WEBVTT
00:00:00.040 --> 00:00:02.470
The following content is
provided under a Creative
00:00:02.470 --> 00:00:03.880
Commons license.
00:00:03.880 --> 00:00:06.920
Your support will help MIT
OpenCourseWare continue to
00:00:06.920 --> 00:00:10.570
offer high quality educational
resources for free.
00:00:10.570 --> 00:00:13.470
To make a donation or view
additional materials from
00:00:13.470 --> 00:00:17.400
hundreds of MIT courses, visit
MIT OpenCourseWare at
00:00:17.400 --> 00:00:18.650
ocw.mit.edu.
00:00:21.980 --> 00:00:23.920
PROFESSOR: Ladies and gentlemen,
welcome to this
00:00:23.920 --> 00:00:27.240
lecture on nonlinear finite
element analysis of solids and
00:00:27.240 --> 00:00:28.430
structures.
00:00:28.430 --> 00:00:31.350
In the previous lectures, we
considered the general
00:00:31.350 --> 00:00:34.100
continuum mechanics formulations
that we use for
00:00:34.100 --> 00:00:36.230
nonlinear finite element
analysis.
00:00:36.230 --> 00:00:39.940
And we also introduced briefly
the finite element matrices.
00:00:39.940 --> 00:00:42.530
In the coming lectures, I would
like now to discuss with
00:00:42.530 --> 00:00:47.470
you these finite element
matrices in more detail.
00:00:47.470 --> 00:00:51.640
Finite element matrices can
generally be categorized as
00:00:51.640 --> 00:00:53.240
continuum elements.
00:00:53.240 --> 00:00:57.060
We call them sometimes
also solid elements.
00:00:57.060 --> 00:00:58.510
The truss element, for
example, would
00:00:58.510 --> 00:01:00.500
be a continuum element.
00:01:00.500 --> 00:01:04.050
The 2D element, the
3D elements would
00:01:04.050 --> 00:01:05.560
be continuum elements.
00:01:05.560 --> 00:01:10.370
The 2D and 3D elements we may
also call solid elements.
00:01:10.370 --> 00:01:14.300
And as another category, we have
the structural elements.
00:01:14.300 --> 00:01:16.920
Structure elements are beam
elements, plate elements,
00:01:16.920 --> 00:01:18.030
shell elements.
00:01:18.030 --> 00:01:20.490
Of course, a distinguishing
feature between the structural
00:01:20.490 --> 00:01:23.860
elements and the continuum
elements is that continuum
00:01:23.860 --> 00:01:27.880
elements carry only nodal point
displacements as degrees
00:01:27.880 --> 00:01:30.880
of freedom, whereas the
structural elements have also
00:01:30.880 --> 00:01:34.140
rotational degrees of freedom
at their nodes.
00:01:34.140 --> 00:01:37.920
In this lecture, I like to talk
about the 2D continuum
00:01:37.920 --> 00:01:41.820
elements, the 2D plane stress,
plane strain, and
00:01:41.820 --> 00:01:43.700
axisymmetric elements.
00:01:43.700 --> 00:01:47.230
These elements are used very,
very widely in the engineering
00:01:47.230 --> 00:01:50.460
professions for all sorts
of analyses--
00:01:50.460 --> 00:01:54.620
plane stress analyses of plates,
plane strain analysis
00:01:54.620 --> 00:01:58.670
all dams, axisymmetric analysis
of shells, and
00:01:58.670 --> 00:02:01.120
so on and so on.
00:02:01.120 --> 00:02:05.140
The elements are very general,
and can be used for geometric
00:02:05.140 --> 00:02:07.770
and material nonlinear
analyses.
00:02:07.770 --> 00:02:11.290
I also like to then, at the
end of the lecture, talk
00:02:11.290 --> 00:02:14.780
briefly about the 3D elements
that are also very widely
00:02:14.780 --> 00:02:18.950
used, and that are really
formulated in the same way as
00:02:18.950 --> 00:02:20.220
the 2D elements.
00:02:20.220 --> 00:02:24.010
Therefore, once you understand
to 2D elements very well, it
00:02:24.010 --> 00:02:27.030
is fairly easy to generalize
these concepts or use these
00:02:27.030 --> 00:02:31.200
concepts also to construct and
formulate 3D elements.
00:02:31.200 --> 00:02:35.110
Let me now go over to my view
graph and discuss it with you
00:02:35.110 --> 00:02:37.850
the information that I have
on these view graphs.
00:02:37.850 --> 00:02:41.310
Once again, I like to talk
about plane stress, plane
00:02:41.310 --> 00:02:43.130
strain elements, and
axisymmetric
00:02:43.130 --> 00:02:45.410
elements in this lecture.
00:02:45.410 --> 00:02:49.450
And these derivations that we
will be discussing, as I said
00:02:49.450 --> 00:02:53.090
already, are directly applicable
also, or can
00:02:53.090 --> 00:02:56.260
directly be extended to three
dimensional elements.
00:02:56.260 --> 00:02:58.560
Let's look at a typical
2D element,
00:02:58.560 --> 00:03:00.010
two dimensional element.
00:03:00.010 --> 00:03:03.720
This is a nine-node element in
the stationary coordinate
00:03:03.720 --> 00:03:05.960
frame, x1, x2.
00:03:05.960 --> 00:03:09.430
x times 0, we would see
this element here.
00:03:09.430 --> 00:03:12.790
Notice there are nine
nodes, 1 to 9.
00:03:12.790 --> 00:03:14.970
Notice that we will be
talking about the
00:03:14.970 --> 00:03:17.550
isoparametric elements.
00:03:17.550 --> 00:03:22.790
And these have the R and F
auxiliary coordinate system,
00:03:22.790 --> 00:03:27.960
the natural coordinate system,
just like in linear analysis.
00:03:27.960 --> 00:03:30.570
X times 0, to the
element is here.
00:03:30.570 --> 00:03:33.370
And at time t, the
element is here.
00:03:33.370 --> 00:03:36.800
Notice that the element has
undergone large displacements,
00:03:36.800 --> 00:03:38.730
large rotations.
00:03:38.730 --> 00:03:41.180
You don't see very large
rotations here, but the
00:03:41.180 --> 00:03:43.390
rotations could be very large.
00:03:43.390 --> 00:03:45.820
And certainly also
large strains.
00:03:45.820 --> 00:03:48.920
You can see directly that the
element here has grown from
00:03:48.920 --> 00:03:50.660
its size, so certainly
it must have been
00:03:50.660 --> 00:03:53.300
subjected to large strains.
00:03:53.300 --> 00:03:55.730
So we consider really a
very general motion.
00:03:55.730 --> 00:03:59.950
But remember, once again, that
the coordinate frame, x1, x2,
00:03:59.950 --> 00:04:03.120
the Cartesian coordinate frame,
remains stationary, as
00:04:03.120 --> 00:04:07.070
we have discussed in the
previous lectures.
00:04:07.070 --> 00:04:10.610
Because the elements are
isoparametric elements, we can
00:04:10.610 --> 00:04:12.710
directly write these
expressions.
00:04:12.710 --> 00:04:17.310
That 0x1's, the coordinates of
the material points in the
00:04:17.310 --> 00:04:21.740
elements at time 0, are given
via this interpolation.
00:04:21.740 --> 00:04:25.730
The hk are the interpolation
functions that we also use in
00:04:25.730 --> 00:04:27.190
linear analysis.
00:04:27.190 --> 00:04:31.370
The 0x1k are the nodal
point coordinates.
00:04:31.370 --> 00:04:34.120
k refers to the nodal point.
00:04:34.120 --> 00:04:36.995
1 refers to the coordinate
direction.
00:04:36.995 --> 00:04:40.010
0 refers to the fact that
they're looking at the
00:04:40.010 --> 00:04:41.900
configuration at time 0.
00:04:41.900 --> 00:04:44.750
This 0 is, of course, the
same 0 that we see here.
00:04:44.750 --> 00:04:47.720
This one is the same one
that we see there.
00:04:47.720 --> 00:04:49.830
We are summing, of course,
of all the nodes.
00:04:49.830 --> 00:04:52.020
And for the element that I just
had on the previous view
00:04:52.020 --> 00:04:56.770
graph, it would be nine nodes,
so n is equal to 9.
00:04:56.770 --> 00:04:58.830
We have a similar expression
for the
00:04:58.830 --> 00:05:02.420
x2 coordinate direction.
00:05:02.420 --> 00:05:05.630
In other words, 0x2 is
given like that.
00:05:05.630 --> 00:05:08.410
This is the x2 coordinate of a
material particle, and it's
00:05:08.410 --> 00:05:11.900
expressed in terms of the nodal
point coordinates of the
00:05:11.900 --> 00:05:16.540
elements expressed in terms of
the nodal point coordinates of
00:05:16.540 --> 00:05:18.030
the element.
00:05:18.030 --> 00:05:21.820
The same expression is also
applicable at time t.
00:05:21.820 --> 00:05:27.596
Notice all we have exchanged
is the 0 to a t, 0 to a t.
00:05:27.596 --> 00:05:31.090
And similarly here for
the x2 coordinate.
00:05:31.090 --> 00:05:33.900
0 to t, and similar
here, 0 to t.
00:05:36.740 --> 00:05:38.800
Let us look at an example.
00:05:38.800 --> 00:05:41.890
Here we have depicted
a four-node element.
00:05:41.890 --> 00:05:43.950
The original element
lies here.
00:05:43.950 --> 00:05:45.400
It's black.
00:05:45.400 --> 00:05:47.780
The R and F system,
of course, is the
00:05:47.780 --> 00:05:49.480
natural coordinate system.
00:05:49.480 --> 00:05:52.590
This is a configuration of
the element at time 0.
00:05:52.590 --> 00:05:57.200
It moves into that configuration
to time t, or it
00:05:57.200 --> 00:06:00.620
is at, in this configuration,
at time t.
00:06:00.620 --> 00:06:04.530
Notice that these are the
interpolations that I just
00:06:04.530 --> 00:06:06.620
introduced you to.
00:06:06.620 --> 00:06:08.830
The hk functions
are, of course,
00:06:08.830 --> 00:06:10.250
interpolation functions.
00:06:10.250 --> 00:06:15.800
Once again, these are here, the
nodal point coordinates.
00:06:15.800 --> 00:06:21.200
k is the nodal point, t is the
time that we're looking at, xi
00:06:21.200 --> 00:06:23.090
means the i's direction.
00:06:23.090 --> 00:06:26.610
i, of course, in this particular
case, 1 or 2.
00:06:26.610 --> 00:06:30.240
Similarly, for the original
geometry of the element.
00:06:30.240 --> 00:06:34.640
The hk's are listed out here
on the right-hand side.
00:06:34.640 --> 00:06:37.490
Notice these are the same
interpolation functions that
00:06:37.490 --> 00:06:39.540
we are using in linear
analysis.
00:06:39.540 --> 00:06:40.720
No difference there.
00:06:40.720 --> 00:06:44.270
For example, h1 is
given right here.
00:06:44.270 --> 00:06:46.980
And this is, of course, the
interpolation function
00:06:46.980 --> 00:06:53.410
corresponding to nodal point
1, as shown right there.
00:06:53.410 --> 00:06:55.570
You are probably very familiar
with these interpolation
00:06:55.570 --> 00:06:58.790
functions, and I don't need
to go into details there.
00:06:58.790 --> 00:07:03.130
But let us look now at the
following, namely what had
00:07:03.130 --> 00:07:07.440
happened in the motion
to a nodal point.
00:07:07.440 --> 00:07:11.670
A typical nodal point would be
the second nodal point here.
00:07:11.670 --> 00:07:16.630
The original coordinates
are as shown.
00:07:16.630 --> 00:07:21.500
And these original coordinates
have grown, or have become
00:07:21.500 --> 00:07:26.370
larger as shown here, because
this node here has moved to
00:07:26.370 --> 00:07:27.650
that new position.
00:07:27.650 --> 00:07:33.030
Notice this is now here, the
coordinate of node 2, of
00:07:33.030 --> 00:07:34.840
course, x1 coordinate.
00:07:34.840 --> 00:07:40.480
This superscript 2 means node
2, this 1 means 1 direction.
00:07:40.480 --> 00:07:42.560
t means time t.
00:07:42.560 --> 00:07:46.940
Here we have 1 and
2 and time 0.
00:07:46.940 --> 00:07:50.690
Notice here, 2, 2 times 0.
00:07:50.690 --> 00:07:56.330
This 2 here, the bottom 2, means
coordinate direction 2.
00:07:56.330 --> 00:07:59.210
This top 2 means
nodal point 2.
00:07:59.210 --> 00:08:02.600
This is a convention that
we want to use.
00:08:02.600 --> 00:08:07.730
It is a bit heavy, but we have
to somehow use a convention to
00:08:07.730 --> 00:08:13.040
label our coordinates, and this
is the one that I chose
00:08:13.040 --> 00:08:14.740
some time ago.
00:08:14.740 --> 00:08:19.810
Similar here for the
2 coordinate, x2
00:08:19.810 --> 00:08:23.210
coordinate, at time t.
00:08:23.210 --> 00:08:27.120
tx2 at nodal point 2.
00:08:27.120 --> 00:08:31.050
That upper 2, once again,
be the nodal point 2.
00:08:31.050 --> 00:08:35.450
And of course, this would
also apply for all
00:08:35.450 --> 00:08:36.740
the other nodal points.
00:08:39.760 --> 00:08:44.179
If we look at the motion of a
material particle that is in
00:08:44.179 --> 00:08:50.775
the element, we would obtain
that motion from the motion of
00:08:50.775 --> 00:08:52.050
the nodal points.
00:08:52.050 --> 00:08:55.950
Here we have now, in a nine-node
element that is
00:08:55.950 --> 00:09:00.500
originality here, and moves
into this position.
00:09:00.500 --> 00:09:03.310
At time t it is in
this position.
00:09:03.310 --> 00:09:08.200
Let's look at one particular
particle within the element.
00:09:08.200 --> 00:09:14.120
Here we have one particle
right there.
00:09:14.120 --> 00:09:18.050
Notice that this particle
here is given via this
00:09:18.050 --> 00:09:20.112
relationship here.
00:09:20.112 --> 00:09:23.810
r and s are both 0.5--
00:09:23.810 --> 00:09:27.050
you see r positive means this
direction, s positive means
00:09:27.050 --> 00:09:31.170
that direction, r and
s 0.5 is there.
00:09:31.170 --> 00:09:38.220
We would use r and s equal to
0.5, substitute into hk.
00:09:38.220 --> 00:09:45.670
And then, of course, we have 9
such hk's, substitute r and s
00:09:45.670 --> 00:09:48.820
equal to 0.5 into each
of these hk's.
00:09:48.820 --> 00:09:54.410
And sum out, this right-hand
side, to get the coordinate,
00:09:54.410 --> 00:09:55.380
the coordinates--
00:09:55.380 --> 00:09:56.980
there are two, of course--
00:09:56.980 --> 00:10:02.390
of this point here, this
material particle, at time 0.
00:10:02.390 --> 00:10:05.655
That's how we would obtain
the coordinates of
00:10:05.655 --> 00:10:07.180
that material particle.
00:10:07.180 --> 00:10:10.950
Now, at time t, we proceed
much in the same way.
00:10:10.950 --> 00:10:13.790
Here we have the equation.
00:10:13.790 --> 00:10:19.850
We would, again, take hk at r
equal to 0.5, s equal to 0.5,
00:10:19.850 --> 00:10:27.500
for all case, and multiply
these hk values by these
00:10:27.500 --> 00:10:32.100
values here, which of course,
are given because we must know
00:10:32.100 --> 00:10:36.210
where these nodal points
have arrived at.
00:10:36.210 --> 00:10:39.240
So we can evaluate the
right-hand side to directly
00:10:39.240 --> 00:10:43.470
get these two values, there are
two such values, tx1 and
00:10:43.470 --> 00:10:48.600
tx2, which gives then the
position of this material
00:10:48.600 --> 00:10:51.330
particle at time t.
00:10:51.330 --> 00:10:55.110
Notice that the isoparametric
coordinates of a material
00:10:55.110 --> 00:10:57.210
particle never change.
00:10:57.210 --> 00:11:00.510
You put that here in red because
that is very important
00:11:00.510 --> 00:11:02.160
to keep in mind.
00:11:02.160 --> 00:11:05.680
Of course, the actual
coordinates of the particle
00:11:05.680 --> 00:11:08.300
change, because that particle
moves through space in a
00:11:08.300 --> 00:11:11.150
stationary coordinate
frame, x1 and x2.
00:11:11.150 --> 00:11:13.290
But we are in s coordinates.
00:11:13.290 --> 00:11:17.490
The natural coordinates
do not change.
00:11:17.490 --> 00:11:22.020
Well a major advantage of the
isoparametric finite element
00:11:22.020 --> 00:11:26.040
analysis is that we can directly
write, of course, at
00:11:26.040 --> 00:11:30.600
the displacements, are given
as shown here via the nodal
00:11:30.600 --> 00:11:32.460
point displacements.
00:11:32.460 --> 00:11:36.180
tU1 is the displacement of
the material particle
00:11:36.180 --> 00:11:37.810
into the one direction.
00:11:37.810 --> 00:11:41.260
The hk's are the isoparametric
interpolation functions.
00:11:41.260 --> 00:11:44.340
And these are the nodal
point displacements.
00:11:44.340 --> 00:11:46.970
k being the nodal point
displacements
00:11:46.970 --> 00:11:49.680
of nodal point k.
00:11:49.680 --> 00:11:53.110
Similarly, for the
2 direction.
00:11:53.110 --> 00:11:57.390
And this holds at time t, and
it also holds for the
00:11:57.390 --> 00:12:02.220
incremental displacements from
time t to time t plus delta t,
00:12:02.220 --> 00:12:03.470
as written right here.
00:12:05.880 --> 00:12:09.430
That this is, in fact, 2 can
easily be shown from the
00:12:09.430 --> 00:12:11.160
coordinate interpolations.
00:12:11.160 --> 00:12:15.850
You see we had already these two
interpolations, and all we
00:12:15.850 --> 00:12:20.150
need to do now is subtract on
the left-hand side, and on the
00:12:20.150 --> 00:12:25.020
right-hand side, to obtain
this equation here.
00:12:25.020 --> 00:12:27.910
And what are left with here,
of course, must be the
00:12:27.910 --> 00:12:30.790
displacement of the material
particle that
00:12:30.790 --> 00:12:32.730
we're looking at--
00:12:32.730 --> 00:12:34.850
tUi.
00:12:34.850 --> 00:12:38.450
And here, we must have
the displacement
00:12:38.450 --> 00:12:40.960
of the nodal points.
00:12:40.960 --> 00:12:44.980
And these are denoted as tUik.
00:12:44.980 --> 00:12:48.220
And that is exactly the
relationships that we were
00:12:48.220 --> 00:12:54.110
just dealing with, tUi
is equal to hk tUik.
00:12:54.110 --> 00:12:57.490
There is one very important
point that I like to
00:12:57.490 --> 00:12:58.790
point out to you.
00:12:58.790 --> 00:13:07.250
Namely, that these equations
show directly that if we use a
00:13:07.250 --> 00:13:12.270
finite element mesh that is
originally compatible, in
00:13:12.270 --> 00:13:16.780
other words, compatible in a
linear analysis, than this
00:13:16.780 --> 00:13:20.320
finite element mesh will remain
compatible throughout
00:13:20.320 --> 00:13:24.000
the motion, throughout the
large deformation motion.
00:13:24.000 --> 00:13:25.980
And that is a very
important point.
00:13:25.980 --> 00:13:31.090
That we can say that the mesh
which originally is compatible
00:13:31.090 --> 00:13:33.730
will remain compatible
throughout the analysis.
00:13:33.730 --> 00:13:38.120
That follows directly from
these equations.
00:13:38.120 --> 00:13:41.150
The element matrices that we
need, of course, require
00:13:41.150 --> 00:13:45.930
derivatives, and these are
obtained much in the same way
00:13:45.930 --> 00:13:48.030
as in linear analysis.
00:13:48.030 --> 00:13:53.250
We need this derivative here,
partial tUi, with respect to
00:13:53.250 --> 00:13:56.410
the original coordinates.
00:13:56.410 --> 00:13:58.000
This is the actual derivative.
00:13:58.000 --> 00:14:00.170
This is the abbreviation
that we used
00:14:00.170 --> 00:14:01.840
in the earlier lectures.
00:14:01.840 --> 00:14:05.050
And we obtained this derivative
by taking the
00:14:05.050 --> 00:14:09.260
differentiation of the hk's with
respect to the original
00:14:09.260 --> 00:14:10.190
coordinates.
00:14:10.190 --> 00:14:11.340
Of course, these are numbers.
00:14:11.340 --> 00:14:14.470
These are the nodal point
displacements.
00:14:14.470 --> 00:14:17.800
So it's this one that we really
need to evaluate this
00:14:17.800 --> 00:14:19.020
derivative.
00:14:19.020 --> 00:14:21.720
Similarly, for the incremental
displacements.
00:14:21.720 --> 00:14:24.020
We want to take the
differentiation of the
00:14:24.020 --> 00:14:26.700
incremental displacement with
respect to the original
00:14:26.700 --> 00:14:27.550
coordinates.
00:14:27.550 --> 00:14:29.050
It's achieved this way.
00:14:29.050 --> 00:14:32.600
Once again, here we have an
expression that we need to
00:14:32.600 --> 00:14:36.870
evaluate, which also goes
in here, of course.
00:14:36.870 --> 00:14:40.000
And as we evaluate, as we will
just now see, much in the same
00:14:40.000 --> 00:14:41.870
way as in linear analysis.
00:14:41.870 --> 00:14:45.320
Notice, here we have written
down the partial of Ui with
00:14:45.320 --> 00:14:48.560
respect to the current
coordinates
00:14:48.560 --> 00:14:50.890
obtained as given here.
00:14:50.890 --> 00:14:54.810
Of course, these are the nodal
point displacement increments,
00:14:54.810 --> 00:14:58.010
and here we have the
differentiation of the hk's
00:14:58.010 --> 00:15:00.660
with respect to the current
coordinates now.
00:15:00.660 --> 00:15:06.330
So once we have these evaluated,
these expressions
00:15:06.330 --> 00:15:10.160
evaluated, we can obtain all
of the derivatives that go
00:15:10.160 --> 00:15:15.190
into the strain displacement
matrices that we want to have
00:15:15.190 --> 00:15:17.620
for the element.
00:15:17.620 --> 00:15:22.275
The derivatives are evaluated
using the chain rule, just as
00:15:22.275 --> 00:15:24.570
in linear analysis.
00:15:24.570 --> 00:15:28.760
We are using that partial hk
with respect to r, is given as
00:15:28.760 --> 00:15:33.395
partial hk with respect to x1
times partial x1 with respect
00:15:33.395 --> 00:15:36.060
to r, et cetera.
00:15:36.060 --> 00:15:38.300
Of course, these are here
the derivatives
00:15:38.300 --> 00:15:41.120
that we want to calculate.
00:15:41.120 --> 00:15:44.240
Notice, this is what we
need to calculate in
00:15:44.240 --> 00:15:46.240
order to get these.
00:15:46.240 --> 00:15:50.340
And these here, just like in
linear analysis, go into the
00:15:50.340 --> 00:15:53.660
Jacobian matrix written
down here.
00:15:53.660 --> 00:15:56.460
Here we have the Jacobian
matrix.
00:15:56.460 --> 00:15:58.540
And this is what we want.
00:15:58.540 --> 00:16:01.970
Therefore, we need to invert
the Jacobian matrix, as in
00:16:01.970 --> 00:16:03.820
linear analysis.
00:16:03.820 --> 00:16:07.160
And we obtain via this
relationship here, the
00:16:07.160 --> 00:16:09.860
required derivatives.
00:16:09.860 --> 00:16:13.560
The entries in this
matrix involve
00:16:13.560 --> 00:16:16.410
derivatives of this form.
00:16:16.410 --> 00:16:20.000
Partial x1, 0x1 with
respect to r, with
00:16:20.000 --> 00:16:22.310
respect to s and so on.
00:16:22.310 --> 00:16:27.910
And those are obtained here as
shown on the right-hand side.
00:16:27.910 --> 00:16:31.070
Notice here, of course, we only
need differentiations
00:16:31.070 --> 00:16:33.190
with respect to r, with
respect to the natural
00:16:33.190 --> 00:16:33.830
coordinates.
00:16:33.830 --> 00:16:39.440
And since the functions hk are
a function of r and f, we can
00:16:39.440 --> 00:16:44.660
directly evaluate these kinds of
expressions that go in the
00:16:44.660 --> 00:16:48.520
Jacobian matrix, and the
inverse, of course.
00:16:48.520 --> 00:16:51.420
If we want to take derivatives
with respect to the current
00:16:51.420 --> 00:16:54.390
coordinates, we proceed
much in the same way.
00:16:54.390 --> 00:16:57.070
This is the relationship that
we arrive at by simply
00:16:57.070 --> 00:17:04.900
substituting instead of the
0xi, the txi in to the
00:17:04.900 --> 00:17:09.319
Jacobian matrix, and of course,
into the expressions
00:17:09.319 --> 00:17:11.319
that are in here.
00:17:11.319 --> 00:17:16.599
So here we have a Jacobian
matrix that is giving the
00:17:16.599 --> 00:17:20.260
derivatives of the current
coordinates at time t with
00:17:20.260 --> 00:17:22.859
respect to the natural
coordinates.
00:17:22.859 --> 00:17:25.349
Such an element is obtained
as shown here.
00:17:25.349 --> 00:17:28.240
It involves again only the
differentiation of the
00:17:28.240 --> 00:17:32.100
interpolation functions with
respect to the natural
00:17:32.100 --> 00:17:33.040
coordinates.
00:17:33.040 --> 00:17:36.440
Here, of course with respect
to s, because we want to
00:17:36.440 --> 00:17:38.590
differentiate with
respect to s.
00:17:38.590 --> 00:17:42.200
Of course, these are the nodal
point coordinate at time t,
00:17:42.200 --> 00:17:43.550
which are known.
00:17:43.550 --> 00:17:47.090
We invert this relationship
here to obtain the
00:17:47.090 --> 00:17:51.350
differentiation that
we need to have.
00:17:51.350 --> 00:17:54.620
We are now ready to compute the
required element matrices
00:17:54.620 --> 00:17:57.820
for the total Lagrangian
formulation.
00:17:57.820 --> 00:18:02.100
And the element matrices that we
want to compute, of course,
00:18:02.100 --> 00:18:06.195
our those established in
the early lectures.
00:18:06.195 --> 00:18:11.130
The matrices that go into
evaluating these matrices are
00:18:11.130 --> 00:18:13.860
listed here.
00:18:13.860 --> 00:18:17.020
Here, of course, the tangent
constitutive relationship,
00:18:17.020 --> 00:18:20.260
which we will talk
about much later.
00:18:20.260 --> 00:18:21.690
Not in this lecture.
00:18:21.690 --> 00:18:24.120
Here we have the linear strain
displacement matrix, the
00:18:24.120 --> 00:18:27.570
nonlinear strain displacement
matrix.
00:18:27.570 --> 00:18:30.750
Here again, the linear strain
displacement matrix, a stress
00:18:30.750 --> 00:18:33.650
vector, and a stress matrix.
00:18:33.650 --> 00:18:36.680
Let's see now how these matrices
look for the two
00:18:36.680 --> 00:18:39.010
dimensional case.
00:18:39.010 --> 00:18:43.690
The constitutive relation,
just very briefly--
00:18:43.690 --> 00:18:47.960
again, much more detail will be
given in a later lecture--
00:18:47.960 --> 00:18:51.480
would be relating the increment
in stress to the
00:18:51.480 --> 00:18:52.810
increments in strain.
00:18:52.810 --> 00:18:58.410
Notice there's a 2 here, and
this is the matrix which is
00:18:58.410 --> 00:19:01.080
established from this
relationship here.
00:19:01.080 --> 00:19:02.800
In the early lectures,
we used this
00:19:02.800 --> 00:19:05.800
relationship, the tensor notation.
00:19:05.800 --> 00:19:08.960
Well we have now a matrix
notation, and this is the
00:19:08.960 --> 00:19:11.680
relationship with that
matrix notation.
00:19:11.680 --> 00:19:16.030
C for a linear elastic material
would look as shown
00:19:16.030 --> 00:19:20.210
here, and you are familiar with
this relationship from
00:19:20.210 --> 00:19:21.200
linear analysis.
00:19:21.200 --> 00:19:25.280
It's the same matrix that we
encounter in linear analysis.
00:19:25.280 --> 00:19:27.300
Of course, e being Young's
modulus, nu
00:19:27.300 --> 00:19:28.575
being Poisson's ratio.
00:19:31.700 --> 00:19:35.700
We also derived in the early
lectures the expression for
00:19:35.700 --> 00:19:37.710
the incremental strain terms.
00:19:37.710 --> 00:19:39.510
And here they are listed out.
00:19:39.510 --> 00:19:42.640
We have derived these right-hand
side expressions.
00:19:42.640 --> 00:19:48.110
We notice that what is here
underscored by a blue line was
00:19:48.110 --> 00:19:50.400
the initial displacement
effect.
00:19:50.400 --> 00:19:52.900
And is this initial displacement
effect, of
00:19:52.900 --> 00:19:56.340
course, is a particular in
gradient off the total
00:19:56.340 --> 00:19:58.310
Lagrangian formulation,
as we discussed
00:19:58.310 --> 00:19:59.560
in the early lectures.
00:20:02.560 --> 00:20:05.800
The nonlinear strain terms that
we also derived in the
00:20:05.800 --> 00:20:08.640
earlier lectures, are
listed here for the
00:20:08.640 --> 00:20:12.160
two dimensional case.
00:20:12.160 --> 00:20:15.470
We have seen all these
expressions on the earlier
00:20:15.470 --> 00:20:19.100
view graph, except that we use
then additional notation, in
00:20:19.100 --> 00:20:24.970
other words, a notation that
involved k and j's and i's,
00:20:24.970 --> 00:20:26.945
and we have to sum over k.
00:20:26.945 --> 00:20:30.320
Well if you do so, you directly
arrive at these
00:20:30.320 --> 00:20:31.570
relationships here.
00:20:35.250 --> 00:20:39.935
It's an interesting point to
derive this expression and the
00:20:39.935 --> 00:20:43.540
linear strain part for the hoop
strain-- this is called
00:20:43.540 --> 00:20:44.910
the hoop strain--
00:20:44.910 --> 00:20:48.230
in the axisymmetric case.
00:20:48.230 --> 00:20:50.690
And let us look at
that derivation
00:20:50.690 --> 00:20:52.360
a bit more in detail.
00:20:52.360 --> 00:20:57.500
Here we have an axisymmetric
element in its original
00:20:57.500 --> 00:20:59.260
configuration.
00:20:59.260 --> 00:21:04.830
At time 0 it has moved into this
configuration up to time
00:21:04.830 --> 00:21:06.290
t plus delta t.
00:21:06.290 --> 00:21:10.380
Since we want to get the
incremental strain from time t
00:21:10.380 --> 00:21:13.420
to time t plus delta t, we're
looking at the configuration
00:21:13.420 --> 00:21:17.040
of time t plus delta t
in this derivation.
00:21:17.040 --> 00:21:21.310
Notice that this here is the
axis of revolution, which we
00:21:21.310 --> 00:21:27.140
denote as x2, x1 is this axis.
00:21:27.140 --> 00:21:33.470
And if we look as a plan view
onto this element, we would
00:21:33.470 --> 00:21:39.720
see this x1, x2 coming out of
the view graph, and x3 going
00:21:39.720 --> 00:21:42.950
down like this.
00:21:42.950 --> 00:21:48.980
Notice this here we label as
0ds, the initial length of a
00:21:48.980 --> 00:21:50.980
hoop fiber, so to say.
00:21:50.980 --> 00:21:56.150
In fact, this hoop fiber starts
right there at 0x1,
00:21:56.150 --> 00:21:59.980
which is, and I have to go now
upwards to the upper graph
00:21:59.980 --> 00:22:03.140
again, which is nothing else
than the starting point of
00:22:03.140 --> 00:22:05.470
that fiber.
00:22:05.470 --> 00:22:10.250
In other words, this green dot,
this green dot is nothing
00:22:10.250 --> 00:22:15.230
else than the start
of this arrow.
00:22:15.230 --> 00:22:19.660
I could say let's line
them up like that.
00:22:19.660 --> 00:22:25.590
So this arrow here goes really
into the view graph up there.
00:22:25.590 --> 00:22:28.800
If you look further to the right
here we see that we have
00:22:28.800 --> 00:22:33.300
also a blue arrow, of
course, curved.
00:22:33.300 --> 00:22:36.310
We call it the hoop, it's a
circle really, the radius
00:22:36.310 --> 00:22:41.430
around this origin that we're
looking at, like that.
00:22:41.430 --> 00:22:47.130
And notice that the start of
this arrow here is this blue
00:22:47.130 --> 00:22:48.380
dot right there.
00:22:50.650 --> 00:22:53.505
A material fiber, a material
particle, let's put it this
00:22:53.505 --> 00:22:58.310
way, a material particle that
has moved from here to there
00:22:58.310 --> 00:23:05.270
causes, in axisymmetric
analysis, this fiber here to
00:23:05.270 --> 00:23:07.090
take on this length.
00:23:10.650 --> 00:23:14.330
And of course, the change
in this length
00:23:14.330 --> 00:23:17.960
gives us the hoop strain.
00:23:17.960 --> 00:23:22.440
Let us look at this relationship
here, because
00:23:22.440 --> 00:23:26.070
that gives us a relationship
between the change in the
00:23:26.070 --> 00:23:31.960
length of the fiber, green
fiber, blue fiber, so to say,
00:23:31.960 --> 00:23:34.020
in plan view.
00:23:34.020 --> 00:23:40.420
And that is nothing else than
0x1 dividing t plus delta x1.
00:23:40.420 --> 00:23:41.590
Why is that the case?
00:23:41.590 --> 00:23:46.210
Well you see it by geometry
from this picture.
00:23:46.210 --> 00:23:48.930
And we will use this expression
here now to
00:23:48.930 --> 00:23:50.700
evaluate the actual strain.
00:23:54.520 --> 00:23:57.800
We find that the
Green-Lagrangian strain can be
00:23:57.800 --> 00:24:01.160
written as in this form.
00:24:01.160 --> 00:24:05.820
We substitute the expression
that we just obtained for this
00:24:05.820 --> 00:24:08.110
relationship.
00:24:08.110 --> 00:24:11.430
We substitute the
displacements.
00:24:11.430 --> 00:24:14.750
Displacements to configuration
t, and incremental
00:24:14.750 --> 00:24:20.430
displacement from t to t plus
delta t divided by the 0x1, of
00:24:20.430 --> 00:24:22.300
course, still here.
00:24:22.300 --> 00:24:26.690
We then go through a number of
steps of arithmetic, I'm sure
00:24:26.690 --> 00:24:31.300
you can easily do those, and
you arrive at this result.
00:24:31.300 --> 00:24:35.820
If we look at this, we find that
this here expression only
00:24:35.820 --> 00:24:40.330
involves the displacements
to time t.
00:24:40.330 --> 00:24:42.950
And that must, therefore,
be the Green-Lagrangian
00:24:42.950 --> 00:24:46.590
strain at time t.
00:24:46.590 --> 00:24:51.070
Notice that this expression
here involves incremental
00:24:51.070 --> 00:24:54.330
displacements, linear
incremental displacement, no
00:24:54.330 --> 00:24:57.000
products of them,
and the initial
00:24:57.000 --> 00:24:59.610
displacement at time t.
00:24:59.610 --> 00:25:02.990
This is the initial displacement
effect, which we
00:25:02.990 --> 00:25:06.160
always have in the total
Lagrangian formulation.
00:25:06.160 --> 00:25:10.570
This is a linear strain term
involving only the incremental
00:25:10.570 --> 00:25:11.540
displacement.
00:25:11.540 --> 00:25:16.370
And this the total linear
strain increment.
00:25:16.370 --> 00:25:20.140
Notice this is a total nonlinear
strain incremental.
00:25:20.140 --> 00:25:24.380
It involves the incremental
displacement u1 squared, and
00:25:24.380 --> 00:25:27.420
that's why it is a nonlinear
strain increment.
00:25:27.420 --> 00:25:32.650
Of course, this total here is
the incremental strain.
00:25:32.650 --> 00:25:36.310
This total is the incremental
strain.
00:25:36.310 --> 00:25:41.590
It's a nice derivation that
gives some insight into how
00:25:41.590 --> 00:25:44.270
these expressions that I had
already on the earlier view
00:25:44.270 --> 00:25:46.450
graph are arrived at.
00:25:46.450 --> 00:25:50.135
Well we are now ready to
construct the B matrix.
00:25:52.750 --> 00:25:54.300
On the left-hand side,
we would have,
00:25:54.300 --> 00:25:56.590
of course, the strains.
00:25:56.590 --> 00:26:00.850
If you look at the linear strain
displacement matrix, we
00:26:00.850 --> 00:26:03.090
have the linear strains here.
00:26:03.090 --> 00:26:08.660
Notice there's a 2 here because
of the 0e12 being
00:26:08.660 --> 00:26:10.540
equal to 0e21.
00:26:10.540 --> 00:26:15.000
We simply put 2 times
0e12 in here.
00:26:15.000 --> 00:26:18.580
And this is a total linear
strain increment.
00:26:18.580 --> 00:26:22.670
Here we have a sum of two
matrices giving us a total
00:26:22.670 --> 00:26:25.380
linear strain displacement
matrix.
00:26:25.380 --> 00:26:27.760
This is the one that does
not include the initial
00:26:27.760 --> 00:26:28.910
displacement effect.
00:26:28.910 --> 00:26:31.310
That one includes initial
displacement effect.
00:26:31.310 --> 00:26:35.680
And here we have the nodal point
incremental displacement
00:26:35.680 --> 00:26:37.860
the way we defined them
in an earlier lecture.
00:26:40.540 --> 00:26:48.030
Well the entries in t0BL0 are
shown here, involving of
00:26:48.030 --> 00:26:52.200
course, four-nodal point
k derivatives of the
00:26:52.200 --> 00:26:54.080
interpolation functions
corresponding
00:26:54.080 --> 00:26:55.330
to nodal point k.
00:26:57.680 --> 00:27:02.290
Notice that this last row is
only included if we are
00:27:02.290 --> 00:27:06.420
dealing with an axisymmetric
analysis.
00:27:06.420 --> 00:27:11.990
And notice that these are then
exactly the terms that we also
00:27:11.990 --> 00:27:15.490
have in linear analysis.
00:27:15.490 --> 00:27:18.590
So no surprises here.
00:27:18.590 --> 00:27:20.980
No new entries, as a matter
of fact, when
00:27:20.980 --> 00:27:23.300
compared to linear analysis.
00:27:23.300 --> 00:27:27.110
Except that we see a 0 here,
meaning, of course, that we're
00:27:27.110 --> 00:27:29.440
taking a differentiation with
respect to the original
00:27:29.440 --> 00:27:31.660
coordinates.
00:27:31.660 --> 00:27:34.590
Generally, in linear analysis,
if you look at the book, of
00:27:34.590 --> 00:27:37.700
course, you would not see this
0 here because it's not
00:27:37.700 --> 00:27:39.330
necessary to have
that 0 there.
00:27:39.330 --> 00:27:41.290
We always take differentiations
with respect
00:27:41.290 --> 00:27:43.770
to the original geometry.
00:27:43.770 --> 00:27:48.140
Maybe a quick word also how
we want to read this here.
00:27:48.140 --> 00:27:52.630
Notice these two entries here,
of course, nothing else in
00:27:52.630 --> 00:27:54.360
these two blue entries.
00:27:54.360 --> 00:27:58.540
Because these two entries
multiply these two columns,
00:27:58.540 --> 00:28:02.780
for readability I put this entry
there, put this entry
00:28:02.780 --> 00:28:05.470
there, so that you directly
see this column here
00:28:05.470 --> 00:28:10.420
corresponds to u1k, and this
column corresponds to u2k.
00:28:10.420 --> 00:28:13.430
Of course, both columns
correspond to node k.
00:28:16.700 --> 00:28:21.170
If we look at the matrix t0BL1,
which includes now the
00:28:21.170 --> 00:28:24.500
initial displacement effect,
in fact, introduces the
00:28:24.500 --> 00:28:28.150
initial displacement effect to
the total strain displacement
00:28:28.150 --> 00:28:32.430
matrix, it looks this way.
00:28:32.430 --> 00:28:35.670
Once again, this is a
contribution coming
00:28:35.670 --> 00:28:39.080
corresponding to u1k This
is the contribution
00:28:39.080 --> 00:28:41.270
corresponding to u2k.
00:28:41.270 --> 00:28:44.050
And notice here you have the
initial displacement effect
00:28:44.050 --> 00:28:47.820
appearing right here.
00:28:47.820 --> 00:28:51.270
All initial displacement
effects.
00:28:51.270 --> 00:28:54.120
And similarly here, initial
displacement effect.
00:28:57.070 --> 00:29:01.320
Once again, if you don't have
an axisymmetric analysis, in
00:29:01.320 --> 00:29:04.000
other words, you have a plane
stress, plane strain case, you
00:29:04.000 --> 00:29:06.020
would drop this last row.
00:29:10.400 --> 00:29:16.000
We construct the t0BNL
and T0S matrix.
00:29:16.000 --> 00:29:19.770
Next for the geometric stiffness
matrix, and we
00:29:19.770 --> 00:29:25.470
talked about this expression in
the earlier lecture, notice
00:29:25.470 --> 00:29:30.790
that this here is giving up, of
course, the matrix, the k
00:29:30.790 --> 00:29:34.830
matrix, that we're looking
for, this part here.
00:29:34.830 --> 00:29:37.840
And the F matrix looks
as shown here.
00:29:40.720 --> 00:29:44.810
I pointed out very strongly in
the earlier lecture that we
00:29:44.810 --> 00:29:50.740
construct the S and the BNL such
that when this product is
00:29:50.740 --> 00:29:54.020
taken, we get directly
this one.
00:29:54.020 --> 00:29:55.650
Because this is basic.
00:29:55.650 --> 00:29:57.520
This is obtained from the
continuum mechanics
00:29:57.520 --> 00:29:59.190
formulation.
00:29:59.190 --> 00:30:01.110
And we want to evaluate this.
00:30:01.110 --> 00:30:05.650
Therefore, we construct the
BNL and S such that this
00:30:05.650 --> 00:30:08.970
product, indeed, gives
us this contribution.
00:30:08.970 --> 00:30:11.870
And as is constructed
as shown here.
00:30:11.870 --> 00:30:15.420
The BNL is constructed
as shown here.
00:30:15.420 --> 00:30:19.160
Notice the u2k contributions,
the u1k
00:30:19.160 --> 00:30:21.670
contribution for node k.
00:30:27.840 --> 00:30:33.670
Finally, we also need the
t0S vector, hat vector.
00:30:33.670 --> 00:30:37.380
I pointed out in the earlier
lecture that this vector and
00:30:37.380 --> 00:30:43.140
this matrix are constructed in
such a way that this product
00:30:43.140 --> 00:30:47.260
here gives us exactly this
expression here.
00:30:47.260 --> 00:30:51.720
This is basic coming from
continuum mechanics, and this
00:30:51.720 --> 00:30:53.680
is what we have to capture.
00:30:53.680 --> 00:30:56.110
The entries in F hat
are given here.
00:31:00.040 --> 00:31:05.220
And with this then, we would be
ready to actually calculate
00:31:05.220 --> 00:31:08.630
all the matrices and vectors
for the total Lagrangian
00:31:08.630 --> 00:31:09.860
formulation.
00:31:09.860 --> 00:31:16.220
Let us now look at an example to
reinforce our understanding
00:31:16.220 --> 00:31:21.070
of how all of these matrices
are constructed,
00:31:21.070 --> 00:31:23.030
how they are evaluated.
00:31:23.030 --> 00:31:25.900
Here we have the
following case.
00:31:25.900 --> 00:31:30.330
A four-node element
originally in the
00:31:30.330 --> 00:31:32.230
configuration shown in black.
00:31:35.110 --> 00:31:39.310
Four nodes, as you can see,
one, two, three, four.
00:31:39.310 --> 00:31:44.830
This element moves from time
0 to time t into this
00:31:44.830 --> 00:31:46.570
configuration.
00:31:46.570 --> 00:31:48.370
The RAD configuration.
00:31:48.370 --> 00:31:54.170
Node 0.1 move there, node
0.2 move there.
00:31:54.170 --> 00:32:00.520
Notice that the element has
stretched and sheared over,
00:32:00.520 --> 00:32:04.160
but it has only stretched into
the vertical direction, not
00:32:04.160 --> 00:32:07.200
into the horizontal direction.
00:32:07.200 --> 00:32:13.140
Let us identify the lengths,
values that we
00:32:13.140 --> 00:32:14.190
have to deal with.
00:32:14.190 --> 00:32:16.930
0.2 here, 0.2 there.
00:32:16.930 --> 00:32:20.000
Notice that the displacement
upward is 0.1.
00:32:20.000 --> 00:32:24.640
And the shearing over,
so to say, is 0.1.
00:32:24.640 --> 00:32:27.550
We want to consider plane
strain conditions.
00:32:27.550 --> 00:32:33.030
And the problem that we like to
pose is calculate these two
00:32:33.030 --> 00:32:36.510
matrices, the linear strain,
and the nonlinear strain
00:32:36.510 --> 00:32:41.440
displacement matrix for
this particular case.
00:32:41.440 --> 00:32:45.740
Let's first look now a bit at
what's happening here to the
00:32:45.740 --> 00:32:48.530
material fiber.
00:32:48.530 --> 00:32:54.100
If we look at the horizontal
material fiber, shown here in
00:32:54.100 --> 00:32:59.100
black in the original
configuration, we notice that
00:32:59.100 --> 00:33:06.050
these material fibers are simply
translated rigidly over
00:33:06.050 --> 00:33:07.690
horizontally.
00:33:07.690 --> 00:33:13.160
As shown here via
the red arrows.
00:33:13.160 --> 00:33:18.830
So these black material fibers
lying horizontally are simply
00:33:18.830 --> 00:33:23.870
sheared over rigidly, as shown
by the red arrows.
00:33:23.870 --> 00:33:25.860
2 time t, of course.
00:33:25.860 --> 00:33:31.060
Let's look now at the vertical
fibers in this problem.
00:33:31.060 --> 00:33:33.140
Let's see what has
happened to them.
00:33:33.140 --> 00:33:38.990
Here we have the vertical fiber
shown in black in the
00:33:38.990 --> 00:33:41.020
original configuration.
00:33:41.020 --> 00:33:45.050
Let's see how they end up in
their final configuration.
00:33:45.050 --> 00:33:49.290
We see that these fibers have
actually stretched, they've
00:33:49.290 --> 00:33:52.470
become longer, and they have
also rotated over.
00:33:56.570 --> 00:34:00.040
Well this is the kinematics
that we are looking at.
00:34:00.040 --> 00:34:05.080
And let us now attack the
problem of constructing, once
00:34:05.080 --> 00:34:10.090
again, the linear strain
displacement matrix, and the
00:34:10.090 --> 00:34:13.469
nonlinear strain displacement
matrix.
00:34:13.469 --> 00:34:18.520
We do so by first identifying
which is the isoparametric
00:34:18.520 --> 00:34:20.790
coordinate system that we use.
00:34:20.790 --> 00:34:25.780
That coordinate system is
shown here, r and s.
00:34:29.360 --> 00:34:33.670
Let us then now start
this problem.
00:34:33.670 --> 00:34:38.130
And we identify simply by
inspection really because it's
00:34:38.130 --> 00:34:41.540
a fairly simple geometry that
we're dealing with.
00:34:41.540 --> 00:34:45.540
That this differentiation is
nothing else than 0.1, this
00:34:45.540 --> 00:34:50.580
differentiation is 0, that one
is 0, and this one is 0.1.
00:34:50.580 --> 00:34:53.940
Of course, these are the
elements that go into the
00:34:53.940 --> 00:34:57.160
Jacobian matrix, as I
discussed earlier.
00:34:57.160 --> 00:35:02.180
We put these elements into this
2 by 2 matrix, calculate
00:35:02.180 --> 00:35:07.660
the determinant, interesting
to note what the value is.
00:35:07.660 --> 00:35:10.550
That one is needed, of course,
later on in the numerical
00:35:10.550 --> 00:35:14.960
integration of the matrices.
00:35:14.960 --> 00:35:20.330
And we also recognize that to
obtain this derivative of an
00:35:20.330 --> 00:35:22.770
interpolation function,
we need to
00:35:22.770 --> 00:35:24.960
invert this matrix here.
00:35:24.960 --> 00:35:28.400
This 0.1 inverted gives
us 10, and that is
00:35:28.400 --> 00:35:30.310
this 10 right there.
00:35:30.310 --> 00:35:34.780
Therefore, we have now partial
with respect to x1, 0x1 is 10
00:35:34.780 --> 00:35:37.810
times partial with respect to
R. Of course, in the actual
00:35:37.810 --> 00:35:43.110
expression, we would put hk's
in here, and hk's in there.
00:35:43.110 --> 00:35:45.190
Similar we get the
differentiation
00:35:45.190 --> 00:35:47.290
with respect to x2.
00:35:47.290 --> 00:35:50.360
We will use this expression
here, putting the actual hk's
00:35:50.360 --> 00:35:53.280
in there and in there.
00:35:53.280 --> 00:35:56.700
Well this is some preparatory
work to actually complete the
00:35:56.700 --> 00:36:01.520
problem of constructing the
linear strain and nonlinear
00:36:01.520 --> 00:36:03.430
strain displacement matrix.
00:36:03.430 --> 00:36:07.270
And I think this is actually
a very good point where you
00:36:07.270 --> 00:36:11.320
might want to sit back and try
to complete the whole problem.
00:36:44.360 --> 00:36:46.770
Ladies and gentlemen,
welcome again.
00:36:46.770 --> 00:36:49.610
I hope you've had a close look
at the example, and surely I
00:36:49.610 --> 00:36:52.170
would be very interested
in knowing how it went.
00:36:52.170 --> 00:36:54.870
But let us now look
at the solution.
00:36:54.870 --> 00:36:58.420
We already discussed that for
this example, the Jacobian is
00:36:58.420 --> 00:37:01.790
given by this matrix here.
00:37:01.790 --> 00:37:04.770
And therefore, these are the
differentiations that we can
00:37:04.770 --> 00:37:06.710
directly use.
00:37:06.710 --> 00:37:09.130
Notice, of course, we need the
differentiation with respect
00:37:09.130 --> 00:37:13.090
to the x1 coordinate, and with
respect to the x2 coordinate
00:37:13.090 --> 00:37:17.330
of all the interpolation
functions.
00:37:17.330 --> 00:37:22.400
One way to proceed now is to
make a little table where we
00:37:22.400 --> 00:37:27.900
have here in this column the
nodes, we have in this column
00:37:27.900 --> 00:37:31.400
here the differentiations that
we need, partial hk with
00:37:31.400 --> 00:37:33.600
respect to 0x1.
00:37:33.600 --> 00:37:39.260
Further differentiations here,
partial hk with respect to x2.
00:37:39.260 --> 00:37:43.710
Notice that the Jacobian, the
inverse of the Jacobian matrix
00:37:43.710 --> 00:37:48.812
being 10, makes a 1/4
equal to 2.--
00:37:48.812 --> 00:37:52.350
1/4 times 10 being
equal to 2.5.
00:37:52.350 --> 00:37:56.420
So that's why you see
the 2.5 here.
00:37:56.420 --> 00:37:59.700
These are the nodal point
displacements, which, of
00:37:59.700 --> 00:38:02.340
course, are given for this
particular case.
00:38:02.340 --> 00:38:05.420
And with these nodal point
displacements given, and these
00:38:05.420 --> 00:38:09.580
differentiations calculated
as listed here, we can now
00:38:09.580 --> 00:38:12.980
evaluate these products here.
00:38:12.980 --> 00:38:17.620
And that is done as shown
in these columns.
00:38:17.620 --> 00:38:22.870
Notice the sum of these gives
us these differentiations
00:38:22.870 --> 00:38:27.960
here, which go into the initial
displacement effect of
00:38:27.960 --> 00:38:29.210
the strain displacement
matrix.
00:38:31.710 --> 00:38:35.150
Well like that, of course, you
can also calculate the
00:38:35.150 --> 00:38:39.780
differentiations of the
t0u21 and the t0u22.
00:38:42.670 --> 00:38:46.400
And both these terms are also
required in the initial
00:38:46.400 --> 00:38:50.280
displacement effect of the
strain displacement matrix.
00:38:53.000 --> 00:38:56.050
Another way to proceed, to get
this initial displacement
00:38:56.050 --> 00:39:00.700
effect, is to evaluate the
deformation gradient.
00:39:00.700 --> 00:39:03.400
In an earlier lecture, we talked
about the deformation
00:39:03.400 --> 00:39:06.170
gradient and it's listed
right here.
00:39:06.170 --> 00:39:08.700
It's a 2 by 2 matrix because
we're talking about a two
00:39:08.700 --> 00:39:10.660
dimensional motion.
00:39:10.660 --> 00:39:14.100
And here you have, for example,
partial tx2 with
00:39:14.100 --> 00:39:15.450
respect to 0x1.
00:39:15.450 --> 00:39:21.100
Here you have partial tx1
with respect to 0x1.
00:39:21.100 --> 00:39:25.190
0x1, et cetera.
00:39:25.190 --> 00:39:29.960
And if we take this deformation
gradient or the
00:39:29.960 --> 00:39:32.910
elements of that information
gradient, we can directly
00:39:32.910 --> 00:39:36.110
obtain these differentiations
here.
00:39:36.110 --> 00:39:41.190
In other words, t0x11
minus 1 must give us
00:39:41.190 --> 00:39:43.270
this expression here.
00:39:43.270 --> 00:39:50.900
t0x12, being that one here,
gives us tu1 with respect to
00:39:50.900 --> 00:39:53.960
0x2, in shorthand notation
written as
00:39:53.960 --> 00:39:57.400
shown here, et cetera.
00:39:57.400 --> 00:40:00.430
So we can obtain, in other
words, these elements also
00:40:00.430 --> 00:40:01.945
from the deformation gradient.
00:40:05.210 --> 00:40:08.090
Let us now look at how the
columns of the strain
00:40:08.090 --> 00:40:10.240
displacement matrices look.
00:40:10.240 --> 00:40:13.540
We simply substitute into the
general expressions that I
00:40:13.540 --> 00:40:21.780
gave you earlier, and here we
have the t0 be a 0 entry for
00:40:21.780 --> 00:40:23.910
node three.
00:40:23.910 --> 00:40:27.430
Of course there are eight
columns altogether because we
00:40:27.430 --> 00:40:29.230
have a four-node element.
00:40:29.230 --> 00:40:32.400
We just showed two such
columns here for--
00:40:32.400 --> 00:40:34.750
namely, those corresponding
to node three.
00:40:34.750 --> 00:40:39.550
Similarly, for t0BL1, we
get these entries here.
00:40:43.040 --> 00:40:47.330
Once again, also for node
three, of course.
00:40:47.330 --> 00:40:50.280
Similarly, we can also construct
the corresponding
00:40:50.280 --> 00:40:53.600
columns in the t0BNL matrix.
00:40:53.600 --> 00:40:56.170
And these columns are
given right there.
00:40:59.110 --> 00:41:01.060
Once again, for node three.
00:41:01.060 --> 00:41:06.050
This is, of course, also a
matrix that have altogether
00:41:06.050 --> 00:41:09.540
eight columns, because there are
four nodes, eight degrees
00:41:09.540 --> 00:41:12.100
of freedom.
00:41:12.100 --> 00:41:15.870
Let us now, as a next step,
consider the updated
00:41:15.870 --> 00:41:21.340
Lagrangian formulation in the
general plane stress, plane
00:41:21.340 --> 00:41:23.650
strain, axisymmetric case.
00:41:23.650 --> 00:41:26.870
In the updated Lagrangian
formulation, we have
00:41:26.870 --> 00:41:30.290
identified earlier, in an
earlier lecture, that we need
00:41:30.290 --> 00:41:33.290
these matrices and
that vector.
00:41:33.290 --> 00:41:36.860
This is, of course, a linear
strain stiffness matrix,
00:41:36.860 --> 00:41:39.110
that's a nonlinear strain
stiffness matrix, and that is
00:41:39.110 --> 00:41:42.350
the force vector that
corresponds to the internet
00:41:42.350 --> 00:41:44.710
element stresses.
00:41:44.710 --> 00:41:48.930
The matrices that go into the
calculation of these matrices
00:41:48.930 --> 00:41:50.770
are listed here.
00:41:50.770 --> 00:41:54.070
The tangent material
relationship, the linear
00:41:54.070 --> 00:41:56.770
strain displacement
matrix go into the
00:41:56.770 --> 00:41:59.620
calculation of this k matrix.
00:41:59.620 --> 00:42:04.410
This stress matrix, and that
strain displacement matrix,
00:42:04.410 --> 00:42:07.060
the nonlinear strain
displacement matrix, these two
00:42:07.060 --> 00:42:10.720
quantities go into the
calculation of this k matrix.
00:42:10.720 --> 00:42:14.720
And this stress vector,
and that linear strain
00:42:14.720 --> 00:42:16.810
displacement matrix
goes into the
00:42:16.810 --> 00:42:20.050
calculation of that F vector.
00:42:20.050 --> 00:42:24.990
We already derived that all
earlier in an earlier lecture.
00:42:24.990 --> 00:42:30.240
The stress strain matrix
is listed here.
00:42:30.240 --> 00:42:35.870
Notice that we have the
incremental stresses.
00:42:35.870 --> 00:42:38.500
Of course, these are increments
in the second
00:42:38.500 --> 00:42:41.890
Piola-Kirchhoff stresses
from time t onwards.
00:42:41.890 --> 00:42:44.400
That's why you have
the t here.
00:42:44.400 --> 00:42:47.880
That's why this t corresponds,
of course, to the updated
00:42:47.880 --> 00:42:49.680
Lagrangian formulation.
00:42:49.680 --> 00:42:52.270
Here we have the material
tensor, here we have the
00:42:52.270 --> 00:42:53.160
strain terms.
00:42:53.160 --> 00:42:56.160
Once again, the 2 that I pointed
out early already for
00:42:56.160 --> 00:42:58.710
the total Lagrangian
case as well.
00:42:58.710 --> 00:43:04.500
Of course this matrix here,
contains the elements of the
00:43:04.500 --> 00:43:09.810
material tensor, tCijrs, which
we used earlier when we talked
00:43:09.810 --> 00:43:11.495
about the continuum mechanics
formations.
00:43:14.360 --> 00:43:18.550
Well for the linear elastic
case, we would simply use the
00:43:18.550 --> 00:43:22.740
very familiar stress strain law
that we are using also in
00:43:22.740 --> 00:43:23.990
linear analysis.
00:43:23.990 --> 00:43:26.550
Young's modulus, Poisson's
ratio.
00:43:26.550 --> 00:43:31.210
So this is one typical case
for this C matrix.
00:43:31.210 --> 00:43:34.960
Of course, we will discuss later
on in later lectures how
00:43:34.960 --> 00:43:37.620
we construct the C matrix for
all sorts of material
00:43:37.620 --> 00:43:38.870
conditions.
00:43:40.900 --> 00:43:45.330
We need for the B matrix, for
the linear strain displacement
00:43:45.330 --> 00:43:50.410
matrix, these entries here,
meaning we need these
00:43:50.410 --> 00:43:52.830
differentiations.
00:43:52.830 --> 00:43:57.270
Notice partial u1 with respect
to tx1 is in shorthand
00:43:57.270 --> 00:43:59.590
notation written
as shown here.
00:43:59.590 --> 00:44:06.270
te22 is equal to that element,
and similarly we go on.
00:44:06.270 --> 00:44:08.380
Notice this, of course, is
again, the hoop strain.
00:44:11.300 --> 00:44:14.890
For the nonlinear strain
displacement matrix, we need
00:44:14.890 --> 00:44:17.540
to look at the nonlinear
strain terms.
00:44:17.540 --> 00:44:22.580
And the nonlinear strain terms
are listed out here now.
00:44:22.580 --> 00:44:26.890
We identified these strain
terms earlier when we
00:44:26.890 --> 00:44:29.790
discussed the updated Lagrangian
formulation in an
00:44:29.790 --> 00:44:31.080
earlier lecture.
00:44:31.080 --> 00:44:34.150
Of course, at that time, the
strain terms were represented
00:44:34.150 --> 00:44:37.590
in terms of ijk indices.
00:44:37.590 --> 00:44:43.110
Now we have to simply put i and
j equal to 1, for example,
00:44:43.110 --> 00:44:47.530
and k equal to 1, and equal to
2, and you will directly
00:44:47.530 --> 00:44:52.410
obtain from the earlier
expression that I presented to
00:44:52.410 --> 00:44:54.420
you this term here.
00:44:54.420 --> 00:44:58.500
And similarly, you obtain all
the other terms as well.
00:44:58.500 --> 00:45:01.880
By the way, the hoop strain,
the linear, and here we see
00:45:01.880 --> 00:45:05.760
the nonlinear hoop strain,
expression would be derived in
00:45:05.760 --> 00:45:09.565
the same way as we have done
it in this lecture for the
00:45:09.565 --> 00:45:11.720
total Lagrangian formulation.
00:45:11.720 --> 00:45:14.710
It would actually be a good
exercise for you to do so one
00:45:14.710 --> 00:45:18.330
and see how that goes.
00:45:18.330 --> 00:45:24.220
We construct the ttBL matrix to
capture the total strain,
00:45:24.220 --> 00:45:29.670
total linear strain, listed in
this vector here as shown by
00:45:29.670 --> 00:45:30.540
this equation.
00:45:30.540 --> 00:45:34.000
Of course, here we have the
nodal point displacements.
00:45:34.000 --> 00:45:38.160
The nodal point displacements
are in this vector.
00:45:38.160 --> 00:45:41.210
Denote always that these are
the discrete nodal point
00:45:41.210 --> 00:45:43.760
displacements.
00:45:43.760 --> 00:45:48.050
This last term, of course,
we only introduced in
00:45:48.050 --> 00:45:50.740
axisymmetric analysis.
00:45:50.740 --> 00:45:56.370
The entries in this ttBL matrix
are shown here for a
00:45:56.370 --> 00:45:58.530
typical node k.
00:45:58.530 --> 00:46:01.290
We use the same kind of picture
as for the total
00:46:01.290 --> 00:46:02.620
Lagrangian formulation.
00:46:02.620 --> 00:46:05.410
Notice here are the actual
displacements, the way they
00:46:05.410 --> 00:46:07.440
would appear in a vector.
00:46:07.440 --> 00:46:12.110
But since this element here hits
this column so to say, we
00:46:12.110 --> 00:46:14.470
have written it here
once more in blue.
00:46:14.470 --> 00:46:17.230
This element here hits this
column, and we have therefore
00:46:17.230 --> 00:46:19.030
written it here once
more in blue.
00:46:19.030 --> 00:46:25.060
So this is really the column
corresponding to the case node
00:46:25.060 --> 00:46:29.070
and the u1, the 1 direction.
00:46:29.070 --> 00:46:32.470
This is the column corresponding
to the case node
00:46:32.470 --> 00:46:35.510
and the 2 direction.
00:46:35.510 --> 00:46:40.550
If you compare this matrix with
the matrix that we use in
00:46:40.550 --> 00:46:45.040
linear analysis, you would see
that the matrix looks very
00:46:45.040 --> 00:46:48.860
similar, except that in linear
analysis you would not have
00:46:48.860 --> 00:46:53.880
this t, or you don't put that t
there in general because the
00:46:53.880 --> 00:46:56.980
t would, of course, be actually
the 0 configuration
00:46:56.980 --> 00:46:59.090
because all the differentiations
are referred
00:46:59.090 --> 00:47:02.120
to 0 configuration anyway.
00:47:02.120 --> 00:47:05.910
And it is common to not have
an index down here.
00:47:05.910 --> 00:47:09.810
If you look at this textbook,
chapter five, you would, for
00:47:09.810 --> 00:47:13.290
example, see there expressions
such as this
00:47:13.290 --> 00:47:17.090
without this t there.
00:47:17.090 --> 00:47:20.200
So there's really not much
of a surprise right here.
00:47:20.200 --> 00:47:24.120
You will surely see that this
is the right relationship to
00:47:24.120 --> 00:47:28.430
use in the B matrix
based on your
00:47:28.430 --> 00:47:32.480
knowledge of linear analysis.
00:47:32.480 --> 00:47:37.530
The expression of the nonlinear
strain displacement
00:47:37.530 --> 00:47:43.450
matrix and the stress matrix are
these two expressions are
00:47:43.450 --> 00:47:47.950
constructed in such a way as to
have that this product on
00:47:47.950 --> 00:47:52.010
the left-hand side is equal
to that expression there.
00:47:52.010 --> 00:47:57.100
I pointed that out already
earlier, that on the
00:47:57.100 --> 00:48:00.380
right-hand side this is the
continuum mechanics variable,
00:48:00.380 --> 00:48:02.670
and this continuum mechanics
variable is, of course, the
00:48:02.670 --> 00:48:06.610
basic quantity, the basic
quantity that we actually want
00:48:06.610 --> 00:48:07.730
to capture.
00:48:07.730 --> 00:48:12.240
And we construct this matrix
and that matrix
00:48:12.240 --> 00:48:14.880
such as to do so.
00:48:14.880 --> 00:48:19.480
I should just point out there
should be no bar here.
00:48:19.480 --> 00:48:22.840
Of course, this is a tensor
quantity, it's not a matrix,
00:48:22.840 --> 00:48:24.490
and so there should
be no bar there.
00:48:24.490 --> 00:48:27.130
That was a little error
on my part.
00:48:27.130 --> 00:48:30.060
The entries in the t tau
are given right here.
00:48:34.860 --> 00:48:38.430
The last row and column are,
of course, only included if
00:48:38.430 --> 00:48:41.470
you deal with an axisymmetric
analysis.
00:48:41.470 --> 00:48:47.650
ttBNL is shown here.
00:48:47.650 --> 00:48:50.920
Notice again, differentiations
with respect to time t.
00:48:53.810 --> 00:48:59.670
And the last row is only
included if you have an
00:48:59.670 --> 00:49:00.920
axisymmetric analysis.
00:49:04.240 --> 00:49:09.740
Finally, we want to also
construct our stress vector
00:49:09.740 --> 00:49:14.620
such that when it is entered
here with the BL matrix that
00:49:14.620 --> 00:49:19.820
we already have defined, we
capture via this product here
00:49:19.820 --> 00:49:21.450
exactly that term.
00:49:21.450 --> 00:49:25.120
This is the term that is basic,
that we have derived
00:49:25.120 --> 00:49:27.100
from the continuum mechanics
formulation.
00:49:27.100 --> 00:49:30.680
This is what we want to capture,
and we do so by this
00:49:30.680 --> 00:49:32.110
expression.
00:49:32.110 --> 00:49:35.230
And t tau hat is given
right here.
00:49:35.230 --> 00:49:38.490
It lists all the stresses
really, including the hoop
00:49:38.490 --> 00:49:42.100
stress if you have an
axisymmetric analysis.
00:49:42.100 --> 00:49:44.260
This completes what I wanted
to say about the two
00:49:44.260 --> 00:49:45.480
dimensional elements.
00:49:45.480 --> 00:49:48.420
We talked about the total
Lagrangian formulation, and
00:49:48.420 --> 00:49:51.260
the updated Lagrangian
formulation of these elements.
00:49:51.260 --> 00:49:53.880
In other words, we really
presented the matrices
00:49:53.880 --> 00:49:55.680
corresponding to these
formulations
00:49:55.680 --> 00:49:58.370
in quite some detail.
00:49:58.370 --> 00:50:00.550
These details that we discussed
for the two
00:50:00.550 --> 00:50:04.470
dimensional elements are also
almost directly applicable for
00:50:04.470 --> 00:50:06.270
the three dimensional
elements.
00:50:06.270 --> 00:50:11.080
And here we have a typical
eight-node element in a
00:50:11.080 --> 00:50:16.180
stationary coordinate frame,
x1, x2, x3, in its original
00:50:16.180 --> 00:50:17.430
configuration.
00:50:19.300 --> 00:50:22.070
The nodal point coordinates
are listed here.
00:50:22.070 --> 00:50:26.130
Notice k, of course, stands
again for the node k.
00:50:26.130 --> 00:50:29.940
1, 2, 3 stands for the
directions 1, 2, 3.
00:50:29.940 --> 00:50:34.460
And the left superscript here
stands for the time 0, the
00:50:34.460 --> 00:50:36.080
configuration 0.
00:50:36.080 --> 00:50:41.500
The element moves from time
0 to time t up to this
00:50:41.500 --> 00:50:43.240
configuration here.
00:50:43.240 --> 00:50:47.190
And the nodal points
now are tx1k--
00:50:47.190 --> 00:50:49.480
nodal point coordinate
I should have said.
00:50:49.480 --> 00:50:52.410
Now tx1k, tx2k, tx3k.
00:50:55.290 --> 00:51:00.170
The notation is much the same
as we have been using in two
00:51:00.170 --> 00:51:01.310
dimensional analysis.
00:51:01.310 --> 00:51:05.590
Of course, we now have a third
coordinate, the x3 coordinate
00:51:05.590 --> 00:51:08.055
also in the formulation.
00:51:08.055 --> 00:51:13.500
We now use interpolations much
the same way as for the two
00:51:13.500 --> 00:51:16.040
dimensional elements.
00:51:16.040 --> 00:51:18.740
For the original configuration,
we have these
00:51:18.740 --> 00:51:19.990
interpolations.
00:51:21.350 --> 00:51:24.530
Notice, the third direction
enters now.
00:51:24.530 --> 00:51:27.480
Once again, these are the
interpolation functions.
00:51:27.480 --> 00:51:32.050
These are the nodal point
coordinates corresponding to
00:51:32.050 --> 00:51:34.710
the 1 direction.
00:51:34.710 --> 00:51:38.170
These are the nodal point
coordinates corresponding to
00:51:38.170 --> 00:51:40.570
the 2 direction, and so on.
00:51:40.570 --> 00:51:44.490
Of course, we now also have to
introduce the nodal point
00:51:44.490 --> 00:51:47.160
coordinates corresponding
to the 3 direction.
00:51:47.160 --> 00:51:49.340
And if the number of nodes,
if you have an eight-node
00:51:49.340 --> 00:51:52.190
element, and of course is 8.
00:51:52.190 --> 00:51:55.395
These are the interpolations
used for the original geometry
00:51:55.395 --> 00:52:01.290
of the element, and for the
configuration at time t, we
00:52:01.290 --> 00:52:02.540
use these interpolations.
00:52:05.110 --> 00:52:10.240
Notice these are now the nodal
points coordinates
00:52:10.240 --> 00:52:11.820
corresponding to time t.
00:52:14.480 --> 00:52:16.180
We use still the same
interpolation
00:52:16.180 --> 00:52:17.440
functions, of course.
00:52:17.440 --> 00:52:23.570
And N is the same number as for
the 0 interpolation, or
00:52:23.570 --> 00:52:27.230
rather for the interpolation
of the original geometry.
00:52:27.230 --> 00:52:29.410
In other words, for the
eight-node element that we
00:52:29.410 --> 00:52:33.520
briefly considered, N is in
each case equal to 8.
00:52:33.520 --> 00:52:40.440
We use these expressions, we
subtract from tx1 the 0x1 from
00:52:40.440 --> 00:52:45.930
tx2, the 0x2 expression from
tx3, the 0x3 expression, and
00:52:45.930 --> 00:52:49.040
directly obtain the
interpolations for the
00:52:49.040 --> 00:52:51.220
displacement of the elements.
00:52:51.220 --> 00:52:54.390
And once we have the
displacement interpolation,
00:52:54.390 --> 00:52:58.050
we, of course, can directly find
the derivative of these
00:52:58.050 --> 00:53:00.910
displacement interpolations
to obtain the strain
00:53:00.910 --> 00:53:02.220
interpolations.
00:53:02.220 --> 00:53:06.260
And these expressions then,
the derivatives of the
00:53:06.260 --> 00:53:09.920
displacement interpolation,
enter into the construction of
00:53:09.920 --> 00:53:12.760
the strain displacement
matrices.
00:53:12.760 --> 00:53:16.250
So we see that basically, the
same procedure that we
00:53:16.250 --> 00:53:18.840
discussed with two dimensional
analysis is directly
00:53:18.840 --> 00:53:21.130
applicable to the three
dimensional analysis, the only
00:53:21.130 --> 00:53:25.090
difference being that the third
direction, x3, has to be
00:53:25.090 --> 00:53:28.230
interpolated, the displacements
have to be
00:53:28.230 --> 00:53:29.850
carried along corresponding
to the third
00:53:29.850 --> 00:53:32.310
direction, et cetera.
00:53:32.310 --> 00:53:35.210
This then brings me to the end
of what I wanted to say in
00:53:35.210 --> 00:53:36.040
this lecture.
00:53:36.040 --> 00:53:37.800
Thank you very much for
your attention.