1264
01:13:45,160 --> 01:13:49,250
But, these pictures are quite
far fetched, and they really

1265
01:13:49,250 --> 01:13:52,710
don't give as deep a physical
insight as one would like to

1266
01:13:52,710 --> 01:13:54,600
have with a stress measure.

1267
01:13:54,600 --> 01:13:59,130
I personally like to now take
almost the attitude of saying,

1268
01:13:59,130 --> 01:14:03,770
well, I accept that there is
no real strong physical

1269
01:14:03,770 --> 01:14:05,890
interpretation of this
stress measure.

1270
01:14:05,890 --> 01:14:08,810
It is used because it's
a convenient stress

1271
01:14:08,810 --> 01:14:10,070
measure to deal with.

1272
01:14:10,070 --> 01:14:12,240
We would not, in an engineering
analysis, really

1273
01:14:12,240 --> 01:14:14,450
print it out in a computer
program.

1274
01:14:14,450 --> 01:14:17,940
As I said earlier, we would,
of course, want to have the

1275
01:14:17,940 --> 01:14:21,060
Cauchy stress as the stress
measure with which we want to

1276
01:14:21,060 --> 01:14:23,200
design our structure.

1277
01:14:23,200 --> 01:14:27,390
I look at it as a measure that
is convenient to work with in

1278
01:14:27,390 --> 01:14:29,490
the engineering analysis.

1279
01:14:29,490 --> 01:14:31,620
It's interesting, of course,
from a theoretical point of

1280
01:14:31,620 --> 01:14:35,580
view to look at the elements and
to see how large they are,

1281
01:14:35,580 --> 01:14:38,470
how small they are, when
compared to the Cauchy stress,

1282
01:14:38,470 --> 01:14:40,890
the real physical stress that
we're interested in.

1283
01:14:40,890 --> 01:14:45,690
But, we don't really need to
look for a strong physical

1284
01:14:45,690 --> 01:14:47,230
interpretation.

1285
01:14:47,230 --> 01:14:48,980
I know we are engineers.

1286
01:14:48,980 --> 01:14:52,660
Engineers like to see pictures,
like to understand

1287
01:14:52,660 --> 01:14:53,940
physically what's happening.

1288
01:14:53,940 --> 01:14:58,710
And I can assure you that I've
tried myself quite hard to get

1289
01:14:58,710 --> 01:15:03,090
physical interpretation of this
definition, of this 2nd

1290
01:15:03,090 --> 01:15:04,650
Piola-Kirchhoff stress tensor.

1291
01:15:04,650 --> 01:15:09,850
But, I have not arrived with
many good results.

1292
01:15:09,850 --> 01:15:13,400
The best I could do is given
in my textbook, in the book

1293
01:15:13,400 --> 01:15:15,390
that you are using
as a textbook.

1294
01:15:15,390 --> 01:15:18,880
Let's look at an example here.

1295
01:15:18,880 --> 01:15:24,580
The example of a four-node
element being stretched and

1296
01:15:24,580 --> 01:15:33,290
rotated as shown here, and then
rotated as shown here.

1297
01:15:33,290 --> 01:15:39,160
Notice that from time t, shown
as the red configuration, to

1298
01:15:39,160 --> 01:15:43,100
time t plus delta t, we have
only a rigid body motion, a

1299
01:15:43,100 --> 01:15:46,150
rotation of 60 degrees.

1300
01:15:46,150 --> 01:15:49,370
Of course, the material would
be subjected to some Cauchy

1301
01:15:49,370 --> 01:15:53,810
stresses, and Cauchy stresses
in this configuration and

1302
01:15:53,810 --> 01:15:57,050
Cauchy stresses in this
configuration.

1303
01:15:57,050 --> 01:16:00,670
Let us see how the Cauchy
stresses, and how the 2nd

1304
01:16:00,670 --> 01:16:05,490
Piola-Kirchhoff stresses evolve
during this motion.

1305
01:16:05,490 --> 01:16:10,040
And on this viewgraph here,
we have summarized that

1306
01:16:10,040 --> 01:16:11,050
information.

1307
01:16:11,050 --> 01:16:15,680
At time t the deformation
gradient is as shown here.

1308
01:16:15,680 --> 01:16:18,710
The Cauchy stresses,
say, are like this.

1309
01:16:18,710 --> 01:16:21,910
Of course, we would have to
introduce the material law, et

1310
01:16:21,910 --> 01:16:22,840
cetera, et cetera.

1311
01:16:22,840 --> 01:16:25,780
Let us assume that the Cauchy
stresses are like that.

1312
01:16:25,780 --> 01:16:29,100
And then, given the Cauchy
stresses, the deformation

1313
01:16:29,100 --> 01:16:36,400
gradient, we calculate the 2nd
Piola-Kirchhoff stress tensor.

1314
01:16:36,400 --> 01:16:41,380
Let's say that this is now,
has now been calculated.

1315
01:16:41,380 --> 01:16:45,360
And we do the same at
time t plus delta t.

1316
01:16:45,360 --> 01:16:48,590
We know that there has, of
course, been additional motion

1317
01:16:48,590 --> 01:16:51,810
from time t to time t plus
delta t, which means that

1318
01:16:51,810 --> 01:16:54,560
these elements here
will change.

1319
01:16:54,560 --> 01:16:59,050
And you should now be in a
position to calculate this

1320
01:16:59,050 --> 01:17:01,830
tensor, because you know there
is only a rigid body motion

1321
01:17:01,830 --> 01:17:03,580
that is being applied.

1322
01:17:03,580 --> 01:17:08,590
And then, knowing this tensor,
we can directly also calculate

1323
01:17:08,590 --> 01:17:12,870
the 2nd Piola-Kirchhoff
stress tensor.

1324
01:17:12,870 --> 01:17:15,520
And we can calculate the
Cauchy stress tensor.

1325
01:17:15,520 --> 01:17:17,270
So we can really calculate
all of these.

1326
01:17:17,270 --> 01:17:18,400
And what do we see?

1327
01:17:18,400 --> 01:17:20,830
This is the really
important point.

1328
01:17:20,830 --> 01:17:23,650
We see that the 2nd
Piola-Kirchhoff stress tensor

1329
01:17:23,650 --> 01:17:28,550
has not changed from time t
to time t plus delta t.

1330
01:17:28,550 --> 01:17:30,230
This is the important
information

1331
01:17:30,230 --> 01:17:31,690
that I mentioned earlier.

1332
01:17:31,690 --> 01:17:33,350
However, the Cauchy
stress tensor,

1333
01:17:33,350 --> 01:17:35,640
of course, has changed.

1334
01:17:35,640 --> 01:17:38,860
The Cauchy stress tensor
has changed.

1335
01:17:38,860 --> 01:17:42,680
One can, of course, prove that
the 2nd Piola-Kirchhoff stress

1336
01:17:42,680 --> 01:17:46,260
tensor does not change when the
material is only subjected

1337
01:17:46,260 --> 01:17:47,930
to a rigid body motion.

1338
01:17:47,930 --> 01:17:53,610
And such proof is actually
given in the textbook.

1339
01:17:53,610 --> 01:17:57,150
But this is a simple
application, and it

1340
01:17:57,150 --> 01:18:00,810
exemplifies to you what I
mean by that statement.

1341
01:18:00,810 --> 01:18:03,720
It's an important statement that
you should keep in mind

1342
01:18:03,720 --> 01:18:06,970
as we go along with our further
discussion of the

1343
01:18:06,970 --> 01:18:08,920
mathematical basis
of non-linear

1344
01:18:08,920 --> 01:18:10,610
finite element analysis.

1345
01:18:10,610 --> 01:18:14,510
This brings us then to the
end of this lecture.

1346
01:18:14,510 --> 01:18:20,350
I have tried to give you some
of the mathematical bases,

1347
01:18:20,350 --> 01:18:24,300
some of the ingredients, rather,
that we are using in

1348
01:18:24,300 --> 01:18:25,910
finite element analysis.

1349
01:18:25,910 --> 01:18:29,500
We have discussed stress and
strain measures that we will

1350
01:18:29,500 --> 01:18:31,860
be encountering further
in the next lectures.

1351
01:18:31,860 --> 01:18:33,400
Thank you very much for
your attention.