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. 00:00:21.880 --> 00:00:23.780 PROFESSOR: Ladies and gentlemen, welcome to this 00:00:23.780 --> 00:00:26.600 lecture on nonlinear finite element analysis of solids and 00:00:26.600 --> 00:00:27.530 structures. 00:00:27.530 --> 00:00:30.360 In this lecture I would like to discuss with you the 00:00:30.360 --> 00:00:33.240 updated Lagrangian formulation for general incremental 00:00:33.240 --> 00:00:34.980 nonlinear analysis. 00:00:34.980 --> 00:00:37.820 However, before doing so, I thought it would be good to 00:00:37.820 --> 00:00:41.100 take a bit of time and review some of the material that we 00:00:41.100 --> 00:00:43.110 have been discussing already. 00:00:43.110 --> 00:00:46.500 We have been encountering a large number of quite 00:00:46.500 --> 00:00:50.860 complicated equations, some very important basic concepts, 00:00:50.860 --> 00:00:52.480 and some subtleties. 00:00:52.480 --> 00:00:56.480 And to really understand all of this material, it is 00:00:56.480 --> 00:00:59.680 necessary to study it, of course, a lot more than the 00:00:59.680 --> 00:01:01.670 time that we have allocated so far. 00:01:01.670 --> 00:01:04.629 But it might help you to just review now some of that 00:01:04.629 --> 00:01:06.460 material once more. 00:01:06.460 --> 00:01:10.420 I've prepared here on the marker board the material that 00:01:10.420 --> 00:01:13.370 relates to the total Lagrangian formulation, in 00:01:13.370 --> 00:01:17.350 which really you have been seeing some of the very 00:01:17.350 --> 00:01:19.200 important equations. 00:01:19.200 --> 00:01:23.010 And here we have the basic equation that is used, of 00:01:23.010 --> 00:01:25.430 course, in finite element analysis. 00:01:25.430 --> 00:01:30.030 This basic equation states, of course, it is the equation of 00:01:30.030 --> 00:01:34.610 the principle of virtual work, states that an internal 00:01:34.610 --> 00:01:38.800 virtual work, or the internal virtual work, must be equal to 00:01:38.800 --> 00:01:40.230 external virtual work. 00:01:40.230 --> 00:01:44.210 You remember that we talked about the Cauchy stress, the 00:01:44.210 --> 00:01:47.030 stress which is actually force per unit area. 00:01:47.030 --> 00:01:49.490 And that's the engineering stress, that's the stress that 00:01:49.490 --> 00:01:51.440 we want to solve for. 00:01:51.440 --> 00:01:54.460 That we talked about a virtual strain. 00:01:54.460 --> 00:01:57.960 A virtual strain that is referred to time 00:01:57.960 --> 00:01:59.840 t plus delta t. 00:01:59.840 --> 00:02:02.530 And that is why we have this subscript here, 00:02:02.530 --> 00:02:04.220 t plus delta t. 00:02:04.220 --> 00:02:09.220 And we integrate this product here over the volume at time t 00:02:09.220 --> 00:02:10.810 plus delta t. 00:02:10.810 --> 00:02:14.600 Notice that I'm using here a green color for the real 00:02:14.600 --> 00:02:19.710 stress that we want to solve for, and a red color for the 00:02:19.710 --> 00:02:24.390 virtual quantity that we're using basically as a mechanism 00:02:24.390 --> 00:02:26.720 to solve for this stress. 00:02:26.720 --> 00:02:29.440 Of course, on the right-hand side, we identified we have 00:02:29.440 --> 00:02:31.260 the external virtual work. 00:02:31.260 --> 00:02:36.110 And in here, this script R here, denotes contributions 00:02:36.110 --> 00:02:40.790 from the surface forces, body forces that are being applied 00:02:40.790 --> 00:02:43.710 externally to the body. 00:02:43.710 --> 00:02:46.500 This is really the basic equation that we want to 00:02:46.500 --> 00:02:51.210 operate on, and that we want to work with to solve for the 00:02:51.210 --> 00:02:54.660 unknown Cauchy stresses at time t plus delta t. 00:02:54.660 --> 00:02:59.450 Now we recall, of course, that we are here integrating over a 00:02:59.450 --> 00:03:04.050 volume at time t plus delta t, and that volume is unknown. 00:03:04.050 --> 00:03:07.450 Because it is unknown, we cannot very easily deal with 00:03:07.450 --> 00:03:10.300 this stress measure, and we have, therefore, introduced a 00:03:10.300 --> 00:03:11.610 new stress measure. 00:03:11.610 --> 00:03:15.740 A stress measure that we call the second Piola-Kirchhoff 00:03:15.740 --> 00:03:17.710 stress tensor. 00:03:17.710 --> 00:03:19.670 We introduce this one here. 00:03:19.670 --> 00:03:24.320 Notice this is a stress at time t plus delta t, referred 00:03:24.320 --> 00:03:28.660 to the original configuration, 0, of the body. 00:03:28.660 --> 00:03:33.150 The capital S, and we have ij as the components of that 00:03:33.150 --> 00:03:34.520 stress tensor. 00:03:34.520 --> 00:03:37.670 Now, this is a new stress tensor that we know we can 00:03:37.670 --> 00:03:39.210 deal very well with. 00:03:39.210 --> 00:03:42.600 And we have to also introduce a new strain tensor. 00:03:42.600 --> 00:03:44.390 Why do we have to do so? 00:03:44.390 --> 00:03:49.120 Well because using this stress tensor, we have to use an 00:03:49.120 --> 00:03:51.470 energy conjugate strain measure. 00:03:51.470 --> 00:03:55.650 And this Green-Lagrange strain tensor, the Green-Lagrange 00:03:55.650 --> 00:03:59.190 strain tensor is energy conjugate to the second 00:03:59.190 --> 00:04:01.650 Piola-Kirchhoff stress tensor. 00:04:01.650 --> 00:04:05.030 Having introduced these two quantities, and we looked at 00:04:05.030 --> 00:04:07.470 the components, how they're constructed, how they're 00:04:07.470 --> 00:04:11.740 related, for example, in this case, how this one is related 00:04:11.740 --> 00:04:13.410 to the Cauchy stress. 00:04:13.410 --> 00:04:17.000 Having now introduced these two quantities, we can 00:04:17.000 --> 00:04:22.450 directly work basically with this equation here. 00:04:22.450 --> 00:04:28.280 Well to do so, we have decomposed the stress tensor 00:04:28.280 --> 00:04:32.540 into a part that we know, and into an unknown part. 00:04:32.540 --> 00:04:33.950 And that's an important point. 00:04:33.950 --> 00:04:38.280 We can decompose this stress tensor into a stress that we 00:04:38.280 --> 00:04:41.580 know, and that we don't know, by simple addition 00:04:41.580 --> 00:04:43.250 here on this side. 00:04:43.250 --> 00:04:46.880 Notice we simply add here because we had a quantity that 00:04:46.880 --> 00:04:53.040 is referred to the volume at time 0, to another quantity 00:04:53.040 --> 00:04:56.520 that is referred to the volume at time 0. 00:04:56.520 --> 00:04:59.930 In other words, the reference configurations for both of 00:04:59.930 --> 00:05:03.470 these are the same, and that's why we can have a simple 00:05:03.470 --> 00:05:06.120 addition sign here. 00:05:06.120 --> 00:05:09.930 Please keep in mind that this quantity is known in our 00:05:09.930 --> 00:05:11.550 incremental solution. 00:05:11.550 --> 00:05:15.280 Because we, of course, assume that we have solved for the 00:05:15.280 --> 00:05:18.630 configuration of the body up to a time t. 00:05:18.630 --> 00:05:22.810 This one is unknown, and this is the reason why I used here 00:05:22.810 --> 00:05:26.550 black color to show that this is an unknown quantity and 00:05:26.550 --> 00:05:29.210 distinguish it from the one that is known. 00:05:29.210 --> 00:05:30.950 We proceed in the same way with the 00:05:30.950 --> 00:05:32.700 Green-Lagrange strain tensor. 00:05:32.700 --> 00:05:34.010 Here we have the equations. 00:05:34.010 --> 00:05:37.320 The Green-Lagrange strain tensor at time t plus delta t 00:05:37.320 --> 00:05:42.170 is decomposed into a quantity that we know, 00:05:42.170 --> 00:05:44.680 plus an unknown quantity. 00:05:44.680 --> 00:05:46.760 Once again, in black. 00:05:46.760 --> 00:05:49.110 So these are the two quantities that we really want 00:05:49.110 --> 00:05:51.960 to solve for, the increment in the second Piola-Kirchhoff 00:05:51.960 --> 00:05:55.750 stress, and the increment in the Green-Lagrange strain. 00:05:55.750 --> 00:05:58.260 Now we also discussed that the increment in the 00:05:58.260 --> 00:06:01.050 Green-Lagrange strain can actually be decomposed into 00:06:01.050 --> 00:06:03.000 two quantities. 00:06:03.000 --> 00:06:07.300 One that is linear in the particle incremental 00:06:07.300 --> 00:06:12.620 displacements ui, and one that is nonlinear in the particle 00:06:12.620 --> 00:06:14.770 incremental displacements ui. 00:06:14.770 --> 00:06:16.230 Please read it as follows. 00:06:16.230 --> 00:06:20.600 Linear in ui, nonlinear in ui. 00:06:20.600 --> 00:06:24.180 Of course, remember that we have so far only spoken about 00:06:24.180 --> 00:06:25.970 the continuum mechanics. 00:06:25.970 --> 00:06:28.570 In other words, we have not really, at this point, said 00:06:28.570 --> 00:06:31.380 anything about the finite element discretization. 00:06:31.380 --> 00:06:34.100 We also referred, however, in an earlier lecture to the fact 00:06:34.100 --> 00:06:38.800 that if we do interpolate in finite element analysis, then 00:06:38.800 --> 00:06:42.230 using continuum elements for the analysis of solids and 00:06:42.230 --> 00:06:47.650 structures, the incremental nodal point variables are 00:06:47.650 --> 00:06:50.710 linearly related to these incremental particle 00:06:50.710 --> 00:06:51.760 displacements. 00:06:51.760 --> 00:06:56.440 And for that reason, this is a true linear strain increment, 00:06:56.440 --> 00:06:59.910 and that is a true nonlinear strain increment. 00:06:59.910 --> 00:07:03.110 On the other hand, if we use structural elements that also 00:07:03.110 --> 00:07:08.920 include or use nodal point rotations, then the particle 00:07:08.920 --> 00:07:14.360 displacements, these ui's, are linearly related to the 00:07:14.360 --> 00:07:17.520 incremental nodal point displacements, but nonlinearly 00:07:17.520 --> 00:07:20.670 related to the incremental nodal point rotations. 00:07:20.670 --> 00:07:24.820 And in that case, this is here still the true linear 00:07:24.820 --> 00:07:27.260 increment of the Green-Lagrange 00:07:27.260 --> 00:07:29.790 strain for the particle. 00:07:29.790 --> 00:07:33.750 But this one here is not containing all the nonlinear 00:07:33.750 --> 00:07:37.480 strain incremental parts for the particle. 00:07:37.480 --> 00:07:38.910 There has to be a little bit-- 00:07:38.910 --> 00:07:42.520 there has to be some amendment here, some addition here. 00:07:42.520 --> 00:07:46.260 And we briefly talked about the effect of neglecting this 00:07:46.260 --> 00:07:47.870 addition as well. 00:07:47.870 --> 00:07:51.940 Well the point now is that we want to use, of course, these 00:07:51.940 --> 00:07:57.110 two quantities to substitute these into the equation of the 00:07:57.110 --> 00:08:00.900 principle of virtual work, and we then arrive directly at 00:08:00.900 --> 00:08:03.250 this relationship here. 00:08:03.250 --> 00:08:04.480 Now what do we see here? 00:08:04.480 --> 00:08:08.290 We see an integral over the original volume of the second 00:08:08.290 --> 00:08:11.290 Piola-Kirchhoff stress referred to the original 00:08:11.290 --> 00:08:15.740 configuration, and it's a real stress, the one that we want 00:08:15.740 --> 00:08:20.420 to solve for, and that's why I use the color green again. 00:08:20.420 --> 00:08:25.440 This here is a virtual strain, Green-Lagrange strain. 00:08:25.440 --> 00:08:29.230 It's a variation on this Green-Lagrange strain here. 00:08:29.230 --> 00:08:33.770 And this product is integrated over the original volume, 00:08:33.770 --> 00:08:36.049 which is, of course, the volume that we know. 00:08:36.049 --> 00:08:38.190 And this is the reason why we can so well 00:08:38.190 --> 00:08:41.330 deal with this integral. 00:08:41.330 --> 00:08:44.280 On the right-hand side, we have made no modification. 00:08:44.280 --> 00:08:47.840 We still have simply the external virtual work. 00:08:47.840 --> 00:08:53.950 Now, please realize that this equation here is totally 00:08:53.950 --> 00:08:57.720 equivalent to the equation that we discussed earlier. 00:08:57.720 --> 00:08:59.180 This equation here. 00:08:59.180 --> 00:09:02.150 Here we are talking about Cauchy stresses, an 00:09:02.150 --> 00:09:06.190 infinitesimal virtual strain integrated over a volume of 00:09:06.190 --> 00:09:09.520 time t plus delta t that is unknown. 00:09:09.520 --> 00:09:14.550 This equation has simply been re-cast into a new form using 00:09:14.550 --> 00:09:16.190 continuum mechanics. 00:09:16.190 --> 00:09:20.350 No finite elements really, just continuum mechanics, into 00:09:20.350 --> 00:09:23.000 this form here. 00:09:23.000 --> 00:09:27.640 And now we have a form that we can work very well with. 00:09:27.640 --> 00:09:31.070 We substitute into this equation and we linearize, and 00:09:31.070 --> 00:09:33.510 we did that in the earlier lecture. 00:09:33.510 --> 00:09:36.950 The result of that substitution and linearization 00:09:36.950 --> 00:09:39.120 is given right here. 00:09:39.120 --> 00:09:42.800 Now notice that in each of these integrals, of course, we 00:09:42.800 --> 00:09:45.160 are integrating over a volume that is known, 00:09:45.160 --> 00:09:47.160 the original volume. 00:09:47.160 --> 00:09:51.700 Notice that in the first integral here, we involve, 00:09:51.700 --> 00:09:53.550 well a material tensor. 00:09:53.550 --> 00:09:55.750 That has to be defined, it depends on the material that 00:09:55.750 --> 00:09:59.460 you're looking at, that you want to analyze. 00:09:59.460 --> 00:10:02.890 This is here the increment in the Green-Lagrange strain, the 00:10:02.890 --> 00:10:04.550 linear part only. 00:10:04.550 --> 00:10:10.760 Here is a variation on this linear increment in green-- 00:10:10.760 --> 00:10:12.400 sorry, it's in red, of course. 00:10:12.400 --> 00:10:13.420 This one was green. 00:10:13.420 --> 00:10:14.680 This one is in red. 00:10:14.680 --> 00:10:16.690 And this is, of course, a virtual quantity. 00:10:16.690 --> 00:10:18.540 This is the virtual quantity. 00:10:18.540 --> 00:10:20.870 Notice that here we are talking about a 00:10:20.870 --> 00:10:22.380 stress that we know. 00:10:22.380 --> 00:10:24.640 We're incrementally decomposing. 00:10:24.640 --> 00:10:26.130 Remember? 00:10:26.130 --> 00:10:30.260 We're adding to the quantities at time t, the increment. 00:10:30.260 --> 00:10:32.160 The quantities at time t are known. 00:10:32.160 --> 00:10:34.630 So this one is known. 00:10:34.630 --> 00:10:39.880 This one here is a variation on the nonlinear increment of 00:10:39.880 --> 00:10:43.050 the Green-Lagrange strain, total increment. 00:10:43.050 --> 00:10:45.260 Well, in red again. 00:10:45.260 --> 00:10:47.890 And on the right-hand side we have the external virtual 00:10:47.890 --> 00:10:51.410 work, which of course, is assumed to be given. 00:10:51.410 --> 00:10:54.690 Do you still remember that we are talking about deformation 00:10:54.690 --> 00:10:56.020 independent loading. 00:10:56.020 --> 00:10:59.850 So we can calculate this piece here directly. 00:10:59.850 --> 00:11:05.330 And here we have also the stress that is known, the 00:11:05.330 --> 00:11:11.100 stress at time t, and the virtual strain, linear strain. 00:11:11.100 --> 00:11:16.270 This is what we arrived at in our earlier lecture. 00:11:16.270 --> 00:11:19.210 Of course, there was a linearization involved, and we 00:11:19.210 --> 00:11:21.200 know there would be errors if we simply 00:11:21.200 --> 00:11:22.880 operate on this equation. 00:11:22.880 --> 00:11:25.270 We don't want to have these errors in the actual finite 00:11:25.270 --> 00:11:26.510 element solution. 00:11:26.510 --> 00:11:29.520 And we realize that it is good to iterate. 00:11:29.520 --> 00:11:33.720 So we also looked in the earlier lecture how we would 00:11:33.720 --> 00:11:35.290 iterate typically. 00:11:35.290 --> 00:11:38.400 And we developed an equation where there was something on 00:11:38.400 --> 00:11:42.690 the left-hand side, which I simply denote here by dots. 00:11:42.690 --> 00:11:46.120 There were quite some complicated expressions. 00:11:46.120 --> 00:11:48.150 You may want to look them up again. 00:11:48.150 --> 00:11:51.150 But I just denote them by dots because what is really 00:11:51.150 --> 00:11:54.340 important, and that I pointed out also earlier, is what is 00:11:54.340 --> 00:11:55.630 on the right-hand side. 00:11:55.630 --> 00:11:57.490 And what do we see on the right-hand side? 00:11:57.490 --> 00:12:02.200 On the right-hand side we see the external virtual work, and 00:12:02.200 --> 00:12:04.860 we are subtracting from it this integral. 00:12:04.860 --> 00:12:06.920 Let's look at this integral once more. 00:12:06.920 --> 00:12:10.250 We have here the second Piola-Kirchhoff stress at time 00:12:10.250 --> 00:12:12.650 t plus delta t. 00:12:12.650 --> 00:12:16.060 Of course, always refer to the original configuration. 00:12:16.060 --> 00:12:20.530 Corresponding to iteration k minus 1. 00:12:20.530 --> 00:12:23.810 The iteration here written in black. 00:12:23.810 --> 00:12:27.840 We are operating on the Green-Lagrange strain at time 00:12:27.840 --> 00:12:29.710 t plus delta t. 00:12:29.710 --> 00:12:32.230 Refer to the original configuration in 00:12:32.230 --> 00:12:34.190 iteration k minus 1. 00:12:34.190 --> 00:12:38.910 Actually, calculated at the end of iteration k minus 1 is 00:12:38.910 --> 00:12:40.160 better to say. 00:12:40.160 --> 00:12:43.850 And this product here is integrated over the original 00:12:43.850 --> 00:12:44.920 volume once more. 00:12:44.920 --> 00:12:48.870 k runs from 1, 2, 3, onwards. 00:12:48.870 --> 00:12:52.510 Well notice that when k is equal to 1 we have a 0 here, 00:12:52.510 --> 00:12:57.050 and we have to ask ourselves what is this quantity then? 00:12:57.050 --> 00:13:00.170 Well if there's a 0 right here, then we're looking here 00:13:00.170 --> 00:13:02.400 at t 0 Sij. 00:13:02.400 --> 00:13:04.640 The stress, the second Piola-Kirchhoff stress 00:13:04.640 --> 00:13:08.290 corresponding to the configuration at time t. 00:13:08.290 --> 00:13:12.160 And if we put a 0 here, we would have also here, the 00:13:12.160 --> 00:13:15.870 initial condition being the Green-Lagrange strain at time 00:13:15.870 --> 00:13:19.370 t, of course, referred to configuration 0. 00:13:19.370 --> 00:13:22.120 So we have the write initial conditions here, and therefore 00:13:22.120 --> 00:13:24.500 we can proceed with the iteration. 00:13:24.500 --> 00:13:26.520 We discussed that in the earlier lecture. 00:13:26.520 --> 00:13:30.680 The important point now is that once this iteration 00:13:30.680 --> 00:13:35.170 converges, we have to use, of course, schemes, solution 00:13:35.170 --> 00:13:37.920 procedures that makes the iteration converge. 00:13:37.920 --> 00:13:39.920 We will talk about that in later lecture. 00:13:39.920 --> 00:13:42.490 But once we have obtained convergence, and let's assume 00:13:42.490 --> 00:13:48.020 we have converged, then this R, the script R, the external 00:13:48.020 --> 00:13:51.670 virtual work, is equal to this integral. 00:13:51.670 --> 00:13:53.450 That's how we define convergence. 00:13:53.450 --> 00:13:57.720 And this means then that the external virtual work is now 00:13:57.720 --> 00:14:00.420 equal to the internal virtual work. 00:14:00.420 --> 00:14:01.560 What does this mean? 00:14:01.560 --> 00:14:07.460 It means in essence, that this right-hand side is 0, which 00:14:07.460 --> 00:14:11.310 means-- and now I like to go back to our earlier equation-- 00:14:11.310 --> 00:14:15.650 that this equation here is satisfied because notice that 00:14:15.650 --> 00:14:18.980 if you look at this equation and you take this left-hand 00:14:18.980 --> 00:14:21.560 side and bring it to the right-hand side with a minus 00:14:21.560 --> 00:14:24.190 in front, then you have exactly what we 00:14:24.190 --> 00:14:26.450 just looked at earlier. 00:14:26.450 --> 00:14:30.640 In other words, our iteration is performed really on an 00:14:30.640 --> 00:14:35.060 expression that is nothing else than this R, this script 00:14:35.060 --> 00:14:40.230 R, minus this one on the right-hand side. 00:14:40.230 --> 00:14:42.840 There's a minus sign here and you take this and bring it 00:14:42.840 --> 00:14:44.060 right there. 00:14:44.060 --> 00:14:46.040 That's what we just looked at. 00:14:46.040 --> 00:14:49.430 Of course, we also remember now that this expression now 00:14:49.430 --> 00:14:53.570 is really totally equivalent to what we started with. 00:14:53.570 --> 00:14:57.170 Namely, this expression right here. 00:14:57.170 --> 00:15:00.860 And this is, of course, what we really want a solve. 00:15:00.860 --> 00:15:03.170 We mentioned earlier that we introduced the second 00:15:03.170 --> 00:15:05.830 Piola-Kirchhoff stress, and that we would like to get a 00:15:05.830 --> 00:15:08.210 physical understanding possibly of that stress. 00:15:08.210 --> 00:15:11.930 And I mentioned that that is very difficult to obtain. 00:15:11.930 --> 00:15:14.640 In the computer program, we just work with that stress 00:15:14.640 --> 00:15:19.260 measure in order to be able to solve the equation, to operate 00:15:19.260 --> 00:15:21.350 well on the principle of virtual work. 00:15:21.350 --> 00:15:24.730 What we really want to do, of course, is to solve this 00:15:24.730 --> 00:15:27.050 equation right here. 00:15:27.050 --> 00:15:28.240 This is the equation. 00:15:28.240 --> 00:15:32.050 That is the physical stress that we want to solve for. 00:15:32.050 --> 00:15:35.860 Well since this equation is totally equivalent to that 00:15:35.860 --> 00:15:41.810 equation, and looking at this equation, if we take this 00:15:41.810 --> 00:15:45.010 left-hand side, bring it to the right-hand side-- 00:15:45.010 --> 00:15:46.950 of course, there's a minus appearing there-- 00:15:46.950 --> 00:15:53.870 we do obtain what we see right here at convergence of course. 00:15:53.870 --> 00:15:58.630 We operate properly on the principle of virtual work. 00:15:58.630 --> 00:16:02.380 So this really closes, so to say, the loop, and I hope it 00:16:02.380 --> 00:16:05.730 helps you to have this overview in understanding of 00:16:05.730 --> 00:16:07.560 what I discussed earlier with you. 00:16:07.560 --> 00:16:09.850 But let's go one step further. 00:16:09.850 --> 00:16:15.610 In the finite element discretization, we obtain from 00:16:15.610 --> 00:16:17.880 this equation here-- 00:16:17.880 --> 00:16:20.910 I haven't listed what is on the left-hand side, you can 00:16:20.910 --> 00:16:22.770 look it up-- 00:16:22.770 --> 00:16:27.330 we obtain directly this equation here. 00:16:27.330 --> 00:16:30.860 Of course, there's quite a number of steps that we still 00:16:30.860 --> 00:16:31.850 have to discuss. 00:16:31.850 --> 00:16:35.570 How we obtain this matrix, we call this the tangent 00:16:35.570 --> 00:16:37.430 stiffness matrix. 00:16:37.430 --> 00:16:41.130 We have to discuss how we calculate this vector here, 00:16:41.130 --> 00:16:42.480 this vector here. 00:16:42.480 --> 00:16:46.200 In fact, the way we actually introduced it, since it's a 00:16:46.200 --> 00:16:49.800 total Lagrangian formulation, I should really put here a 0. 00:16:55.200 --> 00:16:58.330 We have to discuss how we can calculate this vector for 00:16:58.330 --> 00:17:01.165 different elements, et cetera, and there is a 00:17:01.165 --> 00:17:02.490 lot still to be learned. 00:17:02.490 --> 00:17:07.400 But remember that up to here, we only talked about continuum 00:17:07.400 --> 00:17:11.020 mechanics, and this is where the finite element 00:17:11.020 --> 00:17:12.400 discretization starts. 00:17:12.400 --> 00:17:15.400 And these are then the basic equations that are being 00:17:15.400 --> 00:17:17.170 arrived at. 00:17:17.170 --> 00:17:21.819 At convergence, once again, here we iterate, of course. 00:17:21.819 --> 00:17:24.520 Notice that I'm using now the superscript i. 00:17:24.520 --> 00:17:27.250 Of course, this superscript is a dummy, it could be k, it 00:17:27.250 --> 00:17:29.340 could be j, it doesn't matter. 00:17:29.340 --> 00:17:32.220 We frequently use i. 00:17:32.220 --> 00:17:36.140 I didn't want to use up here i because we have an i already 00:17:36.140 --> 00:17:39.990 as a subscript, that's why I'm using k here. 00:17:39.990 --> 00:17:45.980 But at convergence in this iteration where we calculate 00:17:45.980 --> 00:17:51.120 always new displacements, add them to the previously 00:17:51.120 --> 00:17:54.850 calculated displacements, at convergence the right-hand 00:17:54.850 --> 00:17:59.320 side will be 0, or close to 0, of course. 00:17:59.320 --> 00:18:04.830 Well this means that we arrive at this equation right here. 00:18:04.830 --> 00:18:08.690 Because when the right-hand side is 0, we satisfy that t 00:18:08.690 --> 00:18:12.550 plus delta t R is equal to t plus delta t F And if you 00:18:12.550 --> 00:18:18.180 like, here also, we may want to introduce this little 0, 00:18:18.180 --> 00:18:20.890 signifying that we have used the total Lagrangian 00:18:20.890 --> 00:18:25.530 formulation to calculate that F. 00:18:25.530 --> 00:18:30.140 Now it is important to realize what we have achieved in the 00:18:30.140 --> 00:18:32.590 finite element solution. 00:18:32.590 --> 00:18:37.700 We satisfy compatibility in the finite element solution if 00:18:37.700 --> 00:18:40.850 we use compatible finite elements. 00:18:40.850 --> 00:18:43.690 That is buried in here. 00:18:43.690 --> 00:18:47.330 We use compatible finite elements to calculate this F, 00:18:47.330 --> 00:18:50.260 therefore we satisfy compatibility. 00:18:50.260 --> 00:18:54.610 We satisfy the stress strain law properly if we calculate 00:18:54.610 --> 00:19:01.810 this F here accurately and properly through the stress 00:19:01.810 --> 00:19:02.850 strain law. 00:19:02.850 --> 00:19:05.290 Of course, there is much to be discussed still. 00:19:05.290 --> 00:19:09.390 But let's assume we do that properly and then we satisfy 00:19:09.390 --> 00:19:11.040 the stress strain law. 00:19:11.040 --> 00:19:16.430 Equilibrium we also satisfy, but at the nodal points. 00:19:16.430 --> 00:19:20.180 We talked about the fact, in an earlier lecture, that in a 00:19:20.180 --> 00:19:22.910 finite element analysis, displacement based finite 00:19:22.910 --> 00:19:26.040 element analysis, we always satisfy equilibrium at the 00:19:26.040 --> 00:19:26.940 nodal point. 00:19:26.940 --> 00:19:29.140 Of course, assuming that the elements have been properly 00:19:29.140 --> 00:19:31.400 formulated. 00:19:31.400 --> 00:19:35.650 We would only satisfy local equilibrium if the mesh is 00:19:35.650 --> 00:19:36.440 fine enough. 00:19:36.440 --> 00:19:39.940 And we talked earlier, in an earlier lecture, about how we 00:19:39.940 --> 00:19:42.460 can measure whether local equilibrium is well enough 00:19:42.460 --> 00:19:46.980 satisfied using stress jumps, using pressure band plots. 00:19:46.980 --> 00:19:50.660 We talked about that and I gave you some examples. 00:19:50.660 --> 00:19:53.920 The important point though is to recognize that if we use a 00:19:53.920 --> 00:20:02.190 fine enough mesh, then equilibrium is satisfied, and 00:20:02.190 --> 00:20:04.660 the stress strain law would be satisfied if we calculated 00:20:04.660 --> 00:20:07.090 stresses properly from the given strains. 00:20:07.090 --> 00:20:11.070 And compatibility is satisfied if we use a compatible finite 00:20:11.070 --> 00:20:11.830 element mesh. 00:20:11.830 --> 00:20:14.870 In other words, the three basic important requirements 00:20:14.870 --> 00:20:19.640 of mechanics are fulfilled when we are satisfying this 00:20:19.640 --> 00:20:22.410 relationship right here. 00:20:22.410 --> 00:20:25.750 This basically then rounds up what we have been 00:20:25.750 --> 00:20:27.770 discussing so far. 00:20:27.770 --> 00:20:32.870 And we use the total Lagrangian formulation in this 00:20:32.870 --> 00:20:34.800 discussion. 00:20:34.800 --> 00:20:38.050 I now like to go over to discuss this, use the updated 00:20:38.050 --> 00:20:40.100 Lagrangian formulation. 00:20:40.100 --> 00:20:45.480 The interesting point here is that we use a stress measure 00:20:45.480 --> 00:20:48.885 that is referred to the original configuration, and so 00:20:48.885 --> 00:20:51.930 a strain measure that is referred to the original 00:20:51.930 --> 00:20:53.180 configuration. 00:20:54.860 --> 00:20:58.730 I mentioned earlier already that we want to deal with a 00:20:58.730 --> 00:21:00.750 certain configuration that is known. 00:21:00.750 --> 00:21:04.030 The reason, of course, being that we want to incrementally 00:21:04.030 --> 00:21:10.980 decompose this stress into a quantity that we know, plus a 00:21:10.980 --> 00:21:13.930 quantity that we don't know. 00:21:13.930 --> 00:21:17.070 And that incremental decomposition should involve 00:21:17.070 --> 00:21:21.010 stress measures that are referred to a configuration 00:21:21.010 --> 00:21:22.250 that is known. 00:21:22.250 --> 00:21:24.210 The same for the strains. 00:21:24.210 --> 00:21:28.320 Well since we have calculated all the configurations from 00:21:28.320 --> 00:21:33.830 time 0 to time t already, a natural question to ask is why 00:21:33.830 --> 00:21:38.010 don't you use a configuration other than 0, say, the 00:21:38.010 --> 00:21:42.740 configuration at time delta t to delta t, or t. 00:21:42.740 --> 00:21:47.240 well in the updated Lagrangian formulation, we, indeed, use 00:21:47.240 --> 00:21:51.320 the configuration at time t as a reference configuration. 00:21:51.320 --> 00:21:55.670 And so we deal then with the second Piola-Kirchhoff stress 00:21:55.670 --> 00:21:58.140 referred to the configuration at time t. 00:22:06.570 --> 00:22:09.170 This is the stress that we're dealing with. 00:22:09.170 --> 00:22:13.430 Notice t plus delta t here, still the stress at time t 00:22:13.430 --> 00:22:17.030 plus delta t, but now the stress is referred to the 00:22:17.030 --> 00:22:19.260 configuration at time t. 00:22:19.260 --> 00:22:22.910 Similarly, we also deal with a strain. 00:22:29.240 --> 00:22:32.940 The Green-Lagrange strain at time t plus delta t, but 00:22:32.940 --> 00:22:37.760 referred to configuration at time t. 00:22:37.760 --> 00:22:39.900 We could, of course, ask, well why use a 00:22:39.900 --> 00:22:41.310 configuration at time t? 00:22:41.310 --> 00:22:44.825 Why not use some configuration in between? 00:22:44.825 --> 00:22:47.340 The answer is, and I gave it to you already in an earlier 00:22:47.340 --> 00:22:51.710 lecture, the answer is that you are losing all of the 00:22:51.710 --> 00:22:56.750 advantages of the total Lagrangian formulation, and 00:22:56.750 --> 00:22:59.490 you're losing the advantages of the updated Lagrangian 00:22:59.490 --> 00:23:00.680 formulation. 00:23:00.680 --> 00:23:02.500 You are left only with the disadvantages. 00:23:02.500 --> 00:23:06.930 So there's very little point really in trying to solve the 00:23:06.930 --> 00:23:10.752 principle of virtual work using another configuration 00:23:10.752 --> 00:23:14.950 other than 0 or t. 00:23:14.950 --> 00:23:17.680 Let us now look at the details of the updated Lagrangian 00:23:17.680 --> 00:23:18.430 formulation. 00:23:18.430 --> 00:23:23.470 And here I have now prepared some view graphs that show all 00:23:23.470 --> 00:23:26.450 the equations that we're operating on. 00:23:26.450 --> 00:23:30.660 The principle of virtual work, once again, is given right on 00:23:30.660 --> 00:23:31.980 this first view graph. 00:23:31.980 --> 00:23:36.480 We always start with the basic principle of virtual work, and 00:23:36.480 --> 00:23:39.050 that is written here. 00:23:39.050 --> 00:23:41.873 This is, once again, the Cauchy stress, the stress that 00:23:41.873 --> 00:23:45.110 we are actually interested in solving for. 00:23:45.110 --> 00:23:49.010 This is the infinitesimal virtual strain. 00:23:49.010 --> 00:23:52.790 Internal virtual work is equal to external virtual work. 00:23:52.790 --> 00:23:57.460 Now, with the stress measure that I just mentioned to you, 00:23:57.460 --> 00:24:02.370 and the corresponding strain measure, substituted for these 00:24:02.370 --> 00:24:06.490 stress and strain measures, we obtain directly this equation. 00:24:06.490 --> 00:24:09.260 Notice that here on the left-hand side, we have an 00:24:09.260 --> 00:24:14.240 integral that is exactly equal to this integral here. 00:24:14.240 --> 00:24:16.010 And that we are not talking here about 00:24:16.010 --> 00:24:17.780 finite elements yet. 00:24:17.780 --> 00:24:21.660 It's all continuum mechanics at this point. 00:24:21.660 --> 00:24:25.680 The second Piola-Kirchhoff stress here refer to time t 00:24:25.680 --> 00:24:28.775 is, of course, defined as the same way as the second 00:24:28.775 --> 00:24:32.560 Piola-Kirchhoff stress as before when it was 00:24:32.560 --> 00:24:33.900 referred to time 0. 00:24:33.900 --> 00:24:38.610 But what we have to do now is introduce this t whenever we 00:24:38.610 --> 00:24:44.360 use the 0 before in the equation linking up the Cauchy 00:24:44.360 --> 00:24:46.880 stress with the second Piola-Kirchhoff stress. 00:24:49.470 --> 00:24:52.610 We already then know the solution at time t. 00:24:52.610 --> 00:24:56.380 So we know basically these quantities here. 00:24:56.380 --> 00:24:59.910 And we can, therefore, now decompose directly the 00:24:59.910 --> 00:25:06.130 stresses into a stress that we know, and an increment. 00:25:06.130 --> 00:25:09.350 Similarly, we do for the strains. 00:25:09.350 --> 00:25:14.580 It's now interesting to observe that this stress here 00:25:14.580 --> 00:25:17.150 is nothing else than the Cauchy stress. 00:25:17.150 --> 00:25:19.730 If you were to plug in to the formula that I gave you 00:25:19.730 --> 00:25:24.280 earlier, which once again, expresses the second 00:25:24.280 --> 00:25:27.080 Piola-Kirchhoff stress in terms of the Cauchy stress, 00:25:27.080 --> 00:25:30.555 you would simply see that this quantity is nothing else than 00:25:30.555 --> 00:25:31.730 the Cauchy stress. 00:25:31.730 --> 00:25:34.270 And therefore, we have that now here. 00:25:34.270 --> 00:25:39.130 And of course, we carry this part in that equation here. 00:25:39.130 --> 00:25:45.840 The Green-Lagrange strain, if written out, one directly 00:25:45.840 --> 00:25:50.120 identifies that this part here is 0, because we are using 00:25:50.120 --> 00:25:53.690 only the increments in displacements from time t to 00:25:53.690 --> 00:25:56.310 time t plus delta t. 00:25:56.310 --> 00:26:02.020 And we are left with this part here, which is unknown. 00:26:02.020 --> 00:26:04.310 Let's look at the Green-Lagrange strain here. 00:26:04.310 --> 00:26:07.830 Notice t plus delta t, t epsilon ij is 00:26:07.830 --> 00:26:09.910 written as shown here. 00:26:09.910 --> 00:26:16.600 Where I should point out that we are looking here at the 00:26:16.600 --> 00:26:21.250 displacements from time t to time t plus delta t. 00:26:21.250 --> 00:26:26.180 In other words, if I want to write this out here, I'm doing 00:26:26.180 --> 00:26:27.750 it as follows. 00:26:27.750 --> 00:26:32.140 I'm saying that it's a partial of the displacements, t plus 00:26:32.140 --> 00:26:42.720 delta t, ui minus tui with respect to txj. 00:26:42.720 --> 00:26:46.680 Notice that here, we are subtracting from the 00:26:46.680 --> 00:26:49.090 displacement at time t plus delta t, the 00:26:49.090 --> 00:26:51.440 displacements tui. 00:26:51.440 --> 00:26:53.490 So it's the incremental displacements that we're 00:26:53.490 --> 00:26:55.460 dealing with. 00:26:55.460 --> 00:26:58.420 And we're differentiating with respect to the coordinate at 00:26:58.420 --> 00:27:02.760 the time t. 00:27:02.760 --> 00:27:07.700 Similarly, of course, we define these expressions here, 00:27:07.700 --> 00:27:11.930 and those expressions, and those expressions. 00:27:11.930 --> 00:27:16.470 Notice, just to emphasize this point, that this here is 00:27:16.470 --> 00:27:20.600 nothing else than ui. 00:27:20.600 --> 00:27:25.010 And if we now substitute from here into there, we directly 00:27:25.010 --> 00:27:29.650 recognize that this term is, in fact, the same as that one, 00:27:29.650 --> 00:27:33.090 which I expressed already on the earlier view graph. 00:27:33.090 --> 00:27:37.570 Notice, linear part here, nonlinear part there. 00:27:40.730 --> 00:27:44.140 This part here, of course, I like to just write out 00:27:44.140 --> 00:27:44.950 for you once more. 00:27:44.950 --> 00:27:49.590 It's nothing else than partial ui with respect to txj. 00:27:53.260 --> 00:27:57.540 Notice, here we have a product incremental displacement, and 00:27:57.540 --> 00:28:00.260 that's why it is, of course, nonlinear. 00:28:00.260 --> 00:28:02.980 It is interesting to compare this expression for the 00:28:02.980 --> 00:28:05.400 Green-Lagrange strain, for the incremental Green-Lagrange 00:28:05.400 --> 00:28:08.060 strain, to the incremental Green-Lagrange strain of the 00:28:08.060 --> 00:28:10.250 total Lagrangian formulation. 00:28:10.250 --> 00:28:14.600 And if you do so, you'll find that the major difference is 00:28:14.600 --> 00:28:17.100 that in the total Lagrangian formulation you have an 00:28:17.100 --> 00:28:19.050 initial displacement effect. 00:28:19.050 --> 00:28:22.720 Here, we do not have an initial displacement effect. 00:28:22.720 --> 00:28:26.640 And in fact, that is one of the advantages of the updated 00:28:26.640 --> 00:28:27.890 Lagrangian formulation. 00:28:30.570 --> 00:28:35.230 We can now define, just as in the total Lagrangian 00:28:35.230 --> 00:28:38.660 formulation, a linear strain increment, and a nonlinear 00:28:38.660 --> 00:28:39.410 strain increment. 00:28:39.410 --> 00:28:43.380 Of course, linear in the incremental displacement of 00:28:43.380 --> 00:28:44.420 the particles. 00:28:44.420 --> 00:28:48.280 Nonlinear in the incremental displacement of the particles. 00:28:48.280 --> 00:28:50.580 And this means that directly we obtain 00:28:50.580 --> 00:28:52.180 this expression here. 00:28:52.180 --> 00:28:56.260 And the variation on here, of course, on this total 00:28:56.260 --> 00:28:58.660 Green-Lagrange strain increment is simply given by 00:28:58.660 --> 00:28:59.910 these two additions. 00:29:03.060 --> 00:29:07.270 The equation of the principle of virtual work now becomes 00:29:07.270 --> 00:29:13.180 substituting from what I just discussed with you, directly 00:29:13.180 --> 00:29:14.540 this equation. 00:29:14.540 --> 00:29:18.220 Notice we have here now, the external virtual work, just as 00:29:18.220 --> 00:29:23.490 before, and we have here the internal virtual work 00:29:23.490 --> 00:29:27.640 corresponding to the stresses at time t. 00:29:27.640 --> 00:29:28.950 And these are actually the Cauchy 00:29:28.950 --> 00:29:31.520 stresses, the real stresses. 00:29:31.520 --> 00:29:33.540 This is here again our out of balance 00:29:33.540 --> 00:29:36.380 virtual work, so to say. 00:29:36.380 --> 00:29:41.980 On the left-hand side, we have one part that includes an 00:29:41.980 --> 00:29:43.490 increment in the stress. 00:29:43.490 --> 00:29:47.870 Refer to time t now, because the t configuration is our 00:29:47.870 --> 00:29:50.890 reference configuration. 00:29:50.890 --> 00:29:55.600 And we are multiplying it by the variation on the total 00:29:55.600 --> 00:29:58.200 Green-Lagrange strain increment-- 00:29:58.200 --> 00:29:59.340 that's one integral-- 00:29:59.340 --> 00:30:01.280 over the volume at time t. 00:30:01.280 --> 00:30:06.290 And here we have the Cauchy stress times the variation on 00:30:06.290 --> 00:30:08.140 the nonlinear part of the 00:30:08.140 --> 00:30:09.390 Green-Lagrange strain increment. 00:30:11.940 --> 00:30:15.490 Given a variation in displacement data ui, of 00:30:15.490 --> 00:30:19.750 course, we can calculate the right-hand side. 00:30:19.750 --> 00:30:23.560 But we recognize that on the left-hand side, we have 00:30:23.560 --> 00:30:26.420 unknown displacement increments and there would be 00:30:26.420 --> 00:30:30.390 nonlinearities occurring on the left-hand side. 00:30:30.390 --> 00:30:34.430 Remember, that so far no approximations have been made. 00:30:34.430 --> 00:30:37.290 This is a statement that is valid for any amount of 00:30:37.290 --> 00:30:40.160 deformation, any amount of displacements, 00:30:40.160 --> 00:30:41.410 rotations, and strains. 00:30:44.040 --> 00:30:49.110 Just as in the total Lagrangian formulation, the 00:30:49.110 --> 00:30:51.790 equation of the principle of virtual work in the updated 00:30:51.790 --> 00:30:54.240 Lagrangian formulation is, of course, a very complicated 00:30:54.240 --> 00:30:57.280 equation and highly nonlinear in the incremental 00:30:57.280 --> 00:30:58.340 displacement. 00:30:58.340 --> 00:31:01.770 And we somehow have to linearize. 00:31:01.770 --> 00:31:05.790 Because we want to obtain a finite element solution, we 00:31:05.790 --> 00:31:10.090 linearize first the principle of virtual work equation, and 00:31:10.090 --> 00:31:14.580 then we are ready to obtain an equation as shown here, which 00:31:14.580 --> 00:31:18.530 is very much like the equation which we are operating on 00:31:18.530 --> 00:31:20.550 using the total Lagrangian formulation. 00:31:20.550 --> 00:31:23.960 Notice, because we're dealing with the updated Lagrangian 00:31:23.960 --> 00:31:30.410 formulation, we now have a t here, and we have a t there, 00:31:30.410 --> 00:31:33.400 meaning the reference configuration is the time t 00:31:33.400 --> 00:31:38.010 configuration, and here as well. 00:31:38.010 --> 00:31:42.290 We begin to linearize the terms just as we have done it 00:31:42.290 --> 00:31:44.940 earlier in the total Lagrangian formulation. 00:31:44.940 --> 00:31:48.060 And we recognize that this term here is 00:31:48.060 --> 00:31:50.970 already linear in ui. 00:31:50.970 --> 00:31:54.140 That is quite simply seen because the stress does not 00:31:54.140 --> 00:31:57.886 contain ui, it's a known quantity. 00:31:57.886 --> 00:32:04.480 The variation on this part here gives us this expression. 00:32:04.480 --> 00:32:09.200 And we recognize that this is linear in ui because here we 00:32:09.200 --> 00:32:11.290 have the unknown ui appearing. 00:32:11.290 --> 00:32:14.790 This is going to be a constant for a given variation in 00:32:14.790 --> 00:32:15.860 displacement. 00:32:15.860 --> 00:32:18.260 And this is here also linear and ui. 00:32:18.260 --> 00:32:20.810 This is going to be a constant for a given variation in 00:32:20.810 --> 00:32:21.770 displacement. 00:32:21.770 --> 00:32:25.460 And so we have a linear part here in ui. 00:32:25.460 --> 00:32:27.900 No linearization required here. 00:32:27.900 --> 00:32:30.440 Let us now look at the next term. 00:32:30.440 --> 00:32:32.910 Here we have the increment in the second Piola-Kirchhoff 00:32:32.910 --> 00:32:35.680 stress times the variation on the increment in the 00:32:35.680 --> 00:32:38.160 Green-Lagrange strain integrated over the 00:32:38.160 --> 00:32:40.000 volume at time t. 00:32:40.000 --> 00:32:43.930 We recognize that tSij is a nonlinear function in general 00:32:43.930 --> 00:32:47.110 of the increment in the Green-Lagrange strain. 00:32:47.110 --> 00:32:49.000 If we take the variation on that increment and 00:32:49.000 --> 00:32:52.440 Green-Lagrange strain, we obtain these two terms, and we 00:32:52.440 --> 00:32:57.140 recognize that this is also a linear function of ui. 00:32:57.140 --> 00:33:00.350 Therefore, in this product, we have to neglect now all higher 00:33:00.350 --> 00:33:02.830 order terms in ui. 00:33:02.830 --> 00:33:06.690 First we recognize now that tSij can be written as a 00:33:06.690 --> 00:33:10.660 Taylor series expansion, as shown here. 00:33:10.660 --> 00:33:14.610 Notice that tSij is equal to one first term plus higher 00:33:14.610 --> 00:33:15.610 order terms. 00:33:15.610 --> 00:33:17.980 These higher order terms we neglect. 00:33:17.980 --> 00:33:21.340 If you look at this term here, we recognize that we can 00:33:21.340 --> 00:33:26.440 substitute for this term, as shown here. 00:33:26.440 --> 00:33:29.830 And this one here is, of course, the constitutive 00:33:29.830 --> 00:33:31.320 relation, the tangent constitutive 00:33:31.320 --> 00:33:33.580 relation for the material. 00:33:33.580 --> 00:33:37.550 Recognizing that this one here is quadratic in ui, we drop 00:33:37.550 --> 00:33:40.760 that term and we get this linearized result. 00:33:40.760 --> 00:33:44.260 Notice that here we have a constitutive tensor referred 00:33:44.260 --> 00:33:46.240 to the configuration at time t. 00:33:46.240 --> 00:33:49.490 In the total Lagrangian formulation we had here a 0 00:33:49.490 --> 00:33:51.240 instead of a t. 00:33:51.240 --> 00:33:55.820 Well if we now look at the product that we wanted to 00:33:55.820 --> 00:34:00.380 linearize again, we have this product on the left-hand side, 00:34:00.380 --> 00:34:03.780 which is now approximately equal to this term here. 00:34:03.780 --> 00:34:08.310 This already we derived as a linearization to that term. 00:34:08.310 --> 00:34:12.500 Here we are substituting for that term here. 00:34:12.500 --> 00:34:17.199 Notice that if we multiply out, we directly obtain this 00:34:17.199 --> 00:34:18.960 relationship here. 00:34:18.960 --> 00:34:22.210 This one here does not contain ui of course. 00:34:22.210 --> 00:34:25.199 And this one here is linear in ui. 00:34:25.199 --> 00:34:29.020 This means that since this one also is linear in ui, that we 00:34:29.020 --> 00:34:31.600 have here a quadratic term in ui. 00:34:31.600 --> 00:34:35.280 Notice this term now, totally linear, this one quadratic, we 00:34:35.280 --> 00:34:39.870 drop that one, and we are left with the linearized result, 00:34:39.870 --> 00:34:43.050 the result that we wanted to obtain. 00:34:43.050 --> 00:34:46.260 The final linearized equation is then given 00:34:46.260 --> 00:34:47.420 on this view graph. 00:34:47.420 --> 00:34:50.210 Notice we have here the constitutive relation times 00:34:50.210 --> 00:34:53.830 the linear strain increment in the total Green-Lagrange 00:34:53.830 --> 00:34:57.360 strain increment, the variation on it. 00:34:57.360 --> 00:34:59.370 And of course, this one again integrated over the 00:34:59.370 --> 00:35:01.520 volume at time t. 00:35:01.520 --> 00:35:04.530 Here we have a term where there was no 00:35:04.530 --> 00:35:07.450 linearization necessary. 00:35:07.450 --> 00:35:09.830 And here we have a term also there was no 00:35:09.830 --> 00:35:11.210 linearization necessary. 00:35:11.210 --> 00:35:15.870 These two terms were obtained already earlier. 00:35:15.870 --> 00:35:20.430 These here, this term be obtained by linearization. 00:35:20.430 --> 00:35:24.330 Notice that when we now use finite element interpolations, 00:35:24.330 --> 00:35:30.020 we will express this total term here via this term here 00:35:30.020 --> 00:35:37.330 where we now have a tangent finite element matrix, which 00:35:37.330 --> 00:35:42.870 is denoted by the t up there at time t referred to the 00:35:42.870 --> 00:35:45.180 configuration at time t. 00:35:45.180 --> 00:35:48.820 This t here refers to the configuration at time t. 00:35:48.820 --> 00:35:51.460 There's an increment in the displacement that we want to 00:35:51.460 --> 00:35:54.000 solve for, the real displacement increment. 00:35:54.000 --> 00:35:58.140 And this is here, the virtual displacement vector. 00:35:58.140 --> 00:36:02.580 The right-hand side gives us a virtual displacement vector, 00:36:02.580 --> 00:36:06.830 of course, transposed times this vector, and here we have 00:36:06.830 --> 00:36:09.670 the externally applied loads coming from the external 00:36:09.670 --> 00:36:14.940 virtual work minus the nodal point force vector 00:36:14.940 --> 00:36:19.060 corresponding to the internal element stresses. 00:36:19.060 --> 00:36:23.180 That is this term here results into this nodal point force 00:36:23.180 --> 00:36:26.820 vector corresponding to the internal element stresses. 00:36:26.820 --> 00:36:31.890 So here we have the continuum mechanics relation in black, 00:36:31.890 --> 00:36:36.400 linearized from the general principle of virtual work, and 00:36:36.400 --> 00:36:40.130 here we have the finite element expression of that. 00:36:40.130 --> 00:36:42.590 Of course, the actual finite element solution is then 00:36:42.590 --> 00:36:46.990 obtained by actually setting this vector here equal to the 00:36:46.990 --> 00:36:51.830 identity matrix so that we are left with K times delta U is 00:36:51.830 --> 00:36:54.470 equal to t plus delta tR minus tF. 00:36:57.930 --> 00:37:01.540 An important point in this development is to recognize 00:37:01.540 --> 00:37:05.400 that this term here is, of course, the virtual work due 00:37:05.400 --> 00:37:08.980 to the element internal stresses at time t. 00:37:08.980 --> 00:37:13.460 And we interpret this part here as the out of balance 00:37:13.460 --> 00:37:16.560 virtual work just in the same way as we have been doing it 00:37:16.560 --> 00:37:17.825 in the total Lagrangian formulation. 00:37:21.680 --> 00:37:24.440 Let's look now at the total solution using the updated 00:37:24.440 --> 00:37:26.800 Lagrangian formulation. 00:37:26.800 --> 00:37:30.000 The displacement iteration is much the same way as in the 00:37:30.000 --> 00:37:31.970 total Lagrangian formulation. 00:37:31.970 --> 00:37:37.530 We take the old value corresponding to k minus 1 00:37:37.530 --> 00:37:40.490 iteration, and add an increment that we are solving 00:37:40.490 --> 00:37:44.650 for to obtain a new estimate on the displacements. 00:37:44.650 --> 00:37:46.930 k being the iteration counter. 00:37:46.930 --> 00:37:49.640 Notice, of course, in this iteration, we have to have 00:37:49.640 --> 00:37:52.330 initial conditions, and these initial conditions are given 00:37:52.330 --> 00:37:53.060 right here. 00:37:53.060 --> 00:37:57.800 In other words, when k is equal to 1, we would have here 00:37:57.800 --> 00:38:03.690 t plus delta t ui 0, and therefore we need this value, 00:38:03.690 --> 00:38:08.110 and that value is given right here as t plus delta tui, 0 00:38:08.110 --> 00:38:10.360 equal to tui. 00:38:10.360 --> 00:38:11.970 In other words, the displacement at the 00:38:11.970 --> 00:38:15.460 configuration t. 00:38:15.460 --> 00:38:19.020 If you look now at the total continuum mechanics equations 00:38:19.020 --> 00:38:24.120 that we are operating on, I should say one continuum 00:38:24.120 --> 00:38:26.770 mechanics equation really, which, of course, gives us 00:38:26.770 --> 00:38:29.870 then in the finite element discretization, the set of 00:38:29.870 --> 00:38:34.770 simultaneous equations that we will be working with. 00:38:34.770 --> 00:38:38.120 This continuum mechanics equation looks as follows. 00:38:38.120 --> 00:38:41.550 On the left-hand side we have the tangent constitutive 00:38:41.550 --> 00:38:47.770 relationship times an increment in real strains that 00:38:47.770 --> 00:38:50.990 we want to solve for in iteration k. 00:38:50.990 --> 00:38:54.030 These are the virtual strains. 00:38:54.030 --> 00:38:57.650 We integrate that product over the volume at time t. 00:38:57.650 --> 00:39:00.590 That, of course, holds for any one of these integrals. 00:39:00.590 --> 00:39:03.500 And here we have now the actual stresses, the real 00:39:03.500 --> 00:39:06.880 stresses that we know already, which correspond to the 00:39:06.880 --> 00:39:10.020 configuration at time t. 00:39:10.020 --> 00:39:15.150 Then a variation on the incremental Green-Lagrange 00:39:15.150 --> 00:39:19.610 strain, but I should add its only the nonlinear increment 00:39:19.610 --> 00:39:24.750 in the Green-Lagrange strain corresponding to iteration k. 00:39:24.750 --> 00:39:29.290 Once again, integrated over the volume at time t. 00:39:29.290 --> 00:39:33.520 And on the right-hand side, we have the external virtual work 00:39:33.520 --> 00:39:35.290 and this expression. 00:39:35.290 --> 00:39:37.770 Let's look at this expression carefully. 00:39:37.770 --> 00:39:41.845 Here we have an integral over the volume at time t plus 00:39:41.845 --> 00:39:44.930 delta t in the iteration k minus 1. 00:39:44.930 --> 00:39:48.490 Therefore, we are updating that volume as we go along in 00:39:48.490 --> 00:39:50.120 the iteration. 00:39:50.120 --> 00:39:53.730 Notice that here we have the stresses corresponding to 00:39:53.730 --> 00:39:55.500 iteration k minus 1. 00:39:55.500 --> 00:39:58.040 We are updating these stresses, of course, also in 00:39:58.040 --> 00:39:59.280 the iteration. 00:39:59.280 --> 00:40:02.580 Here we have the virtual strains corresponding to 00:40:02.580 --> 00:40:07.870 iteration at the end of iteration k minus 1. 00:40:07.870 --> 00:40:09.600 We are updating this as well. 00:40:09.600 --> 00:40:13.350 And this product here, once again, is integrated over the 00:40:13.350 --> 00:40:17.510 volume at time t plus delta t corresponding to the end of 00:40:17.510 --> 00:40:22.060 iteration k minus 1 or the beginning of iteration k. 00:40:22.060 --> 00:40:27.290 Well it's important to realize that when this right-hand side 00:40:27.290 --> 00:40:34.140 is 0, we have fulfilled, or we have basically solved, the 00:40:34.140 --> 00:40:37.070 equation of the principle of virtual work, because the 00:40:37.070 --> 00:40:42.530 external virtual work is equal to the internal virtual work. 00:40:42.530 --> 00:40:44.790 And then, of course, we have obtained the solution. 00:40:44.790 --> 00:40:49.290 So we iterate with k equals 1, 2, 3, and so on, until the 00:40:49.290 --> 00:40:54.370 right-hand side is 0, and then we are satisfying that the 00:40:54.370 --> 00:40:59.880 external virtual work is equal to the internal virtual work. 00:40:59.880 --> 00:41:05.180 When we discretize this continuum mechanics equation 00:41:05.180 --> 00:41:10.410 via finite elements, we obtain this equation here. 00:41:10.410 --> 00:41:14.590 ttK, once again, the tangent stiffness matrix. 00:41:14.590 --> 00:41:17.590 Here, the increment in the nodal point displacements 00:41:17.590 --> 00:41:19.420 corresponding to iteration k. 00:41:19.420 --> 00:41:21.930 On the right-hand side the R vector of 00:41:21.930 --> 00:41:23.730 externally applied loads. 00:41:23.730 --> 00:41:27.370 And here, the nodal point force vector corresponding to 00:41:27.370 --> 00:41:31.510 the internal element stresses at time t plus delta t in 00:41:31.510 --> 00:41:34.050 iteration k minus 1. 00:41:34.050 --> 00:41:36.080 At the end of iteration k minus 1 is 00:41:36.080 --> 00:41:37.670 more precisely said. 00:41:37.670 --> 00:41:40.810 Notice that we are having a t plus delta t down here because 00:41:40.810 --> 00:41:42.440 we are always updating the actual 00:41:42.440 --> 00:41:44.920 configuration of reference. 00:41:44.920 --> 00:41:47.160 Here we are updating the configuration of reference 00:41:47.160 --> 00:41:51.470 because we are working with the Cauchy stresses at time t 00:41:51.470 --> 00:41:55.390 plus delta t, and in iteration at the end of 00:41:55.390 --> 00:41:57.540 iteration k minus 1. 00:41:57.540 --> 00:42:01.010 Of course, this vector is calculated from the 00:42:01.010 --> 00:42:06.080 displacement at time t plus delta t, and 00:42:06.080 --> 00:42:07.540 iteration k minus 1. 00:42:07.540 --> 00:42:10.160 These, of course, are known. 00:42:10.160 --> 00:42:14.710 This is a known quantity, whereas the unknown quantity 00:42:14.710 --> 00:42:16.840 are the increments in the displacements. 00:42:16.840 --> 00:42:19.020 These we want to calculate. 00:42:19.020 --> 00:42:23.560 We iterate until, with k equals 1, 2, 3, and so on, 00:42:23.560 --> 00:42:26.130 until the right-hand side is 0. 00:42:26.130 --> 00:42:30.380 Of course, this is nothing else than an expression of, in 00:42:30.380 --> 00:42:33.960 finite element analysis, of the principle of virtual work 00:42:33.960 --> 00:42:36.540 where we had also the right-hand side, this was the 00:42:36.540 --> 00:42:39.620 external virtual work, and this was the internal virtual 00:42:39.620 --> 00:42:44.850 work corresponding to the current element stresses. 00:42:44.850 --> 00:42:50.738 Notice that the displacements that we are talking about here 00:42:50.738 --> 00:42:56.500 are obtained by adding all the incremental displacement that 00:42:56.500 --> 00:42:58.815 we're dealing with here. 00:42:58.815 --> 00:43:02.840 We are taking these and add them all up as expressed in 00:43:02.840 --> 00:43:08.660 this summation sign here. j goes from 1 to k to get, and 00:43:08.660 --> 00:43:11.800 we take that sum, of course, and add it to tU, the 00:43:11.800 --> 00:43:15.820 displacement corresponding to configuration time t, to get 00:43:15.820 --> 00:43:17.310 the displacements 00:43:17.310 --> 00:43:21.750 corresponding to the last iteration. 00:43:21.750 --> 00:43:24.878 The total iterative process is summarized on this view graph. 00:43:27.510 --> 00:43:31.060 We start with a given tU, the displacement at time t are 00:43:31.060 --> 00:43:35.340 given, and the external loads are also given corresponding 00:43:35.340 --> 00:43:37.260 to time t plus delta t. 00:43:37.260 --> 00:43:41.470 We compute the tangent stiffness matrix corresponding 00:43:41.470 --> 00:43:45.770 to time t, and the nodal point force vector corresponding to 00:43:45.770 --> 00:43:50.640 the internal element stresses at time t. 00:43:50.640 --> 00:43:54.440 This nodal point force vector provides initial conditions 00:43:54.440 --> 00:43:57.510 for this vector, because that is the vector which will enter 00:43:57.510 --> 00:43:59.550 actually into the iteration. 00:43:59.550 --> 00:44:01.600 This displacement vector provides the initial 00:44:01.600 --> 00:44:03.890 conditions for that vector. 00:44:03.890 --> 00:44:07.900 We set the iteration counter k equal to 1. 00:44:07.900 --> 00:44:13.690 We go in here, and notice now we are solving this equation, 00:44:13.690 --> 00:44:16.680 which we just discussed. 00:44:16.680 --> 00:44:20.410 When k is equal to 1, of course, we have here F0, and 00:44:20.410 --> 00:44:24.520 that F0 is nothing else than that vector here. 00:44:24.520 --> 00:44:28.340 Because that's how we initialized it. 00:44:28.340 --> 00:44:30.030 So we can calculate this incremental 00:44:30.030 --> 00:44:31.880 displacement vector. 00:44:31.880 --> 00:44:34.580 We add that increment and displacement vector to the 00:44:34.580 --> 00:44:36.270 displacement which we had already. 00:44:36.270 --> 00:44:40.160 When k is equal to 1, of course, this vector here is 00:44:40.160 --> 00:44:45.525 nothing else than that vector, which we are given already. 00:44:45.525 --> 00:44:48.130 We add on and get a new estimate for the 00:44:48.130 --> 00:44:49.650 displacements. 00:44:49.650 --> 00:44:51.670 We then check for convergence. 00:44:51.670 --> 00:44:54.330 This means we are checking whether equilibrium is 00:44:54.330 --> 00:44:57.280 satisfied sufficiently, meaning whether the right-hand 00:44:57.280 --> 00:45:04.080 side is 0 with this new displacement estimate. 00:45:04.080 --> 00:45:10.010 If we have not converged, we go into the next calculation 00:45:10.010 --> 00:45:14.150 of the force vector, the nodal point force vector 00:45:14.150 --> 00:45:17.440 corresponding to the current internal element stresses. 00:45:17.440 --> 00:45:25.060 And notice that, of course, now this force vector, these 00:45:25.060 --> 00:45:29.520 elements stresses that go in here, are computed based on 00:45:29.520 --> 00:45:33.560 the displacement at time t plus delta t at the end of 00:45:33.560 --> 00:45:35.400 iteration k. 00:45:35.400 --> 00:45:38.700 In other words, these are the displacements that we have 00:45:38.700 --> 00:45:42.380 just been calculating right there. 00:45:42.380 --> 00:45:45.355 We increase the iteration counter, and keep 00:45:45.355 --> 00:45:47.420 on looping as before. 00:45:47.420 --> 00:45:52.750 So this is the loop that we go through in the iteration until 00:45:52.750 --> 00:45:56.760 we converge, until we converge. 00:45:56.760 --> 00:46:02.370 If you compare this view graph with the view graph that we 00:46:02.370 --> 00:46:05.220 discussed in the total Lagrangian formulation, you 00:46:05.220 --> 00:46:07.350 will find a great similarity. 00:46:07.350 --> 00:46:13.260 You will find that basically all of these quantities have 00:46:13.260 --> 00:46:15.760 already been discussed in the total Lagrangian formulation, 00:46:15.760 --> 00:46:18.440 except that in the total Lagrangian formulation we 00:46:18.440 --> 00:46:25.350 always carry it down here and down there as 0 instead of t, 00:46:25.350 --> 00:46:28.280 t plus delta t here. 00:46:28.280 --> 00:46:29.830 We carry it as 0. 00:46:29.830 --> 00:46:33.510 Well because the reference configuration was the 0 00:46:33.510 --> 00:46:34.690 configuration. 00:46:34.690 --> 00:46:36.370 Now the reference configuration for the 00:46:36.370 --> 00:46:39.770 stiffness matrix is the t configuration. 00:46:39.770 --> 00:46:43.010 The reference configuration for the F vector is the t plus 00:46:43.010 --> 00:46:48.700 delta t configuration after iteration k minus 1. 00:46:48.700 --> 00:46:52.970 So there's a great similarity in the iteration process, in 00:46:52.970 --> 00:46:56.260 the finite element matrices that we are setting up. 00:46:56.260 --> 00:47:01.345 And of course, we have to ask ourselves what comparison is 00:47:01.345 --> 00:47:03.650 there between the T.L. and U.L. formulation? 00:47:03.650 --> 00:47:06.960 Why would one want to use the U.L. formulation if we have 00:47:06.960 --> 00:47:09.680 the T.L. formulation, for example. 00:47:09.680 --> 00:47:14.130 Well in the T.L. formulation, all the derivatives are 00:47:14.130 --> 00:47:17.710 defined with respect to the original configuration. 00:47:17.710 --> 00:47:22.000 In the U.L. formulation, they are defined with respect to 00:47:22.000 --> 00:47:25.030 the current configuration. 00:47:25.030 --> 00:47:27.330 And that will provide differences, that does provide 00:47:27.330 --> 00:47:28.180 differences. 00:47:28.180 --> 00:47:32.510 For example, in the U.L. formulation, we will have to 00:47:32.510 --> 00:47:37.280 calculate the derivatives newly in each iteration, 00:47:37.280 --> 00:47:41.040 because the configuration is continuously updated. 00:47:41.040 --> 00:47:43.940 We don't need to do it, of course, for the k matrix if we 00:47:43.940 --> 00:47:46.860 keep the k matrix constant on the left-hand side. 00:47:46.860 --> 00:47:49.660 But for the F vector on the right-hand side, we will have 00:47:49.660 --> 00:47:53.795 to always calculate new derivatives because the F 00:47:53.795 --> 00:47:57.200 vector involves the Cauchy stresses times the 00:47:57.200 --> 00:48:01.620 infinitesimal virtual strains referred to the current 00:48:01.620 --> 00:48:07.320 configuration, t plus delta t, iteration k minus 1, which is 00:48:07.320 --> 00:48:08.180 being updated. 00:48:08.180 --> 00:48:11.240 And so we would always calculate new derivatives 00:48:11.240 --> 00:48:13.370 right there. 00:48:13.370 --> 00:48:15.270 There, of course, is an expense involved. 00:48:15.270 --> 00:48:18.790 On the other hand, the U.L. formulation does not have this 00:48:18.790 --> 00:48:21.600 initial displacement effect that I mentioned earlier, so 00:48:21.600 --> 00:48:23.460 there's an advantage. 00:48:23.460 --> 00:48:25.630 In the U.L. formulation, we also directly 00:48:25.630 --> 00:48:27.440 deal with Cauchy stresses. 00:48:27.440 --> 00:48:29.430 These are, of course, the physical stresses that we are 00:48:29.430 --> 00:48:32.080 actually interested in, and that we want to get printed 00:48:32.080 --> 00:48:33.880 out from the computer program. 00:48:33.880 --> 00:48:37.810 Hence, there's no transformation necessary from 00:48:37.810 --> 00:48:39.700 some stress measure, and I'm thinking of the second 00:48:39.700 --> 00:48:41.690 Piola-Kirchhoff stress that we use in the total Lagrangian 00:48:41.690 --> 00:48:44.850 formulation, to the Cauchy stresses that we actually want 00:48:44.850 --> 00:48:47.400 to obtain in the computer program, and that we as 00:48:47.400 --> 00:48:50.750 engineers want to get, of course, printed out. 00:48:50.750 --> 00:48:55.420 Well if we now look at one other important point, one 00:48:55.420 --> 00:48:59.990 important point, regarding the T.L. and U.L. formulation, we 00:48:59.990 --> 00:49:02.490 recognize that if we make the assumption, the same 00:49:02.490 --> 00:49:05.020 assumptions, in both of these two formulations in the 00:49:05.020 --> 00:49:08.520 linearization, as we, of course, have done, really, the 00:49:08.520 --> 00:49:10.160 way I've presented it. 00:49:10.160 --> 00:49:15.210 Then the U.L. and T.L. formulation, provided the 00:49:15.210 --> 00:49:18.960 appropriate transformations are made, certain 00:49:18.960 --> 00:49:22.610 transformation rules are followed for the stresses, the 00:49:22.610 --> 00:49:24.040 strains, and so on. 00:49:24.040 --> 00:49:26.740 Those rules we will be talking about further later on in 00:49:26.740 --> 00:49:27.830 later lectures. 00:49:27.830 --> 00:49:33.130 Then, indeed, exactly the same stiffness matrix and force 00:49:33.130 --> 00:49:37.580 vector, F vector, vectors I should say, are calculated. 00:49:37.580 --> 00:49:42.040 In other words, provided we follow certain transformation 00:49:42.040 --> 00:49:48.000 rules, indeed, the same stiffness matrices and force 00:49:48.000 --> 00:49:53.340 vectors are calculated in both of these formulations. 00:49:53.340 --> 00:49:56.230 Of course, the actual numerical arithmetic required 00:49:56.230 --> 00:49:58.730 to obtain those matrices and vectors 00:49:58.730 --> 00:50:00.020 are, of course, different. 00:50:00.020 --> 00:50:03.090 But the final result is the same. 00:50:03.090 --> 00:50:05.900 We can demonstrate that in the general case, as I will do 00:50:05.900 --> 00:50:09.720 later on in a lecture, and we can also demonstrate that on 00:50:09.720 --> 00:50:11.410 simple examples. 00:50:11.410 --> 00:50:14.020 In fact, we will look in one lecture at the truss element 00:50:14.020 --> 00:50:17.850 where we can directly show that the T.L. and U.L. 00:50:17.850 --> 00:50:21.020 formulation really give exactly the same result for 00:50:21.020 --> 00:50:23.120 the k matrix, as well as the F vector. 00:50:23.120 --> 00:50:25.890 So there we have an example, but as I mentioned, we can 00:50:25.890 --> 00:50:28.380 also show it in the general case, and that we do 00:50:28.380 --> 00:50:30.480 as well later on. 00:50:30.480 --> 00:50:32.180 Thank you very much for your attention.